diff --git "a/2tE2T4oBgHgl3EQf5gjq/content/tmp_files/load_file.txt" "b/2tE2T4oBgHgl3EQf5gjq/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/2tE2T4oBgHgl3EQf5gjq/content/tmp_files/load_file.txt" @@ -0,0 +1,780 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf,len=779 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='04192v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='AG] 10 Jan 2023 QUANTIZATIONS OF LOCAL CALABI–YAU THREEFOLDS AND THEIR MODULI OF VECTOR BUNDLES E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' BALLICO, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' GASPARIM, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' RUBILAR, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' SUZUKI Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' We describe the geometry of noncommutative deformations of local Calabi–Yau threefolds, showing that the choice of Poisson structure strongly influences the geometry of the quantum moduli space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Contents 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Introduction 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Noncommutative deformations 2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Vector bundles on noncommutative deformations 3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Moduli of bundles on noncommutative deformations 6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Quantum moduli of bundles on W1 8 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Quantum moduli of bundles on W2 11 Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Computations of H1 15 References 16 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Introduction We discuss moduli of vector bundles on those noncommutative local Calabi–Yau threefolds that occur in noncommutative crepant resolutions of the generalised conifolds xy − znwm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Such crepant resolutions require lines of type (−1, −1) and (−2, 0), that is, those locally modelled by W1 := Tot(OP1(−1) ⊕ OP1(−1)) or W2 := Tot(OP1(−2) ⊕ OP1(0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Their appearance is balanced in a precise sense described in [GKMR] so that no particular configuration of such lines is more likely to occur in a crepant resolution than any other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Our results show that the structure of the quantum moduli space (Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='3) of vector bundles over a noncommutative deformation varies drastically depending on the choice of a Poisson structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' In the 2-dimensional case, [BG] described the geometry of noncommutative deformations of the local surfaces Zk := Tot(OP1(−k)), showing that the quantum moduli space of instantons over a noncommutative deformation (Zk, σ) can be viewed as the ´etale space of a constructible sheaf over the classical moduli space of instantons on Zk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' While in 2 dimensions vector bundles occur as mathematical representations of instantons, in the 3-dimensional case vector bundles occur as mathematical descriptions of BPS states, with W1 and W2 appearing as building blocks, as described in [GKMR, GSTV, OSY].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' 1 QUANTIZATION OF CALABI–YAU THREEFOLDS 2 In this work, we describe the geometry of noncommutative deformations W of a Calabi–Yau threefold W, showing that the quantum moduli space of vector bundles on together with the map taking a vector bundle on W to its classical limit Mℏ j(W, σ) Mj(W) has the structure of a constructible sheaf, whose rank and singularity set depend explicitly on the choice of noncommutative deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' In particular, we describe the geometry of noncom- mutative deformations of some crepant resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' It is at this point yet unclear how these compare with Van den Bergh’s noncommutative crepant resolutions [V].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' To each Poisson structure σ on Wk, with k = 1 or k = 2 there corresponds a noncommutative deformation (Wk, Aσ) with Aσ = (O[[ℏ]], ⋆σ) where ⋆σ is the star product corresponding to σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' All of these Poisson structures were described in [BGKS] in terms of generators over global functions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' when σ is one of such generators, we refer to it as a basic Poisson structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' There exist Poisson structures for which all brackets vanish on the first formal neighbourhood of P1 ⊂ Wk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' we call them extremal Poisson structures, they behave very differently from the basic ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Our main results are: Theorem (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='4,6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Let k = 1 or 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' If σ is an extremal Poisson structure on Wk, then the quantum moduli space Mℏ j(Wk, σ) can be viewed as the ´etale space of a constructible sheaf Ek of generic rank 2j − k − 1 over the classical moduli space Mj(Wk) with singular stalks of all ranks up to 4j − k − 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' If σ′ is another Poisson structure on Wk, then the corresponding sheaf E′ k is a subsheaf of Ek, with the smallest possible sheaf occurring for basic Poisson structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Theorem (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='2,6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Let k = 1 or 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' If σ is a basic Poisson structure on Wk, then the quantum moduli space Mℏ j(Wk, σ) and its classical limit are isomorphic: Mℏ j(Wk, σ) ≃ Mj(Wk) ≃ P4j−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Therefore, comparing these results, we see that the choice of Poisson structure has a strong influence on the geometry of the quantum moduli space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Noncommutative deformations A holomorphic Poisson structure on a complex manifold (or smooth complex algebraic variety) X is given by a holomorphic bivector field σ ∈ H0(X, Λ2TX) whose Schouten–Nijenhuis bracket [σ, σ] ∈ H0(X, Λ3TX) is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' The associated Poisson bracket is then given by the pairing ⟨ · , · ⟩ between vector fields and forms {f, g}σ = ⟨σ, df ∧ dg⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' To obtain a noncommutative deformation of X one must first promote the Poisson structure to a star product on X, that is, a C[[ℏ]]-bilinear associative product ⋆: OX[[ℏ]] × OX[[ℏ]] → OX[[ℏ]] which is of the form f ⋆ g = fg + �∞ n=1 Bn(f, g) ℏn where the Bn are bidifferential operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' The pair (X, ⋆σ) is called a deformation quantization of (X, σ) when the star product on X satisfies B1(f, g) = {f, g}σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' For a holomorphic Poisson manifold (X, σ) with associated Poisson bracket { · , · }σ, the sheaf of formal functions with holomorphic coefficients on the quantization (X, ⋆σ) is Aσ := (O[[ℏ]], ⋆σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' QUANTIZATION OF CALABI–YAU THREEFOLDS 3 We call Wk(σ) = (Wk, Aσ) a noncommutative deformation of Wk, and a vector bundle on a noncommutative deformation is by definition a locally free sheaf of Aσ-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' These vector bundles and their moduli are our objects of study here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' When we work with a fixed Poisson structure, we use the abbreviated notations A, { · , · } and ⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' We also use the cut to order n represented as A(n) = O[[ℏ]]/ℏn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' The existence of star products on Poisson manifolds was proven in the seminal papers of Kontse- vich [Ko1, Ko2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' For a complex algebraic variety X with structure sheaf OX, if both H1(X, OX) and H2(X, OX) vanish, then there is a bijection {Poisson deformations of OX}/∼ ↔ {associative deformations of OX}/∼ where ∼ denotes gauge equivalence [Y, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' These cohomological hypothesis are verified in the cases of W1 and W2 (but not for W3, see App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' We now recall the basic properties of Poisson structures on Wk or k = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' All Poisson structures on Wk may be described by giving their generators over global functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' This is a consequence of the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' [BGKS, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' 1] Let X be a smooth complex threefold and σ a Poisson structure on X, then fσ is integrable for all f ∈ O(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Local Calabi–Yau threefolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' For k ≥ 1, we set Wk = Tot(OP1(−k) ⊕ OP1(k − 2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' The canonical charts for the complex manifold structure of Wk is obtained by gluing the open sets U = C3 {z,u1,u2} and V = C3 {ξ,v1,v2} by the relation (ξ, v1, v2) = (z−1, zku1, z−k+2u2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' All Poisson structures on W1 can be obtained using the following generators [BGKS, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='2] σ1 = ∂z ∧ ∂u1, σ2 = ∂z ∧ ∂u2, σ3 = u1∂u1 ∧ ∂u2 − z∂z ∧ ∂u2, σ4 = u2∂u1 ∧ ∂u2 + z∂z ∧ ∂u1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' The W1-Poisson structures σ1, σ2, σ3, σ4 are pairwise isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' All Poisson structures on W2 can be obtained using the following generators [BGKS, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' 3] σ1 = ∂z ∧ ∂u1, σ2 = ∂z ∧ ∂u2, σ3 = z∂z ∧ ∂u2, σ4 = u1∂u1 ∧ ∂u2, σ5 = 2zu1∂u1 ∧ ∂u2 − z2∂z ∧ ∂u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' The W2-Poisson structures σ2 and σ5 on are isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Moreover, the Poisson structures σ1, σ2, σ3, σ4 on W2 are pairwise inequivalent, giving 4 distinct Poisson manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Vector bundles on noncommutative deformations To discuss moduli of vector bundles on noncommutative deformations of Wk, for k = 1 or 2 we will consider those bundles that are formally algebraic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' We say that p = � pnℏn ∈ O[[ℏ]] is formally algebraic if pn is a polynomial for every n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' We say that a vector bundle over (Wk, σ) is formally algebraic if it is isomorphic to a vector bundle given by formally algebraic transition functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' In addition, if there exists N such that pn = 0 for all n > N, we then say that p is algebraic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Let A be a deformation quantization of O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Then an A-module S is acyclic if and only if S = S/ℏS is acyclic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' QUANTIZATION OF CALABI–YAU THREEFOLDS 4 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Consider the short exact sequence 0 −→ S ℏ −→ S −→ S −→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' It gives, for j > 0 surjections Hj(X, S) ℏ −→ Hj(X, S) −→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' This immediately implies that Hj(X, S) = 0 for j > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' The converse is immediate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' □ Notation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Let Wk be a noncommutative deformation of Wk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Denote by A(j) the line bundle over Wk with transition function z−j, hence the pull back of O(j) on P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' For k = 1, 2 any line bundle on Wk is isomorphic to A(j) for some j ∈ Z, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=', Pic(Wk) = Z when k = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Let f = f0 + �∞ n=1 �fn ℏn ∈ A∗(U ∩ V ) be the transition function for the line bundle L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Then there exist functions a0 ∈ O∗(U) and α0 ∈ O∗(V ) such that α0f0a0 = z−j and viewing a0 resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' α0 as elements in A∗(U) resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' A∗(V ) one has α0 ⋆ f ⋆ a0 = z−j + �∞ n=1 fnℏn for some fn ∈ O(U ∩ V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' We may thus assume that the transition function of L is z−j + �∞ n=1 fnℏn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' To give an isomorphism L ≃ A(j) it suffices to define functions an ∈ O(U) and αn ∈ O(V ) satisfying �1 + �∞ n=1 αnℏn� ⋆ �z−j + �∞ n=1 fnℏn� ⋆ �1 + �∞ n=1 anℏn� = z−j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='5) Collecting terms by powers of ℏ, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='5) is equivalent to the system of equations Sn + z−jan + z−jαn = 0 n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' where Sn is a finite sum involving fi, Bi for i ≤ n, but only ai, αi for i < n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' The first terms are S1 = f1 S2 = f2 + α1f1 + a1f1 + B1 �α1, z−j� + B1 �z−j, a1 � + α1z−ja1 S3 = f3 + B2 �α1, z−j� + B2 �z−j, a1 � + B1 ���2, z−j� + B1 �z−j, a2 � + B1 �α1, f1 � + B1 �α1, z−ja1 � + B1 �z−j, a1 � + α2f1 + α2z−ja1 + α1f2 + α1f1a1 + α1z−ja2 + f2a1 + f1a2 Since by Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='1 we have H1(Wk, O) = 0 when k = 1, 2, we can solve these equations recur- sively, by defining an to cancel out all terms of zjSn having positive powers of z and setting αn = zjSn − an.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' □ Note that this is essentially the same proof as [BG, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='7], and it does not work for k ≥ 3, in fact Pic(W3) is much larger, see Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' We now consider vector bundles of higher rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' For k = 1, 2, vector bundles over Wk(σ) are filtrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' This is a generalisation of Ballico–Gasparim–K¨oppe [BGK1, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='2] to the noncom- mutative case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Let E be a sheaf of A-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='2 gives that the classical limit E0 = E/ℏE is acyclic as a sheaf of A-modules (and equivalently as a sheaf of O-modules) if and only if E is acyclic as a sheaf of A-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Filtrability for a bundle E over Wk, for k = 1, 2 was proved in [K] and is obtained from the vanishing of cohomology groups Hi(Wk, E ⊗ SymnN ∗) for i = 1, 2, where N ∗ is the conormal QUANTIZATION OF CALABI–YAU THREEFOLDS 5 bundle of ℓ ⊂ Wk and n > 0 are integers, the proof proceeds by induction on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' In the noncommutative case, let S denote the kernel of the projection A(n) → A(n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' By construction we have that S/ℏS = SymnN ∗ and the required vanishing of cohomologies is guaranteed by Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' □ The analogous proof does not work for W3, see [K, Rem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' It is unknown whether bundles on Wk are filtrable when k ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' There are also some particular features happening only when k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Every holomorphic vector bundle on W1 is algebraic [K, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='10], and W1 is formally rigid [GKRS, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' In contrast, if k > 1, then Wk has as infinite-dimensional family of deformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' In particular, a deformation family for W2 can be given by (ξ, v1, v2) = � z−1, z2u1 + z � j>0 tjuj 2, u2 � [GKRS, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' 13] and this family contains infinitely many distinct manifolds [BGS, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Furthermore, for k > q > 0, Wk can be deformed to Wq [BGS, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' For each Poisson manifold (Wk, σ), we want to study moduli spaces of vector bundles over (Wk, ⋆) where ⋆ is the corresponding star product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' [K, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='1] showed that a rank 2 bundle E on Wk with first Chern class c1(E) = 0 is deter- mined by a canonical transition matrix �zj p 0 z−j � where, using ǫ = 0, 1 we have: p = 2j−2 � s=ǫ 2j−2−s � i=1−ǫ j−1 � l=i+s−j+1 pliszlui 1us 2 for k = 1, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='8) and p = ∞ � s=ǫ j−1 � i=1−ǫ j−1 � l=2i−j+1 pliszlui 1us 2 for k = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='9) Accordingly, for a noncommutative deformation (Wk, σ) we define the notion of canonical tran- sition matrix as: T = �zj p 0 z−j � with p = ∞ � n=0 pnℏn ∈ Ext1(A(j), A(−j)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='10) Where we have that each pn can be given the same canonical form of the classical case, which can be seen using: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Let A be a deformation quantization of OWk with k = 1 or 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' There is an injective map of C-vector spaces Ext1 A(A(j), A(−j)) ∞ � n=0 Ext1 O(O(j), O(−j))ℏn ≃ Ext1 O(O(j), O(−j))[[ℏ]] p = p0 + ∞ � n=1 pnℏn (p0, p1ℏ, p2ℏ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' ) where pi ∈ Ext1(O(j), O(−j)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Ext1 A(A(j), A(−j)) is the quotient of Ext1 O(O(j), O(−j))[[ℏ]] by the relations qn ≃ qn + � pipn−i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' □ QUANTIZATION OF CALABI–YAU THREEFOLDS 6 We wish to describe the structure of moduli spaces of vector bundles on Wk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Using the results of this section, we may proceed analogously to the classical (commutative) setup, to extract moduli spaces out of extension groups of line bundles, by considering extension classes up to bundle isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Moduli of bundles on noncommutative deformations We recall the notion of isomorphism of vector bundles on a noncommutative deformation of Wk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Let E and E′ be vector bundles over (Wk, σ) defined by transition matrices T and T ′ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' An isomorphism between E and E′ is given by a pair of matrices AU and AV with entries in Aσ(U) and Aσ(V ), respectively, which are invertible with respect to ⋆ and such that T ′ = AV ⋆ T ⋆ AU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Notation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Denoting by Ext1 Alg(A(j), A(−j)) the subset of formally algebraic extension classes, we denote by Mj(Wk) the quotient Mj(Wk) := Ext1 Alg(A(j), A(−j))/∼ consisting of those classes of formally algebraic vector bundles (Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='1), whose classical limit is a stable vector bundle of charge j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Here ∼ denotes bundle isomorphism as in Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='1 and following [BGK2] stability means that the classical limit does not split on the 0-th formal neighbourhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' We denote by Mℏn j (Wk, σ) the moduli of bundles obtained by imposing the cut-off ℏn+1 = 0, that is, the superscript ℏn means quantised to level n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Note that Mj(Wk, σ) := Mℏ0 j (Wk, σ) = Mj(Wk) recovers the classical moduli space obtained when ℏ = 0, while Mℏ j(Wk, σ) denotes the moduli on the first order quantization, which will be the focus of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Accordingly: Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' We call Mj(Wk, σ) the classical moduli space and Mℏ j(Wk, σ) the quantum moduli space of bundles on Wk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' [BGS, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='7] The classical moduli spaces of vector bundles of rank 2 and splitting type j on Wk has dimension 4j − 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' The splitting type of a vector bundle E on (Wk, σ) is the one of its classical limit [BG, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Hence, when the classical limit is an SL(2, C) bundle, the splitting type of E is the smallest integer j such that E can be written as an extension of A(j) by A(−j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' We fix a splitting type j and look at rank 2 bundles on the first formal neighbourhood ℓ(1) of ℓ ≃ P1 ⊂ W1 together with their extensions up to first order in ℏ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' We now calculate isomorphism classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Let p + p′ℏ and q + q′ℏ be two extension classes in Ext1 A(A(j), A(−j)) which are of splitting type j, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' in canonical U-coordinates p, p′, q, q′ are multiples of u1, u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' According to Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='1 bundles defined by p + p′ℏ and q + q′ℏ are isomorphic, if there exist invertible matrices �a + a′ℏ b + b′ℏ c + c′ℏ d + d′ℏ � and �α + α′ℏ β + β′ℏ γ + γ′ℏ δ + δ′ℏ � whose entries are holomorphic on U and V , respectively, such that �α + α′ℏ β + β′ℏ γ + γ′ℏ δ + δ′ℏ � ⋆ �zj q + q′ℏ 0 z−j � = �zj p + p′ℏ 0 z−j � ⋆ �a + a′ℏ b + b′ℏ c + c′ℏ d + d′ℏ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='6) QUANTIZATION OF CALABI–YAU THREEFOLDS 7 We wish to determine the constraints such an isomorphism imposes on the coefficients of q and q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' This is more conveniently rewritten by multiplying by the right-inverse of � zj q+q′ℏ 0 z−j � , which (modulo ℏ2) is �z−j −q − q′ℏ + 2z−j{zj, q}ℏ 0 zj � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' We have that the zero section ℓ ≃ P1 is cut out inside Wk by u1 = u2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Hence, the n-th formal neighbourhood of ℓ is by definition ℓ(n) = OW1 In+1 where I =< u1, u2 >.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' So, on ℓ(1) we have that u2 1 = u2 2 = u1u2 = 0 and therefore we may write a = a0 + a1 1u1 + a2 1u2, α = α0 + α1 1u1 + α2 1u2, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=', where ai 1, αi 1, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' are holomorphic functions of z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Following the details of the proof of [G, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='3] we assume in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='6) that a0 = α0, d0 = δ0 are constant and b = β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Since we already know that on the classical limit the only equivalence on ℓ(1) is projectivization [G, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='2], we assume p = q, keeping in mind a projectivization to be done in the end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' We may also assume that the determinants of the changes of coordinates on the classical limit are 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Accordingly, we rewrite (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='6) as: �α + α′ℏ β′ℏ γ + γ′ℏ δ + δ′ℏ � = �zj p + p′ℏ 0 z−j � ⋆ �a + a′ℏ b′ℏ c + c′ℏ d + d′ℏ � ⋆ �z−j −p − q′ℏ + 2{zj, p}z−jℏ 0 zj � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='7) where a0 = d0 = α0 = δ0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Since we already know the moduli in the classical limit, we only need to study terms containing ℏ, which after multiplying are: (1, 1) = a′ + {zja, z−j} + {zj, a}z−j + {pc, z−j} + {p, c}z−j + (pc′ + p′c)z−j (1, 2) = {p, d}zj − {a, p}zj − {zj, a}p + {zj, p}a + {pd, zj} + 2z−j{zj, p}pc + z2jb′ − (pa′ + q′a)zj + (pd′ + p′d)zj − (pc′ + p′c + q′c)p (2, 1) = z−2jc′ (2, 2) = d′ + {z−jd, zj} + {z−j, d}zj − {z−jc, p} − {z−j, c}p − (pc′ + q′c)z−j + 2{zj, p}z−2jc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' All four terms must be adjusted using the free variables to only contain expressions which are holomorphic on V to satisfy (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' For example, in the (2, 1) term this condition is satisfied precisely when c′ is a section of O(2j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Computing Poisson brackets, we see that the (1, 1) and (2, 2) terms can always be made holomorphic on V by appropriate choices of c and d′, leaving the coefficients of a′ free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' We will need to use these free coefficients for the next step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' It remains to analyse the (1, 2) term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Because we are working on the first formal neighbourhood of ℓ, terms in u2 1, u1u2, u2 2 or higher vanish (recall that we assume that p, p′, q′ are multiples of u1 or u2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Since z2jb′ is there to cancel out any possible terms having power of z greater or equal to 2j, we remove it from the expression, keeping in mind that we only need to cancel out the coefficients of the monomials ziu1 and ziu2 with i ≤ 2j − 1 in the expression: (1, 2) = {p, d+a}zj −{zj, a}p+{zj, p}a+{pd, zj}+2z−j{zj, p}pc+p(d′−a′)zj +(p′−q′)zj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' (�) To determine the quantum moduli spaces, we must verify what restrictions are imposed on q′ so that p′ and q′ define isomorphic bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Since this requires computing brackets, the analysis must be carried out separately for each noncommutative deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' QUANTIZATION OF CALABI–YAU THREEFOLDS 8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Quantum moduli of bundles on W1 The Calabi–Yau threefold we consider in this section is the crepant resolution of the conifold singularity xy − zw = 0, that is, W1 := Tot(OP1(−1) ⊕ OP1(−1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' We will carry out calculations using the canonical coordinates W1 = U ∪ V where U ≃ C3 ≃ V with U = {z, u1, u2}, V = {ξ, v1, v2}, and change of coordinates on U ∩ V ≃ C∗ × C × C given by �ξ = z−1 , v1 = zu1 , v2 = zu2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Consequently, global functions on W1 are generated over C by the monomials 1, u1, zu1, u2, zu2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' For each specific noncommutative deformation (W1, Aσ), we wish to compare the quantum and classical moduli spaces of vector bundles, see Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' This is part of the general quest to understand how deformations of a variety affect moduli of bundles on it, and it is worth noting that no commutative deformation of W1 is known to exits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' For a rank 2 bundle E on a noncommutative deformation W1 with a canonical matrix � zj p 0 z−j � as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='10) where p = �∞ n=0 pnℏn, expression (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='8) gives us the general form of the coefficients pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' In particular, on the first formal neighbourhood, we have: p = j−1 � l=−j+2 pl10zlu1 + j−1 � l=−j+2 pl01zlu2, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='1) where p = 0 if j = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Each noncommutative deformation comes from some Poisson structure which determines the first order terms of the corresponding star product, see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' The most basic Poisson structures σ on W1 are those which generate all others over global functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' We call these generators the basic Poisson structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' If σ is a basic Poisson structure on W1, then the quantum moduli space Mℏ j(Wk, σ) and its classical limit are isomorphic: Mℏ j(W1, σ) ≃ Mj(W1) ≃ P4j−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' We perform the computations using the bracket σ1 = ∂z ∧ ∂u1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' the choice of such a generator is irrelevant, since all the 4 generators give pairwise isomorphic Poisson manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' To obtain an isomorphism, we need to cancel out all coefficients of the terms z2u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' , z2j−1u1 and z2u2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' , z2j−1u2 appearing in expression �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Calculating σ1 brackets, we have {zj, f} = jzj−1 ∂f ∂u1 , and following expressions for a and d coming from the classical part a = 1 + a1 1u1 + a2 1u2, d = 1 − a1 1u1 − a2 1u2, where ai 1 and di 1 are functions of z, gives ∂a ∂u1 = a1 1, ∂d ∂u1 = −a1 1, so that {p, d} − {a, p}zj = −2 �∂p ∂za1 1 − ∂a ∂z ∂p ∂u1 � zj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Therefore, expression � becomes � = −2 �∂p ∂z a1 1 − ∂a ∂z ∂p ∂u1 � zj + 2j � ∂p ∂u1 (a1 1u1 + a2 1u2 + pc) � zj−1 + p(d′ − a′)zj + (p′ − q′)zj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Now we need to cancel out separately the coefficients of each monomial ziu1 and ziu2 for 2 ≤ i ≤ 2j − 1, that is, all those terms potentially giving nonholomorphic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' To determine QUANTIZATION OF CALABI–YAU THREEFOLDS 9 the classes in the moduli space we need to verify what constraints are imposed on q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Take for instance the monomial ziu1 in (p′d − q′a)zj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Since a′ remains free we can always choose its corresponding coefficient in order to cancel out the term in ziu1 in the entire expression of (1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Indeed, notice that the expressions p(d′ − a′)zj and (p′d − q′a)zj contain monomials of the same orders, all of which may be adjusted to zero by choosing a′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Moreover the first three summands in � also contain the same list of monomials, hence may also be absorbed by the appropriate choices of coefficients of a, a′ and c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Since this process can be independently carried out for each monomial, we then conclude that the expression � can be made holomorphic on V for any choice of q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Hence, there are no restrictions on q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Thus, we obtain an equivalence p + p′ℏ ∼ p + q′ℏ for all q′ and the projection onto the classical limit (the first coordinate) π1 : Mℏ j(W1, σ) → Mj(W1) taking (p, p′) to p is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' The isomorphism type of the moduli space is given in [BGS, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='2] as P4j−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' □ We now calculate the quantum moduli space for the particular choice of splitting type j = 2 and for a different choice of Poisson structure on W1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' We use the notation p ∈ Mj(W1) to refer to a point in the classical moduli space, that is, a rank 2 bundle is labelled by its extension class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='3 (j = 2 and σ = u1σ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Here we write p = p0zu1 + p1u1 + p2zu2 + p3u2, p′ = p′ 0zu1 + p′ 1u1 + p′ 2zu2 + p′ 3u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' for the first order part of the extension class, where we have renamed the coefficients to simplify notation ( p0 := p110, p1 := p010, p2 := p101, p3 := p001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='1 implies that σ = u1σ1 is also a Poisson structure on W1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' With this choice, all brackets acquire an extra u1 in comparison to the bracket σ1 used in the proof of Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='2, so that in the first formal neighbourhood the (1, 2)-term described in � simplifies to just: � = z2p(d′ − a′) + z2(p′ − q′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Here a′ = a′ 0 + a′1u1 + a′2u2, d′ = d′ 0 − d′1u1 − d′2u2, so that d′ − a′ = (d′ − a′)0 + (d′ − a′)1u1 + (d′ − a′)2u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Hence, the total expression of � is � = (p0z3u1 + p1z2u1 + p2z3u2 + p3z2u2)((d′ − a′)0 + (d′ − a′)1u1 + (d′ − a′)2u2)) +(p′ 0 − q′ 0)z3u1 + (p′ 1 − q′ 1)z2u1 + (p′ 2 − q′ 2)z3u2 + (p′ 3 − q′ 3)z2u2, where we canceled out all the monomials containing u2 1, u1u2, and u2 2, since we work on the first formal neighbourhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' We rename (d′ − a′)0(z) = λ0 + λ1z + λ2z2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' to simplify notation, and since all terms in (1, 2) having powers of z equal to 4 and higher can be cancelled out by the appropriate choice of the z2jb′, it suffices to analyse the expression � = (p0z3u1 + p1z2u1 + p2z3u2 + p3z2u2)(λ0 + λ1z) +(q′ 0 − p′ 0)z3u1 + (q′ 1 − p′ 1)z2u1 + (q′ 2 − p′ 2)z3u2 + (q′ 3 − p′ 3)z2u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' To have an isomorphism q′ ∼ p′, we need to cancel out the coefficients of z3u1, z2u1, z3u2, z2u2 in � with appropriate choices of λi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Consequently, q′ ∼ p′ if and only if the following equality holds for some choice of λ0 and λ1: \uf8eb \uf8ec \uf8ec \uf8ed q′ 0 − p′ 0 q′ 1 − p′ 1 q′ 2 − p′ 2 q′ 3 − p′ 3 \uf8f6 \uf8f7 \uf8f7 \uf8f8 = λ0 \uf8eb \uf8ec \uf8ec \uf8ed p0 p1 p2 p3 \uf8f6 \uf8f7 \uf8f7 \uf8f8 + λ1 \uf8eb \uf8ec \uf8ec \uf8ed p1 0 p3 0 \uf8f6 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' QUANTIZATION OF CALABI–YAU THREEFOLDS 10 When the vectors v1 = (p0, p1, p2, p3) and v2 = (0, p1, 0, p3) are linearly independent, the point q′ belongs to the plane that passes through the point p′ with v1 and v2 as direction vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Therefore, whenever v1 and v2 are linearly independent vectors, the fibre over p = (p0, p1, p2, p3) is a copy of C4 foliated by 2-planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' The leaf containing a point p′ forms the equivalence class of p′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Thus, the moduli space over the fibre over p is parametrised by the 2-plane through the origin in the direction perpendicular to v1, v2 over the point p, except when p1 = p3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' In contrast, the fibre over a point p = (p0, 0, p2, 0) is a copy of C4 foliated by lines in the direction of v1 = (p0, 0, p2, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' In this case, the moduli space over p is parametrised by a copy of C3 perpendicular to v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' We conclude that Mℏ 2(W1, σ) → M2(W1) ≃ P3 (where the isomorphism is given by Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='4) is the ´etale space of a constructible sheaf, whose stalks have dimension 2 over the Zariski open set (p1, p3) ̸= (0, 0), and dimension 3 over the P1 cut out by p1 = p3 = 0 in P3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' The same techniques readily generalise to give a description of the quantum moduli spaces for other choices of noncommutative deformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' If σ is an extremal Poisson structure on W1, then the quantum moduli space Mℏ j(W1, σ) can be viewed as the ´etale space of a constructible sheaf of generic rank 2j − 2 over the classical moduli space Mj(W1) with singular stalks up to rank 4j − 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' We give the details of the case j = 3, for an extremal Poisson structure, that is, the case when all brackets vanish on the first formal neighbourhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' The general case is clear from these calculations, just notationally more complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' When j = 3 and σ = u1σ1, expression � becomes: � = p(d′ − a′)z3 + (p′d − q′a)z3, and we get a system of equations: � = � p0z5u1 + p1z4u1 + p2z3u1 + p3z2u1 + p4z5u2 + p5z4u2 + p6z3u2 + p7z2u2 � (λ0 + λ1z + λ2z2 + λ3z3 + λ4z4) + +(p′ − q′)0z5u1 + (p′ − q′)1z4u1 + (p′ − q′)2z3u1 + (p′ − q′)3z2u1 +(p′ − q′)4z5u2 + (p′ − q′)5z4u2 + (p′ − q′)6z3u2 + (p′ − q′)7z2u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' To have an isomorphism q′ ∼ p′, we need to cancel out the coefficients of z5u1, z4u1, z3u1, z2u1, z5u2, z4u2, z3u2, z2u2 in � with appropriate choices of λi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Consequently,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' q′ ∼ p′ if and only if the following equality holds for some choice of λ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' λ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' λ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' λ3: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='\uf8eb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='\uf8ec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='\uf8ec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='\uf8ec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='\uf8ec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='\uf8ec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='\uf8ec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='\uf8ec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='\uf8ec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='\uf8ec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='\uf8ec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='\uf8ec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='\uf8ed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='q′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='0 − p′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='q′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='1 − p′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='q′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='2 − p′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='q′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='3 − p′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='q′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='4 − p′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='q′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='5 − p′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='q′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='6 − p′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='q′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='7 − p′ ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='\uf8f7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='\uf8f7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='\uf8f7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='\uf8f7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='\uf8f7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='\uf8f7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='\uf8f8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='= λ0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='\uf8eb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='\uf8ec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='\uf8ec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='\uf8ec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='\uf8ec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='\uf8ec ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' QUANTIZATION OF CALABI–YAU THREEFOLDS 11 Consider now the family U of vector spaces over M2(W1) ≃ P7 whose fibre at p is given by Up = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed p0 p1 p2 p3 p1 p2 p3 0 p2 p3 0 0 p3 0 0 0 p4 p5 p6 p7 p5 p6 p7 0 p6 p7 0 0 p7 0 0 0 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Now, the quantum moduli space is obtained from this family after dividing by the equivalence relation ∼ over each point p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Hence Mℏ 2(W1, σ) = U/ ∼ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' We conclude that Mℏ 2(W1, σ) → M2(W1) ≃ P7 (where the isomorphism is given by Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='4) is the ´etale space of a constructible sheaf or rank 4, with stalk at p having dimension equal to the corank of Up, in this case 4 ≤ dim Mℏ 2(W1, σ)p = 8 − rk Up ≤ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' In the general case we have 2j − 2 ≤ dim Mℏ j(W1, σ)p = corank Up =≤ 4j − 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Quantum moduli of bundles on W2 The Calabi–Yau threefold we consider in this section is a crepant resolution of the singularity xy − w2 = 0 in C4, that is W2 := Tot(OP1(−2) ⊕ OP1) = Z2 × C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Similarly to what we did for W1, we will carry out calculations using the canonical coordinates W2 = U ∪V where U ≃ C3 ≃ V with U = {z, u1, u2}, V = {ξ, v1, v2}, and change of coordinates on U ∩ V ≃ C∗ × C × C given by �ξ = z−1 , v1 = z2u1 , v2 = u2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Consequently, global holomorphic functions on W2 are generated by 1, u1, zu1, z2u1, u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' For each specific noncommutative deformation (W2, Aσ), we wish to compare the quantum and classical moduli spaces of vector bundles, see Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' For a rank 2 bundle E on a noncommutative deformation W2 with a canonical matrix � zj p 0 z−j � as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='10) where p = �∞ n=0 pnℏn, expression (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='8) gives us the general form of the coefficients pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' In particular, on the first formal neighbourhood, we have: p = j−1 � l=−j+3 pl10zlu1 + j−1 � l=−j+1 pl01zlu2 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='1) where in case j = 1 we have only p001u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' To describe the quantum moduli for Poisson structures on W2, we consider the expression �: � = {p, d + a}zj − {zj, a}p + {zj, p}a + {pd, zj} + 2z−j{zj, p}pc + p(d′ − a′)zj + (p′d − q′a)zj, where we need to cancel out the coefficients of z3u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' , z2j−1u1 and zu2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' , z2j−1u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' QUANTIZATION OF CALABI–YAU THREEFOLDS 12 Each noncommutative deformation comes from some Poisson structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' The most basic Poisson structures σ on W2 are those which generate all others over global functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' We call these generators the basic Poisson structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Now, we compute the quantum moduli of bundles for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' We observe that the 4 Poisson manifolds (W2, σi) for i = 1, 2, 3, 4, are pairwise nonisomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' This can be verified by the table of their degeneracy loci: W2 Poisson structures bracket degeneracy σ1 σ2 ∅ σ3 σ4 ∪ Nevertheless, the 4 quantum moduli spaces defined by these basic Poisson structures turn out to be all isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' If σ is a basic Poisson structure on W2, then the quantum moduli space Mℏ j(Wk, σ) and its classical limit are isomorphic: Mℏ j(W2, σ) ≃ Mj(W2) ≃ P4j−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' We carry out calculations for the basic bracket σ4 = u1∂u1 ∧ ∂u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' It does turn out that the result is the same for the the basic brackets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' The calculation for σ4 is shorter, since any of the brackets having one entry equal to zj vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Because we work on the first formal neighbourhood, we also remove the expressions that are quadratic in the ui variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' So, the expression � that remains to be analysed simplifies to: � = {p, d + a}zj + p(d′ − a′)zj + (p′d − q′a)zj, where we must cancel out the coefficients of the monomials z3u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' , z2j−1u1 and zu2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' , z2j−1u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' On the first formal neighbourhood, we write a = 1 + a1(z)u1 + a2(z)u2, d = 1 + d1(z)u1 + d2(z)u2, and a′ = a′ 0(z) + a′ 1(z)u1 + a′ 2(z)u2, d′ = d′ 0(z) + d′ 1(z)u1 + d′ 2(z)u2, so that the partials are ∂uia = ai(z) ∂uid = di(z) and ∂u2a = a2(z) ∂u2d = d2(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' The extension class given in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='9) becomes p = j−1 � l=3−j pl10zlu1 + j−1 � l=1−j pl01zlu2, and computing the bracket gives {p, d + a}zj = \uf8eb \uf8ed j−1 � l=3−j pl10zl \uf8f6 \uf8f8 (d2(z) + a2(z))zju1 + \uf8eb \uf8ed j−1 � l=1−j pl01zl \uf8f6 \uf8f8 (d1(z) + a1(z))zju1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' To work with a simpler notation, we present details of � when j = 2, in which case we can express the extension class as p = p0zu1 + p1zu2 + p2u2 + p3z−1u2, QUANTIZATION OF CALABI–YAU THREEFOLDS 13 having renamed the coefficients for simplicity (making p0 := p110, p1 := p101, p2 := p001, p3 := p−101).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' We will point out the steps for generalising to higher j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Assuming j = 2, we have {p, d + a}z2 = p0(d2(z) + a2(z))z3u1 + (p1z3 + p2z2 + p3z)(d1(z) + a1(z))u1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' To obtain equivalence between q′ and p′, we must cancel out coefficients of z3u1, zu2, z2u2, z3u2 in the expression of �, which becomes � = p0(d2(z) + a2(z))z3u1 + (p1z3 + p2z2 + p3z)(d1(z) + a1(z))u1 +(p0z3u1 + p1z3u2 + p2z2u2 + p3zu2)(d′ 0(z) − a′ 0(z)) +(p′ 0z3u1 + p′ 1z3u2 + p′ 2z2u2 + p′ 3zu2) −(q′ 0z3u1 + q′ 1z3u2 + q′ 2z2u2 + q′ 3zu2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Since the highest power of z to be considered is 3, we observe that d2(z) + a2(z) may be chosen conveniently, we cancel out all terms in z3u1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' We may also choose d1(z) + a1(z) = 0, leaving � = (p1z3u2 + p2z2u2 + p3zu2)(d′ 0(z) − a′ 0(z)) +(p′ 1z3u2 + p′ 2z2u2 + p′ 3zu2) −(q′ 1z3u2 + q′ 2z2u2 + q′ 3zu2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Now we may choose d′ 0 − a′ 0 appropriately to cancel out all terms in u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' We conclude that there are no conditions imposed on q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' In other words, here p + p′ℏ is equivalent to p + q′ℏ for any choice of q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Hence, the quantum and classical moduli spaces are isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' The generalisation to higher j works out similarly, we can first choose di + ai for i > 0 to cancel out the coefficients of u1 and then choose d′ 0 −a′ 0 to take care of the coefficients of u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' So, for all j using the bracket σ4 we conclude that the quantum and classical moduli spaces are isomorphic Mℏ j(W2, σ4) ≃ Mj(W2) ≃ P4j−5 where the second isomorphism is proven in [K, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' □ Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Now choose any Poisson structure of W2 for which all brackets in � vanish on neighbourhood 1, for example σ = u1σ4 = u2 1∂u1 ∧ ∂u2 works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' In such a case, the expression for � reduces to: � = p(d′ − a′)zj + (p′d − q′a)zj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Now, consider the case of j = 2, when we have: � = (p0z3u1 + p1z3u2 + p2z2u2 + p3zu2) + (d′ 0(z) − a′ 0(z)) +(p′ 0z3u1 + p′ 1z3u2 + p′ 2z2u2 + p′ 3zu2) −(q′ 0z3u1 + q′ 1z3u2 + q′ 2z2u2 + q′ 3zu2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Setting d′ 0(z) − a′ 0(z) = λ0 + λ1z + λ2z2, we get a system of equations: \uf8eb \uf8ec \uf8ec \uf8ed q′ 0 − p′ 0 q′ 1 − p′ 1 q′ 2 − p′ 2 q′ 3 − p′ 3 \uf8f6 \uf8f7 \uf8f7 \uf8f8 = \uf8eb \uf8ec \uf8ec \uf8ed λ0 0 0 0 0 λ0 λ1 λ2 0 0 λ0 λ1 0 0 0 λ0 \uf8f6 \uf8f7 \uf8f7 \uf8f8 \uf8eb \uf8ec \uf8ec \uf8ed p0 p1 p2 p3 \uf8f6 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Since we can choose λ1 and λ2 to solve the second and third equations, we see that q′ 1 and q′ 2 are free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Hence (q′ 0, q′ 1, q′ 2, q′ 3) ∼ λ0(q′ 0, ∗, ∗, q′ 3), and our system of equations reduces to �q′ 0 − p′ 0 q′ 3 − p′ 3 � = λ0 �p0 p3 � , QUANTIZATION OF CALABI–YAU THREEFOLDS 14 which is the parametric equation of a line in the (q′ 0, q′ 3)-plane whenever (p0, p3) ̸= (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' The entire question of moduli now reduces to the 2-dimensional case, disregarding p1, p2 coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' If (p0, p3) ̸= (0, 0), then the equivalence class of q′ in the fibre over the point p is the 1-dimensional subspace L directed by the vector (p0, p3) and passing through (q′ 0, q′ 3) in the (p′ 0, p′ 3)-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' If p0 = p3 = 0, then we must have the equality (q′ 0, q′ 3) = (p′ 0, p′ 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' So, its the equivalence class consists of a single point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Accordingly, the set of equivalence classes over p can be represented either by the line L⊥ by the origin perpendicular to L (directed by (−p3, p0) when (p0, p3) ̸= (0, 0) or else by the entire (p′ 0, p′ 3)-plane over (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' We conclude that Mℏ 2(W2, σ) → M2(W2) ≃ P3 (where the isomorphism is given by Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='4) is the ´etale space of a constructible sheaf, whose stalks have dimension 1 over the Zariski open set (p0, p3) ̸= (0, 0), and dimension 2 over the P1 cut out by p0 = p3 = 0 in P3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' In fact, we could express this moduli space as a sheaf given by an extension of OP3(+1) by a torsion sheaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' If σ is an extremal Poisson structure on W2, then the quantum moduli space Mℏ j(W2, σ) can be viewed as the ´etale space of a constructible sheaf of generic rank 2j − 3 over the classical moduli space Mj(W2) with singular stalks up to rank 4j − 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Now, for j = 3, we write down the extremal example when the brackets vanish on the first formal neighbourhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' The generalisation of the extremal cases to all j becomes clear from this example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Where, assuming all brackets vanish on the first formal neighbourhood, we need to cancel out the coefficients of z3u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' , z2j−1u1 and zu2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' , z2j−1u2 in � = p(d′ − a′)zj + (p′d − q′a)zj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' For j = 3 we have p = 2 � l=0 pl10zlu1 + 2 � l=−2 pl01zlu2, which we rewrite as p = p0z2u1 + p1zu1 + p2u1 + p3z2u2 + p4z1u2 + p5u2 + p6z–1u2 + p7z−2u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Setting d′ 0(z) − a′ 0(z) = λ0 + λ1z + λ2z2 + λ3z3 + λ4z4, expression � = p(d′ − a′)z3 + (p′d − q′a)z3 becomes � = � p0z5u1 + p1z4u1 + p2z3u1 + p3z5u2 + p4z4u2 + p5z3u2 + p6z2u2 + p7zu2 � (λ0 + λ1z + λ2z2 + λ3z3 + λ4z4) +(p′ − q′)0z5u1 + (p′ − q′)1z4u1 + (p′ − q′)2z3u1 +(p′ − q′)3z5u2 + (p′ − q′)4z4u2 + (p′ − q′)5z3u2 + (p′ − q′)6z2u2 + (p′ − q′)7zu2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' To start with we notice that λ3 and λ4 can always be chosen to solve the equations involving q′ 3 and q′ 4 so that these 2 coordinates can take any value, that is, there are isomorphisms (q′ 0, q′ 1, q′ 2, q′ 3, q′ 4, q′ 5, q′ 6, q′ 7) ∼ (q′ 0, q′ 1, q′ 2, ∗, ∗, q′ 5, q′ 6, q′ 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' QUANTIZATION OF CALABI–YAU THREEFOLDS 15 Consequently,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' we may remove q′ 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' q′ 4 and rewrite the reduced system as: \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed q′ 0 − p′ 0 q′ 1 − p′ 1 q′ 2 − p′ 2 q′ 5 − p′ 5 q′ 6 − p′ 6 q′ 7 − p′ 7 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 = λ0 \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed p0 p1 p2 p5 p6 p7 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 + λ1 \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed p1 p2 0 p6 p7 0 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 + λ2 \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed p2 0 0 p7 0 0 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Here q′ ∼ p′ if and only if the equality holds for some choice of λ0, λ1, λ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Consider now the family U of vector spaces over M2(W2) ≃ P7 whose fibre at p is given by Up = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed p0 p1 p2 p1 p2 0 p2 0 0 p5 p6 p7 p6 p7 0 p7 0 0 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Now, the quantum moduli space is obtained from this family after dividing by the equivalence relation ∼ over each point p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Hence Mℏ 2(W2, σ) = U/ ∼ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' We conclude that Mℏ 2(W2, σ) → M2(W2) ≃ P7 (where the isomorphism is given by Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='4) is the ´etale space of a constructible sheaf, with stalk at p having dimension equal to the corank of Up, in this case 3 ≤ dim Mℏ 2(W2, σ)p = corank Up = 6 − rk Up ≤ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' In the general case we then have 2j − 3 ≤ dim Mℏ j(W2, σ)p = corank Up = 2j − rk Up ≤ 4j − 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' □ Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Computations of H1 Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' H1(W1, O) = H1(W2, O) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' A 1-cocycle τ ∈ O(U ∩ V ) may be written in the form τU = ∞ � l=−∞ ∞ � i=0 ∞ � s=0 τliszlui 1us 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Since terms containing only positive powers of z are holomorphic on the U-chart τU ∼ −1 � l=−∞ ∞ � i=0 ∞ � s=0 τliszlui 1us 2, where ∼ denotes cohomological equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Changing to V coordinates we have τV = −1 � l=−∞ ∞ � i=0 ∞ � s=0 τlisξ−l+ki+(−k+2)svi 1vs 2, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='2) where, for k = 1, 2 exponents of ξ are non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Thus, τV is holomorphic on V , and τ ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' □ Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' H1(W3, O) is infinite dimensional over C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' QUANTIZATION OF CALABI–YAU THREEFOLDS 16 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' As in the proof of Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='1 we arrive at the expression (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content='2) for the 1-cocycle τ on the V -chart, which in the case k = 3, gives τV ∼ −1 � l=−∞ ∞ � i=0 ∞ � s=0 τlisξ−l+3i−svi 1vs 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' The terms that are not holomorphic on V are all of those satisfying −l + 3i − s < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' We conclude that all terms having s > 3i − l, namely all of −1 � l=−∞ ∞ � i=0 ∞ � s=3i−l+1 τliszlui 1us 2 are nontrivial in first cohomology, so that dim H1(W3, O) = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' □ Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Ballico is a member of GNSAGA of INdAM (Italy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Gasparim acknowledges support of Vicerrector´ıa de Investigaci´on y Desarrollo Tecnol´ogico, UCN Chile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Rubilar acknowledges support of ANID-FAPESP cooperation 2019/13204-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE2T4oBgHgl3EQf5gjq/content/2301.04192v1.pdf'} +page_content=' Suzuki was supported by Grant 2021/11750-7 S˜ao Paulo Research Foundation - FAPESP.' metadata={'source': 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