Quantum deformations from toric geometry

Pinansky, Samuel

doi:10.1088/1126-6708/2006/03/055

Cited by 17 publications

(26 citation statements)

References 33 publications

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“…Since On the other hand, X fits into a canonical C * -family {X t } t∈C [15, Proposition 5.1] whose central fibre X 0 is the P 1 -bundle P(K D ⊕ C) with its zero section blown down. Since this is a toric singularity, the methods of [2] yield that {X t } t∈C is a versal deformation of X 0 and provide explicit equations for X t in P 8 , which can be found in [43]. As a result, X = X 1 is cut out by 14 quadrics in P 8 , hence, in particular, is not a complete intersection.…”

Section: From Kähler-einstein To Sasaki-einsteinmentioning

confidence: 99%

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Asymptotically conical Calabi–Yau metrics on quasi-projective varieties

Conlon

Hein

2015

Geom. Funct. Anal.

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Let X be a compact Kähler orbifold without C-codimension-1 singularities. Let D be a suborbifold divisor in X such that D ⊃ Sing(X) and −pKX = q[D] for some p, q ∈ N with q > p. Assume that D is Fano. We prove the following two main results. (1) If D is Kähler-Einstein, then, applying results from our previous paper [15], we show that each Kähler class on X \ D contains a unique asymptotically conical Ricci-flat Kähler metric, converging to its tangent cone at infinity at a rate of O(r −1−ε ) if X is smooth. This provides a definitive version of a theorem of Tian and Yau [54]. (2) We introduce new methods to prove an analogous statement (with rate O(r −0.0128 )) when X = BlpP 3 and D = Blp 1 ,p 2 P 2 is the strict transform of a smooth quadric through p in P 3 . Here D is no longer Kähler-Einstein, but the normal S 1 -bundle to D in X admits an irregular Sasaki-Einstein structure which is compatible with its canonical CR structure. This provides the first example of an affine Calabi-Yau manifold of Euclidean volume growth with irregular tangent cone at infinity.Date: July 4, 2018. 1 arXiv:1301.5312v3 [math.DG] 8 Dec 2014 Comparing the two theorems and their proofs. Many of the refinements in Theorem A (no neat or almost ample condition, all Kähler classes, the parameter c, uniqueness and symmetry) are due to an improvement of general technique in [15], partly based on important earlier contributions by van Coevering [55], whereas asymptotics of the form (1.1) are already implicit in Tian-Yau [54]. Let us point out one useful consequence of our explicit estimate (1.2).Corollary B. If X is smooth, then the best possible convergence rate λ of the Ricci-flat metrics of Theorem A to their tangent cones at infinity is always strictly greater than 1.Proof. Since N q−p D = K −p D and Pic(D) is torsion-free because π 1 (D) = 0, there exists a line bundle L with L p = N D and L q−p = K −1 D . Thus, by [32, p. 32, Corollary], q − p n with equality if and only if D = P n−1 , so that α − 1 n and λ 1 with equality if and only if D = P n−1 , N D = O(1). But in the latter case, the cone, and hence (X \ D, g c ) itself, must be isometric to flat C n .It seems reasonable to expect that α − 1 n even if X is singular. Moreover, equality should still imply that the cone is C n /Γ (see Remark A.2 for some examples where Γ = {1}), so that λ 2n by 1 Cristiano Spotti pointed out to us that this phenomenon was first observed in [41] for the classical elliptic modular curve H/PSL(2, Z), which is isomorphic to C as a variety but whose π orb 1 and Pic orb are nontrivial.

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Section: From Kähler-einstein To Sasaki-einsteinmentioning

confidence: 99%

“…Explicit equations and coordinate charts for the normal bundle to D in X. Again from [43], the following cone in C 8 with its canonical C * -action (t, Z) → tZ is C * -equivariantly isomorphic to K × D , the canonical bundle of D with its zero section blown down:…”

Section: 33mentioning

confidence: 99%

Asymptotically conical Calabi–Yau metrics on quasi-projective varieties

Conlon

Hein

2015

Geom. Funct. Anal.

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“…One is a deformation brane, corresponding to a complex structure deformation of the cone. The corresponding gauge theory was studied in detail in [25], where it was shown that the deformed chiral algebra encodes precisely the complex deformation computed according to Altmann's rules [15]. The second fractional brane allowed by the geometry is a so called supersymmetry breaking (SB) brane, which corresponds in this case to an obstructed complex deformation of the geometry.…”

Section: A3 When Baryonic Branches Are Presentmentioning

confidence: 99%

A quiver of many runaways

Argurio

Closset²

2007

J. High Energy Phys.

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We study the quantum corrections to the moduli space of the quiver gauge theory corresponding to regular and fractional D3-branes at the dP 1 singularity. We find that besides the known runaway behavior at the lowest step of the duality cascade, there is a runaway direction along a mesonic branch at every higher step of the cascade. Moreover, the algebra of the chiral operators which obtain the large expectation values is such that we reproduce Altmann's first order deformation of the dP 1 cone.

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“…I will not attempt an analysis of that here. Examples have been worked out in [20][21][22]. Conifold-like transitions are also related to large-N dualities in the A-model [23,24].…”

Section: Introductionmentioning

confidence: 99%

Deformations and D-branes

Bergman¹

2008

Advances in Theoretical and Mathematical Physics

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I discuss the relation of Hochschild cohomology to the physical states in the closed topological string. This allows a notion of deformation intrinsic to the derived category. I use this to identify deformations of a quiver gauge theory associated to a D-branes at a singularity with generalized deformations of the geometry of the resolution of the singularity. An explicit map is given from noncommutative deformations (i.e., B-fields) to terms in the superpotential.

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Quantum deformations from toric geometry

Cited by 17 publications

References 33 publications

Asymptotically conical Calabi–Yau metrics on quasi-projective varieties

Asymptotically conical Calabi–Yau metrics on quasi-projective varieties

A quiver of many runaways

Deformations and D-branes

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