Showing papers on "Mirror symmetry published in 2012"
Book•
23 Nov 2012TL;DR: In this article, the authors introduce the Matrix Integrals method and the Goulden-Jackson Formula for computing polygonal gluings, and apply it to the problem of "Mirror Symmetry in Dimension One".
Abstract: 0 Introduction: What is This Book About.- 1 Constellations, Coverings, and Maps.- 2 Dessins d'Enfants.- 3 Introduction to the Matrix Integrals Method.- 4 Geometry of Moduli Spaces of Complex Curves.- 5 Meromorphic Functions and Embedded Graphs.- 6 Algebraic Structures Associated with Embedded Graphs.- A.1 Representation Theory of Finite Groups.- A.1.1 Irreducible Representations and Characters.- A.1.2 Examples.- A.1.3 Frobenius's Formula.- A.2 Applications.- A.2.2 Examples.- A.2.3 First Application: Enumeration of Polygon Gluings.- A.2.4 Second Application: the Goulden-Jackson Formula.- A.2.5 Third Application: "Mirror Symmetry" in Dimension One.- References.
800 citations
TL;DR: In this article, a spectral curve describing torus knots and links in the B-model is proposed, which is obtained by exploiting the full Sl(2;Z) symmetry of the spectral curve of the resolved conifold, and should be regarded as the mirror of the topological D-brane associated to torus knot in the large N Gopakumar{Vafa duality.
Abstract: We propose a spectral curve describing torus knots and links in the B-model. In particular, the application of the topological recursion to this curve generates all their colored HOMFLY invariants. The curve is obtained by exploiting the full Sl(2;Z) symmetry of the spectral curve of the resolved conifold, and should be regarded as the mirror of the topological D-brane associated to torus knots in the large N Gopakumar{Vafa duality. Moreover, we derive the curve as the large N limit of the matrix model computing torus knot invariants.
224 citations
TL;DR: In particular, Gukov and Sulkowski as discussed by the authors showed that for all curves that come from geometry, their non-commutative counterparts can be determined just from the first few steps of the topological recursion.
Abstract: Exact solution to many problems in mathematical physics and quantum field theory often can be expressed in terms of an algebraic curve equipped with a meromorphic differential. Typically, the geometry of the curve can be seen most clearly in a suitable semi-classical limit, as \(\hslash \rightarrow 0\), and becomes non-commutative or “quantum” away from this limit. For a classical curve defined by the zero locus of a polynomial A(x, y), we provide a construction of its non-commutative counterpart \(\hat{A}(\hat{x},\hat{y})\) using the technique of the topological recursion. This leads to a powerful and systematic algorithm for computing \(\hat{A}\) that, surprisingly, turns out to be much simpler than any of the existent methods. In particular, as a bonus feature of our approach comes a curious observation that, for all curves that “come from geometry,” their non-commutative counterparts can be determined just from the first few steps of the topological recursion. We also propose a K-theory criterion for a curve to be “quantizable,” and then apply our construction to many examples that come from applications to knots, strings, instantons, and random matrices. The material contained in this chapter was presented at the conference Mirror Symmetry and Tropical Geometry in Cetraro (July 2011) and is based on the work: Gukov and Sulkowski, “A-polynomial, B-model, and quantization”, JHEP 1202 (2012) 070.
167 citations
Posted Content•
TL;DR: Benini et al. as discussed by the authors showed that the two-sphere partition function of such ultraviolet gauge theories yields the exact Kahler potential on the quantum Kahler moduli space for Calabi-Yau threefold target spaces.
Abstract: Many N=(2,2) two-dimensional nonlinear sigma models with Calabi-Yau target spaces admit ultraviolet descriptions as N=(2,2) gauge theories (gauged linear sigma models). We conjecture that the two-sphere partition function of such ultraviolet gauge theories -- recently computed via localization by Benini et al. and Doroud et al. -- yields the exact Kahler potential on the quantum Kahler moduli space for Calabi-Yau threefold target spaces. In particular, this allows one to compute the genus zero Gromov-Witten invariants for any such Calabi-Yau threefold without the use of mirror symmetry. More generally, when the infrared superconformal fixed point is used to compactify string theory, this provides a direct method to compute the spacetime Kahler potential of certain moduli (e.g., vector multiplet moduli in type IIA), exactly in {\alpha}'. We compute these quantities for the quintic and for Rodland's Pfaffian Calabi-Yau threefold and find agreement with existing results in the literature. We then apply our methods to a codimension four determinantal Calabi-Yau threefold in P^7, recently given a nonabelian gauge theory description by the present authors, for which no mirror Calabi-Yau is currently known. We derive predictions for its Gromov-Witten invariants and verify that our predictions satisfy nontrivial geometric checks.
129 citations
TL;DR: In this paper, a simple explicit formula for the partition function of the quiver tail T(SU(N) is given, which opens the way for the analysis of star-shaped quivers and their mirrors.
Abstract: We study $ \mathcal{N} = {4} $
quiver theories on the three-sphere. We compute partition functions using the localisation method by Kapustin et al. solving exactly the matrix integrals at finite N, as functions of mass and Fayet-Iliopoulos parameters. We find a simple explicit formula for the partition function of the quiver tail T(SU(N)). This formula opens the way for the analysis of star-shaped quivers and their mirrors (that are the Gaiotto-type theories arising from M5 branes on punctured Riemann surfaces). We provide non-perturbative checks of mirror symmetry for infinite classes of theories and find the partition functions of the T
N
theory, the building block of generalised quiver theories.
121 citations
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TL;DR: In this paper, the authors considered the topological string realization of SU(N) Chern-Simons theory on S^3 and showed that the classical function A_K(x,p;Q) contains at least as much information as knot homologies.
Abstract: We reconsider topological string realization of SU(N) Chern-Simons theory on S^3. At large N, for every knot K in S^3, we obtain a polynomial A_K(x,p;Q) in two variables x,p depending on the t'Hooft coupling parameter Q=e^{Ng_s}. Its vanishing locus is the quantum corrected moduli space of a special Lagrangian brane L_K, associated to K, probing the large N dual geometry, the resolved conifold. Using a generalized SYZ conjecture this leads to the statement that for every such Lagrangian brane L_K we get a distinct mirror of the resolved conifold given by uv=A_K(x,p;Q). Perturbative corrections of the refined B-model for the open string sector on the mirror geometry capture BPS degeneracies and thus the knot homology invariants. Thus, in terms of its ability to distinguish knots, the classical function A_K(x,p;Q) contains at least as much information as knot homologies. In the special case when N=2, our observations lead to a physical explanation of the generalized (quantum) volume conjecture. Moreover, the specialization to Q=1 of A_K contains the classical A-polynomial of the knot as a factor.
111 citations
Book•
23 Oct 2012
TL;DR: In this article, Calabi-Yau Manifolds and Mirror Symmetry have been studied in the context of Calibrated Geometry and Hyperkahler Moduli.
Abstract: I. Riemannian Holonomy Groups and Calibrated Geometry.- 1 Introduction.- 2 Introduction to Holonomy Groups.- 3 Berger's Classification of Holonomy Groups.- 4 Kahler Geometry and Holonomy.- 5 The Calabi Conjecture.- 6 The Exceptional Holonomy Groups.- 7 Introduction to Calibrated Geometry.- 8 Calibrated Submanifolds in ?n.- 9 Constructions of SL m-folds in ?m.- 10 Compact Calibrated Submanifolds.- 11 Singularities of Special Lagrangian m-folds.- 12 The SYZ Conjecture, and SL Fibrations.- II. Calabi-Yau Manifolds and Mirror Symmetry.- 13 Introduction.- 14 The Classical Geometry of Calabi-Yau Manifolds.- 15 Kahler Moduli and Gromov-Witten Invariants.- 16 Variation and Degeneration of Hodge Structures.- 17 A Mirror Conjecture.- 18 Mirror Symmetry in Practice.- 19 The Strominger-Yau-Zaslow Approach to Mirror Symmetry.- III. Compact Hyperkahler Manifolds.- 20 Introduction.- 21 Holomorphic Symplectic Manifolds.- 22 Deformations of Complex Structures.- 23 The Beauville-Bogomolov Form.- 24 Cohomology of Compact Hyperkahler Manifolds.- 25 Twistor Space and Moduli Space.- 26 Projectivity of Hyperkahler Manifolds.- 27 Birational Hyperkahler Manifolds.- 18 The (Birational) Kahler Cone.- References.
101 citations
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TL;DR: In this article, the BKMP conjecture has been verified to low genus for several toric CY3folds, and proved to all genus only for C^3 in the general case.
Abstract: The BKMP conjecture (2006-2008), proposed a new method to compute closed and open Gromov-Witten invariants for every toric Calabi-Yau 3-folds, through a topological recursion based on mirror symmetry. So far, this conjecture had been verified to low genus for several toric CY3folds, and proved to all genus only for C^3. In this article we prove the general case. Our proof is based on the fact that both sides of the conjecture can be naturally written in terms of combinatorial sums of weighted graphs: on the A-model side this is the localization formula, and on the B-model side the graphs encode the recursive algorithm of the topological recursion. One can slightly reorganize the set of graphs obtained in the B-side, so that it coincides with the one obtained by localization in the A-model.Then it suffices to compare the weights of vertices and edges of graphs on each side, which is done in 2 steps: the weights coincide in the large radius limit, due to the fact that the toric graph is the tropical limit of the mirror curve. Then the derivatives with respect to Kahler radius coincide due to special geometry property implied by the topological recursion.
96 citations
TL;DR: In this paper, the authors investigated mirror symmetry for toric Calabi-Yau manifold from the perspective of the SYZ conjecture and obtained an enumerative meaning for the (inverse) mirror maps.
Abstract: We investigate mirror symmetry for toric Calabi-Yau manifolds from the perspective of the SYZ conjecture. Starting with a non-toric special Lagrangian torus fibration on a toric Calabi-Yau manifold $X$, we construct a complex manifold $\check{X}$ using T-duality modified by quantum corrections. These corrections are encoded by Fourier transforms of generating functions of certain open Gromov-Witten invariants. We conjecture that this complex manifold $\check{X}$, which belongs to the Hori-Iqbal-Vafa mirror family, is inherently written in canonical flat coordinates. In particular, we obtain an enumerative meaning for the (inverse) mirror maps, and this gives a geometric reason for why their Taylor series expansions in terms of the Kahler parameters of X have integral coefficients. Applying the results in "A formula equating open and closed Gromov-Witten invariants and its applications to mirror symmetry," to appear in Pacific J. Math., and "A relation for Gromov-Witten invariants of local Calabi-Yau threefolds," to appear in Math. Res. Lett., we compute the open Gromov-Witten invariants in terms of local BPS invariants and give evidences of our conjecture for several 3-dimensional examples including $K_{\mathbb{P}^2}$ and $K_{\mathbb{P}^1}$.
91 citations
TL;DR: In this article, the authors decompose sphere partition functions and indices of three-dimensional N = 2 gauge theories into a sum of products involving a universal set of "holomorphic blocks".
Abstract: We decompose sphere partition functions and indices of three-dimensional N=2 gauge theories into a sum of products involving a universal set of "holomorphic blocks". The blocks count BPS states and are in one-to-one correspondence with the theory's massive vacua. We also propose a new, effective technique for calculating the holomorphic blocks, inspired by a reduction to supersymmetric quantum mechanics. The blocks turn out to possess a wealth of surprising properties, such as a Stokes phenomenon that integrates nicely with actions of three-dimensional mirror symmetry. The blocks also have interesting dual interpretations. For theories arising from the compactification of the six-dimensional (2,0) theory on a three-manifold M, the blocks belong to a basis of wavefunctions in analytically continued Chern-Simons theory on M. For theories engineered on branes in Calabi-Yau geometries, the blocks offer a non-perturbative perspective on open topological string partition functions.
88 citations
TL;DR: In this article, it was shown that the Gromov-Witten theory of Calabi-Yau hypersurfaces matches, in genus zero and after an analytic continuation, the quantum singularity theory (FJRW theory) recently introduced by Fan, Jarvis and Ruan following ideas of Witten.
Abstract: We show that the Gromov-Witten theory of Calabi-Yau hypersurfaces matches, in genus zero and after an analytic continuation, the quantum singularity theory (FJRW theory) recently introduced by Fan, Jarvis and Ruan following ideas of Witten. Moreover, on both sides, we highlight two remarkable integral local systems arising from the common formalism of Gamma-integral structures applied to the derived category of the hypersurface {W=0} and to the category of graded matrix factorizations of W. In this setup, we prove that the analytic continuation matches Orlov equivalence between the two above categories.
TL;DR: Mirror symmetry of the type II string has a beautiful generalization to the heterotic string as discussed by the authors, known as (0,2) mirror symmetry, is a field still largely in its infancy.
Abstract: Mirror symmetry of the type II string has a beautiful generalization to the heterotic string. This generalization, known as (0,2) mirror symmetry, is a field still largely in its infancy. We describe recent developments including the ideas behind quantum sheaf cohomology, the mirror map for deformations of (2,2) mirrors, the construction of mirror pairs from worldsheet duality, as well as an overview of some of the many open questions. The (0,2) mirrors of Hirzebruch surfaces are presented as a new example.
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TL;DR: In this article, the authors constructed a quantum field theory on a Calabi-Yau variety of arbitrary dimension and defined what it means to give a quantization/analyse the relation Givental's symplectic formalism for Gromov-Witten theory/and prove uniqueness of the quantization on an elliptic curve.
Abstract: Bershadsky-Cecotti-Ooguri-Vafa (BCOV) proposed that the B-model of mirror symmetry should be described by a quantum field theory on a Calabi-Yau variety, which they called the Kodaira-Spenser theory (we call it the BCOV theory). This is the first of three papers in which we construct and analyze the quantum BCOV theory. In this paper, we construct the classical field theory on a Calabi-Yau variety of arbitrary dimension/ define what it means to give a quantization/ analyze the relation Givental's symplectic formalism for Gromov-Witten theory/ prove uniqueness of the quantization on an elliptic curve/ and prove the Virasoro constraints on an elliptic curve. The second paper (arXiv:1112.4063) proves that the partition function of the quantum BCOV theory on the elliptic curve is equivalent to the Gromov-Witten theory of the mirror elliptic curve. The third paper, in progress, constructs the quantum BCOV theory on a general Calabi-Yau.
TL;DR: In this article, the authors studied the four-dimensional N = 1 effective actions of single space-time filling D p -branes in general Type IIA and Type IIB Calabi-Yau orientifold compactifications.
TL;DR: In this paper, the authors studied moduli spaces of a class of three dimensional 3D models of gauge theories which are in one-to-one correspondence with a certain set of ordered pairs of integer partitions.
Abstract: We study moduli spaces of a class of three dimensional $ \mathcal{N} = 4 $
gauge theories which are in one-to-one correspondence with a certain set of ordered pairs of integer partitions. It was found that these theories can be realised on brane intervals in Type IIB string theory and can therefore be described using linear quiver diagrams. Mirror symmetry was known to act on such a theory by exchanging the partitions in the corresponding ordered pair, and hence the quiver diagram of the mirror theory can be written down in a straightforward way. The infrared Coulomb branch of each theory can be studied using moment map equations for a hyperKahler quotient of the Higgs branch of the mirror theory. We focus on three infinite subclasses of these singular hyperKahler spaces which are complete intersections. The Hilbert series of these spaces are computed in order to count generators and relations, and they turn out to be related to the corresponding partitions of the theories. For each theory, we explicitly discuss the generators of such a space and relations they satisfy in detail. These relations are precisely the defining equations of the corresponding complete intersection space.
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TL;DR: In this paper, the mirror of a hypersurface of general type (and more generally varieties of non-negative Kodaira dimension) is described as the critical locus of the zero fibre of a certain Landau-Ginzburg potential.
Abstract: The goal of this paper is to propose a theory of mirror symmetry for varieties of general type. Using Landau-Ginzburg mirrors as motivation, we describe the mirror of a hypersurface of general type (and more generally varieties of non-negative Kodaira dimension) as the critical locus of the zero fibre of a certain Landau-Ginzburg potential. The critical locus carries a perverse sheaf of vanishing cycles. Our main results shows that one obtains the interchange of Hodge numbers expected in mirror symmetry. This exchange is between the Hodge numbers of the hypersurface and certain Hodge numbers defined using a mixed Hodge structure on the hypercohomology of the perverse sheaf. This exchange can be anticipated from an analysis of Hochschild homology of the relevant categories arising in homological mirror symmetry in this case; we also conjecture that a similar, but different, exchange of dimensions arises from Hochschild cohomology, relating the cohomology of sheaves of polyvector fields on the hypersurface to the cohomology of the critical locus.
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TL;DR: In this paper, a refinement of homological mirror symmetry for the 2-torus has been explored, which relates exact symplectic topology to arithmetic algebraic geometry, and establishes a derived equivalence of the Fukaya category of the punctured torus, relative to a basepoint, with the category of perfect complexes of coherent sheaves on the Tate curve over the "formal disc" Spec Z[[q]].
Abstract: This paper explores a refinement of homological mirror symmetry which relates exact symplectic topology to arithmetic algebraic geometry. We establish a derived equivalence of the Fukaya category of the 2-torus, relative to a basepoint, with the category of perfect complexes of coherent sheaves on the Tate curve over the "formal disc" Spec Z[[q]]. It specializes to a derived equivalence, over Z, of the Fukaya category of the punctured torus with perfect complexes on the curve y^2+xy=x^3 over Spec Z, the central fibre of the Tate curve; and, over the "punctured disc" Spec Z((q)), to an integral refinement of the known statement of homological mirror symmetry for the 2-torus. We also prove that the wrapped Fukaya category of the punctured torus is derived-equivalent over Z to bounded complexes of coherent sheaves on the central fiber of the Tate curve.
TL;DR: In this paper, the authors studied the variation of relative cohomology for a pair consisting of a smooth projective hypersurface and an algebraic subvariety in it, and derived an inhomogeneous Picard-Fuchs equation for the Abel-Jacobi map.
Abstract: We study the variation of relative cohomology for a pair consisting of a smooth projective hypersurface and an algebraic subvariety in it. We construct an inhomogeneous Picard-Fuchs equation by applying a Picard-Fuchs operator to the holomorphic top form on a Calabi-Yau hypersurface in toric variety, and deriving a general formula for the $d$-exact form on one side of the equation. We also derive a double residue formula, giving a purely algebraic way to compute the inhomogeneous Picard-Fuchs equations for the Abel-Jacobi map, which has played an important role in the recent study of D-branes (by Morrison and Walcher). Using the variation formalism, we prove that the relative periods of toric B-branes on Calabi-Yau hypersurface satisfy the enhanced GKZ-hypergeometric system proposed in physics literature (by Alim, Hecht, Mayr, and Mertens), and discuss the relations between several works in the recent study of open string mirror symmetry. We also give the general solutions to the enhanced hypergeometric system.
TL;DR: In this article, structural aspects of the Ablowitz-Ladik hierarchy were studied in the light of its realization as a two-component reduction of the two-dimensional Toda hierarchy, and its connection to the Gromov-Witten theory of local C P 1 was established.
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TL;DR: Abouzaid, Hacking, and Keel as discussed by the authors showed that theta functions exist in much greater generality than abelian varieties, using mirror symmetry and the degeneration of mirror varieties.
Abstract: This is a survey covering aspects of varied work of the authors with Mohammed Abouzaid, Paul Hacking, and Sean Keel While theta functions are traditionally canonical sections of ample line bundles on abelian varieties, we motivate, using mirror symmetry, the idea that theta functions exist in much greater generality This suggestion originates with the work of the late Andrei Tyurin We outline how to construct theta functions on the degenerations of varieties constructed in previous work of the authors, and then explain applications of this construction to homological mirror symmetry and constructions of broad classes of mirror varieties
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TL;DR: The Strominger-Yau-Zaslow conjecture (SYZ conjecture) as discussed by the authors has been studied extensively in the literature since its introduction in 1996 and its refinements over the years.
Abstract: This survey was written for the Current Developments in Mathematics conference, 2012, and is an updating of my article "The Strominger-Yau-Zaslow conjecture: From torus fibrations to degenerations," in the Seattle 2005 proceedings We trace progress and thinking about the SYZ conjecture since its introduction in 1996 We begin with the original differential geometric conjecture and its refinements, and explain how it led to the algebro-geometric program developed by myself and Siebert After explaining the overall philosophy, I explain how recent results fit into this program
TL;DR: In this article, it was shown that the partition function of N=(2, 2) gauge theories on the two-sphere which flow to Calabi-Yau sigma models in the infrared computes the exact Kahler potential on the quantum Kahler moduli space of the corresponding Calabi Yau.
Abstract: We prove a recent conjecture that the partition function of N=(2, 2) gauge theories on the two-sphere which flow to Calabi-Yau sigma models in the infrared computes the exact Kahler potential on the quantum Kahler moduli space of the corresponding Calabi-Yau. This establishes the two-sphere partition function as a new method of computation of worldsheet instantons and Gromov-Witten invariants. We also calculate the exact two-sphere partition function for N=(2,2) Landau-Ginzburg models with an arbitrary twisted superpotential W. These results are used to demonstrate that arbitrary abelian gauge theories and their associated mirror Landau-Ginzburg models have identical two-sphere partition functions. We further show that the partition function of non-abelian gauge theories can be rewritten as the partition function of mirror Landau-Ginzburg models.
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TL;DR: In this paper, an open mirror theorem for toric Calabi-Yau 3-orbifolds was proved, which expresses generating functions of orbifold disk invariants in terms of Abel-Jacobi maps of the mirror curves.
Abstract: We study open-closed orbifold Gromov-Witten invariants of 3-dimensional Calabi-Yau smooth toric DM stacks with respect to Lagrangian branes of Aganagic-Vafa type. We prove an open mirror theorem for toric Calabi-Yau 3-orbifolds, which expresses generating functions of orbifold disk invariants in terms of Abel-Jacobi maps of the mirror curves. This generalizes a conjecture by Aganagic-Vafa and Aganagic- Klemm-Vafa, proved by the first and the second authors, on disk invariants of smooth toric Calabi-Yau 3-folds. Open Gromov-Witten invariants of toric Calabi-Yau 3-folds have been studied extensively by both math- ematicians and physicists. They correspond to "A-model topological open string amplitudes" in the physics literature and can be interpreted as intersection numbers of certain moduli spaces of holomorphic maps from bordered Riemann surfaces to the 3-fold with boundaries in a Lagrangian submanifold. The physics prediction of these open Gromov-Witten invariants comes from string dualities: mirror symmetry relates the A-model topological string theory of a Calabi-Yau 3-fold X to the B-model topological string theory of the mirror Calabi-Yau 3-fold X ∨ ; the large N duality relates the A-model topological string theory of
TL;DR: In this article, the A − D − E classification of ALE spaces was used to catalogue dual quiver gauge theories that arise from the M-theory background with an A-type ALE1 and a D-type Alex2, and verify the duality explicitly by a computation of partition functions of the theories on S 3.
Abstract: Mirror Symmetry for a large class of three dimensional $ \mathcal{N} = 4 $
supersymmetric gauge theories has a natural explanation in terms of M-theory compactified on a product of ALE spaces. A pair of such mirror duals can be described as two different deformations of the eleven-dimensional supergravity background $ \mathcal{M} = {\mathbb{R}^{2,1}} \times {\text{AL}}{{\text{E}}_1} \times {\text{AL}}{{\text{E}}_2} $
, to which they flow in the deep IR. Using the A − D − E classification of ALE spaces, we present a neat way to catalogue dual quiver gauge theories that arise in this fashion. In addition to the well-known examples studied in [1, 2], this procedure leads to new sets of dual theories. For a certain subset of dual theories which arise from the aforementioned M-theory background with an A-type ALE1 and a D-type ALE2, we verify the duality explicitly by a computation of partition functions of the theories on S
3, using localization techniques. We derive the relevant mirror map and discuss its agreement with predictions from the Type IIB brane construction for these theories.
TL;DR: The authors discuss possible ways in which Fukaya categories might be considered as "local", which means glued together from simpler pieces in a loosely sheaf-theoretic sense, in a tropical geometry and mirror symmetry context.
Abstract: These are notes from a 2010 talk. They concern possible ways in which Fukaya categories might be considered as "local", which means glued together from simpler pieces in a loosely sheaf-theoretic sense. As the title suggests, this is purely speculative. Much of the motivation comes from tropical geometry and mirror symmetry.
TL;DR: In this article, the authors report on recent progress in un-derstanding mirror symmetry for some non-complete intersection Calabi-Yau threefolds, and show that the Picard-Fuchs equations coincide with the expected Calabi Yau equations.
Abstract: The aim of this article is to report on recent progress in un- derstanding mirror symmetry for some non-complete intersection Calabi- Yau threefolds. We first construct four new smooth non-complete in- tersection Calabi-Yau threefolds with h 1,1 = 1, whose existence was previously conjectured by C. van Enckevort and D. van Straten in (19). We then compute the period integrals of candidate mirror families of F. Tonoli's degree 13 Calabi-Yau threefold and three of the new Calabi- Yau threefolds. The Picard-Fuchs equations coincide with the expected Calabi-Yau equations listed in (18, 19). Some of the mirror families turn out to have two maximally unipotent monodromy points.
01 Jan 2012
TL;DR: The Strominger-Yau-Zaslow conjecture has been extensively studied in the literature since its introduction in 1996 as discussed by the authors, including a survey of recent developments in this conjecture.
Abstract: This survey was written for the Current Developments in Mathematics conference, 2012, and is an updating of my article "The Strominger-Yau-Zaslow conjecture: From torus fibrations to degenerations," in the Seattle 2005 proceedings. We trace progress and thinking about the SYZ conjecture since its introduction in 1996. We begin with the original differential geometric conjecture and its refinements, and explain how it led to the algebro-geometric program developed by myself and Siebert. After explaining the overall philosophy, I explain how recent results fit into this program.
TL;DR: In this paper, the authors investigated the multiple mirror phenomenon arising from Berglund-Hubsh-Krawitz mirror symmetry and proved that the different mirror Calabi-Yau orbifolds which arise in this context are in fact birational to one another.
Abstract: We investigate a multiple mirror phenomenon arising from Berglund-Hubsh-Krawitz mirror symmetry. We prove that the different mirror Calabi-Yau orbifolds which arise in this context are in fact birational to one another.
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TL;DR: In this paper, the authors studied half-BPS vortex loop operators for N = 2 supersymmetric theories on S^3, its deformation S^1 x S^2.
Abstract: Three dimensional field theories admit disorder line operators, dubbed vortex loop operators. They are defined by the path integral in the presence of prescribed singularities along the defect line. We study half-BPS vortex loop operators for N=2 supersymmetric theories on S^3, its deformation S^3_b and S^1 x S^2. We construct BPS vortex loops defined by the path integral with a fixed gauge or flavor holonomy for infinitesimal curves linking the loop. It is also possible to include a singular profile for matter fields. For vortex loops defined by holonomy, we perform supersymmetric localization by calculating the fluctuation modes, or alternatively by applying the index theory for transversally elliptic operators. We clarify how the latter method works in situations without fixed points of relevant isometries. Abelian mirror symmetry transforms Wilson and vortex loops in a specific way. In particular an ordinary Wilson loop transforms into a vortex loop for a flavor symmetry. Our localization results confirm the predictions of abelian mirror symmetry.
TL;DR: In this paper, a general criterion for two toric varieties to appear as fibers in a flat family over P 1 was given, and they applied this to show that certain birational transformations can be used to map toric fibers to flat families.
Abstract: We give a general criterion for two toric varieties to appear as fibers in a flat family over P 1 . We apply this to show that certain birational transformations mapping