We construct a cubic field theory which provides all genus amplitudes of the topological A-model for all non-compact toric Calabi-Yau threefolds. The topology of a given Feynman diagram encodes the topology of a fixed Calabi-Yau, with Schwinger parameters playing the role of Kahler classes of the threefold. We interpret this result as an operatorial computation of the amplitudes in the B-model mirror which is the quantum Kodaira-Spencer theory. The only degree of freedom of this theory is an unconventional chiral scalar on a Riemann surface. In this setup we identify the B-branes on the mirror Riemann surface as fermions related to the chiral boson by bosonization.

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The Topological Vertex

Introduction to lie algebras and representation theory

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/63B506EB85D6875966CE63BA0336BC2F/S0025557200190515a.pdf/div-class-title-span-class-italic-lie-groups-and-lie-algebras-chapters-1-3-span-by-n-bourbaki-pp-450-dm-98-1989-isbn-3-540-50218-1-springer-div.pdf

Lie groups and Lie algebras (chapters 1-3 ), by N. Bourbaki. Pp 450. DM 98. 1989. ISBN 3-540-50218-1 (Springer)

In this paper we discuss two major conjectures in Mirror Symmetry: Strominger-Yau-Zaslow conjecture about torus fibrations, and the homological mirror conjecture (about an equivalence of the Fukaya category of a Calabi-Yau manifold and the derived category of coherent sheaves on the dual Calabi-Yau manifold). Our point of view on the origin of torus fibrations is based on the standard differential-geometric picture of collapsing Riemannian manifolds as well as analogous considerations for Conformal Field Theories. It seems to give a description of mirror manifolds much more transparent than the one in terms of D-branes. 
Also we make an attempt to prove the homological mirror conjecture using the torus fibrations. In the case of abelian varieties, and for a large class of Lagrangian submanifolds, we obtain an identification of Massey products on the symplectic and holomorphic sides. Tools used in the proof are of a mixed origin: not so classical Morse theory, homological perturbation theory and non-archimedean analysis.

Homological mirror symmetry and torus fibrations

We describe a technique which enables one to quickly compute an infinite number of toric geometries and their dual quiver gauge theories. The central object in this construction is a ``brane tiling,'' which is a collection of D5-branes ending on an NS5-brane wrapping a holomorphic curve that can be represented as a periodic tiling of the plane. This construction solves the longstanding problem of computing superpotentials for D-branes probing a singular non-compact toric Calabi-Yau manifold, and overcomes many difficulties which were encountered in previous work. The brane tilings give the largest class of N=1 quiver gauge theories yet studied. A central feature of this work is the relation of these tilings to dimer constructions previously studied in a variety of contexts. We do many examples of computations with dimers, which give new results as well as confirm previous computations. Using our methods we explicitly derive the moduli space of the entire Y^{p,q} family of quiver theories, verifying that they correspond to the appropriate geometries. Our results may be interpreted as a generalization of the McKay correspondence to non-compact 3-dimensional toric Calabi-Yau manifolds.

Brane Dimers and Quiver Gauge Theories

We show that toric geometry can be used rather effectively to translate a brane configuration to geometry. Roughly speaking the skeletons of toric space are identified with the brane configurations. The cases where the local geometry involves hypersurfaces in toric varieties (such as P^2 blown up at more than 3 points) presents a challenge for the brane picture. We also find a simple physical explanation of Batyrev's construction of mirror pairs of Calabi-Yau manifolds using T-duality.

https://www.intlpress.com/site/pub/files/_fulltext/journals/atmp/1998/0002/0001/ATMP-1998-0002-0001-a004.pdf

Branes and toric geometry

We give geometric explanations and proofs of various mirror symmetry conjectures for T -invariant Calabi-Yau manifolds when instanton corrections are absent. This uses a fiberwise Fourier transformation together with a base Legendre transformation. We discuss mirror transformations of (i) moduli spaces of complex structures and complexified symplectic structures, H’s, Yukawa couplings; (ii) sl (2)× sl (2)-actions; (iii) holomorphic and symplectic automorphisms and (iv) Aand B-connections, supersymmetric Aand B-cycles, correlation functions. We also study (ii) for T -invariant hyperkahler manifolds.

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Mirror Symmetry Without Corrections

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 $K_{\mathbb{P}^1}$.

/pdf/syz-mirror-symmetry-for-toric-calabi-yau-manifolds-554jygbbcz.pdf

SYZ mirror symmetry for toric Calabi-Yau manifolds

Mirror symmetry for toric Fano manifolds via SYZ transformations

Let $X$ be a compact complex Calabi-Yau 4-fold. Under certain assumptions, we define Donaldson-Thomas type deformation invariants ($DT_{4}$ invariants) by studying moduli spaces of solutions to the Donaldson-Thomas equations on $X$. We also study sheaves counting problems on local Calabi-Yau 4-folds. We relate $DT_{4}$ invariants of $K_{Y}$ to the Donaldson-Thomas invariants of the associated Fano 3-fold $Y$. When the Calabi-Yau 4-fold is toric, we adapt the virtual localization formula to define the corresponding equivariant $DT_{4}$ invariants. We also discuss the non-commutative version of $DT_{4}$ invariants for quivers with relations. Finally, we compute $DT_{4}$ invariants for certain Calabi-Yau 4-folds when moduli spaces are smooth and find a $DT_{4}/GW$ correspondence for $X$. Examples of wall-crossing phenomenon in $DT_{4}$ theory are also given.

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Naichung Conan Leung

Papers

Branes and toric geometry

Mirror Symmetry Without Corrections

SYZ mirror symmetry for toric Calabi-Yau manifolds

Mirror symmetry for toric Fano manifolds via SYZ transformations

Donaldson-Thomas theory for Calabi-Yau 4-folds