After the appearance of my preprint [T3] (Special Lagrangian geometry and slightly deformed algebraic geometry (spLag and sdAG), Warwick preprint 22/1998, alg-geom/9806006, 54 pp.). I received an e-mail from Cumrun Vafa, who recognized that the subject is closely related to that of his preprint [V] (Extending mirror conjecture to Calabi-Yau with bundles, hep-th/9804131, 7 pp.). This text started out as an e-mail ``reply'' to his letter. All the constructions we propose have well known ``spectral curve'' prototypes (see for example Friedman and other [FMW], Bershadsky and other [BJPS] and a number of others). Roughly speaking, our constructions are the spectral curve construction plus the phase geometry described in [T3]. So this text should really come before [T3], as motivation for the development of the geometry of the phase map in [T3].
The paper consists of two sections. The first section provides a new definition of mirror symmetry of abelian varieties making sense also over $p$-adic fields. The second section introduces and studies quantized theta-functions with two-sided multipliers, which are functions on non-commutative tori. This is an extension of an earlier work by the author. In the Introduction and in the Appendix the constructions of this paper are put into a wider context.
When combined with mirror symmetry, the A-model approach to quantization leads to a fairly simple and tractable problem. The most interesting part of the problem then becomes finding the mirror of the coisotropic brane. We illustrate how it can be addressed in a number of interesting examples related to representation theory and gauge theory, in which mirror geometry is naturally associated with the Langlands dual group. Hyperholomorphic sheaves and (B,B,B) branes play an important role in the B-model approach to quantization.
We prove the genus-one restriction of the all-genus Landau-Ginzburg/Calabi-Yau conjecture of Chiodo and Ruan, stated in terms of the geometric quantization of an explicit symplectomorphism determined by genus-zero invariants. This provides the first evidence supporting the higher-genus Landau-Ginzburg/Calabi-Yau correspondence for the quintic threefold, and exhibits the first instance of the "genus zero controls higher genus" principle, in the sense of Givental's quantization formalism, for non-semisimple cohomological field theories.
We prove that the punctured generalized conifolds and punctured orbifolded conifolds are mirror symmetric under the SYZ program with quantum corrections. This mathematically confirms the gauge-theoretic prediction by Aganagic-Karch-Lüst-Miemiec, and also provides a supportive evidence to Morrison's conjecture that geometric transitions are reversed under mirror symmetry.
Geometric Quantization links holomorphic geometry with real geometry, a relation that is a prototype for the modern development of mirror symmetry. We show how to use this treatment to construct a special basis in every space of conformal blocks. This is a direct generalization of the basis of theta functions with characteristics in every complete linear system on an Abelian variety (see Mumford's "Tata lectures on theta" cite(Mumford)). The same construction generalizes the classical theory of theta functions to vector bundles of higher rank on Abelian varieties and K3 surfaces. We also discu…
The geometric aspects of mirror symmetry are reviewed, with an eye towards future developments. Given a mirror pair (X,Y) of Calabi-Yau threefolds, the best-understood mirror statements relate certain small corners of the moduli spaces of X and of Y. We will indicate how one might go beyond such statements, and relate the moduli spaces more globally. In fact, in the boldest version of mirror symmetry (the Strominger-Yau-Zaslow conjecture), the Calabi-Yau threefolds X and Y should be directly related to each other through a very geometric construction.
Recently, a correspondence has been proposed between spectral theory and topological strings on toric Calabi-Yau manifolds. In this paper we develop in detail this correspondence for mirror curves of higher genus, which display many new features as compared to the genus one case studied so far. Given a curve of genus g, our quantization scheme leads to g different trace class operators. Their spectral properties are encoded in a generalized spectral determinant, which is an entire function on the Calabi-Yau moduli space. We conjecture an exact expression for this spectral determinant in terms …
We study geometric structures on the complement of a toric mirror arrangement associated with a root system. Inspired by those special hypergeometric functions found by Heckman-Opdam, as well as the work of Couwenberg-Heckman-Looijenga on geometric structures on projective arrangement complements, we consider a family of connections on a total space, namely, a $\mathbb{C}^{\times}$-bundle on the complement of a toric mirror arrangement (=finite union of hypertori, determined by a root system). We prove that these connections are torsion free and flat, and hence define a family of affine struct…
Gross, Hacking, and Keel have constructed mirrors of log Calabi-Yau surfaces in terms of counts of rational curves. Using $q$-deformed scattering diagrams defined in terms of higher genus log Gromov-Witten invariants, we construct deformation quantizations of these mirrors and we produce canonical bases of the corresponding noncommutative algebras of functions.
We introduce geometric quantization in the setting of shifted symplectic structures. We define Lagrangian fibrations and prequantizations of shifted symplectic stacks and their geometric quantization. In addition, we study many examples including symplectic groupoids, Hamiltonian spaces and moduli spaces of flat connections. In the case of Hamiltonian spaces we prove a derived analog of the "quantization commutes with reduction" principle.
This paper is about geometric quantization of the Hitchin system. We quantize a Kahler form on the Hitchin moduli space (which is half the first Kahler form defined by Hitchin) by considering the Quillen bundle as the prequantum line bundle and modifying the Quillen metric using the Higgs field so that the curvature is proportional to the Kahler form. We show that this Kahler form is integral and the Quillen bundle descends as a prequantum line bundle on the moduli space. It is holomorphic and hence one can take holomorphic square integrable sections as the Hilbert space of quantization of the…
This paper is a short version of the author habilitation thesis. The main results have been already published but here a lot of details are clarified. As well we add some new results: we discuss some quasi classical limit of ALG(a) - quantization, mention some topological properties of the moduli space of Bohr - Sommerfeld Lagrangian cycles of fixed type and investigate some properties of the Kaehler structure on it.
We re-examine quantization via branes with the goal of understanding its relation to geometric quantization. If a symplectic manifold $M$ can be quantized in geometric quantization using a polarization ${\mathcal P}$, and in brane quantization using a complexification $Y$, then the two quantizations agree if ${\mathcal P}$ can be analytically continued to a holomorphic polarization of $Y$. We also show, roughly, that the automorphism group of $M$ that is realized as a group of symmetries in brane quantization of $M$ is the group of symplectomorphisms of $M$ that can be analytically continued t…
In this note we quantize the usual symplectic (Kähler) form on the vortex moduli space by modifying the Quillen metric of the Quillen determinant line bundle.
Recent developments in string theory have revealed a surprising connection between spectral theory and local mirror symmetry: it has been found that the quantization of mirror curves to toric Calabi-Yau threefolds leads to trace class operators, whose spectral properties are conjecturally encoded in the enumerative geometry of the Calabi-Yau. This leads to a new, infinite family of solvable spectral problems: the Fredholm determinants of these operators can be found explicitly in terms of Gromov-Witten invariants and their refinements; their spectrum is encoded in exact quantization conditions…
In this article we discuss the geometry of moduli spaces of (1) flat bundles over special Lagrangian submanifolds and (2) deformed Hermitian-Yang-Mills bundles over complex submanifolds in Calabi-Yau manifolds. These moduli spaces reflect the geometry of the Calabi-Yau itself like a mirror. Strominger, Yau and Zaslow conjecture that the mirror Calabi-Yau manifold is such a moduli space and they argue that the mirror symmetry duality is a Fourier-Mukai transformation. We review various aspects of the mirror symmetry conjecture and discuss a geometric approach in proving it. The existence of rig…
National Cheng Kung University（国立成功大学）；Kavli Institute for the Physics and Mathematics of the Universe (WPI), The University of Tokyo Institutes for Advanced Study, The University of Tokyo（东京大学宇宙物质研究机构（WPI））
This is a brief summary of our works [arXiv:1112.4063, arXiv:1201.4501] on constructing higher genus B-model from perturbative quantization of BCOV theory. We analyze Givental's symplectic loop space formalism in the context of B-model geometry on Calabi-Yau manifolds, and explain the Fock space construction via the renormalization techniques of gauge theory. We also give a physics interpretation of the Virasoro constraints as the symmetry of the classical BCOV action functional, and discuss the Virasoro constraints in the quantum theory.
The recent result of Strominger, Yau and Zaslow relating mirror symmetry to the quantum field theory notion of T-duality is reinterpreted as providing a way of geometrically characterizing which Calabi-Yau manifolds have mirror partners. The geometric description---that one Calabi-Yau manifold should serve as a compactified, complexified moduli space for special Lagrangian tori on the other Calabi-Yau manifold---is rather surprising. We formulate some precise mathematical conjectures concerning how these moduli spaces are to be compactified and complexified, as well as a definition of geometri…
The geometric Langlands correspondence has been interpreted as the mirror symmetry of the Hitchin fibrations for two dual reductive groups. This mirror symmetry, in turn, reduces to T-duality on the generic Hitchin fibers, which are smooth tori. In this paper we study what happens when the Hitchin fibers on the B-model side develop orbifold singularities. These singularities correspond to local systems with finite groups of automorphisms. In the classical Langlands Program local systems of this type are called endoscopic. They play an important role in the theory of automorphic representations…
A well-motivated conjecture states that the open topological string partition function on toric geometries in the Nekrasov-Shatashvili limit is annihilated by a difference operator called the quantum mirror curve. Recently, the complex structure variables parameterizing the curve, which play the role of eigenvalues for related operators, were conjectured to satisfy a quantization condition non-perturbative in the NS parameter $\hbar$. Here, we argue that this quantization condition arises from requiring single-valuedness of the partition function, combined with the requirement of smoothness in…
