70,758 research outputs found
Symplectomorphism group relations and degenerations of Landau-Ginzburg models
In this paper, we describe explicit relations in the symplectomorphism groups
of toric hypersurfaces. To define the elements involved, we construct a proper
stack of toric hypersurfaces with compactifying boundary representing toric
hypersurface degenerations. Our relations arise through the study of the one
dimensional strata of this stack. The results are then examined from the
perspective of homological mirror symmetry where we view sequences of relations
as maximal degenerations of Landau-Ginzburg models. We then study the B-model
mirror to these degenerations, which gives a new mirror symmetry approach to
the minimal model program.Comment: 100 pages, 24 figure
Lagrangian fibrations on blowups of toric varieties and mirror symmetry for hypersurfaces
We consider mirror symmetry for (essentially arbitrary) hypersurfaces in
(possibly noncompact) toric varieties from the perspective of the
Strominger-Yau-Zaslow (SYZ) conjecture. Given a hypersurface in a toric
variety we construct a Landau-Ginzburg model which is SYZ mirror to the
blowup of along , under a positivity assumption.
This construction also yields SYZ mirrors to affine conic bundles, as well as a
Landau-Ginzburg model which can be naturally viewed as a mirror to . The
main applications concern affine hypersurfaces of general type, for which our
results provide a geometric basis for various mirror symmetry statements that
appear in the recent literature. We also obtain analogous results for complete
intersections.Comment: 83 pages; v2: added appendix discussing the analytic structure on
moduli of objects in the Fukaya category; v3: further clarifications in
response to referee report; v4: further clarifications throughout, especially
sections 4 and 7 and appendix A; added appendix B on the geometry of reduced
space
Parabolic Whittaker Functions and Topological Field Theories I
First, we define a generalization of the standard quantum Toda chain inspired
by a construction of quantum cohomology of partial flags spaces GL(\ell+1)/P, P
a parabolic subgroup. Common eigenfunctions of the parabolic quantum Toda
chains are generalized Whittaker functions given by matrix elements of
infinite-dimensional representations of gl(\ell+1). For maximal parabolic
subgroups (i.e. for P such that GL(\ell+1)/P=\mathbb{P}^{\ell}) we construct
two different representations of the corresponding parabolic Whittaker
functions as correlation functions in topological quantum field theories on a
two-dimensional disk. In one case the parabolic Whittaker function is given by
a correlation function in a type A equivariant topological sigma model with the
target space \mathbb{P}^{\ell}. In the other case the same Whittaker function
appears as a correlation function in a type B equivariant topological
Landau-Ginzburg model related with the type A model by mirror symmetry. This
note is a continuation of our project of establishing a relation between
two-dimensional topological field theories (and more generally topological
string theories) and Archimedean (\infty-adic) geometry. From this perspective
the existence of two, mirror dual, topological field theory representations of
the parabolic Whittaker functions provide a quantum field theory realization of
the local Archimedean Langlands duality for Whittaker functions. The
established relation between the Archimedean Langlands duality and mirror
symmetry in two-dimensional topological quantum field theories should be
considered as a main result of this note.Comment: Section 1 is extended and Appendices are added, 23 page
SYZ mirror symmetry for toric Calabi-Yau manifolds
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 , we construct a complex
manifold 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
, 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 K\"ahler parameters of
have integral coefficients. Applying the results in \cite{Chan10} and
\cite{LLW10}, 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_{\proj^2} and K_{\proj^1\times\proj^1}.Comment: v3: final version, published in JDG 90 (2012), no. 2, 177-250. 71
pages, 14 figures; substantially revised and expande
Three Dimensional Mirror Symmetry and Partition Function on
We provide non-trivial checks of mirror symmetry in a
large class of quiver gauge theories whose Type IIB (Hanany-Witten)
descriptions involve D3 branes ending on orbifold/orientifold 5-planes at the
boundary. From the M-theory perspective, such theories can be understood in
terms of coincident M2 branes sitting at the origin of a product of an A-type
and a D-type ALE (Asymtotically Locally Euclidean) space with G-fluxes.
Families of mirror dual pairs, which arise in this fashion, can be labeled as
, where and are integers. For a large subset of such
infinite families of dual theories, corresponding to generic values of , arbitrary ranks of the gauge groups and varying , we test the
conjectured duality by proving the precise equality of the partition
functions for dual gauge theories in the IR as functions of masses and FI
parameters. The mirror map for a given pair of mirror dual theories can be read
off at the end of this computation and we explicitly present these for the
aforementioned examples. The computation uses non-trivial identities of
hyperbolic functions including certain generalizations of Cauchy determinant
identity and Schur's Pfaffian identity, which are discussed in the paper.Comment: 45 pages, 9 figure
Bow Varieties---Geometry, Combinatorics, Characteristic Classes
Motivated by the study of 3d mirror symmetry from the perspective of characteristic classes, we develop a combinatorial framework for the study of Cherkis Bow Varieties. Bow varieties are believed to be a natural setting where 3d mirror symmetry for characteristic classes can be observed. We take the first steps toward a general theory of mirror symmetry by describing the geometry of bow varieties in terms of brane diagrams, binary contingency tables, and various combinatorial operations on these objects. We then give a conjectural formula for the cohomological stable envelope of a bow variety. An overview of 3d mirror symmetry for characteristic classes in Schubert calculus is also provided.Doctor of Philosoph
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