Related papers: Integral Regulators on Mirror Curves with Mass Par…
Families of conformal field theories are naturally endowed with a Riemannian geometry which is locally encoded by correlation functions of exactly marginal operators. We show that the curvature of such conformal manifolds can be computed…
We address the spectral problem of the normal quantum mechanical operator associated to the quantized mirror curve of the toric (almost) del Pezzo Calabi--Yau threefold called local $\mathbb{P}^2$ in the case of complex values of Planck's…
We present the most complete list of mirror pairs of Calabi-Yau complete intersections in toric ambient varieties and develop the methods to solve the topological string and to calculate higher genus amplitudes on these compact Calabi-Yau…
In [6], a constraint on invariant measures of bi-permutative cellular automata has been observed: fixed values at the positive indices determine almost-surely a uniform conditional probability on the subset of values of positive conditional…
We demonstrate the existence of topological insulators in one dimension protected by mirror and time-reversal symmetries. They are characterized by a nontrivial $\mathbb{Z}_2$ topological invariant defined in terms of the "partial"…
This paper contains an attempt to formulate rigorously and to check predictions in enumerative geometry of curves following from Mirror Symmetry. The main tool is a new notion of stable map. We give an outline of a contsruction of…
We derive a general obstruction to the existence of Riemannian metrics of positive scalar curvature on closed spin manifolds in terms of hypersurfaces of codimension two. The proof is based on coarse index theory for Dirac operators that…
Mirror symmetry relates type IIB string theory on a Calabi-Yau 3-fold to type IIA on the mirror CY manifold, whose complex structure and Kaehler moduli spaces are exchanged. We show that the mirror map is a particular case of a more general…
Spectral measures provide invariants for braided subfactors via fusion modules. In this paper we study joint spectral measures associated to the compact connected rank two Lie group $SO(5)$ and its double cover the compact connected,…
The mirror curves enable us to study B-model topological strings on non-compact toric Calabi--Yau threefolds. One of the method to obtain the mirror curves is to calculate the partition function of the topological string with single brane.…
We study Mirror Symmetry of log Calabi-Yau surfaces. On one hand, we consider the number of ``affine lines'' of each degree in the complement of a smooth cubic in the projective plane. On the other hand, we consider coefficients of a…
We consider the close relation between duality in N=2 SUSY gauge theories and integrable models. Various integrable models ranging from Toda lattices, Calogero models, spinning tops, and spin chains are related to the quantum moduli space…
Let $G$ be a graph with edge ideal $I(G)$. We recall the notions of $\min-match_{\{K_2, C_5\}}(G)$ and $\ind-match_{\{K_2, C_5\}}(G)$ from \cite{sy}. We show that $${\rm reg}(I(G)^s)\leq 2s+\min-match_{\{K_2, C_5\}}(G)-1,$$for all $s\geq…
Let $\gamma$ be a filling curve on a topological surface $\Sigma$ of genus $g \geq 2$. The inf invariant of $\gamma$, denoted $m_{\gamma}$, is the infimum of the length function on the space of marked hyperbolic structures on $\Sigma$. This…
Operators with continuous spectra naturally arise in spectral theory, quantum mechanics, automorphic forms, and noncommutative geometry. However, analyzing such operators, particularly in the non-selfadjoint setting, remains challenging due…
Quantizing the mirror curve of certain toric Calabi-Yau (CY) three-folds leads to a family of trace class operators. The resolvent function of these operators is known to encode topological data of the CY. In this paper, we show that in…
For three-partition triple Hodge integrals related to the topological vertex, we derive Eynard-Orantin type recursion relations from the cut-and-join equation. This establishes a version of local mirror symmetry for the local $C^3$ geometry…
In this paper we prove a mirror symmetry conjecture based on the work of Brini-Eynard-Mari\~no \cite{BEM} and Diaconescu-Shende-Vafa \cite{DSV}. This conjecture relates open Gromov-Witten invariants of the conifold transition of a torus…
(This is a report for the Proceedings of ``Journees Relativistes 1993'' written in September 1993. Containes a short description of the results published elsewhere in the joint paper with A. Ashtekar) Integral calculus on the space of gauge…
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.…