Related papers: Quantum Cohomology under Birational Maps and Trans…
Compactifications with fluxes and branes motivate us to study various enumerative invariants of Calabi-Yau manifolds. In this paper, we study non-perturbative corrections depending on both open and closed string moduli for a class of…
This is the second part in a series of papers on counting surfaces on Calabi-Yau 4-folds. In this paper, we introduce $K$-theoretic $\mathrm{DT}, \mathrm{PT}_0, \mathrm{PT}_1$ invariants and conjecture a $\mathrm{DT}$-$\mathrm{PT}_0$…
I review some recent results on four-manifold invariants which have been obtained in the context of topological quantum field theory. I focus on three different aspects: (a) the computation of correlation functions, which give explicit…
A noncommutative-geometric generalization of the theory of principal bundles is sketched. A differential calculus over corresponding quantum principal bundles is analysed. The formalism of connections is presented. In particular, operators…
These notes contain a brief introduction to the construction of toric Calabi--Yau hypersurfaces and complete intersections with a focus on issues relevant for string duality calculations. The last two sections can be read independently and…
A cohomology theory of weighted Rota-Baxter $3$-Lie algebras is introduced. Formal deformations, abelian extensions, skeletal weighted Rota-Baxter $3$-Lie 2-algebras and crossed modules of weighted Rota-Baxter 3-Lie algebras are interpreted…
These are significantly expanded lecture notes for the author's minicourse at MSRI in June 2012, as published in the MSRI lecture note series, with some minor additional corrections. In these notes, we give an example-motivated review of…
After a self-contained introduction to Lie algebra cohomology, we present some recent applications in mathematics and in physics. Contents: 1. Preliminaries: L_X, i_X, d 2. Elementary differential geometry on Lie groups 3. Lie algebra…
We show for any oriented surface, possibly with a boundary, how to generalize Kramers-Wannier duality to the world of quantum groups. The generalization is motivated by quantization of Poisson-Lie T-duality from the string theory.…
A mathematical framework of cohomological field theories (CohFTs) is formulated in the language of bigraded manifolds. Algebraic properties of operators in CohFTs are studied. Methods of constructing CohFTs, with or without gauge…
We consider the construction of Calabi-Yau varieties recently generalized to where the defining equations may have negative degrees over some projective space factors in the embedding space. Within such "generalized complete intersection"…
A bicovariant calculus of differential operators on a quantum group is constructed in a natural way, using invariant maps from \fun\ to \uqg\ , given by elements of the pure braid group. These operators --- the `reflection matrix' $Y \equiv…
We prove a decomposition theorem of the quantum cohomology D-module of the blowup of a smooth projective variety X along a smooth subvariety Z. The main tools we use are shift operators and Fourier analysis for equivariant quantum…
These notes are based on the three lectures that one of the authors gave at Tsinghua University in the summer of 2023 as part of the workshop on Geometric Representation Theory and Applications. They contain an introduction to the…
This thesis is devoted to the study of Lie bialgebra and Hopf algebra structures related to certain versions of non-commutative geometry constructed on infinite-dimensional Lie algebras that arise in the context of asymptotic symmetries of…
We construct an algebraic variety by resolving singularities of a quintic Calabi-Yau threefold. The middle cohomology of the threefold is shown to contain a piece coming from a pair of elliptic surfaces. The resulting quotient is a…
We define quantum exterior product wedge_h and quantum exterior differential d_h on Poisson manifolds, of which symplectic manifolds are an important class of examples. Quantum de Rham cohomology is defined as the cohomology of d_h. We also…
In this article, the third of three, we analyse how the Weyl quantisation for compact Lie groups presented in the second article of this series fits with the projective-phase space structure of loop quantum gravity-type models. Thus, the…
We show that $A_s(n)$, the coordinate algebra of Wang's quantum permutation group, is Calabi-Yau of dimension $3$ when $n\geq 4$, and compute its Hochschild cohomology with trivial coefficients. We also show that, for a larger class of…
The aim of this paper is to extend the notion of bialgebra for Leibniz algebras (and Lie algebras) to $3$-Leibniz algebras (and $3$-Lie algebras) by use of the cohomology complex of $3$-Leibniz algebras. Also, some theorems about Leibniz…