Related papers: Proof of a magnificent conjecture
We compute the Donaldson-Thomas invariants for two types of Calabi-Yau 3-folds. These invariants are associated to the moduli spaces of rank-2 Gieseker semistable sheaves. None of the sheaves are locally free, and their double duals are…
We conjecture a formula for the generating function of genus one Gromov-Witten invariants of the local Calabi-Yau manifolds which are the total spaces of splitting bundles over projective spaces. We prove this conjecture in several special…
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…
In this paper we study the holomorphic Euler characteristics of determinant line bundles on moduli spaces of rank 2 semistable sheaves on an algebraic surface X, which can be viewed as $K$-theoretic versions of the Donaldson invariants. In…
We consider an "orientifold" generalization of Khovanov-Lauda-Rouquier algebras, depending on a quiver with an involution and a framing. Their representation theory is related, via a Schur-Weyl duality type functor, to Kac-Moody quantum…
By exploiting new mathematical relations between Pandharipande-Thomas (PT) invariants, closely related to Gopakumar-Vafa (GV) invariants, and rank 0 Donaldson-Thomas (DT) invariants counting D4-D2-D0 BPS bound states, we rigorously compute…
Based on the proof of Labastida-Mari{\~n}o-Ooguri-Vafa conjecture \cite{lmov}, we derive an infinite product formula for Chern-Simons partition functions, the generating function of quantum $\fsl_N$ invariants. Some symmetry properties of…
We show that the link invariants derived from 3-dimensional quantum hyperbolic geometry can be defined by means of planar state sums based on link diagrams and a new family of enhanced Yang-Baxteroperators (YBO) that we compute explicitly.…
We derive a $K$-theoretic analogue of the Vafa--Intriligator formula, computing the (virtual) Euler characteristics of vector bundles over the Quot scheme that compactifies the space of degree $d$ morphisms from a fixed projective curve to…
We conjecture that the generating series of Gromov-Witten invariants of the Hilbert schemes of $n$ points on a K3 surface are quasi-Jacobi forms and satisfy a holomorphic anomaly equation. We prove the conjecture in genus $0$ and for at…
In their 2007 paper, Jarvis, Kaufmann, and Kimura defined the full orbifold $K$-theory of an orbifold ${\mathfrak X}$, analogous to the Chen-Ruan orbifold cohomology of ${\mathfrak X}$ in that it uses the obstruction bundle as a quantum…
We compute the value of finitary localizing invariants, including algebraic $K$-theory, on categories of sheaves over stably locally compact spaces $X$. Our formula simultaneously generalizes the cases of locally compact Hausdorff and…
Let $K$ be a field, $\Gamma $ a finite group of field automorphisms of $K$, $F$ the $\Gamma $-fixed field in $K$ and $G\leq $GL$_v(K)$ a finite matrix group. Then the action of $\Gamma $ defines a grading on the symmetric algebra of the…
We describe the integral equivariant cohomology ring of a weighted projective space in terms of piecewise polynomials, and thence by generators and relations. We deduce that the ring is a perfect invariant, and prove a Chern class formula…
We derive formulas and algorithms for Kitaev's invariants in the periodic table for topological insulators and superconductors for finite disordered systems on lattices with boundaries. We find that K-theory arises as an obstruction to…
Given a finite dimensional representation of a semisimple Lie algebra there are two ways of constructing link invariants: 1) quantum group invariants using the R-matrix, 2) the Kontsevich universal link invariant followed by the Lie algebra…
Inspired by the work of Pomoni-Yan-Zhang in String Theory, we introduce the moduli space of tetrahedron instantons as a Quot scheme and describe it as a moduli space of quiver representations. We construct a virtual fundamental class and…
The Gopakumar-Vafa type invariants on Calabi-Yau 4-folds (which are non-trivial only for genus zero and one) are defined by Klemm-Pandharipande from Gromov-Witten theory, and their integrality is conjectured. In a previous work of…
Using reduced Gromov-Witten theory, we define new invariants which capture the enumerative geometry of curves on holomorphic symplectic 4-folds. The invariants are analogous to the BPS counts of Gopakumar and Vafa for Calabi-Yau 3-folds,…
The gauge structure of the four dimensional effective theory originated in a pure five dimensional Yang-Mills theory compactified on the orbifold $S^1/Z_2$, is discussed on the basis of the BRST symmetry. If gauge parameters propagate in…