arXiv research

We conjecture a formula for the refined $\mathrm{SU}(3)$ Vafa-Witten invariants of any smooth surface $S$ satisfying $H_1(S,\mathbb{Z}) = 0$ and $p_g(S)>0$. The unrefined formula corrects a proposal by Labastida-Lozano and involves unexpected algebraic expressions in modular functions. We prove that our formula satisfi…

Simple perturbation of Vafa-Witten equations leads to transversality.

New conjectures link SU(r) Vafa-Witten invariants to Ramanujan's continued fractions.

This paper establishes transversality for perturbed Vafa-Witten moduli spaces on 4-manifolds.

The paper studies Vafa-Witten equations on Kaehler manifolds and identifies obstructions to nontrivial solutions.

The paper smooths out equations on 4-manifolds to show smooth moduli spaces.

We prove a Freed-Uhlenbeck style generic smoothness theorem for the moduli space of solutions to the Vafa--Witten equations on a closed symplectic four-manifold by using a method developed by Feehan for the study of the $PU(2)$-monopole equations on smooth closed four-manifolds. We introduce a set of perturbation terms…

Study deformation invariants of projective surfaces using twisted sheaves and Azumaya algebras.

Revisits Vafa-Witten theory, deriving new invariants and homologies.

Computes Vafa-Witten invariants of 3-manifolds.

On a polarised surface, solutions of the Vafa-Witten equations correspond to certain polystable Higgs pairs. When stability and semistability coincide, the moduli space admits a symmetric obstruction theory and a $\mathbb C^*$ action with compact fixed locus. Applying virtual localisation we define invariants constant …

A theory linking invariants, Floer homologies, and Higgs bundles.

Study of 5D SYM theory on toric surfaces yields refined Vafa-Witten invariants.

We propose a definition of Vafa-Witten invariants counting semistable Higgs pairs on a polarised surface. We use virtual localisation applied to Mochizuki/Joyce-Song pairs. For $K_S\le0$ we expect our definition coincides with an alternative definition using weighted Euler characteristics. We prove this for deg $K_S<0$…

Establish geometric and analytic constraints for Vafa-Witten equations on 4-manifolds, leading to volume bounds and new estimates for Yamabe constants.

Virtual invariants defined from sheaves on surfaces.

Study of holomorphic triples on surfaces leads to Vafa-Witten invariants.

This paper studies the behavior of sequences of solutions to Seiberg-Witten like equations for a pair consisting of a Hermitian connection on a line bundle over a 4-dimensional manifold and a section of the self-dual spinor bundle of a complex Clifford module on the manifold. Examples include the cases where the Cliffo…

The paper modifies Vafa-Witten equations on 4-manifolds for better solution estimates.

The paper connects algebraic geometry to threefold theories.

We conjecture a formula for the generating function of virtual $χ_y$-genera of moduli spaces of rank 2 sheaves on arbitrary surfaces with holomorphic 2-form. Specializing the conjecture to minimal surfaces of general type and to virtual Euler characteristics, we recover (part of) a formula of C. Vafa and E. Witten. The…

This paper discuss an intrinsic relation among congruent relations \cite{CLPZ}, cyclotomic expansion and Volume Conjecture for $SU(n)$ invariants. Motivated by the congruent relations for $SU(n)$ invariants obtained in our previous work \cite{CLPZ}, we study certain limits of the $SU(n)$ invariants at various roots of …

We conjecture a Verlinde type formula for the moduli space of Higgs sheaves on a surface with a holomorphic 2-form. The conjecture specializes to a Verlinde formula for the moduli space of sheaves. Our formula interpolates between $K$-theoretic Donaldson invariants studied by the first named author and Nakajima-Yoshiok…

This paper is the the third part of a series of paper whose aim is to use of the framework of \emph{twisted spectral triples} to study conformal geometry from a noncommutive geometric viewpoint. In this paper we reformulate the inequality of Vafa-Witten \cite{VW:CMP84} in the setting of twisted spectral triples. This i…

Study topological correlators for $SU(2)$ SYM on four-manifolds, deriving explicit formulae and confirming S-duality.

The study finds invariant Einstein metrics on complex Stiefel manifolds and special unitary groups.

Invariant description of SU(2)-structures on 5-manifolds developed.

Study coclosed G2-structures on SU(2)²-invariant manifolds.

Study on Riemannian properties of SU_n using bi-invariant metric.

We show that the SU(3) Casson invariant for spliced sums along certain torus knots equals 16 times the product of their SU(2) Casson knot invariants. The key step is a splitting formula for su(n) spectral flow for closed 3-manifolds split along a torus.

Researchers found the $\hat{Z}$-invariant for $SU(N)/\mathbb{Z}_m$ is constant regardless of $m$.

A perturbative SU(3) Casson invariant $Λ_{SU(3)}(X)$ for integral homology 3-spheres is defined. Besides being fully perturbative, it has nice properties: (1) $4 . Λ_{SU(3)}(X)$ is an integer. (2) It is preseved under orientation change. (3) A connected sum formula holds. Explicit calculations of the invariant for $1/k…

The paper constructs non-convergent solutions to Vafa-Witten equations with specific harmonic 2-form limits.

The Vafa-Witten equations on an oriented Riemannian 4- manifold are first order, non-linear equations for a pair of connection on a principle SO(3) bundle over the 4-manifold and a self-dual 2-form with values in the associated Lie algebra bundle. The main theorem in this paper characterizes in part the behavior of seq…

This paper proves a symplectic formula for SU(n) generalized Casson invariants.

Details of quantum knot invariant calculations using a specific SU(3)_q-module are given which distinguish the Conway and Kinoshita-Teresaka pair of mutant knots. Features of Kuperberg's skein-theoretic techniques for SU(3)_q invariants in the context of mutant knots are also discussed.

Calculates spectral flow bounds for reducible solutions to Vafa-Witten equations.

No left-invariant hypercomplex structures found on compact Lie groups.

Constructs instantons on a specific G_2-manifold limit.

New methods for computing a variety of gauge theoretic invariants for homology 3-spheres are developed. These invariants include the Chern-Simons invariants, the spectral flow of the odd signature operator, and the rho invariants of irreducible SU(2) representations. These quantities are calculated for flat SU(2) conne…

New invariants for 3-manifolds derived from equivariant Cerf theory.

We introduce the $SU(N)$ Casson-Lin invariants for links $L$ in $S^3$ with more than one component. Writing $L = \ell_1 \cup \cdots \cup \ell_n$, we require as input an $n$-tuple $(a_1,\ldots, a_n) \in {\mathbb Z}^n$ of labels, where $a_j$ is associated with $\ell_j$. The $SU(N)$ Casson-Lin invariant, denoted $h_{N,a}(…

We compute the $SU(2)$ Casson-Lin invariant for the Hopf link and determine the sign in the formula of Harper and Saveliev relating this invariant to the linking number.

We derive a gauge theoretic invariant of integral homology 3-spheres which counts gauge orbits of irreducible, perturbed flat SU(3) connections with sign given by spectral flow. To compensate for the dependence of this sum on perturbations, the invariant includes contributions from the reducible, perturbed flat orbits.…

New steady gradient Ricci solitons found on specific four-manifolds.

New gradient Ricci solitons found for $SU(2)$ invariants.

Study of pseudo-Anosov actions on $SU(2)$-character variety for genus 2 surfaces.

An explicit classification of simply connected compact homogeneous CR manifolds G/L of codimension one, with non-degenerate Levi form, is given. There are three classes of such manifolds: a) the standard CR homogeneous manifolds which are homogeneous S^1-bundles over a flag manifold F, with CR structure induced by an i…