(To be published in American Journal of Mathematics)
Abstract: Let S be a projective simply connected complex surface and L be a line bundle on S. We study the moduli space of stable compactly supported 2-dimensional sheaves on the total spaces of L. The moduli space admits a C ∗ – action induced by scaling the fibers of L. We identify certain components of the fixed locus of the moduli space with the moduli space of torsion free sheaves and the nested Hilbert schemes on S. We define the localized Donaldson-Thomas invariants of L by virtual localization in the case that L twisted by the anticanonical bundle of S admits a nonzero global section. When pg(S) > 0, in combination with Mochizuki’s formulas, we are able to express the localized DT invariants in terms of the invariants of the nested Hilbert schemes defined by the authors in [GSY17a], the Seiberg-Witten invariants of S, and the integrals over the products of Hilbert schemes of points on S. When L is the canonical bundle of S, the Vafa-Witten invariants defined recently by Tanaka-Thomas, can be extracted from these localized DT invariants. VW invariants are expected to have modular properties as predicted by S-duality.
(To be published in Notices of the International Chinese Congress of Mathematicians)
Abstract: This article provides the summary of [GSY17a] and [GSY17b] where the authors studied the enumerative geometry of “nested Hilbert schemes” of points and curves on algebraic surfaces and their connections to threefold theories, and in particular relevant Donaldson-Thomas, Vafa-Witten and SeibergWitten theories.
New Publciation by Juven Wang, Xiao-Gang Wen, and Shing-Tung Yau: “Quantum Statistics and Spacetime Topology: Quantum Surgery Formulas”
Abstract: We apply the geometric-topology surgery theory on spacetime manifolds to study the constraints of quantum statistics data in 2+1 and 3+1 spacetime dimensions. First, we introduce the fusion data for worldline and worldsheet operators capable creating anyon excitations of particles and strings, well-defined in gapped states of matter with intrinsic topological orders. Second, we introduce the braiding statistics data of particles and strings, such as the geometric Berry matrices for particle-string Aharonov-Bohm and multi-loop adiabatic braiding process, encoded by submanifold linkings, in the closed spacetime 3-manifolds and 4-manifolds. Third, we derive new quantum surgery constraints analogous to Verlinde formula associating fusion and braiding statistics data via spacetime surgery, essential for defining the theory of topological orders, and potentially correlated to bootstrap boundary physics such as gapless modes, conformal field theories or quantum anomalies.
New paper by Nishanth Gudapati: “A conserved energy for axially symmetric Newman–Penrose–Maxwell scalars on Kerr black holes.” What follows is a brief summary of the findings. A PDF of the paper can be found here.
Abstract: For Laplacians defined by measures on a bounded domain in Rn, we prove analogs of the classical eigenvalue estimates for the standard Laplacian: lower bound of sums of eigenvalues by Li and Yau, and gaps of consecutive eigenvalues by Payne, Polya and Weinberger. This work is motivated by the study of spectral gaps for Laplacians on fractals.