Title: Localized Donaldson-Thomas theory of surfaces
(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.
Title: Hilbert Schemes, Donaldson-Thomas Theory, Vafa-Witten and Seiberg Witten Theories
(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.
A new preprint by Hossein Movasati: Hodge Cycles for Cubic Hypersurfaces
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 publication by Sze-Man Ngai:
Title: Estimates for Sums and Gaps of EigenValues of Laplacians on Measure Spaces
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.
New research by Alex Teytelboym et al. presented at the 2018 Big Data Conference was recently published in Science Advances. The article, “Interpreting Economic Complexity,” explores patterns related the economic complexity index (ECI) and product complexity index (PCI). The paper can be found here.
Abstract: Two network measures known as the economic complexity index (ECI) and product complexity index (PCI) have provided important insights into patterns of economic development. We show that the ECI and PCI are equivalent to a spectral clustering algorithm that partitions a similarity graph into two parts. The measures are also closely related to various dimensionality reduction methods, such as diffusion maps and correspondence analysis. Our results shed new light on the ECI’s empirical success in explaining cross-country differences in gross domestic product per capita and economic growth, which is often linked to the diversity of country export baskets. In fact, countries with high (low) ECI tend to specialize in high-PCI (low-PCI) products. We also find that the ECI and PCI uncover specialization patterns across U.S. states and U.K. regions.
arXiv:1812.11968 [pdf, ps, other]
New Higher Anomalies, SU(N) Yang-Mills Gauge Theory and ℂℙN−1 Sigma Model
Zheyan Wan, Juven Wang
Subjects: High Energy Physics – Theory (hep-th)
We hypothesize a new and more complete set of anomalies of quantum field theories (QFTs) and then give an eclectic proof. First, we propose a set of 4d ‘t Hooft higher anomalies of time-reversal symmetric SU(N)-Yang-Mills (YM) gauge theory at θ=π, specifically at N=2,3,4 and others, by enlisting all possible 5d cobordism invariants (higher symmetry-protected topological states) and selecting the matched terms. Second, we propose a set of 3d ‘t Hooft anomalies of ℂℙN−1-sigma model obtained via compactifying YM theory on a 2-torus, and select the matched 3d cobordism invariants. Based on algebraic/geometric topology, QFT analysis, manifold generator dimensional reduction, condensed matter inputs and additional physical criteria, we derive a correspondence between 5d and 3d new invariants, thus broadly prove a more complete anomaly-matching between 4d YM and 2d ℂℙN−1 models via a 2-torus reduction. We formulate a higher-symmetry analog of “Lieb-Schultz-Mattis theorem” to constrain the low-energy dynamics. Continue reading New Publications by Juven Wang
A new paper by Aghil Alaee:
Abstract: We prove existence of all possible bi-axisymmetric near-horizon geometries of 5-dimensional minimal supergravity. These solutions possess the cross-sectional horizon topology S3, S1×S2, or L(p,q) and come with prescribed electric charge, two angular momenta, and a dipole charge (in the ring case). Moreover, we establish uniqueness of these solutions up to an isometry of the symmetric space G2(2)/SO(4).
A new paper by Sergiy Verstyuk:
Abstract: The modeling of multivariate time series in an agnostic manner, without assumptions about underlying theoretical structure is traditionally conducted using Vector Auto-Regressions. They are well suited for linear and state-independent evolution. A more general methodology of Multivariate Recurrent Neural Networks allows to capture non-linear and state-dependent dynamics. This paper takes a range of small- to large-scale Long Short-Term Memory MRNNs and pits them against VARs in an application to US data on GDP growth, inflation, commodity prices, Fed Funds rate and bank reserves. Even in a small-sample regime, MRNN outperforms VAR in forecasting: its out-of-sample predictions are about 20% more accurate. MRNN also fares better in interpretability by means of impulse response functions: for instance, a temporary shock to the Fed Funds rate variable generates system dynamics that are more plausible according to conventional