New paper by Aghil Alaee et. al.:
Title: Existence and Uniqueness of Stationary Solutions in 5-Dimensional Minimal Supergravity
Abstract: We study the problem of stationary bi-axially symmetric solutions of the 5-dimensional minimal supergravity equations. Essentially all possible solutions with nondegenerate horizons are produced, having the allowed horizon cross-sectional topologies of the sphere S3, ring S1×S2, and lens L(p,q), as well as the three different types of asymptotics. The solutions are smooth apart from possible conical singularities at the fixed point sets of the axial symmetry. This analysis also includes the solutions known as solitons in which horizons are not present but are rather replaced by nontrivial topology called bubbles which are sustained by dipole fluxes. Uniqueness results are also presented which show that the solutions are completely determined by their angular momenta, electric and dipole charges, and rod structure which fixes the topology. Consequently we are able to identify the finite number of parameters that govern a solution. In addition, a generalization of these results is given where the spacetime is allowed to have orbifold singularities.
New Publication from Zheyan Wan, Juven Wang, Yunqin Zheng:
Title: Quantum Yang-Mills 4d Theory and Time-Reversal Symmetric 5d Higher-Gauge TQFT: Anyonic-String/Brane Braiding Statistics to Topological Link Invariants
Abstract: We explore various 4d Yang-Mills gauge theories (YM) living as boundary conditions of 5d gapped short/long-range entangled (SRE/LRE) topological states. Specifically, we explore 4d time-reversal symmetric pure YM of an SU(2) gauge group with a second-Chern-class topological term at θ=π (SU(2)θ=π YM). Its higher ‘t Hooft anomalies of generalized global symmetries indicate that the 4d SU(2)θ=π YM, in order to realize all global symmetries locally, necessarily couples to a 5d higher symmetry-protected topological state (SPTs, as an invertible TQFT, or as a 5d 1-form-center-symmetry-protected interacting “topological superconductor” in condensed matter). We revisit the 4d SU(2)θ=π YM-5d SRE-higher-SPTs coupled systems in [arXiv:1812.11968] and find their “Fantastic Four Siblings” with four sets of new higher anomalies associated with the Kramers singlet/doublet and bosonic/fermionic properties of Wilson lines. Following Weyl’s gauge principle, by dynamically gauging the 1-form center symmetry, we transform a 5d bulk SRE SPTs into an LRE symmetry-enriched topologically ordered state (SETs); thus we obtain the 4d SO(3)θ=π YM-5d LRE-higher-SETs coupled system with dynamical higher-form gauge fields. Apply the tool introduced in [arXiv:1612.09298], we derive new exotic anyonic statistics of extended objects such as 2-worldsheet of strings and 3-worldvolume of branes, which physically characterize the 5d SETs. We discover new triple and quadruple link invariants potentially associated with the underlying 5d higher-gauge TQFTs, hinting a new intrinsic relation between non-supersymmetric 4d pure YM and topological links in 5d. We provide lattice simplicial complex regularizations and “condensed
New publication from Juven Wang, Xiao-Gang Wen, and Edward Witten:
Title: A New SU(2) Anomaly
Abstract: A familiar anomaly affects SU(2) gauge theory in four dimensions: a theory with an odd number of fermion multiplets in the spin 1/2 representation of the gauge group, and more generally in representations of spin 2r+1/2, is inconsistent. We describe here a more subtle anomaly that can affect SU(2) gauge theory in four dimensions under the condition that fermions transform with half-integer spin under SU(2) and bosons with integer spin. Such a theory, formulated in a way that requires no choice of spin structure, and with an odd number of fermion multiplets in representations of spin 4r+3/2, is inconsistent. The theory is consistent if one picks a spin or spin_c structure. Under Higgsing to U(1), the new SU(2) anomaly reduces to a known anomaly of “all-fermion electrodynamics.” Like that theory, an SU(2) theory with an odd number of fermion multiplets in representations of spin 4r+3/2 can provide a boundary state for a five-dimensional gapped theory whose partition function on a closed five-manifold Y is (−1)∫Yw2w3. All statements have analogs with SU(2) replaced by Sp(2N). There is also an analog in five dimensions.
New publication by Aghil Alaee et al.
Title: Asymptotically flat extensions with charge
Abstract: The Bartnik mass is a notion of quasi-local mass which is remarkably difficult to compute. Mantoulidis and Schoen  developed a novel technique to construct asymptotically flat extensions of minimal Bartnik data in such a way that the ADM mass of these extensions is well-controlled, and thus, they were able to compute the Bartnik mass for minimal spheres satisfying a stability condition. In this work, we develop extensions and gluing tools, à la Mantoulidis and Schoen, for time-symmetric initial data sets for the Einstein-Maxwell equations that allow us to compute the value of an ad-hoc notion of charged Barnik mass for suitable charged minimal Bartnik data.
ArXiv Link: 1903.09014
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.