Videos of the lectures can be found in the Youtube playlist below. Links to talks are also available on the schedule on our website.
Journal of Mathematical Physics60, 052301 (2019); https://doi.org/10.1063/1.5082852
Title: A new SU(2) anomaly
Phys. Rev. D 99, 065013
Abstract: Recent work explores the candidate phases of the 4D adjoint quantum chromodynamics (QCD4) with an SU(2) gauge group and two massless adjoint Weyl fermions. Both Cordova-Dumitrescu and Bi-Senthil propose possible low energy 4D topological quantum field theories (TQFTs) to saturate the higher ’t Hooft anomalies of adjoint QCD4 under a renormalization-group flow from high energy. In this work, we generalize the symmetry-extension method of Wang-Wen-Witten [Phys. Rev. X 8, 031048 (2018)] to higher symmetries, and formulate a higher group cohomology and cobordism theory approach to construct higher-symmetric TQFTs. We prove that the symmetry-extension method saturates certain anomalies, but also prove that neitherAP2(B2)norP2(B2) can be fully trivialized, with the background 1-form field A, Pontryagin square P2, and 2-form field B2. Surprisingly, this indicates an obstruction to constructing a fully 1-form center and 0-form chiral symmetry (full discrete axial symmetry) preserving 4D TQFT with confinement, a no-go scenario via symmetry extension for specific higher anomalies. We comment on the implications and constraints on deconfined quantum criticality and ultraviolet-infrared duality in 3+1 spacetime dimensions.
Phys. Rev. D 99, 111501(R) – Published 10 June 2019
Abstract: We show that the 3450 U(1) chiral fermion theory can appear as the low energy effective field theory of a 1+1D local lattice model of fermions, with an on-site U(1) symmetry and finite-range interactions. The on-site U(1) symmetry means that the U(1) symmetry can be gauged (gaugeable for both background probe and dynamical fields), which leads to a nonperturbative definition of chiral gauge theory—a chiral fermion theory coupled to U(1) gauge theory. Our construction can be generalized to regularize any U(1)-anomaly-free 1+1D gauged chiral fermion theory with a zero chiral central charge (thus no gravitational anomaly) by a lattice, thanks to the recently proven “Poincaré dual” equivalence between the U(1) ’t Hooft anomaly-free condition and the U(1) symmetric interaction gapping rule, via a bosonization-fermionization technique.
A new publication from Juven Wang, Xiao-Gang Wen, and Shing-Tung Yau to be published in Annals of Physics:
Abstract: To formulate the universal constraints of quantum statistics data of generic long-range entangled quantum systems, we introduce the geometric-topology surgery theory on spacetime manifolds where quantum systems reside, cutting and gluing the associated quantum amplitudes, specifically in 2+1 and 3+1 spacetime dimensions. First, we introduce the fusion data for worldline and worldsheet operators capable of creating anyonic 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, 3-string, 4-string, or multi-string adiabatic loop braiding process, encoded by submanifold links, in the closed spacetime 3-manifolds and 4-manifolds. Third, we derive new “quantum surgery” formulas and constraints, analogous to Verlinde formula associating fusion and braiding statistics data via spacetime surgery, essential for defining the theory of topological orders, 3d and 4d TQFTs and potentially correlated to bootstrap boundary physics such as gapless modes, extended defects, 2d and 3d conformal field theories or quantum anomalies.
This article is meant to be an extended and further detailed elaboration of our previous work Wang, Wen and Yau (0000) and Chapter 6 of Wang (2015). Our theory applies to general quantum theories and quantum mechanical systems, also applicable to, but not necessarily requiring the quantum field theory description.
On April 9 and 10, 2019 the CMSA hosted two lectures by Mina Aganagic (UC Berkeley). This was the second annual Math Science Lecture Series held in honor of Raoul Bott.
“Two math lessons from string theory”
Lecture 1: “Lesson on Integrability”
Abstract: The quantum Knizhnik-Zamolodchikov (qKZ) equation is a difference generalization of the famous Knizhnik-Zamolodchikov (KZ) equation. The problem to explicitly capture the monodromy of the qKZ equation has been open for over 25 years. I will describe the solution to this problem, discovered jointly with Andrei Okounkov. The solution comes from the geometry of Nakajima quiver varieties and has a string theory origin.
Part of the interest in the qKZ monodromy problem is that its solution leads to integrable lattice models, in parallel to how monodromy matrices of the KZ equation lead to knot invariants. Thus, our solution of the problem leads to a new, geometric approach, to integrable lattice models. There are two other approaches to integrable lattice models, due to Nekrasov and Shatashvili and to Costello, Witten and Yamazaki. I’ll describe joint work with Nikita Nekrasov which explains how string theory unifies the three approaches to integrable lattice models.
Lecture 2: “Lesson on Knot Categorification”
Abstract: An old problem is to find a unified approach to the knot categorification problem. The new string theory perspective on the qKZ equation I described in the first talk can be used to derive two geometric approaches to the problem.The first approach is based on a category of B-type branes on resolutions of slices in affine Grassmannians. The second is based on a category of A-branes in a Landau-Ginzburg theory. The relation between them is two dimensional (equivariant) mirror symmetry. String theory also predicts that a third approach to categorification, based on counting solutions to five dimensional Haydys-Witten equations, is equivalent to the first two.This talk is mostly based on joint work with Andrei Okounkov.
As part of the program on Mathematical Biology a workshop on Invariance and Geometry in Sensation, Action and Cognition took place on April 15-17, 2019.
View the videos in the youtube playlist below:
CMSA Postdoc Zhong-Zhi Xianyu and colleagues’ latest research examines residual radiation from the Big Bang and their relation to the elementary particles in the Standard Model of particle physics.
New publication by Aghil Alaee et al.
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.