Geometry and analysis of the Yang-Mills-Higgs-Dirac model

New Publication from Enno Keßler et al.:

Title: Geometry and analysis of the Yang-Mills-Higgs-Dirac model
Authors: Jürgen Jost, Enno Keßler, Ruijun Wu, Miaomiao Zhu

Abstract: The harmonic sections of the Kaluza-Klein model can be seen as a variant of harmonic maps with additional gauge symmetry. Geometrically, they are realized as sections of a fiber bundle associated to a principal bundle with a connection. In this paper, we investigate geometric and analytic aspects of a model that combines the Kaluza-Klein model with the Yang-Mills action and a Dirac action for twisted spinors. In dimension two we show that weak solutions of the Euler Lagrange system are smooth. For a sequence of approximate solutions on surfaces with uniformly bounded energies we obtain compactness modulo bubbles, namely, energy identities and the no-neck property hold.


More publications by Juven Wang in Mathematical Physics and Physical Review D

Journal of Mathematical Physics60, 052301 (2019);

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 transform 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 spinc 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(−1)∫Yw2w3. All statements have analogs with SU(2) replaced by Sp(2N). There is also an analog in five dimensions.

Phys. Rev. D 99, 065013

Title:Adjoint QCD deconfined critical phenomena, symmetry-enriched topological quantum field theory, and higher symmetry extension

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

Title: Solution to the 1+1 dimensional gauged chiral fermion problem

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.


Quantum statistics and spacetime topology: Quantum surgery formulas

A new publication from Juven Wang, Xiao-Gang Wen, and Shing-Tung Yau to be published in Annals of Physics: 

Title: Quantum statistics and spacetime topology: Quantum surgery formulas

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.

ArXiv: 1901.11537

Science Direct 

Orthogonal Polynomials Defined by Self-Similar Measures with Overlaps

New Preprint by Sze-man Ngai:

Orthogonal Polynomials Defined by Self-Similar Measures with Overlaps


Abstract. We study orthogonal polynomials with respect to self-similar measures, focusing on the class of infinite Bernoulli convolutions, which are defined by iterated function systems with overlaps, especially those defined by the Pisot, Garsia, and Salem numbers. By using an algorithm of Mantica, we obtain graphs of the coeffi- cients of the 3-term recursion relation defining the orthogonal polynomials. We use these graphs to predict whether the singular infinite Bernoulli convolutions belong to the Nevai class. Based on our numerical results, we conjecture that all infinite Bernoulli Convolutions with contraction ratios greater than or equal to 1/2 belong to Nevai’s class, regardless of the probability weights assigned to the self-similar measures.

PDF of the preprint

Positive mass theorem for initial data sets with corners along a hypersurface

New publication from Aghil Alaee and S.T. Yau:

Title: Positive mass theorem for initial data sets with corners along a hypersurface

Abstract: We prove positive mass theorem with angular momentum and charges for axially symmetric, simply connected, maximal, complete initial data sets with two ends, one designated asymptotically flat and the other either (Kaluza-Klein) asymptotically flat or asymptotically cylindrical, for 4-dimensional Einstein-Maxwell theory and 5-dimensional minimal supergravity theory which metrics fail to be C1 and second fundamental forms and electromagnetic fields fail to be C0 across an axially symmetric hypersurface Σ. Furthermore, we remove the completeness and simple connectivity assumptions in this result and prove it for manifold with boundary such that the mean curvature of the boundary is non-positive.


2019 Bott Lectures

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.

Information about last year’s Math Science Bott lecture can be found here. 

Existence and Uniqueness of Stationary Solutions in 5-Dimensional Minimal Supergravity

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.