New Publication by Enno Keßler, Artan Sheshmani and Shing-Tung Yau

Title: Super $J$-holomorphic Curves: Construction of the Moduli Space
Enno Keßler, Artan Sheshmani and Shing-Tung Yau

Abstract: Let $M$ be a super Riemann surface with holomorphic distribution $\mathcal{D}$ and $N$ a symplectic manifold with compatible almost complex structure $J$. We call a map $\Phi\colon M\to N$ a super $J$-holomorphic curve if its differential maps the almost complex structure on $\mathcal{D}$ to $J$. Such a super $J$-holomorphic curve is a critical point for the superconformal action and satisfies a super differential equation of first order. Using component fields of this super differential equation and a transversality argument we construct the moduli space of super $J$-holomorphic curves as a smooth subsupermanifold of the space of maps $M\to N$.

Arxiv: 1911.05607

Recent Publications by Juven Wang

Phys. Rev. D 100, 085012 – Published 21 October 2019

Title: Quantum 4d Yang-Mills theory and time-reversal symmetric 5d higher-gauge topological field theory

Zheyan Wan,  Juven Wang , and Yunqin Zheng

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], by turning on the background fields for both the time reversal (i.e., on unorientable manifolds) and the one-form center global symmetry. We find four siblings of SU(2)θ=π YM with distinct couplings to background fields, labeled by (K1,K2)K1=0, 1 specifies Kramers singlet/doublet Wilson line and new mixed higher ’t Hooft anomalies; K2=0, 1 specifies the boson/fermionic Wilson line and a new Wess-Zumino-Witten–like counterterm. Higher anomalies indicate that to realize all higher n-global symmetries locally on n simplices, the 4d theory becomes a boundary of a 5d higher-symmetry-protected topological state (SPTs, as an invertible topological quantum field theory (iTQFT) or a cobordism invariant in math, or as a 5d higher-symmetric interacting topological superconductor in condensed matter)… Continue reading Recent Publications by Juven Wang

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.

arXiv:1908.00430

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

Journal of Mathematical Physics60, 052301 (2019); https://doi.org/10.1063/1.5082852

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.

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

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

SZE-MAN NGAI, WEI TANG, ANH TRAN, AND SHUAI YUAN

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.

 

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.

Quantum Yang-Mills 4d Theory and Time-Reversal Symmetric 5d Higher-Gauge TQFT: Anyonic-String/Brane Braiding Statistics to Topological Link Invariants

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

arXiv:1904.00994

A New SU(2) Anomaly

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.

arXiv:1810.00844