## Two New Publications by Aghil Alaee

Title: A localized spacetime Penrose inequality and horizon detection with quasi-local mass
Aghil Alaee, Martin Lesourd, Shing-Tung Yau

Abstract: Our setting is a simply connected bounded domain with a smooth connected boundary, which arises as an initial data set for the general relativistic constraint equations satisfying the dominant energy condition. Assuming the domain to be admissible in a certain precise sense, we prove a localized spacetime Penrose inequality for the Liu-Yau and Wang-Yau quasi-local masses and the area of an outermost marginally outer trapped surface (MOTS). On the basis of this inequality, we obtain sufficient conditions for the existence and non-existence of a MOTS (along with outer trapped surfaces) in the domain, and for the existence of a minimal surface in its Jang graph, expressed in terms of various quasi-local mass quantities and the boundary geometry of the domain.

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Title: Stability of a quasi-local positive mass theorem for graphical hypersurfaces of Euclidean space
Aghil Alaee, Armando J. Cabrera Pacheco, Stephen McCormick
Abstract: We present a quasi-local version of the stability of the positive mass theorem. We work with the Brown–York quasi-local mass as it possesses positivity and rigidity properties, and therefore the stability of this rigidity statement can be studied. Specifically, we ask if the Brown–York mass of the boundary of some compact manifold is close to zero, must the manifold be close to a Euclidean domain in some sense?
Here we consider a class of compact n-manifolds with boundary that can be realized as graphs in ℝn+1, and establish the following. If the Brown–York mass of the boundary of such a compact manifold is small, then the manifold is close to a Euclidean hyperplane with respect to the Federer–Fleming flat distance.

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## New Publications by Juven Wang

Title: Higher Anomalies, Higher Symmetries, and Cobordisms III: QCD Matter Phases Anew

Zheyan Wan, Juven Wang

Abstract: We explore QCD4 quark matter, the μ-T (chemical potential-temperature) phase diagram, possible ‘t Hooft anomalies, and topological terms, via non-perturbative tools of cobordism theory and higher anomaly matching. We focus on quarks in 3-color and 3-flavor on bi-fundamentals of SU(3), then analyze the continuous and discrete global symmetries and pay careful attention to finite group sectors. We input constraints from T=CP or CT time-reversal symmetries, implementing QCD on unorientable spacetimes and distinct topology. Examined phases include the high T QGP (quark-gluon plasma/liquid), the low T ChSB (chiral symmetry breaking), 2SC (2-color superconductivity) and CFL (3-color-flavor locking superconductivity) at high density. We introduce a possibly useful but only approximate higher anomaly, involving discrete 0-form axial and 1-form mixed chiral-flavor-locked center symmetries, matched by the above four QCD phases. We also enlist as much as possible, but without identifying all of, ‘t Hooft anomalies and topological terms relevant to Symmetry Protected/Enriched Topological states (SPTs/SETs) of gauged SU(2) or SU(3) QCDd-like matter theories in general in any spacetime dimensions d=2,3,4,5 via cobordism.

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Title: Higher Anomalies, Higher Symmetries, and Cobordisms II: Applications to Quantum Gauge Theories

Zheyan Wan, Juven Wang, Yunqin Zheng

Abstract: We discuss the topological terms, the global symmetries and their ‘t Hooft anomalies of pure gauge theories in various dimensions, with dynamical gauge group G, the Lorentz symmetry group GLorentz, and the internal global symmetry Ge,[1]×Gm,[d−3] which consists of 1-form electric center symmetry Ge,[1] and (d−3) form magnetic symmetry Gm,[d−3]. The topological terms are determined by the cobordism invariants (Ωd)G′ where G′ is the group extension of GLorentz by G, which also characterize the invertible TQFTs or SPTs with global symmetry G′. The ‘t Hooft anomalies are determined by the cobordism invariants (Ωd+1)G′′ where G′′ is the symmetry extension of GLorentz by the higher form symmetry Ge,[1]×Gm,[d−3]. Different symmetry extensions correspond to different fractionalizations of GLorentz quantum numbers on the symmetry defects of Ge,[1]×Gm,[d−3]. We compute the cobordism groups/invariants described above for G= U(1), SU(2) and SO(3) in d≤5, thus systematically classifies all the topological terms and the ‘t Hooft anomalies of d dimensional quantum gauge theories with the above gauge groups.

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Title: Non-Abelian Gauged Fractonic Matter Field Theory: New Sigma Models, Superfluids and Vortices

Juven Wang, Shing-Tung Yau

Abstract: By gauging a higher-moment polynomial global symmetry and a discrete charge conjugation (i.e., particle-hole) symmetry (mutually non-commutative) coupled to matter fields, we derive a new class of higher-rank tensor non-abelian gauge field theory with dynamically gauged matter fields: Non-abelian gauged matters interact with a hybrid class of higher-rank (symmetric or generic non-symmetric) tensor gauge theory and anti-symmetric tensor topological field theory, generalizing [arXiv:1909.13879, 1911.01804]’s theory. We also apply a quantum phase transition similar to that between insulator v.s. superfluid/superconductivity (U(1) symmetry disordered phase described by a topological gauge theory or a disordered Sigma model v.s. U(1) global/gauge symmetry-breaking ordered phase described by a Sigma model with a U(1) target space underlying Goldstone modes): We can regard our tensor gauge theories as disordered phases, and we transient to their new ordered phases by deriving new Sigma models in continuum field theories. While one low energy theory is captured by degrees of freedom of rotor or scalar modes, another side of low energy theory has vortices and superfluids – we explore non-abelian vortices (two types of vortices mutually interacting non-commutatively) beyond an ordinary group structure and their Cauchy-Riemann relation.

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Title: Higher-Rank Non-Abelian Tensor Field Theory: Higher-Moment or Subdimensional Polynomial Global Symmetry, Algebraic Variety, Noether’s Theorem, and Gauge

Juven Wang, Kai Xu, Shing-Tung Yau

Abstract: With a view toward a theory of fracton and embeddon in condensed matter, we introduce a higher-moment polynomial degree-(m-1) global symmetry, acting on complex scalar/vector/tensor fields. We relate this higher-moment global symmetry of n-dimensional space, to a lower degree (either ordinary or higher-moment, e.g., degree-(m-1-ℓ)) subdimensional or subsystem global symmetry on layers of (n−ℓ)-submanifolds. These submanifolds are algebraic affine varieties (i.e., solutions of polynomials). The structure of layers of submanifolds as subvarieties can be studied via mathematical tools of embedding, foliation and algebraic geometry. We also generalize Noether’s theorem for this higher-moment polynomial global symmetry. We can promote the higher-moment global symmetry to a local symmetry, and derive a new family of higher-rank-m symmetric tensor gauge theory by gauging. By further gauging a discrete charge conjugation symmetry, we derive a new more general class of non-abelian rank-m tensor gauge field theory: a hybrid class of (symmetric or non-symmetric) higher-rank-m tensor gauge theory and anti-symmetric tensor topological field theory, generalizing [arXiv:1909.13879]’s theory interplaying between gapless and gapped sectors.

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## 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

## Videos from the Spacetime and Quantum Mechanics Master Class Workshop

Videos from the workshop are contained in the Youtube playlist below. They can also be found here.

## 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.

## 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

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