arXiv:1812.11968 [pdf, ps, other]

**New Higher Anomalies, SU(N) Yang-Mills Gauge Theory and ℂℙN−1 Sigma Model**

Zheyan Wan, Juven Wang

Subjects: High Energy Physics – Theory (hep-th)

We hypothesize a new and more complete set of anomalies of quantum field theories (QFTs) and then give an eclectic proof. First, we propose a set of 4d ‘t Hooft higher anomalies of time-reversal symmetric SU(N)-Yang-Mills (YM) gauge theory at θ=π, specifically at N=2,3,4 and others, by enlisting all possible 5d cobordism invariants (higher symmetry-protected topological states) and selecting the matched terms. Second, we propose a set of 3d ‘t Hooft anomalies of ℂℙN−1-sigma model obtained via compactifying YM theory on a 2-torus, and select the matched 3d cobordism invariants. Based on algebraic/geometric topology, QFT analysis, manifold generator dimensional reduction, condensed matter inputs and additional physical criteria, we derive a correspondence between 5d and 3d new invariants, thus broadly prove a more complete anomaly-matching between 4d YM and 2d ℂℙN−1 models via a 2-torus reduction. We formulate a higher-symmetry analog of “Lieb-Schultz-Mattis theorem” to constrain the low-energy dynamics.

arXiv:1812.11967 [pdf, ps, other]

**Non-Abelian Gauge Theories, Sigma Models, Higher Anomalies, Symmetries, and Cobordisms**

Zheyan Wan, Juven Wang

Comments: See a Primer: arXiv:1711.11587 for introduction

Subjects: High Energy Physics – Theory (hep-th)

We explore the higher-form generalized global symmetries and higher anomalies based on a generalized cobordism theory. Our cobordism calculation guides us to classify higher anomalies and topological terms for Yang-Mills (YM) gauge theories and sigma models in various dimensions. Some of YM gauge theories can be obtained from dynamically gauging the SU(N) time-reversal symmetric cobordism invariants (SU(N)-generalized topological superconductors/insulators [arXiv:1711.11587]). We elaborate the cases of YM theory with a compact Lie gauge group (such as SU(N) with N=2, 3 and others) particularly in a 4d spacetime. We provide the relevant homotopy and cobordism group calculations of higher classifying spaces, based on mathematical tools of algebraic topology, to support the physics stories.

This is a companion article with further detailed calculations supporting other shorter articles.

arXiv:1812.11959 [pdf, other]

**Fermionic Finite-Group Gauge Theories and Interacting** Symmetric/Crystalline Orders via Cobordisms

Meng Guo, Kantaro Ohmori, Pavel Putrov, Zheyan Wan, Juven Wang

Comments: 81 pages, 28 figures

Subjects: High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph); Algebraic Topology (math.AT); Geometric Topology (math.GT); Quantum Algebra (math.QA)

We formulate a family of spin Topological Quantum Filed Theories (spin-TQFTs) as fermionic generalization of bosonic Dijkgraaf-Witten TQFTs. They are obtained by gauging G-equivariant invertible spin-TQFTs, or, in physics language, gauging the interacting fermionic Symmetry Protected Topological states (SPTs) with a finite group G symmetry. We use the fact that the latter are classified by Pontryagin duals to spin-bordism groups of the classifying space BG. We also consider unoriented analogs, that is G-equivariant invertible pin±-TQFTs (fermionic time-reversal-SPTs) and their gauging. We compute these groups for various examples of abelian G using Adams spectral sequence and describe all corresponding TQFTs via certain bordism invariants in dimensions 3, 4, and other. This gives explicit formulas for the partition functions of spin-TQFTs on closed manifolds with possible extended operators inserted. The results also provide an explicit classification of ‘t Hooft anomalies of fermionic QFTs with finite abelian group symmetries in one dimension lower. We construct new anomalous boundary deconfined spin-TQFTs (surface fermionic topological orders). We explore SPT and SET (symmetry enriched topologically ordered) states, and crystalline SPTs protected by space-group (e.g. translation ℤ) or point-group (e.g. reflection, inversion or rotation Cm) symmetries, via the layer-stacking construction.

arXiv:1812.11955 [pdf, other]

**Higher Symmetry-Extension, Symmetry-Enriched TQFT and Adjoint QCD4**

Zheyan Wan, Juven Wang

Subjects: High Energy Physics – Theory (hep-th)

Recent work explores the candidate phases of the adjoint QCD4 with an SU(2) gauge group and two massless adjoint Weyl fermions. Both Bi-Senthil and Cordova-Dumitrescu propose possible 4d TQFTs to saturate higher ‘t Hooft anomalies. We generalize the symmetry-extension methods of Wang-Wen-Witten to higher symmetries and discuss a cobordism theory approach to construct such symmetric TQFTs. We find certain symmetric 4d TQFTs have an obstruction, possibly a no-go scenario for Type I anomaly.