When Does A Three-Dimensional Chern-Simons-Witten Theory Have A Time Reversal Symmetry?

In this paper, we completely characterize time-reversal invariant three-dimensional Chern-Simons gauge theories with torus gauge group. At the level of the Lagrangian, toral Chern-Simons theory is defined by an integral lattice, while at the quantum level, it is entirely determined by a quadratic function on a finite Abelian group and an integer mod 24. We find that quantum time-reversally symmetric theories can be defined by classical Lagrangians defined by integral lattices which have self-perpendicular embeddings into a unimodular lattice. We give an explicit list of finite Abelian groups equipped with quadratic functions such that all T-invariant quantum theories must be direct sums of the elements of that list. In the non-Abelian case, we conjecture that the Chern-Simons theory admits a time-reversal symmetry iff the associated modular tensor category is an order two element in the Witt group of non-degenerate braided fusion categories.

[1]  Liang Kong,et al.  An invitation to topological orders and category theory , 2022, 2205.05565.

[2]  Z. Komargodski,et al.  Higher central charges and topological boundaries in 2+1-dimensional TQFTs , 2021, SciPost Physics.

[3]  S. Ng,et al.  Higher central charges and Witt groups , 2020, Advances in Mathematics.

[4]  Cain EDIE-MICHELL,et al.  Auto-equivalences of the modular tensor categories of type A, B, C and G , 2020, Advances in Mathematics.

[5]  Eric C. Rowell,et al.  The Witt classes of $SO(2r)_{2r}$ , 2021, 2107.06746.

[6]  K. Williams,et al.  Gauss and Jacobi sums , 2021, Mathematical Surveys and Monographs.

[7]  Daniel S. Freed,et al.  Gapped Boundary Theories in Three Dimensions , 2020, Communications in Mathematical Physics.

[8]  P. Schauenburg,et al.  Modular categories are not determined by their modular data , 2017, Letters in Mathematical Physics.

[9]  G. Moore,et al.  Dyson’s classification and real division superalgebras , 2020, 2010.01675.

[10]  J. Gomis,et al.  Symmetries of abelian Chern-Simons theories and arithmetic , 2019, Journal of High Energy Physics.

[11]  S. Ng,et al.  Higher Gauss sums of modular categories , 2018, Selecta Mathematica.

[12]  Zhenghan Wang,et al.  Classification of Super-Modular Categories by Rank , 2017, Algebras and Representation Theory.

[13]  K. Walker,et al.  Reflection and Time Reversal Symmetry Enriched Topological Phases of Matter: Path Integrals, Non-orientable Manifolds, and Anomalies , 2016, Communications in Mathematical Physics.

[14]  Meng Cheng,et al.  Symmetry fractionalization, defects, and gauging of topological phases , 2014, Physical Review B.

[15]  Yuji Tachikawa,et al.  A study of time reversal symmetry of abelian anyons , 2018, Journal of High Energy Physics.

[16]  Eric C. Rowell,et al.  Mathematics of Topological Quantum Computing , 2017, 1705.06206.

[17]  A. Henriques.  What Chern–Simons theory assigns to a point , 2015, Proceedings of the National Academy of Sciences.

[18]  O. Aharony,et al.  Chern-Simons-matter dualities with SO and USp gauge groups , 2016, 1611.07874.

[19]  N. Seiberg,et al.  Level/rank duality and Chern-Simons-matter theories , 2016, Journal of High Energy Physics.

[20]  Tobias J. Hagge,et al.  Fermionic Modular Categories and the 16-fold Way , 2016, 1603.09294.

[21]  E. Witten,et al.  Gapped boundary phases of topological insulators via weak coupling , 2016, 1602.04251.

[22]  Liang Kong.  Anyon condensation and tensor categories , 2013, 1307.8244.

[23]  V. Ostrik,et al.  Level-Rank Duality via Tensor Categories , 2012, 1208.5131.

[24]  A. Vishwanath,et al.  Non-Abelian Topological Order on the Surface of a 3D Topological Superconductor from an Exactly Solved Model , 2013, 1305.5851.

[25]  T. Senthil,et al.  Boson topological insulators: A window into highly entangled quantum phases , 2013, 1302.6234.

[26]  D. Freed,et al.  Twisted Equivariant Matter , 2012, 1208.5055.

[27]  Michael Levin,et al.  Classification and analysis of two-dimensional Abelian fractional topological insulators , 2012, 1205.1244.

[28]  On Wigner's theorem , 2011, 1112.2133.

[29]  V. Ostrik,et al.  On the structure of the Witt group of braided fusion categories , 2011, 1109.5558.

[30]  P. Etingof,et al.  Descent and Forms of Tensor Categories , 2011, 1102.0657.

[31]  A. Kapustin,et al.  Topological boundary conditions in abelian Chern-Simons theory , 2010, 1008.0654.

[32]  A. Kapustin,et al.  Surface operators in 3d Topological Field Theory and 2d Rational Conformal Field Theory , 2010, 1012.0911.

[33]  V. Ostrik,et al.  The Witt group of non-degenerate braided fusion categories , 2010, 1009.2117.

[34]  J. Fuchs,et al.  DEFECT LINES, DUALITIES AND GENERALISED ORBIFOLDS , 2009, 0909.5013.

[35]  V. Drinfeld,et al.  On braided fusion categories I , 2009, 0906.0620.

[36]  D. Freed,et al.  Topological Quantum Field Theories from Compact Lie Groups , 2009, 0905.0731.

[37]  S. D. Stirling.  Abelian Chern-Simons theory with toral gauge group, modular tensor categories, and group categories , 2008, 0807.2857.

[38]  S. Ng,et al.  Frobenius–Schur indicators and exponents of spherical categories , 2005, math/0601012.

[39]  G. Moore,et al.  Classification of abelian spin Chern-Simons theories , 2005, hep-th/0505235.

[40]  E. Witten.  SL(2;Z) Action On Three-Dimensional Conformal Field Theories With Abelian Symmetry , 2003, hep-th/0307041.

[41]  I. Singer,et al.  Quadratic functions in geometry, topology, and M-theory , 2002, math/0211216.

[42]  J. Fuchs,et al.  TFT construction of RCFT correlators I: Partition functions , 2002, hep-th/0204148.

[43]  On fusion categories , 2002, math/0203060.

[44]  Al.  WHICH SINGULAR K3 SURFACES COVER AN ENRIQUES SURFACE , 2002 .

[45]  L. Taylor.  Gauss Sums in Algebra and Topology , 2022, 2208.06319.

[46]  Michael Mueger.  From subfactors to categories and topology. II. The quantum double of tensor categories and subfactors , 2001, math/0111205.

[47]  T. Kohno,et al.  Level-Rank Duality of Witten’s 3-Manifold Invariants , 1996 .

[48]  T. Kerler.  On braided tensor categories , 1994, hep-th/9402018.

[49]  Bruce W. Westbury,et al.  Spherical Categories , 1993, hep-th/9310164.

[50]  S. D. Pietra,et al.  Geometric quantization of Chern-Simons gauge theory , 1991 .

[51]  H. Schnitzer,et al.  Group-level duality in WZW models and Chern-Simons theory , 1990 .

[52]  E. Witten,et al.  Topological gauge theories and group cohomology , 1990 .

[53]  Nathan Seiberg,et al.  Remarks on the canonical quantization of the Chern-Simons-Witten theory , 1989 .

[54]  Nathan Seiberg,et al.  LECTURES ON RCFT , 1989 .

[55]  Edward Witten,et al.  Quantum field theory and the Jones polynomial , 1989 .

[56]  G. Moore,et al.  Classical and quantum conformal field theory , 1989 .

[57]  G. Moore,et al.  Naturality in Conformal Field Theory , 1989 .

[58]  E. Verlinde,et al.  Modular Invariance and the Fusion Algebra , 1988 .

[59]  G. Anderson,et al.  Rationality in conformal field theory , 1988 .

[60]  C. Itzykson,et al.  The A-D-E classification of minimal andA1(1) conformal invariant theories , 1987 .

[61]  R. Miranda.  Nondegenerate symmetric bilinear forms on finite abelian 2-groups , 1984 .

[62]  V V Nikulin,et al.  Integral Symmetric Bilinear Forms and Some of Their Applications , 1980 .

[63]  A. Kawauchi,et al.  Algebraic classification of linking pairings on 3-manifolds , 1980 .

[64]  Alan Durfee.  Bilinear and quadratic forms on torsion modules , 1977 .

[65]  C. Wall.  Quadratic forms on finite groups, and related topics , 1963 .

[66]  Freeman J. Dyson,et al.  The Threefold Way. Algebraic Structure of Symmetry Groups and Ensembles in Quantum Mechanics , 1962 .

[67]  E. Wigner,et al.  Book Reviews: Group Theory. And Its Application to the Quantum Mechanics of Atomic Spectra , 1959 .

[68]  M. Kneser,et al.  Quadratische Formen und Verschlingungsinvarianten von Knoten , 1953 .

[69]  S. Maclane,et al.  Cohomology theory of abelian groups and homotopy theory I. , 1950, Proceedings of the National Academy of Sciences of the United States of America.