$a\times b=c$ in $2+1$D TQFT

Matthew Buican, Linfeng Li, and Rajath Radhakrishnan

CRST and School of Physics and Astronomy, Queen Mary University of London, London E1 4NS, UK

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.


We study the implications of the anyon fusion equation $a\times b=c$ on global properties of $2+1$D topological quantum field theories (TQFTs). Here $a$ and $b$ are anyons that fuse together to give a unique anyon, $c$. As is well known, when at least one of $a$ and $b$ is abelian, such equations describe aspects of the one-form symmetry of the theory. When $a$ and $b$ are non-abelian, the most obvious way such fusions arise is when a TQFT can be resolved into a product of TQFTs with trivial mutual braiding, and $a$ and $b$ lie in separate factors. More generally, we argue that the appearance of such fusions for non-abelian $a$ and $b$ can also be an indication of zero-form symmetries in a TQFT, of what we term "quasi-zero-form symmetries" (as in the case of discrete gauge theories based on the largest Mathieu group, $M_{24}$), or of the existence of non-modular fusion subcategories. We study these ideas in a variety of TQFT settings from (twisted and untwisted) discrete gauge theories to Chern-Simons theories based on continuous gauge groups and related cosets. Along the way, we prove various useful theorems.

► BibTeX data

► References

[1] Gregory W. Moore and N. Read. Nonabelions in the fractional quantum Hall effect. Nucl. Phys. B, 360: 362–396. 10.1016/​0550-3213(91)90407-O.

[2] Edward Witten. Quantum Field Theory and the Jones Polynomial. Commun. Math. Phys., 121: 351–399, 1989. 10.1007/​BF01217730. [233(1988)].

[3] Zhenghan Wang. Topological quantum computation. Number 112. American Mathematical Soc., 2010. 10.1090/​cbms/​112.

[4] Gregory W. Moore and Nathan Seiberg. LECTURES ON RCFT. In 1989 Banff NATO ASI: Physics, Geometry and Topology Banff, Canada, August 14-25, 1989, pages 1–129, 1989. 10.1007/​978-1-4615-3802-8_8. [,1(1989)].

[5] Bojko Bakalov and Alexander A Kirillov. Lectures on tensor categories and modular functors, volume 21. American Mathematical Soc., 2001. 10.1090/​ulect/​021.

[6] Alexei Kitaev. Anyons in an exactly solved model and beyond. Annals of Physics, 321 (1): 2–111, 2006. 10.1016/​j.aop.2005.10.005.

[7] Eric Rowell, Richard Stong, and Zhenghan Wang. On classification of modular tensor categories. Communications in Mathematical Physics, 292 (2): 343–389, 2009. 10.1007/​s00220-009-0908-z.

[8] Paul Bruillard, Julia Plavnik, and Eric Rowell. Modular categories of dimension $p^3m$ with $m$ square-free. Proceedings of the American Mathematical Society, 147 (1): 21–34, 2019. doi.org/​10.1090/​proc/​13776.

[9] Gil Young Cho, Dongmin Gang, and Hee-Cheol Kim. M-theoretic Genesis of Topological Phases. JHEP, 11: 115, 2020. 10.1007/​JHEP11(2020)115.

[10] Michael Müger. On the structure of modular categories. Proceedings of the London Mathematical Society, 87 (2): 291–308, 2003. 10.1112/​S0024611503014187.

[11] Davide Gaiotto, Anton Kapustin, Nathan Seiberg, and Brian Willett. Generalized Global Symmetries. JHEP, 02: 172, 2015. 10.1007/​JHEP02(2015)172.

[12] Po-Shen Hsin, Ho Tat Lam, and Nathan Seiberg. Comments on One-Form Global Symmetries and Their Gauging in 3d and 4d. 2018. 10.21468/​SciPostPhys.6.3.039.

[13] FA Bais and JK Slingerland. Condensate-induced transitions between topologically ordered phases. Physical Review B, 79 (4): 045316, 2009. 10.1103/​PhysRevB.79.045316.

[14] Kenneth A. Intriligator. Bonus Symmetry in Conformal Field Theory. Nucl. Phys. B, 332: 541–565, 1990. 10.1016/​0550-3213(90)90001-T.

[15] A.N. Schellekens and S. Yankielowicz. Simple Currents, Modular Invariants and Fixed Points. Int. J. Mod. Phys. A, 5: 2903–2952, 1990a. 10.1142/​S0217751X90001367.

[16] Robbert Dijkgraaf and Edward Witten. Topological Gauge Theories and Group Cohomology. Commun. Math. Phys., 129: 393, 1990. 10.1007/​BF02096988.

[17] P. Roche, V. Pasquier, and R. Dijkgraaf. QuasiHopf algebras, group cohomology and orbifold models. Nucl. Phys. B Proc. Suppl., 18: 60–72, 1990. 10.1016/​0920-5632(91)90123-V.

[18] Dmitri Nikshych and Brianna Riepel. Categorical lagrangian grassmannians and brauer–picard groups of pointed fusion categories. Journal of Algebra, 411: 191–214, 2014. 10.1016/​j.jalgebra.2014.04.013.

[19] Salman Beigi, Peter W Shor, and Daniel Whalen. The quantum double model with boundary: condensations and symmetries. Communications in mathematical physics, 306 (3): 663–694, 2011. 10.1007/​s00220-011-1294-x.

[20] Maissam Barkeshli, Parsa Bonderson, Meng Cheng, and Zhenghan Wang. Symmetry Fractionalization, Defects, and Gauging of Topological Phases. Phys. Rev. B, 100 (11): 115147, 2019. 10.1103/​PhysRevB.100.115147.

[21] Matthew Buican, Linfeng Li, and Rajath Radhakrishnan. Non-Abelian Anyons and Some Cousins of the Arad-Herzog Conjecture. 12 2020. URL https:/​/​arxiv.org/​abs/​2012.03394v2.

[22] Deepak Naidu. Categorical morita equivalence for group-theoretical categories. Communications in Algebra, 35 (11): 3544–3565, 2007. 10.1080/​00927870701511996.

[23] Yuting Hu, Yidun Wan, and Yong-Shi Wu. Twisted quantum double model of topological phases in two dimensions. Phys. Rev. B, 87 (12): 125114, 2013. 10.1103/​PhysRevB.87.125114.

[24] Pierre Deligne. Catégories tensorielles. Moscow Mathematical Journal, 2 (2): 227–248, 2002. URL https:/​/​publications.ias.edu/​book/​export/​html/​434.

[25] Ilan Zisser. Irreducible products of characters in $a_n$. Israel Journal of Mathematics, 84 (1-2): 147–151, 1993. 10.1007/​BF02761696.

[26] Deepak Naidu, Dmitri Nikshych, and Sarah Witherspoon. Fusion subcategories of representation categories of twisted quantum doubles of finite groups. International Mathematics Research Notices, 2009 (22): 4183–4219, 2009. 10.1093/​imrn/​rnp084.

[27] Shawn X Cui, César Galindo, Julia Yael Plavnik, and Zhenghan Wang. On gauging symmetry of modular categories. Communications in Mathematical Physics, 348 (3): 1043–1064, 2016. 10.1007/​s00220-016-2633-8.

[28] Tom Rudelius and Shu-Heng Shao. Topological Operators and Completeness of Spectrum in Discrete Gauge Theories. 6 2020. 10.1007/​JHEP12(2020)172.

[29] I Martin Isaacs. Character theory of finite groups, volume 69. Courier Corporation, 1994. 10.1090/​chel/​359.

[30] W. Burnside. Theory of groups of finite order (2nd Ed.). Dover Publications, Inc., New York, 1955. 10.1017/​CBO9781139237253.

[31] Dilip Gajendragadkar. A characteristic class of characters of finite $\pi$-separable groups. Journal of algebra, 59 (2): 237–259, 1979. 10.1016/​0021-8693(79)90124-8.

[32] Gabriel Navarro. New properties of the $\pi$-special characters. Journal of Algebra, 187 (1): 203 – 213, 1997. ISSN 0021-8693. 10.1006/​jabr.1997.6798.

[33] Peter Brooksbank and Matthew Mizuhara. On groups with a class-preserving outer automorphism. Involve, a Journal of Mathematics, 7 (2): 171–179, 2013. 10.2140/​involve.2014.7.171.

[34] Shawn X. Cui, Dawei Ding, Xizhi Han, Geoffrey Penington, Daniel Ranard, Brandon C. Rayhaun, and Zhou Shangnan. Kitaev's quantum double model as an error correcting code. 8 2019. 10.22331/​q-2020-09-24-331.

[35] Yuting Hu and Yidun Wan. Electric-magnetic duality in twisted quantum double model of topological orders. arXiv preprint arXiv:2007.15636, 2020. 10.1007/​JHEP11(2020)170.

[36] Matthew Buican and Rajath Radhakrishnan. Galois conjugation and multiboundary entanglement entropy. JHEP, 12: 045, 2020. 10.1007/​JHEP12(2020)045.

[37] Michaël Mignard and Peter Schauenburg. Modular categories are not determined by their modular data. arXiv preprint arXiv:1708.02796, 2017. 10.1007/​s11005-021-01395-0.

[38] Alexei Davydov. Unphysical diagonal modular invariants. Journal of Algebra, 446: 1–18, 2016. 10.1016/​j.jalgebra.2015.09.007.

[39] Tohru Eguchi, Hirosi Ooguri, and Yuji Tachikawa. Notes on the K3 Surface and the Mathieu group $M_{24}$. Exper. Math., 20: 91–96, 2011. 10.1080/​10586458.2011.544585.

[40] Miranda C.N. Cheng, John F.R. Duncan, and Jeffrey A. Harvey. Umbral Moonshine. Commun. Num. Theor. Phys., 08: 101–242, 2014. 10.4310/​CNTP.2014.v8.n2.a1.

[41] Terry Gannon. Much ado about Mathieu. Adv. Math., 301: 322–358, 2016. 10.1016/​j.aim.2016.06.014.

[42] GAP. GAP group: GAP-groups, algorithms, and programming, Version 4.4 (2004). URL http:/​/​www.gap-system.org.

[43] A.D. Berenstein and A.V. Zelevinsky. Tensor Product Multiplicities and Convex Polytopes in Partition Space. J. Geom. Phys., 5: 453, 1989. 10.1016/​0393-0440(88)90033-2.

[44] Doron Gepner and Edward Witten. String Theory on Group Manifolds. Nucl. Phys., B278: 493–549, 1986. 10.1016/​0550-3213(86)90051-9.

[45] P. Di Francesco, P. Mathieu, and D. Senechal. Conformal Field Theory. Graduate Texts in Contemporary Physics. Springer-Verlag, New York, 1997. ISBN 9780387947853, 9781461274759. 10.1007/​978-1-4612-2256-9.

[46] A.N. Kirillov, P. Mathieu, D. Senechal, and M.A. Walton. Can fusion coefficients be calculated from the depth rule? Nucl. Phys. B, 391: 651–674, 1993. 10.1016/​0550-3213(93)90087-6.

[47] A.N. Kirillov, P. Mathieu, D. Senechal, and M.A. Walton. Crystallizing the depth rule for WZNW fusion coefficients. In 19th International Colloquium on Group Theoretical Methods in Physics, 9 1992. URL https:/​/​arxiv.org/​abs/​hep-th/​9209114.

[48] Alex J. Feingold and Stefan Fredenhagen. A New perspective on the Frenkel-Zhu fusion rule theorem. J. Algebra, 320: 2079–2100, 2008. 10.1016/​j.jalgebra.2008.05.026.

[49] Andrew Urichuk and Mark A. Walton. Adjoint affine fusion and tadpoles. J. Math. Phys., 57 (6): 061702, 2016. 10.1063/​1.4954909.

[50] J.M. Isidro, J.M.F. Labastida, and A.V. Ramallo. Coset constructions in Chern-Simons gauge theory. Phys. Lett. B, 282: 63–72, 1992. 10.1016/​0370-2693(92)90480-R.

[51] P. Goddard, A. Kent, and David I. Olive. Virasoro Algebras and Coset Space Models. Phys. Lett. B, 152: 88–92, 1985. 10.1016/​0370-2693(85)91145-1.

[52] P. Ramadevi, T.R. Govindarajan, and R.K. Kaul. Knot invariants from rational conformal field theories. Nucl. Phys. B, 422: 291–306, 1994. 10.1016/​0550-3213(94)00102-2.

[53] P. Goddard, A. Kent, and David I. Olive. Unitary Representations of the Virasoro and Supervirasoro Algebras. Commun. Math. Phys., 103: 105–119, 1986. 10.1007/​BF01464283.

[54] A.N. Schellekens and S. Yankielowicz. Field Identification Fixed Points in the Coset Construction. Nucl. Phys. B, 334: 67–102, 1990b. 10.1016/​0550-3213(90)90657-Y.

Cited by

On Crossref's cited-by service no data on citing works was found (last attempt 2021-06-16 02:01:02). On SAO/NASA ADS no data on citing works was found (last attempt 2021-06-16 02:01:03).