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

Buican, Matthew; Li, Linfeng; Radhakrishnan, Rajath

doi:10.22331/q-2021-06-04-468

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

Dec 29, 2020

65 pages

Published in:

Quantum 5 (2021) 468

Published: Jun 4, 2021

e-Print:

2012.14689 [hep-th]

DOI:

10.22331/q-2021-06-04-468

Report number:

QMUL-PH-20-37

View in:

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Abstract: (Verein zur Foerderung des Open Access Publizierens in den Quantenwissenschaften)

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.

Note:

65 pages; 3 figures; 3 appendices; v2: some clarifications in section 3; reference added; v3: brief additional discussion in introduction and references added; v4: summary of results added, typos fixed, additional discussion--to appear in Quantum

field theory: topological
gauge field theory: discrete
group: Mathieu
fusion
anyon
nonabelian
Chern-Simons term
twist
space-time: dimension: 3

References(54)

Figures(0)

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