Non-abelian anyons and some cousins of the Arad–Herzog conjecture

Buican, Matthew; Li, Linfeng; Radhakrishnan, Rajath

doi:10.1088/1751-8121/ac3623

Cited by 3 publications

(3 citation statements)

References 38 publications

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“…Moreover, we defined Galois action on the TQFT such that it doesn't change the sign of D (we can consider taking D → −D as a second step, supplementing our Galois conjugation, when exploring particular orbits). 47 Explicitly including a D → −D transformation leads to certain simple extensions of our results.…”

Section: Jhep01(2022)004 6 Conclusionmentioning

confidence: 55%

“…Since Galois conjugation fixes rational numbers, it is clear that the space of integral MTCs (i.e., theories whose anyons all have integer quantum dimensions) is closed under it. An important class of integral MTCs are (twisted) discrete gauge theories (see [47,48] for a recent discussion of these theories, their subcategory structure, and their fusion rules). Since there are integral MTCs that are not (twisted) discrete gauge theories [49], we might naively imagine that these theories mix with discrete gauge theories under Galois conjugation.…”

Section: Discrete Gauge Theoriesmentioning

confidence: 99%

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Galois orbits of TQFTs: symmetries and unitarity

Buican

Radhakrishnan

2022

J. High Energ. Phys.

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We study Galois actions on 2+1D topological quantum field theories (TQFTs), characterizing their interplay with theory factorization, gauging, the structure of gapped boundaries and dualities, 0-form symmetries, 1-form symmetries, and 2-groups. In order to gain a better physical understanding of Galois actions, we prove sufficient conditions for the preservation of unitarity. We then map out the Galois orbits of various classes of unitary TQFTs. The simplest such orbits are trivial (e.g., as in various theories of physical interest like the Toric Code, Double Semion, and 3-Fermion Model), and we refer to such theories as unitary “Galois fixed point TQFTs”. Starting from these fixed point theories, we study conditions for preservation of Galois invariance under gauging 0-form and 1-form symmetries (as well as under more general anyon condensation). Assuming a conjecture in the literature, we prove that all unitary Galois fixed point TQFTs can be engineered by gauging 0-form symmetries of theories built from Deligne products of certain abelian TQFTs.

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Section: Jhep01(2022)004 6 Conclusionmentioning

confidence: 55%

Section: Discrete Gauge Theoriesmentioning

confidence: 99%

Galois orbits of TQFTs: symmetries and unitarity

Buican

Radhakrishnan

2022

J. High Energ. Phys.

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“…Both the product of conjugacy classes as well as the tensor product of representations play a crucial role in the fusion rules of line operators in three-dimensional G-TQFTs [9]. In [17], the Arad-Herzog conjecture in finite group theory [18] was studied in the context of three-dimensional topological quantum field theories (TQFTs). It was shown that this conjecture implies special fusion rules for line operators in these TQFTs.…”

Section: Introductionmentioning

confidence: 99%

Row–column duality and combinatorial topological strings

Padellaro,

Radhakrishnan,

Ramgoolam

2024

J. Phys. A: Math. Theor.

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Integrality properties of partial sums over irreducible representations, along columns of character tables of finite groups, were recently derived using combinatorial topological string theories (CTST). These CTST were based on Dijkgraaf-Witten theories of flat G-bundles for finite groups G in two dimensions, denoted G-TQFTs. We define analogous combinatorial topological strings related to two dimensional TQFTs based on fusion coefficients of finite groups. These TQFTs are denoted as R(G)-TQFTs and allow analogous integrality results to be derived for partial row sums of characters over conjugacy classes along fixed rows. This relation between the G-TQFTs and R(G)-TQFTs defines a row-column duality for character tables, which provides a physical framework for exploring the mathematical analogies between rows and columns of character tables. These constructive proofs of integrality are complemented with the proof of similar and complementary results using the more traditional Galois theoretic framework for integrality properties of character tables. The partial row and column sums are used to define generalised partitions of the integer row and column sums, which are of interest in combinatorial representation theory.

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On reconstructing finite gauge group from fusion rules

Radhakrishnan

2024

J. High Energ. Phys.

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Gauging a finite group 0-form symmetry G of a quantum field theory (QFT) results in a QFT with a Rep(G) symmetry implemented by Wilson lines. The group G determines the fusion of Wilson lines. However, in general, the fusion rules of Wilson lines do not determine G. In this paper, we study the properties of G that can be determined from the fusion rules of Wilson lines and surface operators obtained from higher-gauging Wilson lines. This is in the spirit of Richard Brauer who asked what information in addition to the character table of a finite group needs to be known to determine the group. We show that fusion rules of surface operators obtained from higher-gauging Wilson lines can be used to distinguish infinite pairs of groups which cannot be distinguished using the fusion of Wilson lines. We derive necessary conditions for two non-isomorphic groups to have the same surface operator fusion and find a pair of such groups.

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Non-abelian anyons and some cousins of the Arad–Herzog conjecture

Cited by 3 publications

References 38 publications

Galois orbits of TQFTs: symmetries and unitarity

Galois orbits of TQFTs: symmetries and unitarity

Row–column duality and combinatorial topological strings

On reconstructing finite gauge group from fusion rules

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