Galois symmetry induced by Hecke relations in rational conformal field theory and associated modular tensor categories

Harvey, Jeffrey A.; Hu, Yichen; Wu, Yuxiao

doi:10.1088/1751-8121/ab8e03

Cited by 19 publications

(44 citation statements)

References 66 publications

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“…We will call a fusion category unitary if there exists a gauge in which the F symbols are unitary. 18,19 A braided fusion category is unitary if there exists a gauge in which both the F and R symbols are unitary. The condition on R is not an extra constraint, since any set of consistent R symbols obtained from unitary F symbols is unitary [41].…”

Section: Unitary Galois Orbitsmentioning

confidence: 99%

“…In fact, the authors of [42] showed that there is a gauge in which both can be achieved simultaneously. 18 There is a gauge-independent definition of unitarity of a fusion category. But we will use the definition in terms of the F matrices since both are equivalent [39].…”

Section: Unitary Galois Orbitsmentioning

confidence: 99%

“…In other words, Galois conjugation is a systematic way to move around the space of TQFTs. Moreover, Galois conjugation is useful in practice: it has played a role in the classification of low-rank TQFTs [11], proving the rank finiteness theorem [12], finding modular isotopes [13], studying low-dimensional lattice models [14][15][16], in connections between TQFT and other types of QFTs [17][18][19][20][21][22], and in the study of gapped boundaries [23].…”

Section: Introductionmentioning

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: Unitary Galois Orbitsmentioning

confidence: 99%

Section: Unitary Galois Orbitsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Galois orbits of TQFTs: symmetries and unitarity

Buican

Radhakrishnan

2022

J. High Energ. Phys.

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“…One usually first studies the small l case because the solutions to small l give rise to solutions for larger l: for example, when every character of a solution to a [n, l] MLDE is multiplied by the modular-invariant function j(τ ), the resulting characters are solutions to a [n, l + 6n] MLDE (j(τ ) has a zero of third order at τ = e 2πi 3 which contributes 6 to the Wronskian index and there are n factors of j(τ )). The MLDE approach to RCFTs has been covered in both the physics [14,15,[21][22][23][24][25][26][27][40][41][42] and the maths literature [28][29][30][31][32][33][34][35][36]. A status report on the program can be found in [19].…”

Section: Jhep11(2021)195mentioning

confidence: 99%

Classifying three-character RCFTs with Wronskian index equalling 0 or 2

Das

Gowdigere

Santara

2021

J. High Energ. Phys.

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In the modular linear differential equation (MLDE) approach to classifying rational conformal field theories (RCFTs) both the MLDE and the RCFT are identified by a pair of non-negative integers [n,l]. n is the number of characters of the RCFT as well as the order of the MLDE that the characters solve and l, the Wronskian index, is associated to the structure of the zeroes of the Wronskian of the characters. In this paper, we study [3,0] and [3,2] MLDEs in order to classify the corresponding CFTs. We reduce the problem to a “finite” problem: to classify CFTs with central charge 0 < c ≤ 96, we need to perform 6, 720 computations for the former and 20, 160 for the latter. Each computation involves (i) first finding a simultaneous solution to a pair of Diophantine equations and (ii) computing Fourier coefficients to a high order and checking for positivity.In the [3,0] case, for 0 < c ≤ 96, we obtain many character-like solutions: two infinite classes and a discrete set of 303. After accounting for various categories of known solutions, including Virasoro minimal models, WZW CFTs, Franc-Mason vertex operator algebras and Gaberdiel-Hampapura-Mukhi novel coset CFTs, we seem to have seven hitherto unknown character-like solutions which could potentially give new CFTs. We also classify [3,2] CFTs for 0 < c ≤ 96: each CFT in this case is obtained by adjoining a constant character to a [2,0] CFT, whose classification was achieved by Mathur-Mukhi-Sen three decades ago.

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“…However there is no reason to consider this a "typical" case. One of the fascinating developments of the last few years has been the generation of interesting results beyond = 0 [13,14,20,28,32,35]. By substituting the series expansion eq.…”

Section: Jhep05(2020)003mentioning

confidence: 99%

Rational CFT with three characters: the quasi-character approach

2020

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Quasi-characters are vector-valued modular functions having an integral, but not necessarily positive, q-expansion. Using modular differential equations, a complete classification has been provided in arXiv:1810.09472 for the case of two characters. These in turn generate all possible admissible characters, of arbitrary Wronskian index, in order two. Here we initiate a study of the three-character case. We conjecture several infinite families of quasi-characters and show in examples that their linear combinations can generate admissible characters with arbitrarily large Wronskian index. The structure is completely different from the order two case, and the novel coset construction of arXiv:1602.01022 plays a key role in discovering the appropriate families. Using even unimodular lattices, we construct some explicit three-character CFT corresponding to the new admissible characters.

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Galois symmetry induced by Hecke relations in rational conformal field theory and associated modular tensor categories

Cited by 19 publications

References 66 publications

Galois orbits of TQFTs: symmetries and unitarity

Galois orbits of TQFTs: symmetries and unitarity

Classifying three-character RCFTs with Wronskian index equalling 0 or 2

Rational CFT with three characters: the quasi-character approach

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