Rational CFT with three characters: the quasi-character approach

Mukhi, Sunil; Poddar, Rahul; Singh, Palash

doi:10.1007/jhep05(2020)003

“…We note in passing that the existence of modular-invariant partition functions with negative integral coefficients seems to be a non-holomorphic version of the "quasi-characters" extensively studied in[23][24][25]. It seems that negative integrality is a rather widespread feature in the context of both holomorphic vectorvalued modular forms and non-holomorphic modular invariants.…”

mentioning

confidence: 80%

Poincaré series, 3d gravity and averages of rational CFT

Meruliya

¹

,

Mukhi

²

,

Singh

³

2021

J. High Energ. Phys.

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We investigate the Poincaré series approach to computing 3d gravity partition functions dual to Rational CFT. For a single genus-1 boundary, we show that for certain infinite sets of levels, the SU(2)k WZW models provide unitary examples for which the Poincaré series is a positive linear combination of two modular-invariant partition functions. This supports the interpretation that the bulk gravity theory (a topological Chern-Simons theory in this case) is dual to an average of distinct CFT’s sharing the same Kac-Moody algebra. We compute the weights of this average for all seed primaries and all relevant values of k. We then study other WZW models, notably SU(N)1 and SU(3)k, and find that each class presents rather different features. Finally we consider multiple genus-1 boundaries, where we find a class of seed functions for the Poincaré sum that reproduces both disconnected and connected contributions — the latter corresponding to analogues of 3-manifold “wormholes” — such that the expected average is correctly reproduced.

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“…We can repeat the exercise of the previous paragraph for (n, l) = (3, 0) i.e for threecharacter vanishing-Wronskian-index RCFTs, where the final classification has not yet been achieved, even if lots of progress has happened [12,21,24,36]. From our results, we obtain the following partial classification of three-character vanishing-Wronskian-index CFTs:…”

Section: Jhep04(2021)294mentioning

confidence: 66%

Wronskian indices and rational conformal field theories

Das

¹

,

Gowdigere

²

,

Santara

³

2021

J. High Energ. Phys.

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The classification scheme for rational conformal field theories, given by the Mathur-Mukhi-Sen (MMS) program, identifies a rational conformal field theory by two numbers: (n, l). n is the number of characters of the rational conformal field theory. The characters form linearly independent solutions to a modular linear differential equation (which is also labelled by (n, l)); the Wronskian index l is a non-negative integer associated to the structure of zeroes of the Wronskian.In this paper, we compute the (n, l) values for three classes of well-known CFTs viz. the WZW CFTs, the Virasoro minimal models and the $$ \mathcal{N} $$ N = 1 super-Virasoro minimal models. For the latter two, we obtain exact formulae for the Wronskian indices. For WZW CFTs, we get exact formulae for small ranks (upto 2) and all levels and for all ranks and small levels (upto 2) and for the rest we compute using a computer program. We find that any WZW CFT at level 1 has a vanishing Wronskian index as does the $$ {\hat{\mathbf{A}}}_{\mathbf{1}} $$ A ̂ 1 CFT at all levels. We find intriguing coincidences such as: (i) for the same level CFTs with $$ {\hat{\mathbf{A}}}_{\mathbf{2}} $$ A ̂ 2 and $$ {\hat{\mathbf{G}}}_{\mathbf{2}} $$ G ̂ 2 have the same (n, l) values, (ii) for the same level CFTs with $$ {\hat{\mathbf{B}}}_{\mathbf{r}} $$ B ̂ r and $$ {\hat{\mathbf{D}}}_{\mathbf{r}} $$ D ̂ r have the same (n, l) values for all r ≥ 5.Classifying all rational conformal field theories for a given (n, l) is one of the aims of the MMS program. We can use our computations to provide partial classifications. For the famous (2, 0) case, our partial classification turns out to be the full classification (achieved by MMS three decades ago). For the (3, 0) case, our partial classification includes two infinite series of CFTs as well as fifteen “discrete” CFTs; except three all others have Kac-Moody symmetry.

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“…The study of vvmfs has a long history in both physics (e.g. [27][28][29][30][31][32][33][34][35][36][37][38]) and mathematics (e.g. [38][39][40][41][42][43][44][45][46][47][48][49][50][51][52]); for concise recent reviews, see e.g.…”

Section: Twist Spectrum and Vector-valued Modular Formsmentioning

confidence: 99%

“…Indeed, even though the four-dimensional vvmf is not integral, it turns out that it can be decomposed into a direct sum of two twodimensional vvmfs, one of which is integral. 37 We can then simply apply theorem 3.1 to this subrepresentation to conclude that m 0 < 0. So even though the full four-dimensional vvmf is not integral, the fact that a subrepresentation of it is integral is enough to obtain our usual bounds.…”

Section: Jhep02(2021)064mentioning

confidence: 99%

“…As listed in section 5.3, our constructions of non-factorizable, non-holomorphic cuspidal functions satisfying discreteness and integrality can be separated into three categories: 1) 37 That the four-dimensional vvmf can be decomposed in this way can be seen by noting that the T -matrix is diagonal, and that the S-matrix can be put in block-diagonal form [123],…”

Section: Cuspidal Function Constructionsmentioning

confidence: 99%

See 1 more Smart Citation

Discreteness and integrality in Conformal Field Theory

Kaidi

¹

,

Perlmutter

²

2021

J. High Energ. Phys.

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Various observables in compact CFTs are required to obey positivity, discreteness, and integrality. Positivity forms the crux of the conformal bootstrap, but understanding of the abstract implications of discreteness and integrality for the space of CFTs is lacking. We systematically study these constraints in two-dimensional, non-holomorphic CFTs, making use of two main mathematical results. First, we prove a theorem constraining the behavior near the cusp of integral, vector-valued modular functions. Second, we explicitly construct non-factorizable, non-holomorphic cuspidal functions satisfying discreteness and integrality, and prove the non-existence of such functions once positivity is added. Application of these results yields several bootstrap-type bounds on OPE data of both rational and irrational CFTs, including some powerful bounds for theories with conformal manifolds, as well as insights into questions of spectral determinacy. We prove that in rational CFT, the spectrum of operator twists $$ t\ge \frac{c}{12} $$ t ≥ c 12 is uniquely determined by its complement. Likewise, we argue that in generic CFTs, the spectrum of operator dimensions $$ \Delta >\frac{c-1}{12} $$ Δ > c − 1 12 is uniquely determined by its complement, absent fine-tuning in a sense we articulate. Finally, we discuss implications for black hole physics and the (non-)uniqueness of a possible ensemble interpretation of AdS3 gravity.

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Rational CFT with three characters: the quasi-character approach

Cited by 21 publications

References 38 publications

Poincaré series, 3d gravity and averages of rational CFT

Poincaré series, 3d gravity and averages of rational CFT

Wronskian indices and rational conformal field theories

Discreteness and integrality in Conformal Field Theory

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