DILOGARITHM IDENTITIES, FUSION RULES AND STRUCTURE CONSTANTS OF CFTs

Terhoeven, Michael

doi:10.1142/s0217732394000149

Cited by 16 publications

(13 citation statements)

References 10 publications

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“…However, the corresponding fermionic representations are still under investigation. The first such representation discovered was for the Z N parafermionic models [18] and in the last two years a large number of fermionic representations of characters and branching functions for affine Lie algebras of integer level have been found [1]- [6], [25]- [34].…”

Section: Introductionmentioning

confidence: 99%

Continued fractions and fermionic representations for characters of M(p,p?) minimal models

1996

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We present fermionic sum representations of the characters χ (p,p ′ ) r,s of the minimal M (p, p ′ ) models for all relatively prime integers p ′ > p for some allowed values of r and s. Our starting point is binomial (q-binomial) identities derived from a truncation of the state counting equations of the XXZ spin 1 2 chain of anisotropy −∆ = − cos(π p p ′ ). We use the Takahashi-Suzuki method to express the allowed values of r (and s) in terms of the continued fraction decomposition of { p ′ p } (and p p ′ ) where {x} stands for the fractional part of x. These values are, in fact, the dimensions of the hermitian irreducible representations of SU q − (2) (and SU q + (2)) with q − = exp(iπ{ p ′ p }) (and q + = exp(iπ p p ′ )). We also establish the duality relation M (p, p ′ ) ↔ M (p ′ − p, p ′ ) and discuss the action of the Andrews-Bailey transformation in the space of minimal models. Many new identities of the Rogers-Ramanujan type are presented.

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Section: Introductionmentioning

confidence: 99%

Continued fractions and fermionic representations for characters of M(p,p?) minimal models

1996

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“…These are formal power series with integer coefficients in some rational power of q, and are analytic in the unit disk |q| < 1, but they are very seldom modular: apart from the two Rogers-Ramanujan cases (A, B, C) = (2, 0, − ) were known for which f A,B,C is modular, and it was later proved ( [33], [38]) that these are in fact the only ones. Since this list of seven examples is not very enlightening, Nahm introduced also a higher-order version…”

Section: Relation To Quantum Knot Theorymentioning

confidence: 99%

“…(We comment here that there are several definitions of the Bloch group in the literature, all the same up to 6-torsion, and that the specific choice made in Definition 1.1, which forces 3[0] = 0, [X]+[1/X] = 0 and [X]+[1−X] = [0] for any field F and any element X of P 1 (F ), was chosen precisely so that L is well-defined on B(R) and takes values in the full circle group R/ π 2 2 Z rather than just its quotient R/ π 2 6 Z.) Specifically, let A, B and C be as above let X = X A be the distinguished solution of (33) as in (ii) and F the corresponding number field, and for each integer n choose a primitive nth root of unity ζ, set F n = F (ζ) and denote by H = H n the Kummer extension of F n obtained by adjoining the positive nth roots x i of the X i . We are interested in the asymptotic expansion of f A,B,C (ζe −h/n ) as h → 0 + .…”

Section: Relation To Quantum Knot Theorymentioning

confidence: 99%

Bloch groups, algebraic K-theory, units, and Nahm's Conjecture

Calegari¹,

Garoufalidis²,

Zagier³

2017

Preprint

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Given an element of the Bloch group of a number field F and a natural number n, we construct an explicit unit in the field F n = F (e 2πi/n ), well-defined up to n-th powers of nonzero elements of F n . The construction uses the cyclic quantum dilogarithm, and under the identification of the Bloch group of F with the K-group K 3 (F ) gives an explicit formula for a certain abstract Chern class from K 3 (F ). The units we define are conjectured to coincide with numbers appearing in the quantum modularity conjecture for the Kashaev invariant of knots (which was the original motivation for our investigation), and also appear in the radial asymptotics of Nahm sums near roots of unity. This latter connection is used to prove one direction of Nahm's conjecture relating the modularity of certain q-hypergeometric series to the vanishing of the associated elements in the Bloch group of Q. Contents 1. Introduction 1.1. Bloch groups and associated units 1.2. Algebraic K-groups and associated units 1.3. Nahm's Conjecture 1.4. Plan of the paper 2. The maps P ζ and R ζ 2.1. The map P ζ 2.2. The map R ζ 2.3. Reduction to the case of prime powers 2.4. The 5-term relation 2.5. An eigenspace computation 3. Chern Classes for algebraic K-theory 3.1. Definitions 3.2. The relation between étale cohomology and Galois cohomology 3.3. Upgrading from F × n to O Fn [1/S] × 3.4. Upgrading from S-units to units 3.5. Proof of Theorem 1.5 4. Reduction to finite fields 4.1. Local Chern class maps 4.2. The Bloch group of F q 4.3. The local Chern class map c ζ 4.4. The local R ζ map 5. Comparison between the maps c ζ and R ζ 5.1. Proof of Theorem 1.6 5.2. Digression: the mod-p-q dilogarithm 5.3. The Chern class map on n-torsion in Q(ζ) + 6. The connecting homomorphism to K-theory

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“…The first formula (9) originated in the PhD thesis of Michael Terhoeven [11] and calculates C given B. The second formula (10) doesn't seem to be stated explicitly in the literature but was calculated by the authors of this paper using exactly the methods described in [11,12]. Both formulae are used to speed up the search for B-values as follows: for each B-value under consideration, check firstly that it satisfies equation (10), and secondly that it gives rise to a rational value of C using equation (9).…”

Section: Use Asymptotic Formulae To Immediately Eliminate Many Of The...mentioning

confidence: 99%

Nahm’s conjecture and coset models: a systematic search for matching parameters

Keegan¹,

Nahm²

2011

J. Phys. A: Math. Theor.

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When is a q-series modular? This is an interesting open question in mathematics that has deep connections to conformal field theory. In this paper, we define a particular r-fold q-hypergeometric series fA, B, C, with data given by a matrix A, a vector B and a scalar C, all rational, and ask when fA, B, C is modular. In the past much work has been done to predict which values of A give rise to modular fA, B, C; however, there is no straightforward method for calculating corresponding values of B. We approach this problem from the point of view of conformal field theory, by considering (2n + 3, 2)-minimal models, and coset models of the form . By calculating the characters of these models and comparing them to the functions fA, B, C, we succeed in computing appropriate B-values in many cases.

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DILOGARITHM IDENTITIES, FUSION RULES AND STRUCTURE CONSTANTS OF CFTs

Cited by 16 publications

References 10 publications

Continued fractions and fermionic representations for characters of M(p,p?) minimal models

Continued fractions and fermionic representations for characters of M(p,p?) minimal models

Bloch groups, algebraic K-theory, units, and Nahm's Conjecture

Nahm’s conjecture and coset models: a systematic search for matching parameters

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