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

Keegan, Sinéad; Nahm, W.

doi:10.1088/1751-8113/44/50/505204

Cited by 3 publications

(5 citation statements)

References 14 publications

(48 reference statements)

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“…The equations (C.43) have a unique solution with 0 < z i < 1 which corresponds to the vacuum character of the 2d coset CFT G 2 /U(1) [49,51]. Then 6 i L(z i )/π 2 is the central charge of the G 2 /U(1) coset CFT, and…”

Section: Jhep11(2017)013mentioning

confidence: 99%

Superconformal index, BPS monodromy and chiral algebras

Cecotti

Song

Vafa

et al. 2017

J. High Energ. Phys.

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We show that specializations of the 4d N = 2 superconformal index labeled by an integer N is given by Tr M N where M is the Kontsevich-Soibelman monodromy operator for BPS states on the Coulomb branch. We provide evidence that the states enumerated by these limits of the index lead to a family of 2d chiral algebras A N . This generalizes the recent results for the N = −1 case which corresponds to the Schur limit of the superconformal index. We show that this specialization of the index leads to the same integrand as that of the elliptic genus of compactification of the superconformal theory on S 2 × T 2 where we turn on 1 2 N units of U(1) r flux on S 2 .

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Section: Jhep11(2017)013mentioning

confidence: 99%

Superconformal index, BPS monodromy and chiral algebras

Cecotti

Song

Vafa

et al. 2017

J. High Energ. Phys.

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“…The Andrews-Gordon identities are well-known generalizations of the Rogers-Ramanujan identities and they give examples of modular triples with matrix parts of the form A = C(A 1 ) ⊗ C(T n ) −1 . See [KN11] for a detailed discussion of more examples.…”

Section: Introductionmentioning

confidence: 99%

Nahm’s conjecture and $Y$-systems

Lee

2013

Communications in Number Theory and Physics

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Abstract. Nahm's conjecture relates q-hypergeometric modular functions to torsion elements in the Bloch group. An interesting class of such functions can be (conjecturally) obtained from a pair (X, X ′ ) of diagrams, each of which is either a Dynkin diagram of type ADE or a diagram of type T . Using properties of Y-systems, we prove that for a matrix of the form A = C(X) ⊗ C(X ′ ) −1 where C(X) and C(X ′ ) are the corresponding Cartan matrices, every solution of the equationA gives rise to a torsion element of the Bloch group.

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“…This case has stong ties to conformal field theory and was studied experimentally in [KN11]. In particular, Keegan and Nahm considered the case when B is any column vector of −C −1 A k−1 .…”

Section: Chaptermentioning

confidence: 99%

“…Keegan and Nahm ask whether these are the only modular B-vectors? For small values of k, they carry out a computational search in [KN11] for all modular B-vectors within the range −8 ≤ b i ≤ 8. Only vectors in the simple family were obtained.…”

Section: Chaptermentioning

confidence: 99%

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Nahm's conjecture in the Cartan case

Dunn¹

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When is a q-hypergeometric series a modular form? The connection between these two objects dates back to 1913 in Ramanujan's first letter to Hardy. In general, this is a deep question and out of reach at present. In 1994, Nahm conjectured a criterion for this phenomenon that has become a guiding principle in this area of research. The conjecture connects the modularity of a family of Eulerian series to torsion elements in Bloch groups of number fields determined by the Eulerian series. In 2011, Vlasenko and Zwegers exhibited counter-examples to this conjecture. Despite this, a theorem of Lee in 2013 provides strong evidence the conjecture is true in a special case related to models in conformal field theory. These models are parameterised by a pair (X, X ), where X and X are Dynkin diagrams of ADET type. When gcd (n − 1)!, k = 1, we prove Nahm's conjecture holds in the case (X, X ) = (A n−1 , A k−1 ). We make use of string functions coming from the representation theory of affine Kac-Moody algebras. This complements previous results pertaining to (X, X ) = (A 2n , T k−1 ) due to Feigin-Stoyanovsky (n = 1) and Stoyanovsky, and (X, X ) = (A 2n−1 , T 1 ) due to Warnaar-Zudulin. We also conduct computational investigations in classical Andrews-Gordon case i.e. when (X, X ) = (A 1 , T k−1 ). In particular, Keegan and Nahm ask whether a special family of modular B-vectors are the only ones giving rise to a modular Eulerian series. We confirm this for k = 3 and k = 4.Chapter 2 will introduce the basic objects needed to state Nahm's conjecture. This will include details on modular forms, dilogarithms and the Bloch group.

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

Cited by 3 publications

References 14 publications

Superconformal index, BPS monodromy and chiral algebras

Superconformal index, BPS monodromy and chiral algebras

Nahm’s conjecture and $Y$-systems

Nahm's conjecture in the Cartan case

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