Evaluation of state integrals at rational points

Garoufalidis, Stavros; Kashaev, Rinat

doi:10.4310/cntp.2015.v9.n3.a3

Cited by 34 publications

(81 citation statements)

References 13 publications

(16 reference statements)

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“…Under the symmetry enhancement, the Cartan u(1) J for the topological U(1) J symmetry is embedded into su(3) as follows u(1) J = 1 3 diag{1, 1, −2} ∈ su(3) . The F matches the computation in [58]. Note that the C T is less than that of the free theory of 3 chiral multiplets, which also has SU(3) symmetry.…”

Section: (419)supporting

confidence: 73%

Expanding 3d $$ \mathcal{N} $$ = 2 theories around the round sphere

Gang

Yamazaki

2020

J. High Energ. Phys.

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We study a perturbative expansion of the squashed 3-sphere (S 3 b ) partition function of 3d N = 2 gauge theories around the squashing parameter b = 1. Our proposal gives the coefficients of the perturbative expansion as a finite sum over the saddle points of the supersymmetric-localization integral in the limit b → 0 (the so-called Bethe vacua), and the contribution from each Bethe vacua can be systematically computed using saddle-point methods. Our expansion provides an efficient and practical method for computing basic CFT data (F, C T , C JJ and higher-point correlation functions of the stress-energy tensor) of the IR superconformal field theory without performing the localization integrals.1 There is no term linear in b − 1, as expected from the symmetry b → b −1 of the geometry (1.1). 2 There are counterexamples to the conjecture that CT decreases along the RG flow [11].

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Section: (419)supporting

confidence: 73%

Expanding 3d $$ \mathcal{N} $$ = 2 theories around the round sphere

Gang

Yamazaki

2020

J. High Energ. Phys.

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“…Now, when b 2 is of the form M/N , with M and N coprime integers, Faddeev's quantum dilogarithm simplifies into an elementary function [68]-and it is in principle possible to calculate these integrals exactly, by residues. Using this technique, many results for the spectral traces have been obtained in [33].…”

Section: Exact Results For Local Pmentioning

confidence: 99%

“…Using the properties of Faddeev's quantum dilogarithm for rational b 2 established in [68], one finds for example:…”

Section: Exact Results For Local Pmentioning

confidence: 99%

Resurgence matches quantization

Couso-Santamaría¹,

Mariño²,

Schiappa³

2017

J. Phys. A: Math. Theor.

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Abstract:The quest to find a nonperturbative formulation of topological string theory has recently seen two unrelated developments. On the one hand, via quantization of the mirror curve associated to a toric Calabi-Yau background, it has been possible to give a nonperturbative definition of the topological-string partition function. On the other hand, using techniques of resurgence and transseries, it has been possible to extend the string (asymptotic) perturbative expansion into a transseries involving nonperturbative instanton sectors. Within the specific example of the local P 2 toric Calabi-Yau threefold, the present work shows how the Borel-Padé-Ecalle resummation of this resurgent transseries, alongside occurrence of Stokes phenomenon, matches the string-theoretic partition function obtained via quantization of the mirror curve. This match is highly non-trivial, given the unrelated nature of both nonperturbative frameworks, signaling at the existence of a consistent underlying structure.

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“…with Φ b (x) is the standard quantum dilogarithm, as discussed for instance in [78]. This is partly a matter of convention: Φ b (σ) would be precisely the contribution from a chiral multiplet in the "U (1) 1 2 quantization," while we are considering the "U (1) − 1 2 quantization."…”

Section: Part IIImentioning

confidence: 99%

“…(This apparent coincidence between the two subjects is explained by the 3d/3d correspondence [76] in string theory.) The state integral corresponding to an U (1) k supersymmetric CS theory coupled to N f chiral multiplets of unit charge was given an evaluation formula for b 2 rational equivalent to (7.27) in [78]. 67 The evaluation formula (7.27) renders manifest a number of properties of Z S 3 b which are less than obvious from the integral expression.…”

Section: Integral Formula and Its Evaluationmentioning

confidence: 99%

Seifert fibering operators in 3d $$ \mathcal{N}=2 $$ theories

Closset

Kim

Willett

2018

J. High Energ. Phys.

206

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We study 3d N = 2 supersymmetric gauge theories on closed oriented Seifert manifolds-circle bundles over an orbifold Riemann surface-, with a gauge group G given by a product of simply-connected and/or unitary Lie groups. Our main result is an exact formula for the supersymmetric partition function on any Seifert manifold, generalizing previous results on lens spaces. We explain how the result for an arbitrary Seifert geometry can be obtained by combining simple building blocks, the "fibering operators." These operators are half-BPS line defects, whose insertion along the S 1 fiber has the effect of changing the topology of the Seifert fibration. We also point out that most supersymmetric partition functions on Seifert manifolds admit a discrete refinement, corresponding to the freedom in choosing a three-dimensional spin structure. As a strong consistency check on our result, we show that the Seifert partition functions match exactly across infrared dualities. The duality relations are given by intricate (and seemingly new) mathematical identities, which we tested numerically. Finally, we discuss in detail the supersymmetric partition function on the lens space L(p, q) b with rational squashing parameter b 2 ∈ Q, comparing our formalism to previous results, and explaining the relationship between the fibering operators and the threedimensional holomorphic blocks.

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Evaluation of state integrals at rational points

Cited by 34 publications

References 13 publications

Expanding 3d $$ \mathcal{N} $$ = 2 theories around the round sphere

Expanding 3d $$ \mathcal{N} $$ = 2 theories around the round sphere

Resurgence matches quantization

Seifert fibering operators in 3d $$ \mathcal{N}=2 $$ theories

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