On skein relations in class S theories

Tachikawa, Yuji; Watanabe, Noriaki

doi:10.1007/jhep06(2015)186

Cited by 23 publications

(47 citation statements)

References 44 publications

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“…Property 3 confirms the "weak positivity conjecture" in [7], and furthermore Property 4 brings support to the "strong positivity conjecture." To make precise contact with the conjectures of Fock and Goncharov (Conjecture 12.4 in [9] and Conjecture 4.8 in [21]) would require a better understanding of the relation between the space of tropical points of the so-called A-space and the space of networks with junctions (see for example the "higher laminations" of [22,23] and the "charge/network dictionary" in [24,25]). It should also be possible to relate exactly to the SL(2, C) quantum trace of Bonahon and Wong [11] by introducing some normalizations, and to see that their application of Weyl quantum ordering matches the pattern of intersections of detoured paths.…”

Section: Contentsmentioning

confidence: 99%

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Quantum Holonomies from Spectral Networks and Framed BPS States

Gabella¹

2016

Commun. Math. Phys.

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We propose a method for determining the spins of BPS states supported on line defects in 4d N = 2 theories of class S. Via the 2d-4d correspondence, this translates to the construction of quantum holonomies on a punctured Riemann surface C. Our approach combines the technology of spectral networks, which decomposes flat GL(K, C)-connections on C in terms of flat abelian connections on a K-fold cover of C, and the skein algebra in the 3-manifold C × [0, 1], which expresses the representation theory of the quantum group U q (gl K ). With any path on C, the quantum holonomy associates a positive Laurent polynomial in the quantized Fock-Goncharov coordinates of higher Teichmüller space. This confirms various positivity conjectures in physics and mathematics.arXiv:1603.05258v2 [hep-th]

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Section: Contentsmentioning

confidence: 99%

“…Crossings corresponding to theR-matrices can be expressed, via the skein relation, in terms of networks with junctions (see [24,26] for reviews). TheR-matrix for the fundamental representation of U q (gl K ) can indeed be decomposed aŝ…”

Section: Skein Relations and Junctionsmentioning

confidence: 99%

Quantum Holonomies from Spectral Networks and Framed BPS States

Gabella¹

2016

Commun. Math. Phys.

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“…Sikora's construction allows one to recover the construction of quantum sl N invariants previously given by Murakami, Ohtsuki, and Yamada in [55] (a useful summary is given in [59]). This construction uses trivalent graphs with a "flow" built out of the following two types of vertices:…”

Section: Jhep10(2015)143mentioning

confidence: 99%

“…. , N − 1} [55] (see [59] for the normalization): (2.41) with m = min{i, j, N − i, N − j}. For the other ordering at the crossing, one should replace q by q −1 .…”

Section: Jhep10(2015)143mentioning

confidence: 99%

Line operators in theories of class S $$ \mathcal{S} $$ , quantized moduli space of flat connections, and Toda field theory

Coman

Gabella

Teschner³

2015

J. High Energ. Phys.

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Non-perturbative aspects of N = 2 supersymmetric gauge theories of class S are deeply encoded in the algebra of functions on the moduli space M flat of flat SL(N )-connections on Riemann surfaces. Expectation values of Wilson and 't Hooft line operators are related to holonomies of flat connections, and expectation values of line operators in the low-energy effective theory are related to Fock-Goncharov coordinates on M flat . Via the decomposition of UV line operators into IR line operators, we determine their noncommutative algebra from the quantization of Fock-Goncharov Laurent polynomials, and find that it coincides with the skein algebra studied in the context of Chern-Simons theory. Another realization of the skein algebra is generated by Verlinde network operators in Toda field theory. Comparing the spectra of these two realizations provides non-trivial support for their equivalence. Our results can be viewed as evidence for the generalization of the AGT correspondence to higher-rank class S theories.

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“…Another extension of this work regards the study of other N = 2 models as for example class S-theories [16]. It should be interesting apply the results of [17][18][19], like the relation between the twisted character of the chiral algebra and the Lens space index, in order to reproduce the lattices and the S-duality structure obtained in [20][21][22][23][24][25][26][27][28]. Also the results of [29], investigating the geometric origin of the global properties from the 6d N = (1, 0) perspective, may be useful for the study of N = 2 models.…”

Section: Discussionmentioning

confidence: 99%

Lens space index and global properties for 4d $$ \mathcal{N} $$ = 2 models

Amariti¹,

Marcassoli²

2020

J. High Energ. Phys.

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The additional data necessary to univocally fix the gauge group for a given algebra are represented by the same charge lattices of mutually local Wilson and 't Hooft lines for both 4d N = 4 SYM and N = 2 elliptic models. Motivated by this equivalence in this paper we study the Lens space index of these N = 2 elliptic models. The index is indeed sensitive to the global properties and in the N = 4 case it is expected to coincide among S-dual models with different global properties, while it gives different results for models that lie in other S-duality orbits. Here by an explicit calculation we show that the same results hold for the N = 2 elliptic models as well.

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On skein relations in class S theories

Cited by 23 publications

References 44 publications

Quantum Holonomies from Spectral Networks and Framed BPS States

Quantum Holonomies from Spectral Networks and Framed BPS States

Line operators in theories of class S $$ \mathcal{S} $$ , quantized moduli space of flat connections, and Toda field theory

Lens space index and global properties for 4d $$ \mathcal{N} $$ = 2 models

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