Toda Systems, Cluster Characters, and Spectral Networks

Williams, Harold

doi:10.1007/s00220-016-2692-x

Cited by 15 publications

(19 citation statements)

References 62 publications

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“…The work [45] establishes a connection between line defects, quantum discrete integrable systems and cluster algebras. The latter aspect is also discussed in [46,62,63].…”

Section: Introductionmentioning

confidence: 99%

Quivers, Line Defects and Framed BPS Invariants

Cirafici

2017

Ann. Henri Poincaré

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A large class of N = 2 quantum field theories admits a BPS quiver description and the study of their BPS spectra is then reduced to a representation theory problem. In such theories the coupling to a line defect can be modelled by framed quivers. The associated spectral problem characterises the line defect completely. Framed BPS states can be thought of as BPS particles bound to the defect. We identify the framed BPS degeneracies with certain enumerative invariants associated with the moduli spaces of stable quiver representations. We develop a formalism based on equivariant localization to compute explicitly such BPS invariants, for a particular choice of stability condition. Our framework gives a purely combinatorial solution of this problem. We detail our formalism with several explicit examples.Once again to construct the pyramid arrangement we have to look at the relevant terms in the Jacobian algebra J W = ⊕ n≥0 J W,n :Now we write down the Jacobian algebra elements corresponding to cyclic modules (6.32) and summands with higher grading do not contribute to the classification of the fixed points due to the condition dim V f = 1 and equations (6.15) and (6.18) evaluated at B =B = 0. To write down J W,4 we have usedthanks to (6.16) and (6.24) respectively, andWe can write the cyclic elements of the Jacobian algebra as A 2ψ A 1 C 3 v} . (6.44) the cyclic vector v. As usual this is graded J W = n≥0 J W,n and the first few terms are is obtained from the analog situation for SU (3), equation (6.14). To build the pyramid arrangement we have to study modules generated by the action of the Jacobian algebra on a single vector v ∈ V f , with V f one dimensional. The Jacobian algebra thus generated is naturally graded J W = n≥0 J W,n , and the first few terms are easy to work outAt the next levels we have four independent vectorswhere we have used the F-term relations obtained from (7.16) to show7.19) 56 Similar arguments give J W,7 = {ψÃ 2 ψ A 3 φÃ 1C v ,ψ A 2 ψ A 3 φÃ 1C v} , J W,8 = {A 3ψÃ2 ψ A 3 φÃ 1C v ,Ã 3ψÃ2 ψ A 3 φÃ 1C v} , J W,9 = {φ A 3ψÃ2 ψ A 3 φÃ 1C v ,φÃ 3ψÃ2 ψ A 3 φÃ 1C v} , J W,10 = {Ã 1φ A 3ψÃ2 ψ A 3 φÃ 1C v , A 1φ A 3ψÃ2 ψ A 3 φÃ 1C v} . (7.20)

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“…The work [45] establishes a connection between line defects, quantum discrete integrable systems and cluster algebras. The latter aspect is also discussed in [46,62,63].…”

Section: Introductionmentioning

confidence: 99%

Quivers, Line Defects and Framed BPS Invariants

Cirafici

2017

Ann. Henri Poincaré

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“…As hypothesized by [49], the relation between these two systems may be clearer if we start from five dimensions / from the 2-particle relativistic closed Toda chain and take an appropriate limit. According to what we discussed until now, a practical way to realize this suggestion might be to take the four-dimensional limit of our fivedimensional N = 1 SU (2) theory on S 3 ω 1 ,ω 2 ×R 2 and see what happens.…”

Section: (3114)mentioning

confidence: 88%

Exact relativistic Toda chain eigenfunctions from Separation of Variables and gauge theory

Sciarappa

2017

J. High Energ. Phys.

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We provide a proposal, motivated by Separation of Variables and gauge theory arguments, for constructing exact solutions to the quantum Baxter equation associated to the N -particle relativistic Toda chain and test our proposal against numerical results. Quantum Mechanical non-perturbative corrections, essential in order to obtain a sensible solution, are taken into account in our gauge theory approach by considering codimension two defects on curved backgrounds (squashed S 5 and degenerate limits) rather than flat space; this setting also naturally incorporates exact quantization conditions and energy spectrum of the relativistic Toda chain as well as its modular dual structure. arXiv:1706.05142v1 [hep-th]

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“…More precisely, we describe a subclass of networks known as WKB spectral networks. A more general definition can be found in [20,28].…”

Section: Generalized Fenchel-nielsen Networkmentioning

confidence: 99%

“…r,0 is left underdetermined, corresponding to the freedom of adding a multiple of v 28) to fix the boundary condition v The expansion of v (1) (t, ) in terms of hypergeometric functions as in equation (9.23) will be useful to analytically continue to t = ∞.…”

Section: Heun's Differential Equationmentioning

confidence: 99%

Higher length-twist coordinates, generalized Heun’s opers, and twisted superpotentials

Hollands

Kidwai

2018

Advances in Theoretical and Mathematical Physics

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In this paper we study a proposal of Nekrasov, Rosly and Shatashvili that describes the effective twisted superpotential obtained from a class S theory geometrically as a generating function in terms of certain complexified length-twist coordinates, and extend it to higher rank. First, we introduce a higher rank analogue of Fenchel-Nielsen type spectral networks in terms of a generalized Strebel condition. We find new systems of spectral coordinates through the abelianization method and argue that they are higher rank analogues of the Nekrasov-Rosly-Shatashvili Darboux coordinates. Second, we give an explicit parametrization of the locus of opers and determine the generating functions of this Lagrangian subvariety in terms of the higher rank Darboux coordinates in some specific examples. We find that the generating functions indeed agree with the known effective twisted superpotentials. Last, we relate the approach of Nekrasov, Rosly and Shatashvili to the approach using quantum periods via the exact WKB method.It can be shown that any (E, ∇) admits at most one oper structure, and thus we can indeed identify opers with a subspace of M flat (C, SL(K)).More concretely, any SL(K) oper can locally be written as a Kth order linear differential operator 18) 2 Class S geometry Fix a positive integer K, sometimes called the "rank", and a possibly punctured Riemann surface C. We equip the Riemann surface with a collection

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Toda Systems, Cluster Characters, and Spectral Networks

Cited by 15 publications

References 62 publications

Quivers, Line Defects and Framed BPS Invariants

Quivers, Line Defects and Framed BPS Invariants

Exact relativistic Toda chain eigenfunctions from Separation of Variables and gauge theory

Higher length-twist coordinates, generalized Heun’s opers, and twisted superpotentials

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