ADE spectral networks and decoupling limits of surface defects

Longhi, Pietro; Park, Chan Y.

doi:10.1007/jhep02(2017)011

Cited by 10 publications

(28 citation statements)

References 37 publications

(207 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Similar concepts have been pursued in the study of D-branes in Calabi-Yau varieties [22][23][24][25]. There are by now several approaches to determine the BPS degeneracies; for example spectral networks [26][27][28][29][30][31][32][33][34][35], the MPS wall-crossing formula [36][37][38][39][40][41][42], or a direct localization approach [18,43,44].…”

Section: Introductionmentioning

confidence: 99%

Quivers, Line Defects and Framed BPS Invariants

Cirafici

2017

Ann. Henri Poincaré

View full text Add to dashboard Cite

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)

show abstract

Section: Introductionmentioning

confidence: 99%

Quivers, Line Defects and Framed BPS Invariants

Cirafici

2017

Ann. Henri Poincaré

View full text Add to dashboard Cite

show abstract

“…In fact, invariance of U under wall crossing suggests that it should admit a definition even on walls of marginal stability, including the locus B c , we claim that 2d-4d wall crossing provides such an interpretation.Going back to the graph W c , we can explain its relevance to computing U by recalling the physical interpretation of spectral networks. The combinatorial data attached to a spectral network encodes the spectrum of 2d-4d BPS states for a particular type of surface defect, termed canonical defect [14,21,22]. Moreover this data is determined entirely by the topology of the network W(ϑ, u), and exhibits wall crossing behavior simultaneously with the topological degeneration of the network, at the critical phase ϑ c in our setup.…”

mentioning

confidence: 99%

Wall Crossing Invariants from Spectral Networks

Longhi

2017

Ann. Henri Poincaré

Self Cite

View full text Add to dashboard Cite

Abstract.A new construction of BPS monodromies for 4d N = 2 theories of class S is introduced. A novel feature of this construction is its manifest invariance under Kontsevich-Soibelman wall crossing, in the sense that no information on the 4d BPS spectrum is employed. The BPS monodromy is encoded by topological data of a finite graph, embedded into the UV curve C of the theory. The graph arises from a degenerate limit of spectral networks, constructed at maximal intersections of walls of marginal stability in the Coulomb branch of the gauge theory. The topology of the graph, together with a notion of framing, encode equations that determine the monodromy. We develop an algorithmic technique for solving the equations and compute the monodromy in several examples. The graph manifestly encodes the symmetries of the monodromy, providing some support for conjectural relations to specializations of the superconformal index. For A1-type theories, the graphs encoding the monodromy are "dessins d'enfants" on C, the corresponding Strebel differentials coincide with the quadratic differentials that characterize the Seiberg-Witten curve.

show abstract

“…As we vary this phase between 0 and 2π, S-walls move around on the UV curve C, and some of them will swipe across z. The spectrum of 2d-4d BPS solitons on the surface defects is the union of "soliton data" carried by each of these walls [13,34,44].…”

Section: Universal Features Of 2d-4d Bps Spectra From Bps Graphsmentioning

confidence: 99%

“…We further extend this result to higher-vorticity defects, with v > 1. To engineer them, we adopt ideas of [42][43][44], and replace the Seiberg-Witten curve with Hitchin spectral curves in higher symmetric representations (of A 1 ), of dimension N = v + 1. 4 By direct computation we are then able to prove that the 2d-4d BPS monodromy in the presence of a vortex defect of vorticity v is related to the 4d BPS monodromy as follows…”

Section: Introductionmentioning

confidence: 99%

“…In fact, while the 2d-4d BPS spectrum and 4d BPS spectrum are related by invariance of the 4d and 2d-4d BPS spectrum generators, this constraint does not apply a priori to the spectrum of 2d particles, leaving open the possibility for interesting 2d wall-crossing behavior from the "irregular" regime to the "regular" one. We leave this for future investigation 44. One small difference from the case of higher-genus surfaces with one regular puncture, is that the flavor charge in that case was twice the sum of all double-walls.…”

mentioning

confidence: 98%

See 1 more Smart Citation

An infrared bootstrap of the Schur index with surface defects

Fluder

Longhi

2019

J. High Energ. Phys.

Self Cite

View full text Add to dashboard Cite

The infrared formula relates the Schur index of a 4d N = 2 theory to its wallcrossing invariant, a.k.a. BPS monodromy. A further extension of this formula, proposed by Córdova, Gaiotto and Shao, includes contributions by various types of line and surface defects. We study BPS monodromies in the presence of vortex surface defects of arbitrary vorticity for general class S theories of type A 1 engineered by UV curves with at least one regular puncture. The trace of these defect BPS monodromies is shown to coincide with the action of certain q-difference operators acting on the trace of the (pure) 4d BPS monodromy. We use these operators to develop a "bootstrap" (of traces) of BPS monodromies, relying only on their infrared properties, thereby reproducing the very general ultraviolet characterization of the Schur index.A relation between Schur indices and BPS monodromies was conjectured in [23], following 6 See also [45], for some results of an IR formula for the lens index. 9 The "no-exotics" property asserts that BPS states have R = 0 in 4d N = 2 gauge theories [33,54,55]. 10 In practice, BPS spectra at arbitrary moduli can be rather involved, and by moving onto a "simple" sector of the theory, in which one can explicitly compute the protected spin character, the wall-crossing formulae allow to extract the BPS degeneracies in other sectors. the latter reference are precisely the "vortex defects" considered in [31] via the Higgsing procedure (see also Appendix A.3), and in the present paper via the IR formalism.

show abstract

ADE spectral networks and decoupling limits of surface defects

Cited by 10 publications

References 37 publications

Quivers, Line Defects and Framed BPS Invariants

Quivers, Line Defects and Framed BPS Invariants

Wall Crossing Invariants from Spectral Networks

An infrared bootstrap of the Schur index with surface defects

Contact Info

Product

Resources

About