BPS graphs: from spectral networks to BPS quivers

Gabella, Maxime; Longhi, Pietro; Park, Chan Y.; Yamazaki, Masahito

doi:10.1007/jhep07(2017)032

Cited by 28 publications

(45 citation statements)

References 104 publications

(302 reference statements)

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“…In addition, this gives a clear geometric way to study the BPS spectrum, line operators, surface operators, and expectation values of supersymmetric operators in the associated four dimensional theory [47,48,49,50,51,52,53,54,55]. See [56,57,58,44] for a more general review.…”

Section: Pos(tasi2017)015mentioning

confidence: 99%

“…These theories can be constructed for type g = A N−1 13 by compactifying N M5-branes on a Riemann surface Σ × R 4 with the same topological twist, where Σ → C is a multisheeted cover of C. This derives the data of the Seiberg-Witten curve and Seiberg-Witten 1-form, where the curve the Prym variety 14 associated to the map Σ → C [40,41,46]. In addition, this gives a clear geometric way to study the BPS spectrum, line operators, surface operators, and expectation values of supersymmetric operators in the associated four dimensional theory [47,48,49,50,51,52,53,54,55]. See [56,57,58,44] for a more general review.…”

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confidence: 99%

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The String Landscape, the Swampland, and the Missing Corner

Vafa¹,

Brennan²,

Carta³

2018

Proceedings of Theoretical Advanced Study Institute Summer School 2017 "Physics at the Fundamental Frontier" — PoS(TASI2017)

139

209

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We give a brief overview of the string landscape and techniques used to construct string compactifications. We then explain how this motivates the notion of the swampland and review a number of conjectures that attempt to characterize theories in the swampland. We also compare holography in the context of superstrings with the similar, but much simpler case of topological string theory. For topological strings, there is a direct definition of topological gravity based on a sum over a "quantum gravitational foam." In this context, holography is the statement of an identification between a gravity and gauge theory, both of which are defined independently of one another. This points to a missing corner in string dualities which suggests the search for a direct definition of quantum theory of gravity rather than relying on its strongly coupled holographic dual as an adequate substitute (Based on TASI 2017 lectures given by C. Vafa). Theoretical Advanced Study Institute in Elementary PoS(TASI2017)015The String Landscape, the Swampland, and the Missing Corner Cumrun VafaThese lecture notes from TASI 2017 give a brief overview of some of the open problems in string theory. We will be generally motivated by the philosophy that string theory is ultimately supposed to describe the fundamental laws of our universe. String theory is so versatile that it can be used to study a wide array of physical problems such as various topics in condensed matter and quark-gluon plasma or aspects of quantum fields theories in diverse dimensions. Much of the recent work using string theory has been focused on using its properties to solve specific problems rather than developing our understanding of string theory as a fundamental description of our universe. Here we aim to discuss topics which we hope will be useful in bringing string theory closer to observable aspects of fundamental physics.With this philosophy in mind, we will begin these lectures by reviewing some of what we know about string theory and its possible application to the universe by describing some generalities about the space of low energy theories theories coming from string theory compactifications: this is called the "string landscape." Supersymmetry plays a key organizing principle in this context. This will naturally lead us to investigate the question of how we know a priori if a low energy theory is in the landscape or it is not. The set of low energy physics models which look consistent but ultimately are not when coupled to gravity, is called the "swampland." Finding simple criteria to distinguish the swampland from the landscape is of great importance. In particular such criteria can lead to concrete predictions for our universe as we will discuss later. We review a number of conjectures which are aimed at distinguishing the swampland from the landscape.The string landscape and the swampland will be the topic of the first two lectures. In the third lecture, which is on a somewhat disjoint topic, we review critically where we are in our current understanding o...

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Section: Pos(tasi2017)015mentioning

confidence: 99%

mentioning

confidence: 99%

The String Landscape, the Swampland, and the Missing Corner

Vafa¹,

Brennan²,

Carta³

2018

Proceedings of Theoretical Advanced Study Institute Summer School 2017 "Physics at the Fundamental Frontier" — PoS(TASI2017)

139

209

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“…On the other hand, using the invariance under wall-crossing of the quantum spectrum generator allows us to evaluate it in "simpler" chambers. In this paper, we shall take advantage of the existence of a so-called "Roman locus" [22], characterized by u ∈ B, such that the 4d BPS particles have central charges of common phase. One might worry that at this locus, the 4d BPS spectrum is ill-defined, since by definition 4d BPS states would be at marginal stability.…”

Section: )mentioning

confidence: 99%

“…The Schur index I(q) was introduced in [16,17], as a particular limit of the full 4d N = 2 superconformal index [18,19], and -in analogy with the AGT correspondence -it was shown that I(q) arises as a correlator of a 2d TQFT on the UV curve C. The BPS monodromy M (q) arose in the work of Kontsevich and Soibelman as a wall-crossing invariant of BPS spectra [20]. A series of developments for studying BPS spectra through geometric techniques on C [4,13], led to a construction of M (q) based on topological data of a certain ribbon graph G, embedded in C [21,22]. previous related work [24][25][26][27].…”

Section: Introductionmentioning

confidence: 99%

“…When the UV curve has multiple regular punctures, it becomes necessary to establish a criterion to determine which puncture the defect is "close to". An elegant solution to this problem is offered by BPS graphs [22]. In A 1 theories, a BPS graph G is a graph embedded in C, whose edges define different patches of C, each of which contains exactly one puncture.…”

Section: Introductionmentioning

confidence: 99%

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An infrared bootstrap of the Schur index with surface defects

Fluder

Longhi

2019

J. High Energ. Phys.

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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.

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Topological strings, strips and quivers

Panfil

Sułkowski

2019

J. High Energ. Phys.

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We find a direct relation between quiver representation theory and open topological string theory on a class of toric Calabi-Yau manifolds without compact four-cycles, also referred to as strip geometries. We show that various quantities that characterize open topological string theory on these manifolds, such as partition functions, Gromov-Witten invariants, or open BPS invariants, can be expressed in terms of characteristics of the moduli space of representations of the corresponding quiver. This has various deep consequences; in particular, expressing open BPS invariants in terms of motivic Donaldson-Thomas invariants, immediately proves integrality of the former ones. Taking advantage of the relation to quivers we also derive explicit expressions for classical open BPS invariants for an arbitrary strip geometry, which lead to a large set of number theoretic integrality statements. Furthermore, for a specific framing, open topological string partition functions for strip geometries take form of generalized q-hypergeometric functions, which leads to a novel representation of these functions in terms of quantum dilogarithms and integral invariants. We also study quantum curves and A-polynomials associated to quivers, various limits thereof, and their specializations relevant for strip geometries. The relation between toric manifolds and quivers can be regarded as a generalization of the knots-quivers correspondence to more general Calabi-Yau geometries.

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BPS graphs: from spectral networks to BPS quivers

Cited by 28 publications

References 104 publications

The String Landscape, the Swampland, and the Missing Corner

The String Landscape, the Swampland, and the Missing Corner

An infrared bootstrap of the Schur index with surface defects

Topological strings, strips and quivers

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