The Coulomb Branch Formula for Quiver Moduli Spaces

The F-theory realization of the rank-1 Minahan Nemeschansky (MN) E 6 , E 7 and E 8 theories leads to a description of the BPS states on the Coulomb branch in terms of Type IIB (p, q)-string networks. Subject to a simple ansatz for the types of networks which can occur, we study the representations of the flavor symmetry group which occur in the BPS spectrum. The results we find for the E 6 and E 7 theories are in perfect agreement with previous calculations by other methods (in particular, we find that arbitrarily large representations occur), but our scheme is easier to implement and more computationally efficient. The string network picture also gives a possible explanation of the experimental observation that in rank-1 MN theories, BPS states whose charge is n times a primitive charge occur with BPS index divisible by (−1) n+1 n.• quiver quantum mechanics and related ideas, e.g. [4][5][6][7][8][9],

Section: String Junctions and Bps Statesmentioning

confidence: 99%

“…The string network picture also gives a possible explanation of the experimental observation that in rank-1 MN theories, BPS states whose charge is n times a primitive charge occur with BPS index divisible by (−1) n+1 n.• quiver quantum mechanics and related ideas, e.g. [4][5][6][7][8][9],…”

mentioning

confidence: 99%

On the BPS Spectrum of the rank-1 Minahan-Nemeschansky theories

Distler

Martone

Neitzke

2020

“…The general formula is known as the 'Coulomb branch formula' from [35,17] (see §2 below for a precise statement) and the coefficients appearing in front of each χ Q({γ i }) involve a new set of quiver invariantsΩ S (γ i ) known as 'single-centered invariants' or 'intrinsic Higgs invariants' [38,39,17,18], which are independent of the stability conditions, and conjecturally depend only on the variable t conjugate to p − q in (1.2), but not on y [17]. These invariants are currently defined in an indirect, recursive way (see [40] for a concise explanation of the Coulomb branch formula). This conjecture, if true, gives a powerful way of obtaining the the full Hodge polynomial from the knowledge of χ-genus of Q and of its subquivers.…”

Section: Introductionmentioning

confidence: 99%

Quiver indices and Abelianization from Jeffrey-Kirwan residues

Beaujard

Mondal

Pioline

2019

Self Cite

In quiver quantum mechanics with 4 supercharges, supersymmetric ground states are known to be in one-to-one correspondence with Dolbeault cohomology classes on the moduli space of stable quiver representations. Using supersymmetric localization, the refined Witten index can be expressed as a residue integral with a specific contour prescription, originally due to Jeffrey and Kirwan, depending on the stability parameters. On the other hand, the physical picture of quiver quantum mechanics describing interactions of BPS black holes predicts that the refined Witten index of a non-Abelian quiver can be expressed as a sum of indices for Abelian quivers, weighted by 'single-centered invariants'. In the case of quivers without oriented loops, we show that this decomposition naturally arises from the residue formula, as a consequence of applying the Cauchy-Bose identity to the vector multiplet contributions. For quivers with loops, the same procedure produces a natural decomposition of the single-centered invariants, which remains to be elucidated. In the process, we clarify some under-appreciated aspects of the localization formula. Part of the results reported herein have been obtained by implementing the Jeffrey-Kirwan residue formula in a public Mathematica code.

“…See ref. [55] for a compact summary. Their approach deals with the problematic asymptotic direction by unphysical deformation of FI constants in the Abelianized middle steps, and also cannot compute the single center states that would be counted by the quiver invariant [29,52].…”

Section: Rational Invariants and Threshold Bound Statesmentioning

confidence: 99%

“…[29] and subsequently incorporated into the multi-center counting in refs. [53][54][55]. For quivers, the index associated with such states is called the quiver-invariant to distinguish it from Witten index.…”

Section: Jhep06(2016)089mentioning

confidence: 99%

Witten index for noncompact dynamics

Lee¹,

Yi²

2016

Among gauged dynamics motivated by string theory, we find many with gapless asymptotic directions. Although the natural boundary condition for ground states is L 2 , one often turns on chemical potentials or supersymmetric mass terms to regulate the infrared issues, instead, and computes the twisted partition function. We point out how this procedure generically fails to capture physical L 2 Witten index with often misleading results. We also explore how, nevertheless, the Witten index is sometimes intricately embedded in such twisted partition functions. For d = 1 theories with gapless continuum sector from gauge multiplets, such as non-primitive quivers and pure Yang-Mills, a further subtlety exists, leading to fractional expressions. Quite unexpectedly, however, the integral L 2 Witten index can be extracted directly and easily from the twisted partition function of such theories. This phenomenon is tied to the notion of the rational invariant that appears naturally in the wall-crossing formulae, and offers a general mechanism of reading off Witten index directly from the twisted partition function. Along the way, we correct early numerical results for some of N = 4, 8, 16 pure Yang-Mills quantum mechanics, and count threshold bound states for general gauge groups beyond SU(N ).