Coulomb branch and the moduli space of instantons

Abstract

Abstract:The moduli space of instantons on C 2 for any simple gauge group is studied using the Coulomb branch of N = 4 gauge theories in three dimensions. For a given simple group G, the Hilbert series of such an instanton moduli space is computed from the Coulomb branch of the quiver given by the over-extended Dynkin diagram of G. The computation includes the cases of non-simply-laced gauge groups G, complementing the ADHM constructions which are not available for exceptional gauge groups. Even though the Lagrangian description for non-simply laced Dynkin diagrams is not currently known, the prescription for computing the Coulomb branch Hilbert series of such diagrams is very simple. For instanton numbers one and two, the results are in agreement with previous works. New results and general features for the moduli spaces of three and higher instanton numbers are reported and discussed in detail.

S-duality and the prepotential in N = 2 ⋆ $$ \mathcal{N}={2}^{\star } $$ theories (I): the ADE algebras

“…We now show that, in the decoupling limit in which the N = 2 ⋆ theory reduces to the pure N = 2 SYM, the above result agrees with the explicit computations that are present in the literature [14][15][16][17]. In the decoupling limit the mass m is sent to infinity and the instanton weight q to zero, keeping constant the dynamically generated scale Λ defined as…”

Quiver theories for moduli spaces of classical group nilpotent orbits

“…The Hilbert series of these moduli spaces have been calculated [16]. Indeed, many different constructions for the Hilbert series of RSIMS are known, including Coulomb branch and other types [17][18][19].…”

ADHM and the 4d quantum Hall effect

“…Finally, mirror symmetry provides a dual description of the same manifold as the Coulomb branch of yet another gauge theory which lives in three spacetime dimensions. In recent work [20][21][22], this perspective has been used to provide alternative formulae for the Hilbert series of the instanton moduli space. The equality of the resulting Coulomb branch formulae to the dual Higgs branch expression (i.e.…”

“…The Coulomb branch approach has been used to compute the Hilbert series for many different Coulomb branches including, most pertinently, the instanton moduli spaces [22]. The Coulomb branch approach to the partition function does not, at present, allow for the inclusion of the effects of the five-dimensional Chern-Simons level k. We remedy this.…”

“…With this preamble in mind, here we present the prescription of [20,22] for counting monopole operators and, in doing so, computing the partition function of interest. For each charge J, the monopole operators in U(n) can sit in any of the U(1) n Cartan elements.…”