R3Index for Four-DimensionalN=2Field Theories

Abstract

In theories with N ¼ 2 supersymmetry on R 3;1 , supersymmetric bound states can decay across walls of marginal stability in the space of Coulomb branch parameters, leading to discontinuities in the BPS indices Ωðγ; uÞ. We consider a supersymmetric index I which receives contributions from 1=2-BPS states, generalizing the familiar Witten index Trð−1Þ F e −βH . We expect I to be smooth away from loci where massless particles appear, thanks to contributions from the continuum of multiparticle states. Taking inspiration from a similar phenomenon in the hypermultiplet moduli space of N ¼ 2 string vacua, we conjecture a formula expressing I in terms of the BPS indices Ωðγ; uÞ, which is continuous across the walls and exhibits the expected contributions from single particle states at large β. This gives a universal prediction for the contributions of multiparticle states to the index I. This index is naturally a function on the moduli space after reduction on a circle, closely related to the canonical hyperkähler metric and hyperholomorphic connection on this space. DOI: 10.1103/PhysRevLett.114.121601 PACS numbers: 11.25.Mj, 11.25.Uv, 11.30.Pb It has been clear since the work of Seiberg and Witten [1] that extended supersymmetry gives enough control over four-dimensional quantum field theories to produce exact results on the dynamics of the theories, even when these theories are strongly interacting. Remarkably, such results are deeply related to some of the most interesting questions in the mathematics of algebraic geometry and differential geometry. As a significant example, the moduli space of a four-dimensional theory with N ¼ 2 supersymmetry on a circle is a hyperkähler manifold (a special class of manifolds satisfying Einstein's equations), whose metric encodes both instanton corrections to gauge couplings and the spectrum of Bogomol'nyi-Prasad-Sommerfield (BPS) states in the four-dimensional theory [2]. In this Letter, we reinforce this connection, and construct a canonical function on the aforementioned moduli space, which, on the one hand, generates a solution to the self-dual Yang-Mills equations on this manifold, and, on the other hand, purportedly encodes interactions of BPS states in four dimensions.BPS indices and the Witten index.-In four-dimensional field theories on R 3;1 with N ¼ 2 supersymmetry, the spectrum of BPS states in general strongly depends on the value of the Coulomb branch parameters. Part of this dependence can be removed by considering the BPS indexwhere H 1 ðγ; uÞ is the Hilbert space of one-particle states with electromagnetic charge γ ∈ Γ in the Coulomb vacuum u, J 3 is a component of the rotation group along a fixed axis, and ð−1Þ 2J 3 is the fermionic parity by virtue of the spin statistics theorem. The BPS index Ωðγ; uÞ, being sensitive only to short multiplets saturating the BPS bound [3], is a locally constant, integer valued function of u, but it is discontinuous across certain walls in moduli space, where some of the BPS bound states with charge γ decay into multiparticle B...

Wall-crossing made smooth

“…In [1], we matched this result with the index of a relativistic particle of charge γ and mass |Z γ |. To define the index, we regulated the infrared divergences by switching on a chemical potential θ for the rotations J 3 in the xy plane, restricting the z direction to a finite interval of length L, and removing the regulators as follows,…”

“…The recent work [1] proposed a general formula for the index I(R, u, C) in four-dimensional field theories on R 3,1 with N = 2 supersymmetry. This index can be understood as the partition function on R 3 times an Euclidean circle of radius R, with periodic boundary conditions for fermions, chemical potentials C conjugate to the electromagnetic charge γ, and with an insertion of a suitable four-fermion vertex so as to saturate fermionic zeromodes.…”

“…In [1], using insight from the study of the hyperkähler metric on the Coulomb branch in the theory compactified down to three dimensions [7], and by analogy with a similar…”

Hypermultiplet metric and D-instantons

“…A quantity, which is needed for evaluation of the metric and plays an important role in this story [38,39], is the contact potential appearing in the relation (2.9). It was explicitly evaluated in [13], again in terms of the solution of (2.12), and is given by…”

Rigid limit for hypermultiplets and five-dimensional gauge theories

“…This partially explains the important role played by this function both in physics and mathematics [84]. As for the dilaton, we will also use for it the notation r = e φ .…”