Wall-crossing made smooth

Boris Pioline

doi:10.1007/jhep04(2015)092

Abstract

In $D=4,N=2$ theories on $R^{3,1}$, the index receives contributions not only from single-particle BPS states, counted by the BPS indices, but also from multi-particle states made of BPS constituents. In a recent work [arXiv:1406.2360], a general formula expressing the index in terms of the BPS indices was proposed, which is smooth across walls of marginal stability and reproduces the expected single-particle contributions. In this note, I analyze the two-particle contributions predicted by this formula, and show agreement with the spectral asymmetry of the continuum of scattering states in the supersymmetric quantum mechanics of two non-relativistic, mutually non-local dyons. This provides a physical justification for the error function profile used in the mathematics literature on indefinite theta series, and in the physics literature on black hole partition functions.

Highlights

JHEP04(2015)092 construction in the context of the hypermultiplet moduli space in string vacua [8, 9], we proposed a general formula for the index1
In a recent work [1], a general formula expressing the index in terms of the BPS indices was proposed, which is smooth across walls of marginal stability and reproduces the expected single-particle contributions
I have shown that the general formula for the index (1.2) in N = 2, D = 4 gauge theories correctly reproduces the contribution of the continuum of two-particle states, in the vicinity of a wall of marginal stability where the constituents can be treated as nonrelativistic BPS particles

Summary

Two-particle contribution to the index

According to the conjecture (1.2), the contribution of a two-particle state with charges {γ, γ } to the index is obtained by inserting the one-particle approximation to (1.3) in (1.2), Iγ(2,γ). The two-particle contribution is discontinuous across the wall: as ψγγ goes from negative to positive, Iγ(2,γ) jumps by. The one-particle contribution Iγ(1,γ) is discontinuous across the wall, due to the fact that the one-particle index Ω (γ + γ ) jumps [18]:3. Contributions is continuous, and differentiable across the wall (see figure 1 for illustration), which acquires a finite width of order 1/ R mγ,γ as a function of the relative phase ψγγ between the central charges Zγ and Zγ. It would be interesting to generalize this computation to the case of non-primitive wall-crossing, and to relax the non-relativistic limit R → ∞

Supersymmetric electron-monopole quantum mechanics

Discussion

A Robustness of the spectral asymmetry