Spontaneous supersymmetry breaking in matrix models from the viewpoints of localization and Nicolai mapping

Abstract

In the previous work, it was shown that, in supersymmetric (matrix) discretized quantum mechanics, inclusion of an external field twisting the boundary condition of fermions enables us to discuss spontaneous breaking of supersymmetry (SUSY) in the path-integral formalism in a well-defined way. In the present work, we continue investigating the same systems from the points of view of localization and Nicolai mapping. The localization is studied by changing of integration variables in the path integral, which is applicable whether or not SUSY is explicitly broken. We examine in detail how the integrand of the partition function with respect to the integral over the auxiliary field behaves as the auxiliary field vanishes, which clarifies a mechanism of the localization. In SUSY matrix models, we obtain a matrix-model generalization of the localization formula. In terms of eigenvalues of matrix variables, we observe that eigenvalues' dynamics is governed by balance of attractive force from the localization and repulsive force from the Vandermonde determinant. The approach of the Nicolai mapping works even in the presence of the external field. It enables us to compute the partition function of SUSY matrix models for finite N (N is the rank of matrices) with arbitrary superpotential at least in the leading nontrivial order of an expansion with respect to the small external field. We confirm the restoration of SUSY in the large-N limit of a SUSY matrix model with a double-well scalar potential observed in the previous work.

Tracy-Widom distribution as instanton sum of 2D IIA superstrings

“…7 That issue is discussed in the context of trans-series and resurgence in [35,36]. 8 • In the double scaling limit of our model, aspects of nonperturbative SUSY breaking for the region s > 0 carry over to the region s < 0 where SUSY is broken at the classical level. In fact, the order parameter of spontaneous SUSY breaking 1 N tr(iB) α , which is proportional to the first s-derivative of the free energy, also crosses smoothly over from s > 0 to s < 0.…”

“…by filling fraction (ν + , ν − ), 2 and transition between the phases (I) and (II) is of the third order [7,8]. As discussed in [10][11][12], various correlation functions of the two-dimensional type IIA superstring theory compactified on R × S 1 at the selfdual radius, with the string coupling g s and the Liouville coupling ω (multiplied by the tachyon operator), coincide with their counterparts in this matrix model through the identification g s = N −1 and 4ω = µ 2 − 2, in the double scaling limit…”

“…With the help of (3.22), s-derivatives of F (1, s) at the origin are expressed in terms of q(0) and q ′ (0) of the Hastings-McLeod solution as: • While the third-order phase transition is found in the planar limit for this model [8], it turns into a crossover in the double scaling limit and the phases (I) and (II) are smoothly connected without any phase transition.…”

“…6 Since the planar solution for µ 2 < 2 has a symmetric eigenvalue distribution with the support of a single interval [8], physics of the region s < 0 should connect to that of the region s > 0 with the filling fraction (1/2, 1/2) rather than (1, 0). However, concerning the free energy the argument in the text is valid, because as seen from the relation above eq.…”

“…Although various matrix models have been investigated as nonperturbative formulations of superstring/M theory [1][2][3][4][5][6], it is still difficult to elucidate whether these models do break SUSY and derive our four-dimensional world. In this situation, a simple double-well SUSY matrix model had been recently considered in [7,8], and its connection to two-dimensional type IIA superstring theory [9] on a nontrivial Ramond-Ramond background had been explored from the viewpoint of symmetries [10] and from direct comparison of scattering amplitudes at the tree and one-loop orders [11]. Interestingly, in a double scaling limit that realizes the type IIA superstring theory, instanton effects of the matrix model survive and break the SUSY spontaneously [12].…”