New critical behavior in a supersymmetric double-well matrix model

Abstract

We compute various correlation functions at the planar level in a simple supersymmetric matrix model, whose scalar potential is in shape of a double-well. The model has infinitely degenerate vacua parametrized by filling fractions \nu_\pm representing the numbers of matrix eigenvalues around the two minima of the double-well. The computation is done for general filling fractions corresponding to general two-cut solutions for the eigenvalue distribution. The model is mapped to the O(n) model on a random surface with n=-2, and some sector of the model is described by two-dimensional quantum gravity with c=-2 matter or (2,1) minimal string theory. For the other sector in which such description is not possible, we find new critical behavior of powers of logarithm for correlation functions. We regard the matrix model as a supersymmetric analog of the Penner model, and discuss correspondence of the matrix model to two-dimensional type IIA superstring theory from the viewpoint of symmetry and spectrum. In particular, single-trace operators in the matrix model are naturally interpreted as vertex operators in the type IIA theory. Also, the result of the correlation functions implies that the corresponding type IIA theory has a nontrivial Ramond-Ramond background.Comment: 45 pages, no figure; (v2) some explanation added; (v3) 40 pages, appendix D moved to section 6.2, two references added, typos fixed, version to be published in Nuclear Physics

Resurgence of one-point functions in a matrix model for 2D type IIA superstrings

“…In this paper, we concentrate on one-point functions of powers of matrix φ: 1 N tr φ n . In the previous work [50], it is shown that the operators with n even (n ∈ 2N ) are essentially supersymmetric, and 1/N or genus expansions of their one-point functions are polynomials in the parameter µ 2 , terminating at some genus, which do not lead to any nonanalytic behavior in the double scaling limit. In [47], taking account of effects nonperturbative in 1/N, we have calculated the one-point function 1 N tr (φ 2 − µ 2 ) (or equivalently 1 N tr B ) as an order parameter of the spontaneous supersymmetry breaking.…”

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

“…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…”

“…This observation leads to identifying F k−inst (s) as a k-instanton contribution to the free energy, thereby justifying the notation. 10 Alternatively, one might as well employ asymptotic expansion of the Painlevé II transcendent q(s) for s → ∞ in (3.21) as presented in [48,49]. Concretely, one substitutes a trans-series…”

“…10 It may be possible to obtain the result of F k−inst by integrating k eigenvalues in the region of [2, 2 + N −2/3 s] and the remaining (N − k) in the region [−2, 2] in (2.6) as discussed in section 3 of [12]. However, it seems a technically formidable task for general k, although the technique of an isomonodromic system [47] would manage to deal with the cases of small k.…”

“…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].…”