Abstract
We present an analytic expression of the nonperturbative free energy of a double-well supersymmetric matrix model in its double scaling limit, which corresponds to two-dimensional type IIA superstring theory on a nontrivial Ramond-Ramond background. To this end we draw upon the wisdom of random matrix theory developed by Tracy and Widom, which expresses the largest eigenvalue distribution of unitary ensembles in terms of a Painleve II transcendent. Regularity of the result at any value of the string coupling constant shows that the third-order phase transition between a supersymmetry-preserving phase and a supersymmetry-broken phase, previously found at the planar level, becomes a smooth crossover in the double scaling limit. Accordingly, the supersymmetry is always broken spontaneously as its order parameter stays nonzero for the whole region of the coupling constant. Coincidence of the result with the unitary one-matrix model suggests that one-dimensional type 0 string theories partially correspond to the type IIA superstring theory. Our formulation naturally allows for introduction of an instanton chemical potential, and reveals the presence of a novel phase transition, possibly interpreted as condensation of instantons.
Highlights
In this paper, the nonperturbative computation of the free energy of the SUSY matrix model is completed by drawing upon the result of Tracy and Widom [13, 14] on the distribution of the largest eigenvalue in random matrix theory.1 we shall find that the full nonperturbative free energy is expressed in terms of a Painleve II transcendent, in coincidence with the unitary one-matrix model [24, 25]
We present an analytic expression of the nonperturbative free energy of a double-well supersymmetric matrix model in its double scaling limit, which corresponds to two-dimensional type IIA superstring theory on a nontrivial Ramond-Ramond background
As discussed in [10,11,12], various correlation functions of the two-dimensional type IIA superstring theory compactified on R × S1 at the selfdual radius, with the string coupling gs and the Liouville coupling ω, coincide with their counterparts in this matrix model through the identification gs = N −1 and 4ω = μ2 − 2, in the double scaling limit
Summary
The SUSY double-well matrix model is defined by the zero-dimensional reduction of a. The integration region of each eigenvalue is divided into the positive and negative real axes, and the partition function associated with the filling fraction (ν+, ν−), denoted by Z(ν+,ν−)(μ2), is defined by integrations over the positive real axis for the first ν+N eigenvalues and over the negative real axis for the remaining ν−N. 2 ν± are nonnegative fractional numbers such that ν+ +ν− = 1, corresponding to ν+N (ν−N ) eigenvalues of φ located around the minimum x. Using this expression, one- and two-instanton effects to the one-point function and Z(1,0)(μ2) are analytically obtained in [12], from which spontaneous breaking of SUSY by instantons is concluded. One of the aims of this article is to present an analytic form of full nonperturbative contributions to Z(1,0)(μ2) by recalling results in random matrix theory
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