Multi-Instanton Calculus in c = 1 String Theory

Abstract

We formulate a strategy for computing the complete set of non-perturbative corrections to closed string scattering in c = 1 string theory from the worldsheet perspective. This requires taking into account the effect of multiple ZZ-instantons, including higher instantons constructed from ZZ boundary conditions of type (m, 1), with a careful treatment of the measure and contour in the integration over the instanton moduli space. The only a priori ambiguity in our prescription is a normalization constant N m that appears in the integration measure for the (m, 1)-type ZZ instanton, at each positive integer m. We investigate leading corrections to the closed string reflection amplitude at the n-instanton level, i.e. of order e −n/gs , and find striking agreement with our recent proposal on the non-perturbative completion of the dual matrix quantum mechanics, which in turn fixes N m for all m.

Normalization of Type IIB D-instanton Amplitudes

“…Many D-instanton induced terms were predicted using S-duality invariances of various compactified string theories , but except for the early attempts [2], the direct systematic computation of these amplitudes from first principles has not been carried out. However during the last two years progress was made in the context of two dimensional string theory [26][27][28][29][30][31]. In particular, [26,27] computed the precise contributions to the amplitudes from D-instantons in terms of some constants that appear to be divergent in the world-sheet formalism.…”

“…However during the last two years progress was made in the context of two dimensional string theory [26][27][28][29][30][31]. In particular, [26,27] computed the precise contributions to the amplitudes from D-instantons in terms of some constants that appear to be divergent in the world-sheet formalism. It was then found that string field theory gives finite, unambiguous values of these constants [28][29][30][31].…”

Normalization of D-instanton Amplitudes

“…Let us now focus on the specific case of (1,1) D-instanton in two dimensional string theory. In this case we have [28]:…”

“…1 The constant N given in (2.1) is the overall multiplicative factor that appears in the instanton induced effective action of the closed string fields [27]. This is related to the normalization constant N introduced in [3,28], appearing as a multiplicative factor in the S-matrix, via the relation N = i N. In our analysis, we shall not be careful in fixing the sign of N, since this will be fixed at the end using separate considerations.…”

Muti-instanton Amplitudes in Type IIB String Theory

“…1 One did not need to do this in the analysis of multi-instanton amplitudes in [5], since there was only one set of ξ a µ 's associated with the Euclidean time direction, and ξ a 0 T a was gauge equivalent to a diagonal matrix.…”