Normalization of ZZ instanton amplitudes in minimal string theory

Abstract

We use insights from string field theory to analyze and cure the divergences in the cylinder diagram in minimal string theory with both boundaries lying on a ZZ brane. We focus on theories with worldsheet matter consisting of the (2, p) minimal model plus Liouville theory, with total central charge 26, together with the usual bc-ghosts. The string field theory procedure gives a finite, purely imaginary normalization constant for non-perturbative effects in minimal string theory, or doubly non-perturbative effects in JT gravity. We find precise agreement with the prediction from the dual double-scaled one-matrix integral. We also make a few remarks about the extension of this result to the more general (p′, p) minimal string.

Multi-instantons in minimal string theory and in matrix integrals

“…In particular, this happens because the usual U(1) gauge symmetry on the worldvolume of D-branes is a rigid U(1) symmetry in the case of D-instantons. Reverting back to a non gauge-fixed form of the path integral then yields meaningful results which agree with the matrix integral [9,16].…”

“…Motivated by the study of instanton effects in the c = 1 string theory [7][8][9][10], as well as critical (superstring) theories [11][12][13][14][15], in our previous work [16] we used tools from string field theory to compute the one-loop normalization constant that multiplies these instanton contributions to various physical quantities. In particular, the genus expansion of the free energy F reads…”

“…The dual matrix integral, on the other hand, predicts a finite value for N [4,[19][20][21][22][23][24][25][26]. In [16], we resolved this tension by noting that the exponentiated ZZ annulus is, in fact, finite. This is done by identifying the source of the divergence as a pair of Majorana zero modes, which are zero modes because Siegel gauge breaks down in the presence of D-instantons [9].…”

“…Since the perturbation series given in (1.1) is not Borel summable, the significance of non-perturbative instanton corrections is not a priori clear. In [16] we addressed this issue by noting that while the coefficients F g are all real, the normalization constant N is purely imaginary. Therefore instanton corrections give the leading imaginary contribution to the free energy, and this can be reliably computed using D-instanton physics.…”

“…Eventually, the complete non-perturbative result is obtained by expressing the actual integration contour as sum over the Lefschetz thimbles of different saddle points with appropriate weights, but this involves a separate analysis. From this view point, the analysis of [16] involved the contribution from the integration contour where all but one of the eigenvalues are integrated over the Lefschetz thimble of the JHEP10(2022)065…”