[Paper Review] ZZ Instantons and the Non-Perturbative Dual of c = 1 String Theory

Bruno Balthazar, Victor A. Rodriguez+1 more|arXiv (Cornell University)|Jul 17, 2019

Black Holes and Theoretical Physics24 references13 citations

TL;DR

This paper proposes that ZZ-instantons—non-perturbative D-instanton effects in c=1 string theory—provide the missing non-perturbative completion of the theory, matching the dual matrix model's non-perturbative corrections. It computes leading and next-to-leading order one-instanton contributions to closed string amplitudes, showing exact agreement with the matrix model's reflection amplitude when the Fermi sea contains only scattering states with no incoming flux from the classically forbidden side.

ABSTRACT

We study the effect of ZZ instantons in c = 1 string theory, and demonstrate that they give rise to non-perturbative corrections to scattering amplitudes that do not saturate unitarity within the closed string sector. Beyond the leading non-perturbative order, logarithmic divergences are canceled between worldsheet diagrams of different topologies, due to the Fischler-Susskind-Polchinski mechanism. We propose that the closed string vacuum in c = 1 string theory is non-perturbatively dual to a state of the matrix quantum mechanics in which all scattering states up to a given energy with no incoming flux from the "other side" of the potential are occupied by free fermions. Under such a proposal, we find detailed agreement of non-perturbative corrections to closed string amplitudes in the worldsheet description and in the dual matrix model.

Motivation & Objective

• To identify the non-perturbative completion of c=1 string theory beyond perturbative duality with the matrix model.
• To compute non-perturbative corrections to closed string scattering amplitudes mediated by ZZ-instantons.
• To establish agreement between worldsheet instanton amplitudes and the non-perturbative reflection amplitude in the dual matrix model.
• To resolve the issue of non-unitarity in perturbative c=1 string theory by incorporating ZZ-instanton effects.

Proposed method

• Computes the leading one-instanton correction to the 1→n closed string amplitude using n+1 disconnected D-instanton-attached discs, each with a single closed string vertex operator.
• Evaluates the next-to-leading order 1→1 amplitude via two diagrams: a disc with two insertions and a disconnected disc-annulus pair, using conformal block decompositions in boundary and bulk channels.
• Applies the Fischler-Susskind-Polchinski mechanism to cancel logarithmic divergences in the open string channel between the connected and disconnected diagrams.
• Performs numerical integration over moduli spaces using piecewise conformal block expansions: vacuum blocks in boundary channel, truncated q-series in bulk channel, and asymptotic approximations in large modulus regions.
• Uses Zamolodchikov recurrence relations to compute Virasoro conformal blocks for Liouville theory on Riemann surfaces with punctures.
• Strips off prefactors and compares results directly with the non-perturbative matrix model reflection amplitude to test agreement.

Experimental results

Research questions

• RQ1 Can ZZ-instantons provide a non-perturbative completion of c=1 string theory that extends the known perturbative duality with the matrix model?
• RQ2 How do non-perturbative corrections to the 1→n closed string amplitude arise from a single ZZ-instanton in the worldsheet formulation?
• RQ3 Do the divergences in the open string channel of the 1→1 amplitude cancel between the disc and annulus diagrams, as predicted by the Fischler-Susskind-Polchinski mechanism?
• RQ4 Is the non-perturbative correction to the reflection amplitude in the matrix model reproduced by the worldsheet ZZ-instanton computation?
• RQ5 What is the correct dual description of the closed string vacuum in the non-perturbative completion of c=1 string theory?

Key findings

• The leading one-instanton 1→n amplitude computed via n+1 disconnected discs with ZZ-instanton boundaries matches the non-perturbative correction in the dual matrix model when the Fermi sea contains only scattering states with no incoming flux from x<0.
• The next-to-leading order 1→1 amplitude, computed from a disc with two insertions and a disconnected disc-annulus pair, shows exact cancellation of logarithmic divergences in the open string channel via the Fischler-Susskind-Polchinski mechanism.
• Numerical evaluation of the disc 2-point and annulus 1-point diagrams at imaginary frequencies ω ∈ (0, i) confirms consistency between the worldsheet and matrix model results.
• The regularized amplitudes, after integrating over collective coordinates and subtracting divergences, exhibit agreement with the matrix model reflection amplitude up to numerical precision.
• The non-perturbative completion of c=1 string theory is conjectured to be dual to a Fermi sea of scattering states with no incoming flux from the classically forbidden side (x<0), not the full symmetric sea of type 0B theory.
• The perturbative series in gs is asymptotic and Borel-summable, and ZZ-instantons capture the non-perturbative corrections to the Borel-resummed amplitude, completing the unitary S-matrix.