Sub-Critical Closed String Field Theory in D Less Than 26

TL;DR

This paper develops a second quantized action for sub-critical closed string field theory in dimensions less than 26, demonstrating gauge invariance, anomaly cancellation, and reproducing known amplitudes in two dimensions.

Contribution

It generalizes the non-polynomial closed string field theory to sub-critical dimensions and explicitly shows gauge invariance and anomaly cancellation.

Findings

Gauge invariance is maintained in sub-critical dimensions.

Anomaly contributions cancel only under a specific dimension-dependent condition.

The four-point function matches the shifted Shapiro-Virasoro amplitude.

Abstract

We construct the second quantized action for sub-critical closed string field theory with zero cosmological constant in dimensions $ 2 \leq D < 26$, generalizing the non-polynomial closed string field theory action proposed by the author and the Kyoto and MIT groups for $D = 26$. The proof of gauge invariance is considerably complicated by the presence of the Liouville field $\phi$ and the non-polynomial nature of the action. However, we explicitly show that the polyhedral vertex functions obey BRST invariance to all orders. By point splitting methods, we calculate the anomaly contribution due to the Liouville field, and show in detail that it cancels only if $D - 26 + 1 + 3 Q ^ 2 = 0 $, in both the bosonized and unbosonized polyhedral vertex functions. We also show explicitly that the four point function generated by this action reproduces the shifted Shapiro-Virasoro amplitude found from $c=1$ matrix models and Liouville theory in two dimensions. LATEX file.