Gauge Invariance and Equations of Motion for Closed String Modes

TL;DR

This paper develops background covariant, gauge-invariant equations of motion for closed string modes using the exact renormalization group, highlighting the role of background curvature and providing explicit equations for massive modes and the graviton.

Contribution

It introduces a background covariant approach to derive gauge-invariant equations of motion for closed string modes using the exact renormalization group, including massive modes and mixed symmetry tensors.

Findings

Equations are gauge invariant only in flat spacetime.

The RG provides quadratic equations of motion for all modes.

Explicit equations for level (2,2) massive fields are derived.

Abstract

We continue earlier discussions on the exact renormalization group and loop variables on the string world sheet for closed and open string backgrounds. The world sheet action with a UV regulator is written in a generally background covariant way by introducing a background metric. It is shown that the renormalization group gives background covariant equations of motion - this is the gauge invariance of the graviton. Interaction is written in terms of gauge invariant and generally covariant field strength tensors. The basic idea is to work in Riemann normal coordinates and covariantize the final equation. It turns out that the equations for massive modes are gauge invariant only if the space time curvature of the (arbitrary) background is zero. The exact RG equations give quadratic equations of motion for all the modes {\em including} the physical graviton. The level $(2,\bar 2)$ massive field equations are used to illustrate the techniques. At this level there are mixed symmetry tensors. Gauge invariant interacting equations can be written down. In flat space an action can also be written for the free theory.