Non-linear sigma model in string field theory

We revisit the non-linear sigma model approach to string theory with the closed superstring field theory. We construct the string field theory around the non-linear sigma model background with the patch-by-patch description. We show that our string field theory action is invariant under the gauge transformation and solves the BV master equation, thereby providing a tool to study quantum gravitational effects in curved backgrounds in small ${\alpha }^{\mathrm{\prime }}$ and $g_s$ approximation. We illustrate how to use our results to study curved backgrounds in ${\alpha }^{\mathrm{\prime }}$ expansion by studying Calabi-Yau compactification in ${\alpha }^{\mathrm{\prime }}$ expansion. We draw connections between Tseytlin’s approach to the non-linear sigma model and the string field theoretic approach. We comment on future directions.