Advances in Theoretical and Mathematical Physics

Volume 24 (2020)

Number 7

The first of two special issues in honor of Cumrun Vafa’s 60th birthday

Anomaly cancellation in the topological string

Pages: 1723 – 1771



Kevin Costello (Perimeter Institute of Theoretical Physics, Waterloo, Ontario, Canada)

Si Li (Department of Mathematical Sciences and Yau Mathematical Sciences Center, Tsinghua University, Beijing, China)


We describe the coupling of holomorphic Chern–Simons theory at large $N$ with Kodaira–Spencer gravity. We explain a new anomaly cancellation mechanism at all loops in perturbation theory for open-closed topological $B$‑model. At one loop this anomaly cancellation is analogous to the Green–Schwarz mechanism.

As an application, we introduce a type I version of Kodaira–Spencer theory in complex dimensions $3$ and $5$. In complex dimension $5$, we show that it can only be coupled consistently at the quantum level to holomorphic Chern–Simons theory with gauge group $SO(32)$. This is analogous to the Green–Schwarz mechanism for the physical type I string. This coupled system is conjectured to be a supersymmetric localization of type I string theory. In complex dimension 3, the required gauge group is $SO(8)$.

The work of K.C. was supported by the NSERC Discovery Grant program, and by the Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada, and by the Province of Ontario through the Ministry of Research and Innovation.

The work of S. Li was partially supported by the National Key Research and Development Program of China (No. 2020YFA0713000), by the Beijing Municipal Natural Science Foundation (No. Z180003), and by the National Natural Science Foundation of China (No. 11801300).

Published 8 September 2021