The Nahm transform of multi-fractional instantons

Abstract

We embed the multi-fractional instantons of $SU(N)$ gauge theories on ${\mathbb{T}}^4$ with 't Hooft twisted boundary conditions into $U(N)$ bundles and use the Nahm transform to study the corresponding configurations on the dual ${\hat{\mathbb{T}}}^4$. We first show that $SU(N)$ fractional instantons of topological charge $Q=\frac{r}{N}$, $r\in \{1,2,...,N-1\}$, are mapped to fractional instantons of $SU(\hat{N})$ of charge $\hat{Q}=\frac{r}{\hat{N}}$, where $\hat{N}=N{q}_1{q}_3-r{q}_3+q_1$ and $q_{1,3}$ are integer-quantized $U(1)$ fluxes. We then explicitly construct the Nahm transform of constant field strength fractional instantons of $SU(N)$ and find the $SU(\hat{N})$ configurations they map to. Both the ${\mathbb{T}}^4$ instantons and their ${\hat{\mathbb{T}}}^4$ images are self-dual for appropriately tuned torus periods. The Nahm duality can be extended to tori with detuned periods, with detuning parameter $\Delta$, mapping solutions with $\Delta >0$ on ${\mathbb{T}}^4$ to ones with $\hat{\Delta }<0$ on ${\hat{\mathbb{T}}}^4$. We also recall that fractional instantons appear in string theory precisely via the $U(N)$ embedding, suggesting that studying the end point of tachyon condensation for $\Delta \ne 0$ is needed -- and is perhaps feasible in a small-$\Delta$ expansion, as in field theory studies -- in order to understand the appearance and role of fractional instantons in $D$-brane constructions.

Comments: 17 pages+appendices