The Nahm transform of multi-fractional instantons

Abstract

We embed the multi-fractional instantons of SU(N) gauge theories on \( {\mathbbm{T}}^4 \) with ’t Hooft twisted boundary conditions into U(N) bundles and use the Nahm transform to study the corresponding configurations on the dual \( {\hat{\mathbbm{T}}}^4 \). We first show that SU(N) fractional instantons of topological charge \( Q=\frac{r}{N},r\in \left\{1,2,\dots, N-1\right\} \), are mapped to fractional instantons of SU(\( \hat{N} \)) of charge \( \hat{Q}=\frac{r}{\hat{N}} \), where \( \hat{N} \) = Nq₁q₃ − rq₃ + q₁ 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 \( {\mathbbm{T}}^4 \) instantons and their \( {\hat{\mathbbm{T}}}^4 \) images are self-dual for appropriately tuned torus periods. The Nahm duality can be extended to tori with detuned periods, with detuning parameter ∆, mapping solutions with ∆ > 0 on \( {\mathbbm{T}}^4 \) to ones with \( \hat{\Delta } \) < 0 on \( {\hat{\mathbbm{T}}}^4 \). 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 ∆ ≠ 0 is needed — and is perhaps feasible in a small-∆ expansion, as in field theory studies — in order to understand the appearance and role of fractional instantons in D-brane constructions.