The Nahm transform of multi-fractional instantons

Abstract We embed the multi-fractional instantons of SU(N) gauge theories on $$ {\mathbbm{T}}^4 $$ 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 $$ {\hat{\mathbbm{T}}}^4 $$ T ̂ 4 . We first show that SU(N) fractional instantons of topological charge $$ Q=\frac{r}{N},r\in \left\{1,2,\dots, N-1\right\} $$ Q = r N , r ∈ 1 2 … N − 1 , are mapped to fractional instantons of SU( $$ \hat{N} $$ N ̂ ) of charge $$ \hat{Q}=\frac{r}{\hat{N}} $$ Q ̂ = r N ̂ , where $$ \hat{N} $$ N ̂ = Nq 1 q 3 − rq 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( $$ \hat{N} $$ N ̂ ) configurations they map to. Both the $$ {\mathbbm{T}}^4 $$ T 4 instantons and their $$ {\hat{\mathbbm{T}}}^4 $$ 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 $$ {\mathbbm{T}}^4 $$ T 4 to ones with $$ \hat{\Delta } $$ ∆ ̂ < 0 on $$ {\hat{\mathbbm{T}}}^4 $$ 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.