The Nahm transform of multi-fractional instantons

We embed the multi-fractional instantons of SU(N) gauge theories on T4 with ’t Hooft twisted boundary conditions into U(N) bundles and use the Nahm transform to study the corresponding configurations on the dual T̂4. We first show that SU(N) fractional instantons of topological charge Q=rN, r∈12…N−1, are mapped to fractional instantons of SU(N̂) of charge Q̂=rN̂, where N̂ = Nq1q3− rq3 + q1 and q1, 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(N̂) configurations they map to. Both the T4 instantons and their T̂4 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 T4 to ones with ∆̂ < 0 on T̂4. 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.