The Nahm transform of multi-fractional instantons

A bstract We embed the multi-fractional instantons of SU( N ) gauge theories on $$ {\mathbbm{T}}^4 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>T</mml:mi> <mml:mn>4</mml:mn> </mml:msup> </mml:math> with ’t Hooft twisted boundary conditions into U( N ) bundles and use the Nahm transform to study the corresponding configurations on the dual $$ {\hat{\mathbbm{T}}}^4 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mover> <mml:mi>T</mml:mi> <mml:mo>̂</mml:mo> </mml:mover> <mml:mn>4</mml:mn> </mml:msup> </mml:math> . We first show that SU( N ) fractional instantons of topological charge $$ Q=\frac{r}{N},r\in \left\{1,2,\dots, N-1\right\} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Q</mml:mi> <mml:mo>=</mml:mo> <mml:mfrac> <mml:mi>r</mml:mi> <mml:mi>N</mml:mi> </mml:mfrac> <mml:mo>,</mml:mo> <mml:mi>r</mml:mi> <mml:mo>∈</mml:mo> <mml:mfenced> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> <mml:mo>…</mml:mo> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:mfenced> </mml:math> , are mapped to fractional instantons of SU( $$ \hat{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>N</mml:mi> <mml:mo>̂</mml:mo> </mml:mover> </mml:math> ) of charge $$ \hat{Q}=\frac{r}{\hat{N}} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>Q</mml:mi> <mml:mo>̂</mml:mo> </mml:mover> <mml:mo>=</mml:mo> <mml:mfrac> <mml:mi>r</mml:mi> <mml:mover> <mml:mi>N</mml:mi> <mml:mo>̂</mml:mo> </mml:mover> </mml:mfrac> </mml:math> , where $$ \hat{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>N</mml:mi> <mml:mo>̂</mml:mo> </mml:mover> </mml:math> = 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( $$ \hat{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>N</mml:mi> <mml:mo>̂</mml:mo> </mml:mover> </mml:math> ) configurations they map to. Both the $$ {\mathbbm{T}}^4 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>T</mml:mi> <mml:mn>4</mml:mn> </mml:msup> </mml:math> instantons and their $$ {\hat{\mathbbm{T}}}^4 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mover> <mml:mi>T</mml:mi> <mml:mo>̂</mml:mo> </mml:mover> <mml:mn>4</mml:mn> </mml:msup> </mml:math> 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 ∆ &gt; 0 on $$ {\mathbbm{T}}^4 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>T</mml:mi> <mml:mn>4</mml:mn> </mml:msup> </mml:math> to ones with $$ \hat{\Delta } $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mtext>∆</mml:mtext> <mml:mo>̂</mml:mo> </mml:mover> </mml:math> &lt; 0 on $$ {\hat{\mathbbm{T}}}^4 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mover> <mml:mi>T</mml:mi> <mml:mo>̂</mml:mo> </mml:mover> <mml:mn>4</mml:mn> </mml:msup> </mml:math> . 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.