D-branes and fractional instantons on a twisted $$ {\mathbbm{T}}^4 $$: the moduli space as an $$ \mathcal{N} $$ = 2 supersymmetric Higgs branch

A bstract We study self-dual instantons of topological charge $$ Q=\frac{r}{N} $$ <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:math> , ∀ r ∈ ℕ , in SU ( N ) Yang-Mills theory 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 twists, by embedding them into worldvolume theories of D -branes. To study their moduli, we construct the wrapped intersecting brane configurations dual to general constant field strength instanton backgrounds. We show that, locally, the moduli space is identified with the Higgs branch of an $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> =2 supersymmetric theory. This parameterization of the moduli space is equivalent to one recently found in field theory, but is obtained with significantly less effort and has manifest hyper-Kähler structure. Our hope is that combining different perspectives on instantons on the twisted torus will help understand the still unknown global structure of the moduli space for general solutions with $$ Q=\frac{r}{N} $$ <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:math> as well as the nature of instantons with all moduli turned on — when some Q &lt;1 and all Q ≥1 instantons become space-time dependent. For integer Q , these are expected to match the ADHM solution in an appropriately taken infinite volume limit.