Musings

del Pezzo

Seiberg Duality is one of the mysterious and wonderful features of strongly-coupled $N=1$ supersymmetric gauge theories to have emerged from the interplay between string theory and gauge theory. In the purely gauge-theoretic context, it’s a bit of a black art to construct Seiberg dual pairs of gauge theories. A stringy context in which a large class of examples can be found, and hence where one can hope to find a systematic understanding, is D-branes on local del Pezzo surfaces.

Let $X= \{K\to S\}$ be the noncompact Calabi-Yau 3-fold, which is the total space of the canonical bundle of a del Pezzo surface, $S$ ($\mathcal{P}^2$, with $k=0,\dots 8$ points blown up). $X$ has a minimal-sized surface, $S_0$ ($S$, embedded via the zero section). “Compactify” Type IIB on $X$, and consider space-filling D3-branes and D5-branes wrapped on cycles of $S_0$. Varying the Kähler moduli of $X$ is an irrelevant deformation of the resulting 4D gauge theory. So, studying the different D-brane descriptions which arise as one moves in the Kähler moduli space gives a concrete description of Seiberg Duality. (I’m lying slightly, here, but part of the mystery of the subject is understanding exactly when that’s a lie.)

At certain loci in the moduli space, a nonabelian gauge invariance is manifest, and one has a quiver gauge theory with massless bifundamentals and, typically, some gauge-invariant superpotential for them. There’s a close relation between, $D^b(X)$, the derived category of sheaves on $X$ (in which the aforementioned D-branes are objects) and the derived category of quiver representations (with relations given by the derivatives of the superpotential).

There’s a rich literature on this subject, but two recent papers provide a good entrée into it for those (like yours truly) who haven’t been following the literature in much detail. Chris Herzog argues that admissible mutations of strongly exceptional collection of coherent sheaves (which, in turn, for a basis of objects in the derived category) is Seiberg Duality. Aspinwall and Melnikov discuss the same issue from the point of view of tilting equivalences between the derived categories of quiver representations (an approach pioneered by Berenstein and Douglas).