01308nas a2200133 4500008004100000245005300041210005000094260001300144520090800157100001601065700002001081700002201101856005101123 2014 en d00aN = 2 Quiver Gauge Theories on A-type ALE Spaces0 aN 2 Quiver Gauge Theories on Atype ALE Spaces bSpringer3 aWe survey and compare recent approaches to the computation of the partition functions and correlators of chiral BPS observables in N = 2 gauge theories on ALE spaces based on quiver varieties and the minimal resolution Xk of the Ak-1 toric singularity C2/Zk, in light of their recently conjectured duality with two-dimensional coset conformal field theories. We review and elucidate the rigorous constructions of gauge theories for a particular family of ALE spaces, using their relation to the cohomology of moduli spaces of framed torsion-free sheaves on a suitable orbifold compactification of Xk. We extend these computations to generic N = 2 superconformal quiver gauge theories, obtaining in these instances new constraints on fractional instanton charges, a rigorous proof of the Nekrasov master formula, and new quantizations of Hitchin systems based on the underlying Seibergâ€“Witten geometry.1 aBruzzo, Ugo1 aSala, Francesco1 aSzabo, Richard J. uhttp://urania.sissa.it/xmlui/handle/1963/3471902234nas a2200109 4500008004100000245004000041210004000081520191700121100001602038700002002054856005002074 2013 en d00aFramed sheaves on projective stacks0 aFramed sheaves on projective stacks3 aGiven a normal projective irreducible stack $\mathscr X$ over an algebraically closed field of characteristic zero we consider {\em framed sheaves} on $\mathscr X$, i.e., pairs $(\mathcal E,\phi_{\mathcal E})$, where $\mathcal E$ is a coherent sheaf on $\mathscr X$ and $\phi_{\mathcal E}$ is a morphism from $\mathcal E$ to a fixed coherent sheaf $\mathcal F$.
After introducing a suitable notion of (semi)stability, we construct a projective scheme, which is a moduli space for semistable framed sheaves with fixed Hilbert polynomial, and an open subset of it, which is a fine moduli space for stable framed sheaves. If $\mathscr X$ is a projective irreducible orbifold of dimension two and $\mathcal F$ a locally free sheaf on a smooth divisor $\mathscr D\subset \mathscr X$ satisfying certain conditions, we consider {\em $(\mathscr{D}, \mathcal{F})$-framed sheaves}, i.e., framed sheaves $(\mathcal E,\phi_{\mathcal E})$ with $\mathcal E$ a torsion-free sheaf which is locally free in a neighborhood of $\mathscr D$, and ${\phi_{\mathcal{E}}}_{\vert \mathscr{D}}$ an isomorphism. These pairs are $\mu$-stable for a suitable choice of a parameter entering the (semi)stability condition, and of the polarization of $\mathscr X$. This implies the existence of a fine moduli space parameterizing isomorphism classes of $(\mathscr{D}, \mathcal{F})$-framed sheaves on $\mathscr{X}$ with fixed Hilbert polynomial, which is a quasi-projective scheme. In an appendix we develop the example of stacky Hirzebruch surfaces.
This is the first paper of a project aimed to provide an algebro-geometric approach to the study of gauge theories on a wide class of 4-dimensional Riemannian manifolds by means of framed sheaves on ``stacky" compactifications of them. In particular, in a subsequent paper \cite{art:bruzzopedrinisalaszabo2013} these results are used to study gauge theories on ALE spaces of type $A_k$.1 aBruzzo, Ugo1 aSala, Francesco uhttp://urania.sissa.it/xmlui/handle/1963/7438