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Register a new userModuli spaces of Type~$mathcal{B}$ surfaces with torsion
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Authors
Peter B Gilkey
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We examine moduli spaces of locally homogeneous surfaces of Type~$mathcal{B}$ with torsion where the symmetric Ricci tensor is non-degenerate. We also determine the space of affine Killing vector fields in this context.
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We examine the moduli spaces of Type~A connections on oriented and unoriented surfaces both with and without torsion in relation to the signature of the associated symmetric Ricci tensor. If the signature of the symmetric Ricci tensor is (1,1) or (0,2), the moduli spaces are smooth. If the signature is (2,0), there is an orbifold singularity.
We analyze the moduli space of non-flat homogeneous affine connections on surfaces. For Type $mathcal{A}$ surfaces, we write down complete sets of invariants that determine the local isomorphism type depending on the rank of the Ricci tensor and examine the structure of the associated moduli space. For Type $mathcal{B}$ surfaces which are not Type $mathcal{A}$ we show the corresponding moduli space is a simply connected real analytic 4-dimensional manifold with second Betti number equal to $1$.
We show that for many moduli spaces M of torsion sheaves on K3 surfaces S, the functor D(S) -> D(M) induced by the universal sheaf is a P-functor, hence can be used to construct an autoequivalence of D(M), and that this autoequivalence can be factored into geometrically meaningful equivalences associated to abelian fibrations and Mukai flops. Along the way we produce a derived equivalence between two compact hyperkaehler 2g-folds that are not birational, for every g >= 2. We also speculate about an approach to showing that birational moduli spaces of sheaves on K3 surfaces are derived-equivalent.
We give a new proof of the uniformization theorem of the leaves of a lamination by surfaces of hyperbolic conformal type. We use a laminated version of the Ricci flow to prove the existence of a laminated Riemannian metric (smooth on the leaves, transversaly continuous) with leaves of constant Gaussian curvature equal to -1, which is conformally equivalent to the original metric.
Let $X$ be a compact manifold, $D$ a real elliptic operator on $X$, $G$ a Lie group, $Pto X$ a principal $G$-bundle, and ${mathcal B}_P$ the infinite-dimensional moduli space of all connections $ abla_P$ on $P$ modulo gauge, as a topological stack. For each $[ abla_P]in{mathcal B}_P$, we can consider the twisted elliptic operator $D^{ abla_{Ad(P)}}$ on X. This is a continuous family of elliptic operators over the base ${mathcal B}_P$, and so has an orientation bundle $O^D_Pto{mathcal B}_P$, a principal ${mathbb Z}_2$-bundle parametrizing orientations of Ker$D^{ abla_{Ad(P)}}oplus$Coker$D^{ abla_{Ad(P)}}$ at each $[ abla_P]$. An orientation on $({mathcal B}_P,D)$ is a trivialization $O^D_Pcong{mathcal B}_Ptimes{mathbb Z}_2$. In gauge theory one studies moduli spaces $mathcal M$ of connections $ abla_P$ on $P$ satisfying some curvature condition, such as anti-self-dual instantons on Riemannian 4-manifolds $(X, g)$. Under good conditions $mathcal M$ is a smooth manifold, and orientations on $({mathcal B}_P,D)$ pull back to orientations on $mathcal M$ in the usual sense under the inclusion ${mathcal M}hookrightarrow{mathcal B}_P$. This is important in areas such as Donaldson theory, where one needs an orientation on $mathcal M$ to define enumerative invariants. We explain a package of techniques, some known and some new, for proving orientability and constructing canonical orientations on $({mathcal B}_P,D)$, after fixing some algebro-topological information on $X$. We use these to construct canonical orientations on gauge theory moduli spaces, including new results for moduli spaces of flat connections on 2- and 3-manifolds, instantons, Kapustin-Witten and Vafa-Witten equations on 4-manifolds, and the Haydys-Witten equations on 5-manifolds. Two sequels arXiv:1811.02405, arXiv:1811.09658 discuss orientations in 7 and 8 dimensions.
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