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تسجيل مستخدم جديدA Note on Semidensities in Antisymplectic Geometry
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تأليف
K. Bering
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We revisit Khudaverdians geometric construction of an odd nilpotent operator Delta_E that sends semidensities to semidensities on an antisymplectic manifold. We find a local formula for the Delta_E operator in arbitrary coordinates and we discuss its connection to Batalin-Vilkovisky quantization.
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We consider Khudaverdians geometric version of a Batalin-Vilkovisky (BV) operator Delta_E in the case of a degenerate anti-Poisson manifold. The characteristic feature of such an operator (aside from being a Grassmann-odd, nilpotent, second-order dif
ferential operator) is that it sends semidensities to semidensities. We find a local formula for the Delta_E operator in arbitrary coordinates. As an important application of this setup, we consider the Dirac antibracket on an antisymplectic manifold with antisymplectic second-class constraints. We show that the entire Dirac construction, including the corresponding Dirac BV operator Delta_{E_D}, exactly follows from conversion of the antisymplectic second-class constraints into first-class constraints on an extended manifold.
We introduce an antisymplectic Dirac operator and antisymplectic gamma matrices. We explore similarities between, on one hand, the Schroedinger-Lichnerowicz formula for spinor bundles in Riemannian spin geometry, which contains a zeroth-order term pr
oportional to the Levi-Civita scalar curvature, and, on the other hand, the nilpotent, Grassmann-odd, second-order Delta operator in antisymplectic geometry, which in general has a zeroth-order term proportional to the odd scalar curvature of an arbitrary antisymplectic and torsionfree connection that is compatible with the measure density. Finally, we discuss the close relationship with the two-loop scalar curvature term in the quantum Hamiltonian for a particle in a curved Riemannian space.
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Let $X$ be a compact connected Riemann surface and $D$ an effective divisor on $X$. Let ${mathcal N}_H(r,d)$ denote the moduli space of $D$-twisted stable Higgs bundles (a special class of Hitchin pairs) on $X$ of rank $r$ and degree $d$. It is known
that ${mathcal N}_H(r,d)$ has a natural holomorphic Poisson structure which is in fact symplectic if and only if $D$ is the zero divisor. We prove that ${mathcal N}_H(r,d)$ admits a natural enhancement to a holomorphic symplectic manifold which is called here ${mathcal M}_H(r,d)$. This ${mathcal M}_H(r,d)$ is constructed by trivializing, over $D$, the restriction of the vector bundles underlying the $D$-twisted Higgs bundles; such objects are called here as framed Higgs bundles. We also investigate the symplectic structure on the moduli space ${mathcal M}_H(r,d)$ of framed Higgs bundles as well as the Hitchin system associated to it.
Let $X$ be a compact connected Riemann surface, $D, subset, X$ a reduced effective divisor, $G$ a connected complex reductive affine algebraic group and $H_x, subsetneq, G_x$ a Zariski closed subgroup for every $x, in, D$. A framed principal $G$--bun
dle is a pair $(E_G,, phi)$, where $E_G$ is a holomorphic principal $G$--bundle on $X$ and $phi$ assigns to each $x, in, D$ a point of the quotient space $(E_G)_x/H_x$. A framed $G$--Higgs bundle is a framed principal $G$--bundle $(E_G,, phi)$ together with a section $theta, in, H^0(X,, text{ad}(E_G)otimes K_Xotimes{mathcal O}_X(D))$ such that $theta(x)$ is compatible with the framing $phi$ for every $x, in, D$. We construct a holomorphic symplectic structure on the moduli space $mathcal{M}_{FH}(G)$ of stable framed $G$--Higgs bundles. Moreover, we prove that the natural morphism from $mathcal{M}_{FH}(G)$ to the moduli space $mathcal{M}_{H}(G)$ of $D$-twisted $G$--Higgs bundles $(E_G,, theta)$ that forgets the framing, is Poisson. These results generalize cite{BLP} where $(G,, {H_x}_{xin D})$ is taken to be $(text{GL}(r,{mathbb C}),, {text{I}_{rtimes r}}_{xin D})$. We also investigate the Hitchin system for $mathcal{M}_{FH}(G)$ and its relationship with that for $mathcal{M}_{H}(G)$.
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