12 research outputs found

    On transversally elliptic operators and the quantization of manifolds with ff-structure

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    An ff-structure on a manifold MM is an endomorphism field \phi\in\Gamma(M,\End(TM)) such that ϕ3+ϕ=0\phi^3+\phi=0. Any ff-structure ϕ\phi determines an almost CR structure E_{1,0}\subset T_\C M given by the +i+i-eigenbundle of ϕ\phi. Using a compatible metric gg and connection ∇\nabla on MM, we construct an odd first-order differential operator DD, acting on sections of §=ΛE0,1∗\S=\Lambda E_{0,1}^*, whose principal symbol is of the type considered in arXiv:0810.0338. In the special case of a CR-integrable almost §\S-structure, we show that when ∇\nabla is the generalized Tanaka-Webster connection of Lotta and Pastore, the operator DD is given by D = \sqrt{2}(\dbbar+\dbbar^*), where \dbbar is the tangential Cauchy-Riemann operator. We then describe two "quantizations" of manifolds with ff-structure that reduce to familiar methods in symplectic geometry in the case that ϕ\phi is a compatible almost complex structure, and to the contact quantization defined in \cite{F4} when ϕ\phi comes from a contact metric structure. The first is an index-theoretic approach involving the operator DD; for certain group actions DD will be transversally elliptic, and using the results in arXiv:0810.0338, we can give a Riemann-Roch type formula for its index. The second approach uses an analogue of the polarized sections of a prequantum line bundle, with a CR structure playing the role of a complex polarization.Comment: 31 page

    On Non-Abelian Symplectic Cutting

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    We discuss symplectic cutting for Hamiltonian actions of non-Abelian compact groups. By using a degeneration based on the Vinberg monoid we give, in good cases, a global quotient description of a surgery construction introduced by Woodward and Meinrenken, and show it can be interpreted in algebro-geometric terms. A key ingredient is the `universal cut' of the cotangent bundle of the group itself, which is identified with a moduli space of framed bundles on chains of projective lines recently introduced by the authors.Comment: Various edits made, to appear in Transformation Groups. 28 pages, 8 figure

    Equivariant volumes of non-compact quotients and instanton counting

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    Motivated by Nekrasov's instanton counting, we discuss a method for calculating equivariant volumes of non-compact quotients in symplectic and hyper-K\"ahler geometry by means of the Jeffrey-Kirwan residue-formula of non-abelian localization. In order to overcome the non-compactness, we use varying symplectic cuts to reduce the problem to a compact setting, and study what happens in the limit that recovers the original problem. We implement this method for the ADHM construction of the moduli spaces of framed Yang-Mills instantons on R4\R^{4} and rederive the formulas for the equivariant volumes obtained earlier by Nekrasov-Shadchin, expressing these volumes as iterated residues of a single rational function.Comment: 34 pages, 2 figures; minor typos corrected, to appear in Comm. Math. Phy

    Localization for Yang-Mills Theory on the Fuzzy Sphere

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    We present a new model for Yang-Mills theory on the fuzzy sphere in which the configuration space of gauge fields is given by a coadjoint orbit. In the classical limit it reduces to ordinary Yang-Mills theory on the sphere. We find all classical solutions of the gauge theory and use nonabelian localization techniques to write the partition function entirely as a sum over local contributions from critical points of the action, which are evaluated explicitly. The partition function of ordinary Yang-Mills theory on the sphere is recovered in the classical limit as a sum over instantons. We also apply abelian localization techniques and the geometry of symmetric spaces to derive an explicit combinatorial expression for the partition function, and compare the two approaches. These extend the standard techniques for solving gauge theory on the sphere to the fuzzy case in a rigorous framework.Comment: 55 pages. V2: references added; V3: minor corrections, reference added; Final version to be published in Communications in Mathematical Physic

    ADMISSIBLE COADJOINT ORBITS FOR COMPACT LIE GROUPS

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