Dirac operators on quasi-Hamiltonian <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:mi>G</mml:mi></mml:math>-spaces

Song, Yanli

doi:10.1016/j.geomphys.2016.01.012

Cited by 8 publications

(5 citation statements)

References 10 publications

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“…A different approach to quantization for Hamiltonian LH-spaces due to the second author appeared in [42]. In this approach, an operator acting on sections of an infinite dimensional bundle over the quasi-Hamiltonian space M was constructed, whose index was directly a positive energy representation of LH (strictly speaking a formal difference of two such).…”

Section: Appendix B Schrödinger-type Operatorsmentioning

confidence: 99%

“…Thus the intermediaries appearing in the other two approaches (the Freed-Hopkins-Teleman Theorem in [31], and multiplication by the Weyl-Kac denominator in Definition 4.9) were not needed. Unfortunately the definition of the operator in [42] was quite complicated; for example, the operator was constructed locally in cross-sections, and patched together using a partition of unity. The definition also involved combining geometric Dirac operators on submanifolds of M with Kostant's algebraic relative 'cubic' Dirac operator for loop groups.…”

Section: Appendix B Schrödinger-type Operatorsmentioning

confidence: 99%

“…We discuss this relationship briefly in Section 4.8. There is a third approach [42], due to the second author, that we hope to return to in the future.…”

Section: Introductionmentioning

confidence: 99%

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Quantization of Hamiltonian loop group spaces

Loizides¹,

Song²

2018

Preprint

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We prove a Fredholm property for spin-c Dirac operators D on non-compact manifolds satisfying a certain condition with respect to the action of a semi-direct product group K ⋉ Γ, with K compact and Γ discrete. We apply this result to an example coming from the theory of Hamiltonian loop group spaces. In this context we prove that a certain index pairing [X ] ∩ [D] yields an element of the formal completion R −∞ (T ) of the representation ring of a maximal torus T ⊂ H; the resulting element has an additional antisymmetry property under the action of the affine Weyl group, indicating [X ] ∩ [D] corresponds to an element of the ring of projective positive energy representations of the loop group.

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Section: Appendix B Schrödinger-type Operatorsmentioning

confidence: 99%

Section: Appendix B Schrödinger-type Operatorsmentioning

confidence: 99%

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Quantization of Hamiltonian loop group spaces

Loizides¹,

Song²

2018

Preprint

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“…We begin with the quantization problem of Hamiltonian loop group spaces [AMM,Mei1,Son,LMS,LS]. A Hamiltonian loop group space is an infinite-dimensional symplectic manifold equipped with a loop group action and a proper moment map taking values in the dual Lie algebra of the loop group, where the loop group action on it is the gauge action.…”

Section: Introductionmentioning

confidence: 99%

“…There are many results on substitutes for "L 2 -spaces" and "Dirac operators" on infinite-dimensional spaces related to loop groups, for example [FHT2,Lan,Son,Was]. These results are based on representation theoretical ideas: The L 2 -space on a compact group can be written using its representation theory by the Peter-Weyl theorem, and direction derivatives are written as infinitesimal representations.…”

Section: Introductionmentioning

confidence: 99%

Topological Aspects of the Equivariant Index Theory of Infinite-Dimensional LT-Manifolds

Takata

2020

Preprint

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Let T be the circle group and let LT be its loop group. We formulate and investigate several topological aspects of the LT -equivariant index theory for proper LT -spaces, where proper LT -spaces are infinite-dimensional manifolds equipped with "proper cocompact" LT -actions. Concretely, we introduce "RKK-theory for infinite-dimensional manifolds", and by using it, we formulate an infinite-dimensional version of the KK-theoretical Poincaré duality homomorphism, and an infinite-dimensional version of the RKK-theory counterpart of the assembly map, for proper LT -spaces.The left hand side of the Poincaré duality homomorphism is formulated by the "C * -algebra of a Hilbert manifold" introduced by Guoliang Yu. Thus, the result of this paper suggests that this construction carries some topological information of Hilbert manifolds. In order to formulate the assembly map in a classical way, we need crossed products, which require an invariant measure of a group. However, there is an alternative formula to define them using generalized fixed-point algebras. We will adopt it as the definition of "crossed products by LT ".

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Noncommutative Geometry and An Index Theory of Infinite-Dimensional Manifolds

Takata

2018

Springer Proceedings in Mathematics &Amp; Statistics

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Dirac operators on quasi-Hamiltonian G-spaces

Cited by 8 publications

References 10 publications

Quantization of Hamiltonian loop group spaces

Quantization of Hamiltonian loop group spaces

Topological Aspects of the Equivariant Index Theory of Infinite-Dimensional LT-Manifolds

Noncommutative Geometry and An Index Theory of Infinite-Dimensional Manifolds

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