Hitchin's connection, Toeplitz operators, and symmetry invariant deformation quantization

Publikation: Bidrag til tidsskriftTidsskriftartikelForskningpeer review

Resumé

We introduce the notion of a rigid family of Kähler structures on a symplectic manifold. We then prove that a Hitchin connection exists for any rigid holomorphic family of Kähler structures on any compact pre-quantizable symplectic manifold which satisfies certain simple topological constraints. Using Toeplitz operators we prove that the Hitchin connection induces a unique formal connection on smooth functions on the symplectic manifold. Parallel transport of this formal connection produces equivalences between the corresponding Berezin–Toeplitz deformation quantizations. In the cases where the Hitchin connection is projectively flat, the formal connections will be flat and we get a symmetry-invariant formal quantization. If a certain cohomological condition is satisfied a global trivialization of this algebra bundle is constructed. As a corollary we get a symmetry-invariant deformation quantization.

Finally, these results are applied to the moduli space situation in which Hitchin originally constructed his connection. First we get a proof that the Hitchin connection in this case is the same as the connection constructed by Axelrod, Della Pietra, and Witten. Second we obtain in this way a mapping class group invariant formal quantization of the smooth symplectic leaves of the moduli space of flat SU(n)-connections on any compact surface.
OriginalsprogEngelsk
TidsskriftQuantum Topology
Vol/bind3
Udgave nummer3/4
Sider (fra-til)293-325
Antal sider33
ISSN1663-487X
DOI
StatusUdgivet - 2012
Udgivet eksterntJa

Fingeraftryk

Deformation Quantization
Symplectic Manifold
Toeplitz Operator
Symmetry
Moduli Space
Invariant
Quantization
Mapping Class Group
Smooth function
Bundle
Corollary
Leaves
Equivalence
Algebra
Family

Citer dette

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title = "Hitchin's connection, Toeplitz operators, and symmetry invariant deformation quantization",
abstract = "We introduce the notion of a rigid family of K{\"a}hler structures on a symplectic manifold. We then prove that a Hitchin connection exists for any rigid holomorphic family of K{\"a}hler structures on any compact pre-quantizable symplectic manifold which satisfies certain simple topological constraints. Using Toeplitz operators we prove that the Hitchin connection induces a unique formal connection on smooth functions on the symplectic manifold. Parallel transport of this formal connection produces equivalences between the corresponding Berezin–Toeplitz deformation quantizations. In the cases where the Hitchin connection is projectively flat, the formal connections will be flat and we get a symmetry-invariant formal quantization. If a certain cohomological condition is satisfied a global trivialization of this algebra bundle is constructed. As a corollary we get a symmetry-invariant deformation quantization. Finally, these results are applied to the moduli space situation in which Hitchin originally constructed his connection. First we get a proof that the Hitchin connection in this case is the same as the connection constructed by Axelrod, Della Pietra, and Witten. Second we obtain in this way a mapping class group invariant formal quantization of the smooth symplectic leaves of the moduli space of flat SU(n)-connections on any compact surface.",
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Hitchin's connection, Toeplitz operators, and symmetry invariant deformation quantization. / Andersen, Jørgen Ellegaard.

I: Quantum Topology, Bind 3, Nr. 3/4, 2012, s. 293-325.

Publikation: Bidrag til tidsskriftTidsskriftartikelForskningpeer review

TY - JOUR

T1 - Hitchin's connection, Toeplitz operators, and symmetry invariant deformation quantization

AU - Andersen, Jørgen Ellegaard

PY - 2012

Y1 - 2012

N2 - We introduce the notion of a rigid family of Kähler structures on a symplectic manifold. We then prove that a Hitchin connection exists for any rigid holomorphic family of Kähler structures on any compact pre-quantizable symplectic manifold which satisfies certain simple topological constraints. Using Toeplitz operators we prove that the Hitchin connection induces a unique formal connection on smooth functions on the symplectic manifold. Parallel transport of this formal connection produces equivalences between the corresponding Berezin–Toeplitz deformation quantizations. In the cases where the Hitchin connection is projectively flat, the formal connections will be flat and we get a symmetry-invariant formal quantization. If a certain cohomological condition is satisfied a global trivialization of this algebra bundle is constructed. As a corollary we get a symmetry-invariant deformation quantization. Finally, these results are applied to the moduli space situation in which Hitchin originally constructed his connection. First we get a proof that the Hitchin connection in this case is the same as the connection constructed by Axelrod, Della Pietra, and Witten. Second we obtain in this way a mapping class group invariant formal quantization of the smooth symplectic leaves of the moduli space of flat SU(n)-connections on any compact surface.

AB - We introduce the notion of a rigid family of Kähler structures on a symplectic manifold. We then prove that a Hitchin connection exists for any rigid holomorphic family of Kähler structures on any compact pre-quantizable symplectic manifold which satisfies certain simple topological constraints. Using Toeplitz operators we prove that the Hitchin connection induces a unique formal connection on smooth functions on the symplectic manifold. Parallel transport of this formal connection produces equivalences between the corresponding Berezin–Toeplitz deformation quantizations. In the cases where the Hitchin connection is projectively flat, the formal connections will be flat and we get a symmetry-invariant formal quantization. If a certain cohomological condition is satisfied a global trivialization of this algebra bundle is constructed. As a corollary we get a symmetry-invariant deformation quantization. Finally, these results are applied to the moduli space situation in which Hitchin originally constructed his connection. First we get a proof that the Hitchin connection in this case is the same as the connection constructed by Axelrod, Della Pietra, and Witten. Second we obtain in this way a mapping class group invariant formal quantization of the smooth symplectic leaves of the moduli space of flat SU(n)-connections on any compact surface.

U2 - 10.4171/QT/30

DO - 10.4171/QT/30

M3 - Journal article

VL - 3

SP - 293

EP - 325

JO - Quantum Topology

JF - Quantum Topology

SN - 1663-487X

IS - 3/4

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