scholarly journals Quantization commutes with singular reduction: Cotangent bundles of compact Lie groups

2019 ◽  
Vol 31 (06) ◽  
pp. 1950016
Author(s):  
Jord Boeijink ◽  
Klaas Landsman ◽  
Walter van Suijlekom
Keyword(s):  
Hilbert Space ◽  
Dirac Operator ◽  
Cotangent Bundle ◽  
Geometric Quantization ◽  
Compact Lie Groups ◽  
Kähler Structure ◽  
Momentum Map ◽  
Cotangent Bundles ◽  
Reduction Problem ◽  
Mathematical Literature

We analyze the ‘quantization commutes with reduction’ problem (first studied in physics by Dirac, and known in the mathematical literature also as the Guillemin–Sternberg Conjecture) for the conjugate action of a compact connected Lie group [Formula: see text] on its own cotangent bundle [Formula: see text]. This example is interesting because the momentum map is not proper and the ensuing symplectic (or Marsden–Weinstein quotient) [Formula: see text] is typically singular.In the spirit of (modern) geometric quantization, our quantization of [Formula: see text] (with its standard Kähler structure) is defined as the kernel of the Dolbeault–Dirac operator (or, equivalently, the spin[Formula: see text]–Dirac operator) twisted by the pre-quantum line bundle. We show that this quantization of [Formula: see text] reproduces the Hilbert space found earlier by Hall (2002) using geometric quantization based on a holomorphic polarization. We then define the quantization of the singular quotient [Formula: see text] as the kernel of the twisted Dolbeault–Dirac operator on the principal stratum, and show that quantization commutes with reduction in the sense that either way one obtains the same Hilbert space [Formula: see text].

scholarly journals A Study on the Para-Holomorphic Sectional Curvature of Para-Kähler Cotangent Bundles

2015 ◽  
Vol 61 (1) ◽  
pp. 253-262
Author(s):  
S. L. Druţă-Romaniuc
Keyword(s):  
Sectional Curvature ◽  
Cotangent Bundle ◽  
Total Space ◽  
Holomorphic Sectional Curvature ◽  
Kähler Structure ◽  
Nonzero Constant ◽  
Cotangent Bundles

Abstract We obtain the conditions under which the total space T *M of the cotangent bundle, endowed with a natural diagonal para-Kähler structure (G, P), has constant para-holomorphic sectional curvature. Moreover we prove that (T *M,G, P) cannot have nonzero constant para-holomorphic sectional curvature.

scholarly journals The dimension of the Hilbert space of geometric quantization of vortices on a Riemann surface

2017 ◽  
Vol 14 (10) ◽  
pp. 1750144
Author(s):  
Rukmini Dey ◽  
Saibal Ganguli
Keyword(s):  
Hilbert Space ◽  
Riemann Surface ◽  
Line Bundle ◽  
Euler Characteristic ◽  
Moduli Space ◽  
Geometric Quantization

In this paper, we calculate the dimension of the Hilbert space of Kähler quantization of the moduli space of vortices on a Riemann surface. This dimension is given by the holomorphic Euler characteristic of the quantum line bundle.

scholarly journals Hyperkähler metrics associated to compact Lie groups

1996 ◽  
Vol 120 (1) ◽  
pp. 61-69 ◽  
Author(s):  
Andrew Dancer ◽  
Andrew Swann
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Cotangent Bundle ◽  
Symplectic Structure ◽  
Compact Lie Group ◽  
Compact Lie Groups ◽  
Left And Right ◽  
Hyperkähler Metrics

It is well known that the cotangent bundle of any manifold has a canonical symplectic structure. If we specialize to the case when the manifold is a compact Lie group G, then this structure is preserved by the actions of G on T*G induced by left and right translation on G. We refer to these as the left and right actions of G on T*G.

scholarly journals The Connes–Lott Program on the Sphere

1997 ◽  
Vol 09 (06) ◽  
pp. 689-717 ◽  
Author(s):  
J. A. Mignaco ◽  
C. Sigaud ◽  
F. J. Vanhecke ◽  
A. R. Da Silva
Keyword(s):  
Hilbert Space ◽  
Type Model ◽  
Projective Modules ◽  
Induced Representation ◽  
Kähler Structure ◽  
Schwinger Model ◽  
Field Action ◽  
De Rham ◽  
Complex Valued ◽  
Dixmier Trace

We describe a classical Schwinger-type model as a study of the projective modules over the algebra of complex-valued functions on the sphere. On these modules, classified by π2(S2), we construct hermitian connections with values in the universal differential envelope. Instead of describing matter by the usual Dirac spinors yielding the standard Schwinger model on the sphere, we apply the Connes–Lott program to the Hilbert space of complexified inhomogeneous forms with its Atiyah–Kähler structure. This Hilbert space splits in two minimal left ideals of the Clifford algebra preserved by the Dirac–Kähler operator D=i(d-δ). The induced representation of the universal differential envelope, in order to recover its differential structure, is divided by the unwanted differential ideal and the obtained quotient is the usual complexified de Rham exterior algebra with Clifford action on the "spinors" of the Hilbert space. The subsequent steps of the Connes–Lott program allow to define a matter action, and the field action is obtained using the Dixmier trace which reduces to the integral of the curvature squared.

scholarly journals STRINGS IN SPACE-TIME COTANGENT BUNDLE AND T-DUALITY

1995 ◽  
Vol 10 (04) ◽  
pp. 323-329 ◽  
Author(s):  
C. KLIMČÍK ◽  
P. ŠEVERA
Keyword(s):  
Cotangent Bundle ◽  
Space Time ◽  
Geometric Description ◽  
Cotangent Bundles

A simple geometric description of T-duality is given by identifying the cotangent bundles of the original and the dual manifold. Strings propagate naturally in the cotangent bundle and the original and the dual string phase spaces are obtained by different projections. Buscher's transformation follows readily and it is literally projective. As an application of the formalism, we prove that the duality is a symplectomorphism of the string phase spaces.

scholarly journals Invariance of symplectic cohomology and twisted cotangent bundles over surfaces

2020 ◽  
Vol 31 (09) ◽  
pp. 2050070
Author(s):  
Gabriele Benedetti ◽  
Alexander F. Ritter
Keyword(s):  
Symplectic Form ◽  
Cotangent Bundle ◽  
Local System ◽  
Symplectic Manifolds ◽  
Existence Results ◽  
Open Convex ◽  
Floer Theory ◽  
Cotangent Bundles ◽  
Sample Application ◽  
Symplectic Cohomology

We prove that symplectic cohomology for open convex symplectic manifolds is invariant when the symplectic form undergoes deformations which may be nonexact and noncompactly supported, provided one uses the correct local system of coefficients in Floer theory. As a sample application beyond the Liouville setup, we describe in detail the symplectic cohomology for disc bundles in the twisted cotangent bundle of surfaces, and we deduce existence results for periodic magnetic geodesics on surfaces. In particular, we show the existence of geometrically distinct orbits by exploiting properties of the BV-operator on symplectic cohomology.

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