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Journal articles on the topic 'Equivariant quantization'

Author: Grafiati

Published: 16 November 2024

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1

Bieliavsky, Pierre, Victor Gayral, Sergey Neshveyev, and Lars Tuset. "On deformations of C∗-algebras by actions of Kählerian Lie groups." International Journal of Mathematics 27, no. 03 (March 2016): 1650023. http://dx.doi.org/10.1142/s0129167x16500233.

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We show that two approaches to equivariant strict deformation quantization of C[Formula: see text]-algebras by actions of negatively curved Kählerian Lie groups, one based on oscillatory integrals and the other on quantizations maps defined by dual 2-cocycles, are equivalent.

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2

Lecomte, Pierre B. A. "Towards Projectively Equivariant Quantization." Progress of Theoretical Physics Supplement 144 (December 1, 2001): 125–32. http://dx.doi.org/10.1143/ptps.144.125.

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3

Poncin, N., F. Radoux, and R. Wolak. "Equivariant quantization of orbifolds." Journal of Geometry and Physics 60, no. 9 (September 2010): 1103–11. http://dx.doi.org/10.1016/j.geomphys.2010.04.003.

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4

PFLAUM, M. J., H. B. POSTHUMA, X. TANG, and H. H. TSENG. "ORBIFOLD CUP PRODUCTS AND RING STRUCTURES ON HOCHSCHILD COHOMOLOGIES." Communications in Contemporary Mathematics 13, no. 01 (February 2011): 123–82. http://dx.doi.org/10.1142/s0219199711004142.

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In this paper, we study the Hochschild cohomology ring of convolution algebras associated to orbifolds, as well as their deformation quantizations. In the first case, the ring structure is given in terms of a wedge product on twisted polyvectorfields on the inertia orbifold. After deformation quantization, the ring structure defines a product on the cohomology of the inertia orbifold. We study the relation between this product and an S1-equivariant version of the Chen–Ruan product. In particular, we give a de Rham model for this equivariant orbifold cohomology.

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5

Hawkins, Eli. "Quantization of Equivariant Vector Bundles." Communications in Mathematical Physics 202, no. 3 (May 1, 1999): 517–46. http://dx.doi.org/10.1007/s002200050594.

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6

Tang, Xiang, and Yi-Jun Yao. "K -theory of equivariant quantization." Journal of Functional Analysis 266, no. 2 (January 2014): 478–86. http://dx.doi.org/10.1016/j.jfa.2013.10.005.

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7

Rogers, Alice. "Equivariant BRST quantization and reducible symmetries." Journal of Physics A: Mathematical and Theoretical 40, no. 17 (April 11, 2007): 4649–63. http://dx.doi.org/10.1088/1751-8113/40/17/016.

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8

Michel, Jean-Philippe. "Conformally Equivariant Quantization for Spinning Particles." Communications in Mathematical Physics 333, no. 1 (December 16, 2014): 261–98. http://dx.doi.org/10.1007/s00220-014-2229-0.

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9

Duval, Christian, Pierre Lecomte, and Valentin Ovsienko. "Conformally equivariant quantization: existence and uniqueness." Annales de l’institut Fourier 49, no. 6 (1999): 1999–2029. http://dx.doi.org/10.5802/aif.1744.

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10

Donin, J., and A. Mudrov. "Reflection equation, twist, and equivariant quantization." Israel Journal of Mathematics 136, no. 1 (December 2003): 11–28. http://dx.doi.org/10.1007/bf02807191.

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11

Robinson, P. L. "Equivariant prequantization and admissible coadjoint orbits." Mathematical Proceedings of the Cambridge Philosophical Society 114, no. 1 (July 1993): 131–42. http://dx.doi.org/10.1017/s0305004100071462.

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The orbit method has as its primary goal the construction and parametrization of the irreducible unitary representations of a (simply-connected) Lie group in terms of its coadjoint orbits. This goal was achieved with complete success for nilpotent groups by Kirillov[8] and for type I solvable groups by Auslander and Kostant[l] but is known to encounter difficulties when faced with more general groups. Geometric quantization can be viewed as an outgrowth of the orbit method aimed at providing a geometric passage from classical mechanics to quantum mechanics. Whereas the original geometric quantization scheme due to Kostant[9] and Souriau[14] enabled such a passage in a variety of situations, it too encounters difficulties in broader contexts.

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12

Grigorescu, M. "Energy and time as conjugate dynamical variables." Canadian Journal of Physics 78, no. 11 (November 1, 2000): 959–67. http://dx.doi.org/10.1139/p00-082.

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The energy and time variables of the elementary classical dynamical systems are described geometrically, as canonically conjugate coordinates of an extended phase-space. It is shown that the Galilei action of the inertial equivalence group on this space is canonical, but not Hamiltonian equivariant. Although it has no effect at a classical level, the lack of equivariance makes the Galilei action inconsistent with the canonical quantization. A Hamiltonian equivariant action can be obtained by assuming that the inertial parameter in the extended phase-space is quasi-isotropic. This condition leads naturally to the Lorentz transformations between moving frames as a particular case of symplectic transformations. The limit speed appears as a constant factor relating the two additional canonical coordinates to the energy and time. Its value is identified with the speed of light by using the relationship between the electromagnetic potentials and the symplectic form of the extended phase-space. PACS Nos.: 45.20Jj, 11.30Cp, 03.50De

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13

Donin, J., and A. Mudrov. "Method of Quantum Characters in Equivariant Quantization." Communications in Mathematical Physics 234, no. 3 (March 1, 2003): 533–55. http://dx.doi.org/10.1007/s00220-002-0771-7.

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14

Karolinsky, E., A. Stolin, and V. Tarasov. "Irreducible highest weight modules and equivariant quantization." Advances in Mathematics 211, no. 1 (May 2007): 266–83. http://dx.doi.org/10.1016/j.aim.2006.08.004.

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15

Galasso, Andrea, and Mauro Spera. "Remarks on the geometric quantization of Landau levels." International Journal of Geometric Methods in Modern Physics 13, no. 10 (October 26, 2016): 1650122. http://dx.doi.org/10.1142/s021988781650122x.

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In this note, we resume the geometric quantization approach to the motion of a charged particle on a plane, subject to a constant magnetic field perpendicular to the latter, by showing directly that it gives rise to a completely integrable system to which we may apply holomorphic geometric quantization. In addition, we present a variant employing a suitable vertical polarization and we also make contact with Bott’s quantization, enforcing the property “quantization commutes with reduction”, which is known to hold under quite general conditions. We also provide an interpretation of translational symmetry breaking in terms of coherent states and index theory. Finally, we give a representation theoretic description of the lowest Landau level via the use of an [Formula: see text]-equivariant Dirac operator.

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16

Bichr, Taher, Jamel Boujelben, and Khaled Tounsi. "Bilinear differential operators: Projectively equivariant symbol and quantization maps." Tohoku Mathematical Journal 67, no. 4 (December 2015): 481–93. http://dx.doi.org/10.2748/tmj/1450798067.

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17

Radoux, F. "Non-uniqueness of the natural and projectively equivariant quantization." Journal of Geometry and Physics 58, no. 2 (February 2008): 253–58. http://dx.doi.org/10.1016/j.geomphys.2007.11.002.

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18

Hansoul, Sarah. "Projectively Equivariant Quantization for Differential Operators Acting on Forms." Letters in Mathematical Physics 70, no. 2 (November 2004): 141–53. http://dx.doi.org/10.1007/s11005-004-4293-4.

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19

Radoux, F. "Explicit Formula for the Natural and Projectively Equivariant Quantization." Letters in Mathematical Physics 78, no. 2 (October 13, 2006): 173–88. http://dx.doi.org/10.1007/s11005-006-0116-0.

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20

Karolinsky, E., A. Stolin, and V. Tarasov. "Equivariant quantization of Poisson homogeneous spaces and Kostant's problem." Journal of Algebra 409 (July 2014): 362–81. http://dx.doi.org/10.1016/j.jalgebra.2014.03.033.

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21

Giselsson, Olof. "q-Independence of the Jimbo–Drinfeld Quantization." Communications in Mathematical Physics 376, no. 3 (January 7, 2020): 1737–65. http://dx.doi.org/10.1007/s00220-019-03660-9.

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AbstractLet $${\mathrm {G}}$$G be a connected semi-simple compact Lie group and for $$0<q<1$$0<q<1, let $$({\mathbb {C}}[\mathrm {G]_q},\varDelta _q)$$(C[G]q,Δq) be the Jimbo–Drinfeld q-deformation of $${\mathrm {G}}$$G. We show that the $$C^*$$C∗-completions of $$\mathrm {C}[\mathrm {G]_q}$$C[G]q are isomorphic for all values of q. Moreover, these isomorphisms are equivariant with respect to the right-actions of the maximal torus.

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22

PAOLETTI, ROBERTO. "LOCAL TRACE FORMULAE AND SCALING ASYMPTOTICS IN TOEPLITZ QUANTIZATION." International Journal of Geometric Methods in Modern Physics 07, no. 03 (May 2010): 379–403. http://dx.doi.org/10.1142/s021988781000435x.

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A trace formula for Toeplitz operators was proved by Boutet de Monvel and Guillemin in the setting of general Toeplitz structures. Here, we give a local version of this result for a class of Toeplitz operators related to continuous groups of symmetries on quantizable compact symplectic manifolds. The local trace formula involves certain scaling asymptotics along the clean fixed locus of the Hamiltonian flow of the symbol, reminiscent of the scaling asymptotics of the equivariant components of the Szegö kernel along the diagonal.

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23

PAOLETTI, ROBERTO. "SCALING ASYMPTOTICS FOR QUANTIZED HAMILTONIAN FLOWS." International Journal of Mathematics 23, no. 10 (October 2012): 1250102. http://dx.doi.org/10.1142/s0129167x12501029.

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In recent years, the near diagonal asymptotics of the equivariant components of the Szegö kernel of a positive line bundle on a compact symplectic manifold have been studied extensively by many authors. As a natural generalization of this theme, here we consider the local scaling asymptotics of the Toeplitz quantization of a Hamiltonian symplectomorphism, and specifically how they concentrate on the graph of the underlying classical map.

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24

Gargoubi, Hichem, Najla Mellouli, and Valentin Ovsienko. "Differential Operators on Supercircle: Conformally Equivariant Quantization and Symbol Calculus." Letters in Mathematical Physics 79, no. 1 (November 30, 2006): 51–65. http://dx.doi.org/10.1007/s11005-006-0129-8.

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25

Cirio, Lucio S., Giovanni Landi, and Richard J. Szabo. "Instantons and vortices on noncommutative toric varieties." Reviews in Mathematical Physics 26, no. 09 (October 2014): 1430008. http://dx.doi.org/10.1142/s0129055x14300088.

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We elaborate on the quantization of toric varieties by combining techniques from toric geometry, isospectral deformations and noncommutative geometry in braided monoidal categories, and the construction of instantons thereon by combining methods from noncommutative algebraic geometry and a quantized twistor theory. We classify the real structures on a toric noncommutative deformation of the Klein quadric and use this to derive a new noncommutative four-sphere which is the unique deformation compatible with the noncommutative twistor correspondence. We extend the computation of equivariant instanton partition functions to noncommutative gauge theories with both adjoint and fundamental matter fields, finding agreement with the classical results in all instances. We construct moduli spaces of noncommutative vortices from the moduli of invariant instantons, and derive corresponding equivariant partition functions which also agree with those of the classical limit.

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26

Ostapenko, Vadim. "On Uħ (ℊ, r)-equivariant quantization of non-orbit homogeneous varieties." Reports on Mathematical Physics 61, no. 2 (April 2008): 303–10. http://dx.doi.org/10.1016/s0034-4877(08)80018-3.

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27

Duval, C., and G. Valent. "A new integrable system on the sphere and conformally equivariant quantization." Journal of Geometry and Physics 61, no. 8 (August 2011): 1329–47. http://dx.doi.org/10.1016/j.geomphys.2011.02.020.

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28

Brylinski, Ranee. "Equivariant deformation quantization for the cotangent bundle of a flag manifold." Annales de l’institut Fourier 52, no. 3 (2002): 881–97. http://dx.doi.org/10.5802/aif.1905.

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29

ASCHIERI, PAOLO. "TWISTING ALL THE WAY: FROM ALGEBRAS TO MORPHISMS AND CONNECTIONS." International Journal of Modern Physics: Conference Series 13 (January 2012): 1–19. http://dx.doi.org/10.1142/s201019451200668x.

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Given a Hopf algebra H and an algebra A that is an H-module algebra we consider the category of left H-modules and A-bimodules [Formula: see text], where morphisms are just right A-linear maps (not necessarily H-equivariant). Given a twist [Formula: see text] of H we then quantize (deform) H to [Formula: see text], A to A⋆ and correspondingly the category [Formula: see text] to [Formula: see text]. If we consider a quasitriangular Hopf algebra H, a quasi-commutative algebra A and quasi-commutative A-bimodules, we can further construct and study tensor products over A of modules and of morphisms, and their twist quantization. This study leads to the definition of arbitrary (i.e., not necessarily H-equivariant) connections on quasi-commutative A-bimodules, to extend these connections to tensor product modules and to quantize them to A⋆-bimodule connections. Their curvatures and those on tensor product modules are also determined.

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30

SEMENOFF, GORDON W., and RICHARD J. SZABO. "EQUIVARIANT LOCALIZATION, SPIN SYSTEMS AND TOPOLOGICAL QUANTUM THEORY ON RIEMANN SURFACES." Modern Physics Letters A 09, no. 29 (September 21, 1994): 2705–18. http://dx.doi.org/10.1142/s0217732394002550.

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We study equivariant localization formulas for phase space path-integrals when the phase space is a multiply connected compact Riemann surface. We consider the Hamiltonian systems to which the localization formulas are applicable and show that the localized partition function for such systems is a topological invariant which represents the non-trivial homology classes of the phase space. We explicitly construct the coherent states in the canonical quantum theory and show that the Hilbert space is finite-dimensional with the wave functions carrying a projective representation of the discrete homology group of the phase space. The corresponding coherent state path-integral then describes the quantum dynamics of a novel spin system given by the quantization of a nonsymmetric coadjoint Lie group orbit. We also briefly discuss the geometric structure of these quantum systems.

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31

Rădulescu, Florin. "The Γ-equivariant form of the Berezin quantization of the upper half plane." Memoirs of the American Mathematical Society 133, no. 630 (1998): 0. http://dx.doi.org/10.1090/memo/0630.

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32

Spera, Mauro. "Moment map and gauge geometric aspects of the Schrödinger and Pauli equations." International Journal of Geometric Methods in Modern Physics 13, no. 04 (March 31, 2016): 1630004. http://dx.doi.org/10.1142/s021988781630004x.

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In this paper we discuss various geometric aspects related to the Schrödinger and the Pauli equations. First we resume the Madelung–Bohm hydrodynamical approach to quantum mechanics and recall the Hamiltonian structure of the Schrödinger equation. The probability current provides an equivariant moment map for the group [Formula: see text] of volume-preserving diffeomorphisms of [Formula: see text] (rapidly approaching the identity at infinity) and leads to a current algebra of Rasetti–Regge type. The moment map picture is then extended, mutatis mutandis, to the Pauli equation and to generalized Schrödinger equations of the Pauli–Thomas type. A gauge theoretical reinterpretation of all equations is obtained via the introduction of suitable Maurer–Cartan gauge fields and it is then related to Weyl geometric and pilot wave ideas. A general framework accommodating Aharonov–Bohm and Aharonov–Casher effects is presented within the gauge approach. Furthermore, a kind of holomorphic geometric quantization can be performed and yields natural “coherent state” representations of [Formula: see text]. The relationship with the covariant phase space and density manifold approaches is then outlined. Comments on possible extensions to nonlinear Schrödinger equations, on Fisher-information theoretic aspects and on stochastic mechanics are finally made.

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33

Varshovi, Amir Abbass. "⋆-cohomology, third type Chern character and anomalies in general translation-invariant noncommutative Yang–Mills." International Journal of Geometric Methods in Modern Physics 18, no. 06 (February 24, 2021): 2150089. http://dx.doi.org/10.1142/s0219887821500894.

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A representation of general translation-invariant star products ⋆ in the algebra of [Formula: see text] is introduced which results in the Moyal–Weyl–Wigner quantization. It provides a matrix model for general translation-invariant noncommutative quantum field theories in terms of the noncommutative calculus on differential graded algebras. Upon this machinery a cohomology theory, the so-called ⋆-cohomology, with groups [Formula: see text], [Formula: see text], is worked out which provides a cohomological framework to formulate general translation-invariant noncommutative quantum field theories based on the achievements for the commutative fields, and is comparable to the Seiberg–Witten map for the Moyal case. Employing the Chern–Weil theory via the integral classes of [Formula: see text] a noncommutative version of the Chern character is defined as an equivariant form which contains topological information about the corresponding translation-invariant noncommutative Yang–Mills theory. Thereby, we study the mentioned Yang–Mills theories with three types of actions of the gauge fields on the spinors, the ordinary, the inverse, and the adjoint action, and then some exact solutions for their anomalous behaviors are worked out via employing the homotopic correlation on the integral classes of ⋆-cohomology. Finally, the corresponding consistent anomalies are also derived from this topological Chern character in the ⋆-cohomology.

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34

Boniver, F., and P. Mathonet. "IFFT-equivariant quantizations." Journal of Geometry and Physics 56, no. 4 (April 2006): 712–30. http://dx.doi.org/10.1016/j.geomphys.2005.04.014.

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35

Bouarroudj, S., and M. Iadh AYARI. "On (2)-equivariant quantizations." Journal of Nonlinear Mathematical Physics 14, no. 2 (January 2007): 179–87. http://dx.doi.org/10.2991/jnmp.2007.14.2.4.

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36

Mathonet, P. "Equivariant quantizations and Cartan connections." Bulletin of the Belgian Mathematical Society - Simon Stevin 13, no. 5 (January 2007): 857–74. http://dx.doi.org/10.36045/bbms/1170347809.

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37

Čap, Andreas, and Josef Šilhan. "Equivariant quantizations for AHS-structures." Advances in Mathematics 224, no. 4 (July 2010): 1717–34. http://dx.doi.org/10.1016/j.aim.2010.01.016.

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38

Zwicknagl, Sebastian. "Equivariant quantizations of symmetric algebras." Journal of Algebra 322, no. 12 (December 2009): 4247–82. http://dx.doi.org/10.1016/j.jalgebra.2009.08.007.

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39

Mathonet, P., and F. Radoux. "On natural and conformally equivariant quantizations." Journal of the London Mathematical Society 80, no. 1 (June 12, 2009): 256–72. http://dx.doi.org/10.1112/jlms/jdp024.

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40

Hansoul, Sarah. "Existence of natural and projectively equivariant quantizations." Advances in Mathematics 214, no. 2 (October 2007): 832–64. http://dx.doi.org/10.1016/j.aim.2007.03.007.

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41

Bieliavsky, Pierre, Victor Gayral, Sergey Neshveyev, and Lars Tuset. "Addendum: On deformations of C∗-algebras by actions of Kählerian Lie groups." International Journal of Mathematics 30, no. 11 (October 2019): 1992002. http://dx.doi.org/10.1142/s0129167x19920022.

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We show that two approaches to equivariant deformation of C[Formula: see text]-algebras by actions of negatively curved Kählerian Lie groups, one based on oscillatory integrals and the other on quantizations maps defined by dual [Formula: see text]-cocycles, are still equivalent despite the nonunitarity of our [Formula: see text]-cocycles.

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42

Mathonet, P., and F. Radoux. "Cartan connections and natural and projectively equivariant quantizations." Journal of the London Mathematical Society 76, no. 1 (August 2007): 87–104. http://dx.doi.org/10.1112/jlms/jdm030.

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43

Boniver, F., and P. Mathonet. "Maximal subalgebras of vector fields for equivariant quantizations." Journal of Mathematical Physics 42, no. 2 (2001): 582. http://dx.doi.org/10.1063/1.1332782.

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44

Leuther, Thomas, Pierre Mathonet, and Fabian Radoux. "On osp(p+1,q+1|2r)-equivariant quantizations." Journal of Geometry and Physics 62, no. 1 (January 2012): 87–99. http://dx.doi.org/10.1016/j.geomphys.2011.09.003.

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45

Mathonet, P., and F. Radoux. "Natural and Projectively Equivariant Quantizations by means of Cartan Connections." Letters in Mathematical Physics 72, no. 3 (June 2005): 183–96. http://dx.doi.org/10.1007/s11005-005-6783-4.

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46

Mathonet, Pierre, and Fabian Radoux. "Projectively Equivariant Quantizations over the Superspace $${\mathbb{R}^{p|q}}$$." Letters in Mathematical Physics 98, no. 3 (February 25, 2011): 311–31. http://dx.doi.org/10.1007/s11005-011-0474-0.

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47

Lecomte, Pierre B. A. "On Martin Bordemann's proof of the existence of projectively equivariant quantizations." Central European Journal of Mathematics 2, no. 5 (October 2004): 793–800. http://dx.doi.org/10.2478/bf02475977.

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48

Nguyen, Hans, Alexander Schenkel, and Richard J. Szabo. "Batalin–Vilkovisky quantization of fuzzy field theories." Letters in Mathematical Physics 111, no. 6 (December 2021). http://dx.doi.org/10.1007/s11005-021-01490-2.

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AbstractWe apply the modern Batalin–Vilkovisky quantization techniques of Costello and Gwilliam to noncommutative field theories in the finite-dimensional case of fuzzy spaces. We further develop a generalization of this framework to theories that are equivariant under a triangular Hopf algebra symmetry, which in particular leads to quantizations of finite-dimensional analogues of the field theories proposed recently through the notion of ‘braided $$L_\infty $$ L ∞ -algebras’. The techniques are illustrated by computing perturbative correlation functions for scalar and Chern–Simons theories on the fuzzy 2-sphere, as well as for braided scalar field theories on the fuzzy 2-torus.

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49

Michel, Jean-Philippe. "Conformally Equivariant Quantization - a Complete Classification." Symmetry, Integrability and Geometry: Methods and Applications, April 15, 2012. http://dx.doi.org/10.3842/sigma.2012.022.

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50

Genolini, Pietro Benetti, Jerome P. Gauntlett, and James Sparks. "Equivariant localization for AdS/CFT." Journal of High Energy Physics 2024, no. 2 (February 1, 2024). http://dx.doi.org/10.1007/jhep02(2024)015.

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Abstract We explain how equivariant localization may be applied to AdS/CFT to compute various BPS observables in gravity, such as central charges and conformal dimensions of chiral primary operators, without solving the supergravity equations. The key ingredient is that supersymmetric AdS solutions with an R-symmetry are equipped with a set of equivariantly closed forms. These may in turn be used to impose flux quantization and compute observables for supergravity solutions, using only topological information and the Berline-Vergne-Atiyah-Bott fixed point formula. We illustrate the formalism by considering AdS5 × M6 and AdS3 × M8 solutions of D = 11 supergravity. As well as recovering results for many classes of well-known supergravity solutions, without using any knowledge of their explicit form, we also compute central charges for which explicit supergravity solutions have not been constructed.

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