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Journal articles on the topic 'Symplectic groupoids'

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Published: 8 February 2024

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1

MACKENZIE, K. C. H. "ON SYMPLECTIC DOUBLE GROUPOIDS AND THE DUALITY OF POISSON GROUPOIDS." International Journal of Mathematics 10, no. 04 (June 1999): 435–56. http://dx.doi.org/10.1142/s0129167x99000185.

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We prove that the cotangent of a double Lie groupoid S has itself a double groupoid structure with sides the duals of associated Lie algebroids, and double base the dual of the Lie algebroid of the core of S. Using this, we prove a result outlined by Weinstein in 1988, that the side groupoids of a general symplectic double groupoid are Poisson groupoids in duality. Further, we prove that any double Lie groupoid gives rise to a pair of Poisson groupoids (and thus of Lie bialgebroids) in duality. To handle the structures involved effectively we extend to this context the dualities and canonical isomorphisms for tangent and cotangent structures of the author and Ping Xu.

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2

Cattaneo, Alberto S., Benoit Dherin, and Giovanni Felder. "Formal Lagrangian Operad." International Journal of Mathematics and Mathematical Sciences 2010 (2010): 1–36. http://dx.doi.org/10.1155/2010/643605.

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Given a symplectic manifoldM, we may define an operad structure on the the spacesOkof the Lagrangian submanifolds of(M¯)k×Mvia symplectic reduction. IfMis also a symplectic groupoid, then its multiplication space is an associative product in this operad. Following this idea, we provide a deformation theory for symplectic groupoids analog to the deformation theory of algebras. It turns out that the semiclassical part of Kontsevich's deformation ofC∞(ℝd) is a deformation of the trivial symplectic groupoid structure ofT∗ℝd.

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3

XU, PING. "ON POISSON GROUPOIDS." International Journal of Mathematics 06, no. 01 (February 1995): 101–24. http://dx.doi.org/10.1142/s0129167x95000080.

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Some important properties of Poisson groupoids are discussed. In particular, we obtain a useful formula for the Poisson tensor of an arbitrary Poisson groupoid, which generalizes the well-known multiplicativity condition for Poisson groups. Morphisms between Poisson groupoids and between Lie bialgebroids are also discussed. In particular, for a special class of Lie bialgebroid morphisms, we give an explicit lifting construction. As an application, we prove that a Poisson group action on a Poisson manifold lifts to a Poisson action on its α-simply connected symplectic groupoid.

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4

Ševera, Pavol, and Michal Širaň. "Integration of Differential Graded Manifolds." International Mathematics Research Notices 2020, no. 20 (February 15, 2019): 6769–814. http://dx.doi.org/10.1093/imrn/rnz004.

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Abstract We consider the problem of integration of $L_\infty $-algebroids (differential non-negatively graded manifolds) to $L_\infty $-groupoids. We first construct a “big” Kan simplicial manifold (Fréchet or Banach) whose points are solutions of a (generalized) Maurer–Cartan equation. The main analytic trick in our work is an integral transformation sending the solutions of the Maurer–Cartan equation to closed differential forms. Following the ideas of Ezra Getzler, we then impose a gauge condition that cuts out a finite-dimensional simplicial submanifold. This “smaller” simplicial manifold is (the nerve of) a local Lie $\ell $-groupoid. The gauge condition can be imposed only locally in the base of the $L_\infty $-algebroid; the resulting local $\ell $-groupoids glue up to a coherent homotopy, that is, we get a homotopy coherent diagram from the nerve of a good cover of the base to the (simplicial) category of local $\ell $-groupoids. Finally, we show that a $k$-symplectic differential non-negatively graded manifold integrates to a local $k$-symplectic Lie $\ell$-groupoid; globally, these assemble to form an $A_\infty$-functor. As a particular case for $k=2$, we obtain integration of Courant algebroids.

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5

Cattaneo, Alberto S., and Ivan Contreras. "Relational Symplectic Groupoids." Letters in Mathematical Physics 105, no. 5 (April 22, 2015): 723–67. http://dx.doi.org/10.1007/s11005-015-0760-3.

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6

Gualtieri, Marco, and Songhao Li. "Symplectic Groupoids of Log Symplectic Manifolds." International Mathematics Research Notices 2014, no. 11 (March 1, 2013): 3022–74. http://dx.doi.org/10.1093/imrn/rnt024.

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7

Mehta, Rajan Amit, and Xiang Tang. "Constant symplectic 2-groupoids." Letters in Mathematical Physics 108, no. 5 (November 15, 2017): 1203–23. http://dx.doi.org/10.1007/s11005-017-1026-z.

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8

戴, 远莉. "Symplectic Reduction for Cotangent Groupoids." Pure Mathematics 11, no. 03 (2021): 323–29. http://dx.doi.org/10.12677/pm.2021.113043.

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9

Weinstein, Alan. "Symplectic groupoids and Poisson manifolds." Bulletin of the American Mathematical Society 16, no. 1 (January 1, 1987): 101–5. http://dx.doi.org/10.1090/s0273-0979-1987-15473-5.

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10

Li, Songhao, and Dylan Rupel. "Symplectic groupoids for cluster manifolds." Journal of Geometry and Physics 154 (August 2020): 103688. http://dx.doi.org/10.1016/j.geomphys.2020.103688.

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11

Xu, Ping. "Flux homomorphism on symplectic groupoids." Mathematische Zeitschrift 226, no. 4 (December 1997): 575–97. http://dx.doi.org/10.1007/pl00004355.

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12

Şahin, Fulya. "Generalized symplectic Golden manifolds and Lie Groupoids." Filomat 36, no. 5 (2022): 1663–74. http://dx.doi.org/10.2298/fil2205663s.

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By considering the notion of Golden manifold and natural symplectic form on a generalized tangent bundle, we introduce generalized symplectic Golden structures on manifolds and obtain integrability conditions in terms of bivector fields, 2-forms, 1-forms and endomorphisms on manifolds and investigate isotropic subbundles. We also find certain relations between the integrability conditions of generalized symplectic Golden manifolds and Lie Groupoids which are important in mechanics as configuration space.

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13

Mayrand, Maxence. "Hyperkähler metrics near Lagrangian submanifolds and symplectic groupoids." Journal für die reine und angewandte Mathematik (Crelles Journal) 2022, no. 782 (October 26, 2021): 197–218. http://dx.doi.org/10.1515/crelle-2021-0059.

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Abstract The first part of this paper is a generalization of the Feix–Kaledin theorem on the existence of a hyperkähler metric on a neighborhood of the zero section of the cotangent bundle of a Kähler manifold. We show that the problem of constructing a hyperkähler structure on a neighborhood of a complex Lagrangian submanifold in a holomorphic symplectic manifold reduces to the existence of certain deformations of holomorphic symplectic structures. The Feix–Kaledin structure is recovered from the twisted cotangent bundle. We then show that every holomorphic symplectic groupoid over a compact holomorphic Poisson surface of Kähler type has a hyperkähler structure on a neighborhood of its identity section. More generally, we reduce the existence of a hyperkähler structure on a symplectic realization of a holomorphic Poisson manifold of any dimension to the existence of certain deformations of holomorphic Poisson structures adapted from Hitchin’s unobstructedness theorem.

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14

Mikami, Kentaro, and Alan Weinstein. "Moments and reduction for symplectic groupoids." Publications of the Research Institute for Mathematical Sciences 24, no. 1 (1988): 121–40. http://dx.doi.org/10.2977/prims/1195175328.

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15

LU, JIANG-HUA, and VICTOR MOUQUIN. "DOUBLE BRUHAT CELLS AND SYMPLECTIC GROUPOIDS." Transformation Groups 23, no. 3 (August 8, 2017): 765–800. http://dx.doi.org/10.1007/s00031-017-9437-6.

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16

Cañez, Santiago. "Double groupoids and the symplectic category." Journal of Geometric Mechanics 10, no. 2 (2018): 217–50. http://dx.doi.org/10.3934/jgm.2018009.

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17

Carlos Marrero, Juan, David Martín de Diego, and Ari Stern. "Symplectic groupoids and discrete constrained Lagrangian mechanics." Discrete & Continuous Dynamical Systems - A 35, no. 1 (2015): 367–97. http://dx.doi.org/10.3934/dcds.2015.35.367.

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18

Karabegov, Alexander V. "Fedosov’s formal symplectic groupoids and contravariant connections." Journal of Geometry and Physics 56, no. 10 (October 2006): 1985–2009. http://dx.doi.org/10.1016/j.geomphys.2005.11.004.

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19

ŞAHİN, Fulya. "Symplectic groupoids and generalized almost subtangent manifolds." TURKISH JOURNAL OF MATHEMATICS 39 (2015): 156–67. http://dx.doi.org/10.3906/mat-1305-67.

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20

Martinez, Nicolas. "Poly-symplectic Groupoids and Poly-Poisson Structures." Letters in Mathematical Physics 105, no. 5 (February 5, 2015): 693–721. http://dx.doi.org/10.1007/s11005-015-0746-1.

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21

Mikami, Kentaro. "Symplectic Double Groupoids Over Poisson (ax + b)-Groups." Transactions of the American Mathematical Society 324, no. 1 (March 1991): 447. http://dx.doi.org/10.2307/2001517.

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22

Karabegov, Alexander. "Lagrangian fields, Calabi functions, and local symplectic groupoids." Differential Geometry and its Applications 85 (December 2022): 101933. http://dx.doi.org/10.1016/j.difgeo.2022.101933.

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23

Mikami, Kentaro. "Symplectic double groupoids over Poisson $(ax+b)$-groups." Transactions of the American Mathematical Society 324, no. 1 (January 1, 1991): 447–63. http://dx.doi.org/10.1090/s0002-9947-1991-1025757-x.

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24

Bursztyn, Henrique, and Olga Radko. "Gauge equivalence of Dirac structures and symplectic groupoids." Annales de l’institut Fourier 53, no. 1 (2003): 309–37. http://dx.doi.org/10.5802/aif.1945.

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25

Cueca, Miquel, and Chenchang Zhu. "Shifted symplectic higher Lie groupoids and classifying spaces." Advances in Mathematics 413 (January 2023): 108829. http://dx.doi.org/10.1016/j.aim.2022.108829.

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26

Bunk, Severin. "Gerbes in Geometry, Field Theory, and Quantisation." Complex Manifolds 8, no. 1 (January 1, 2021): 150–82. http://dx.doi.org/10.1515/coma-2020-0112.

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Abstract This is a mostly self-contained survey article about bundle gerbes and some of their recent applications in geometry, field theory, and quantisation. We cover the definition of bundle gerbes with connection and their morphisms, and explain the classification of bundle gerbes with connection in terms of differential cohomology. We then survey how the surface holonomy of bundle gerbes combines with their transgression line bundles to yield a smooth bordism-type field theory. Finally, we exhibit the use of bundle gerbes in geometric quantisation of 2-plectic as well as 1- and 2-shifted symplectic forms. This generalises earlier applications of gerbes to the prequantisation of quasi-symplectic groupoids.

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27

BOS, ROGIER. "GEOMETRIC QUANTIZATION OF HAMILTONIAN ACTIONS OF LIE ALGEBROIDS AND LIE GROUPOIDS." International Journal of Geometric Methods in Modern Physics 04, no. 03 (May 2007): 389–436. http://dx.doi.org/10.1142/s0219887807002077.

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We construct Hermitian representations of Lie algebroids and associated unitary representations of Lie groupoids by a geometric quantization procedure. For this purpose, we introduce a new notion of Hamiltonian Lie algebroid actions. The first step of our procedure consists of the construction of a prequantization line bundle. Next, we discuss a version of Kähler quantization suitable for this setting. We proceed by defining a Marsden–Weinstein quotient for our setting and prove a "quantization commutes with reduction" theorem. We explain how our geometric quantization procedure relates to a possible orbit method for Lie groupoids. Our theory encompasses the geometric quantization of symplectic manifolds, Hamiltonian Lie algebra actions, actions of bundles of Lie groups, and foliations, as well as some general constructions from differential geometry.

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28

SÄMANN, CHRISTIAN, and RICHARD J. SZABO. "GROUPOIDS, LOOP SPACES AND QUANTIZATION OF 2-PLECTIC MANIFOLDS." Reviews in Mathematical Physics 25, no. 03 (April 2013): 1330005. http://dx.doi.org/10.1142/s0129055x13300057.

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We describe the quantization of 2-plectic manifolds as they arise in the context of the quantum geometry of M-branes and non-geometric flux compactifications of closed string theory. We review the groupoid approach to quantizing Poisson manifolds in detail, and then extend it to the loop spaces of 2-plectic manifolds, which are naturally symplectic manifolds. In particular, we discuss the groupoid quantization of the loop spaces of ℝ3, 𝕋3and S3, and derive some interesting implications which match physical expectations from string theory and M-theory.

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29

Kotov, Alexei, and Thomas Strobl. "Lie algebroids, gauge theories, and compatible geometrical structures." Reviews in Mathematical Physics 31, no. 04 (April 17, 2019): 1950015. http://dx.doi.org/10.1142/s0129055x19500156.

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The construction of gauge theories beyond the realm of Lie groups and algebras leads one to consider Lie groupoids and algebroids equipped with additional geometrical structures which, for gauge invariance of the construction, need to satisfy particular compatibility conditions. This paper is supposed to analyze these compatibilities from a mathematical perspective.In particular, we show that the compatibility of a connection with a Lie algebroid that one finds is the Cartan condition, introduced previously by A. Blaom. For the metric on the base [Formula: see text] of a Lie algebroid equipped with any connection, we show that the compatibility suggested from gauge theories implies that the foliation induced by the Lie algebroid becomes a Riemannian foliation. Building upon a result of del Hoyo and Fernandes, we prove, furthermore, that every Lie algebroid integrating to a proper Lie groupoid admits a compatible Riemannian base. We also consider the case where the base is equipped with a compatible symplectic or generalized Riemannian structure.

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30

Contreras, Ivan, and Nicolas Martinez Alba. "Poly-Poisson sigma models and their relational poly-symplectic groupoids." Journal of Mathematical Physics 59, no. 7 (July 2018): 072901. http://dx.doi.org/10.1063/1.5016851.

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31

Suzuki, Haruo. "Central S1-extensions of symplectic groupoids and the Poisson classes." Pacific Journal of Mathematics 203, no. 2 (April 1, 2002): 489–501. http://dx.doi.org/10.2140/pjm.2002.203.489.

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32

Frejlich, Pedro, and Ioan Mărcuț. "Normal forms for Poisson maps and symplectic groupoids around Poisson transversals." Letters in Mathematical Physics 108, no. 3 (October 4, 2017): 711–35. http://dx.doi.org/10.1007/s11005-017-1007-2.

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33

FIORENZA, DOMENICO, CHRISTOPHER L. ROGERS, and URS SCHREIBER. "A HIGHER CHERN–WEIL DERIVATION OF AKSZ σ-MODELS." International Journal of Geometric Methods in Modern Physics 10, no. 01 (November 15, 2012): 1250078. http://dx.doi.org/10.1142/s0219887812500788.

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Chern–Weil theory provides for each invariant polynomial on a Lie algebra 𝔤 a map from 𝔤-connections to differential cocycles whose volume holonomy is the corresponding Chern–Simons theory action functional. Kotov and Strobl have observed that this naturally generalizes from Lie algebras to dg-manifolds and to dg-bundles and that the Chern–Simons action functional associated this way to an n-symplectic manifold is the action functional of the AKSZ σ-model whose target space is the given n-symplectic manifold (examples of this are the Poisson σ-model or the Courant σ-model, including ordinary Chern–Simons theory, or higher-dimensional Abelian Chern–Simons theory). Here we show how, within the framework of the higher Chern–Weil theory in smooth ∞-groupoids, this result can be naturally recovered and enhanced to a morphism of higher stacks, the same way as ordinary Chern–Simons theory is enhanced to a morphism from the stack of principal G-bundles with connections to the 3-stack of line 3-bundles with connections.

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34

Cattaneo, Alberto S., Benoit Dherin, and Giovanni Felder. "Formal Symplectic Groupoid." Communications in Mathematical Physics 253, no. 3 (October 20, 2004): 645–74. http://dx.doi.org/10.1007/s00220-004-1199-z.

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35

Ding, Hao. "On quasi-symplectic groupoid reduction." Journal of Geometry and Physics 124 (January 2018): 311–24. http://dx.doi.org/10.1016/j.geomphys.2017.11.011.

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36

Karabegov, Alexander V. "Formal Symplectic Groupoid of a Deformation Quantization." Communications in Mathematical Physics 258, no. 1 (April 12, 2005): 223–56. http://dx.doi.org/10.1007/s00220-005-1336-3.

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37

Karabegov, Alexander. "Infinitesimal Deformations of a Formal Symplectic Groupoid." Letters in Mathematical Physics 97, no. 3 (May 10, 2011): 279–301. http://dx.doi.org/10.1007/s11005-011-0495-8.

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38

Cattaneo, Alberto S., Nima Moshayedi, and Konstantin Wernli. "Relational symplectic groupoid quantization for constant poisson structures." Letters in Mathematical Physics 107, no. 9 (April 28, 2017): 1649–88. http://dx.doi.org/10.1007/s11005-017-0959-6.

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39

Losev, Ivan. "On categories for quantized symplectic resolutions." Compositio Mathematica 153, no. 12 (September 7, 2017): 2445–81. http://dx.doi.org/10.1112/s0010437x17007382.

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In this paper we study categories ${\mathcal{O}}$ over quantizations of symplectic resolutions admitting Hamiltonian tori actions with finitely many fixed points. In this generality, these categories were introduced by Braden et al. We establish a family of standardly stratified structures (in the sense of the author and Webster) on these categories ${\mathcal{O}}$. We use these structures to study shuffling functors of Braden et al. (called cross-walling functors in this paper). Most importantly, we prove that all cross-walling functors are derived equivalences that define an action of the Deligne groupoid of a suitable real hyperplane arrangement.

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40

Rybicki, Tomasz. "On the existence of a Hofer type metric for Poisson manifolds." International Journal of Mathematics 27, no. 09 (August 2016): 1650075. http://dx.doi.org/10.1142/s0129167x16500750.

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An analogue of the Hofer metric [Formula: see text] on the Hamiltonian group [Formula: see text] of a Poisson manifold [Formula: see text] can be defined, but there is the problem of its nondegeneracy. First, we observe that [Formula: see text] is a genuine metric on [Formula: see text], when the union of all proper leaves of the corresponding symplectic foliation is dense. Next, we deal with the important class of integrable Poisson manifolds. Recall that a Poisson manifold is called integrable, if it can be realized as the space of units of a symplectic groupoid. Our main result states that [Formula: see text] is a Hofer type metric for every Poisson manifold, which admits a Hausdorff integration.

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41

Martínez Torres, David. "The Diffeomorphism Type of Canonical Integrations of Poisson Tensors on Surfaces." Canadian Mathematical Bulletin 58, no. 3 (September 1, 2015): 575–79. http://dx.doi.org/10.4153/cmb-2015-011-7.

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AbstractA surface ∑ endowed with a Poisson tensor π is known to admit a canonical integration, 𝒢(π), which is a 4-dimensional manifold with a (symplectic) Lie groupoid structure. In this short note we show that if π is not an area form on the 2-sphere, then 𝒢(π) is diffeomorphic to the cotangent bundle T*∑. This extends results by the author and by Bonechi, Ciccoli, Staffolani, and Tarlini.

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42

CATTANEO, ALBERTO S., and GIOVANNI FELDER. "POISSON SIGMA MODELS AND DEFORMATION QUANTIZATION." Modern Physics Letters A 16, no. 04n06 (February 28, 2001): 179–89. http://dx.doi.org/10.1142/s0217732301003255.

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This is a review aimed at the physics audience on the relation between Poisson sigma models on surfaces with boundary and deformation quantization. These models are topological open string theories. In the classical Hamiltonian approach, we describe the reduced phase space and its structures (symplectic groupoid), explaining in particular the classical origin of the noncommutativity of the string endpoint coordinates. We also review the perturbative Lagrangian approach and its connection with Kontsevich's star product. Finally we comment on the relation between the two approaches.

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43

Rybicki, Tomasz. "On the group of lagrangian bisections of a symplectic groupoid." Banach Center Publications 54 (2001): 235–47. http://dx.doi.org/10.4064/bc54-0-13.

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44

Bondal, A. I. "A symplectic groupoid of triangular bilinear forms and the braid group." Izvestiya: Mathematics 68, no. 4 (August 31, 2004): 659–708. http://dx.doi.org/10.1070/im2004v068n04abeh000495.

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45

Bonechi, F., N. Ciccoli, N. Staffolani, and M. Tarlini. "The quantization of the symplectic groupoid of the standard Podle s ̀ sphere." Journal of Geometry and Physics 62, no. 8 (August 2012): 1851–65. http://dx.doi.org/10.1016/j.geomphys.2012.04.001.

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46

karabegov, Alexander V. "On the Inverse Mapping of the Formal Symplectic Groupoid of a Deformation Quantization." Letters in Mathematical Physics 70, no. 1 (October 2004): 43–56. http://dx.doi.org/10.1007/s11005-004-0610-1.

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47

Lu, Jiang-Hua, Victor Mouquin, and Shizhuo Yu. "Configuration Poisson Groupoids of Flags." International Mathematics Research Notices, November 26, 2022. http://dx.doi.org/10.1093/imrn/rnac321.

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Abstract Let $G$ be a connected complex semi-simple Lie group and ${\mathcal {B}}$ its flag variety. For every positive integer $n$, we introduce a Poisson groupoid over ${{\mathcal {B}}}^n$, called the $n$th total configuration Poisson groupoid of flags of $G$, which contains a family of Poisson sub-groupoids whose total spaces are generalized double Bruhat cells and bases generalized Schubert cells in ${\mathcal {B}}^n$. Certain symplectic leaves of these Poisson sub-groupoids are then shown to be symplectic groupoids over generalized Schubert cells. We also give explicit descriptions of symplectic leaves in three series of Poisson varieties associated to $G$.

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48

Cabrera, Alejandro, and Miquel Cueca. "Dimensional reduction of Courant sigma models and Lie theory of Poisson groupoids." Letters in Mathematical Physics 112, no. 5 (October 2022). http://dx.doi.org/10.1007/s11005-022-01596-1.

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AbstractWe show that the 2d Poisson Sigma Model on a Poisson groupoid arises as an effective theory of the 3d Courant Sigma Model associated with the double of the underlying Lie bialgebroid. This field-theoretic result follows from a Lie-theoretic one involving a coisotropic reduction of the odd cotangent bundle by a generalized space of algebroid paths. We also provide several examples, including the case of symplectic groupoids in which we relate the symplectic realization construction of Crainic–Marcut to a particular gauge fixing of the 3d theory.

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49

Bailey, Michael, and Marco Gualtieri. "Integration of generalized complex structures." Journal of Mathematical Physics 64, no. 7 (July 1, 2023). http://dx.doi.org/10.1063/5.0091245.

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We solve the integration problem for generalized complex manifolds, obtaining as the natural integrating object a holomorphic stack with a shifted symplectic structure; in other words, a real symplectic groupoid with a compatible complex structure is defined only up to Morita equivalence. We explain how such objects differentiate to give generalized complex manifolds, and we show that a generalized complex manifold is integrable in this sense if and only if its underlying real Poisson structure is integrable. We describe several concrete examples of these integrations. Crucial to our solution are new technical tools, which are of independent interest, namely, a reduction procedure for Lie groupoid actions on Courant algebroids, as well as certain local-to-global extension results for multiplicative forms on local Lie groupoids.

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50

Cosserat, Oscar. "Symplectic groupoids for Poisson integrators." Journal of Geometry and Physics, January 2023, 104751. http://dx.doi.org/10.1016/j.geomphys.2023.104751.

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