BOOK OF ABSTRACTS

Minicourses, Invited Lectures, Contributed Talks, Posters

Index

A. Minicourses

1. Janusz Grabowski 9“Brackets”

2. Mariano Santander 10“Spaces: a perspective view”

B. Invited Lectures

1. Henrique Bursztyn 13“B-fields and Morita equivalence in Poisson geometry and quantization”

2. Mark J. Gotay and Marco Castrillon Lopez 14“Covariantizing Classical Field Theories”

3. Yolanda Lozano 15“Duality in String Theory”

4. Catherine Meusburger and Torsten Schonfeld 16“Gauge fixing and dynamical r-matrices in Chern-Simons theory”

5. Chris Rogers 17“Higher symplectic geometry”

C. Contributed Talks

1. Paulo Antunes and Joana M. Nunes da Costa 21“Hypersymplectic structures on Courant algebroids”

2. Anna M. Candela, A. Romero and M. Sanchez 22“Dynamics of particles under the action of general force fields”

3. Leonardo Colombo, Sebastian Ferraro and David Martın de Diego 23“On the variational discretization of optimal control problems”

4. Javier de Lucas and Janusz Grabowski 24“Mixed superposition rules: theory and applications”

5. Marıa Pilar Garcıa del Moral 25“T-duality in the supermembrane”

6. Ignazio Lacirasella, J.C. Marrero, and E. Padron 26“Reduction of time-dependent mechanical systems and Poisson structureswith magnetic terms”

3

7. Alice Rogers and Simon Pelling 27“Multisymplectic BRST”

8. Carlos Tejero 28“Yang-Mills functional on Kahler manifolds and generalized Bogomolov typeinequalities”

9. Miguel Vaquero, S. Ferraro, M. de Leon, Juan C. Marreroand David Martın de Diego 29“On the geometry of the Hamilton-Jacobi equation for generating functions”

10. Raquel Villacampa, M. Ceballos, A. Otal and L. Ugarte 30“Strongly Gauduchon metrics”

11. Andres Vina 31“Invariants under deformation of Hamiltonian actions”

12. Marco Zambon and Iakovos Androulidakis 32“Holonomy and singular foliations”

D. Posters

1. Isabel G. Ayala, T. Garcıa, N.A. Cordero and E. Romera 35“Wave packet revivals in graphene nanostructures”

2. Angel Ballesteros, Francisco J. Herranz and Fabio Musso 36“The anisotropic oscillator on the sphere and the hyperbolic plane”

3. Alfonso Blasco, Angel Ballesteros and Fabio Musso 37“Hamiltonian formulation for the generalized ladder system”

4. Rutwig Campoamor-Stursberg 38“Subalgebras of the Witt-Virasoro algebra and dynamical symmetries ofintegrable systems”

5. Raquel Caseiro 39“The modular class of Dirac structures”

6. Laurent Delisle, Veronique Hussin and Wojtek J. Zakrzewski 40“Holomorphic maps of Grassmannian sigma models”

7. Antonio de Nicola, Beniamino Cappelletti Montano and Ivan Yudin 41“Topology of 3-quasi-Sasakian manifolds”

8. Stanislaw Ewert-Krzemieniewski 42“On Killing vector fields on tangent bundle with g-natural metric”

9. Manuel Gadella and Osvaldo Civitarese 43“Mathematical and physical description of resonances”

10. I. Alvaro Gutierrez 44“The Lie algebroid of the symplectic grupoid T ∗G”

4

11. Francisco J. Herranz, Angel Ballesteros, Alberto Enciso,Orlando Ragnisco and Danilo Riglioni 45“Superintegrable Lorentzian metrics and Bertrand spacetimes”

12. Micha l Jozwikowski 46“Jacobi vector fields on algebroids”

13. Ignacio Lujan and M. Castrillon Lopez 47“Strongly degenerate homogeneous pseudo-Kahler structures of linear typeand “complex” plane waves”

14. Antonino Marciano, Stephon Alexander and Leonardo Modesto 48“The hidden quantum groups symmetry of super-renormalizable gravity”

15. Fabio Musso 49“Integrable deformations of bihamiltonian systems through Poisson-Lie groups”

16. Jose Navarro and J.B. Sancho 50“Electromagnetic energy tensor for charged p-branes”

17. Javier Negro, R. Campoamor-Stursberg, M. Gadella and S. Kuru 51“Action-angle variables, ladder operators and coherent states”

18. Joana M. Nunes da Costa and Paulo Antunes 52“Nijenhuis and compatible tensors on exact Courant algebroids”

19. Leszek Pysiak, Wiesaw Sasin and Joanna Tarka 53“Derivations of some groupoid algebra”

20. Narciso Roman-Roy, M. Salgado and S. Vilarino 54“Higher-order Noether symmetries in k-symplectic field theories”

21. Juan Jesus Salamanca, A. Romero and R.M. Rubio 55“Parabolicity of spacelike hypersurfaces in generalized Robertson-Walker spacetimes.Applications to uniqueness results”

22. Larissa Sbitneva 56“Differential equations of smooth loops related to some space-time models:integrability conditions and geometry”

5

A. MINICOURSES

Brackets

Janusz Grabowski

Institute of Mathematics, Polish Academy of SciencesSniadecich 8, 00-956 Warszawa, POLAND

E-mail: jagrab@impan.pl

Abstract

In algebra, ‘brackets’ are usually understood as non-associative operations on vector spacesor modules, with Lie brackets as the main example.The aim of this mini-course is to review origins and main properties of the most importantbracket operations appearing canonically in differential geometry and mathematical physicsas well as relations between them.The list of structures we want to discuss includes the following.

• Lie brackets - canonical examples.

• Quasi-derivations (derivative endomorphisms) and QD-algebroids.

• Lie algebroids of Poisson, Jacobi, and Kirillov brackets.

• Lie and general algebroids in Geometric Mechanics.

• Nambu and Loday brackets.

• Nijenhuis tensors and contractions.

• Courant brackets and Dirac structures.

• Loday algebroids.

• Superbrackets and graded brackets.

References

[1] J. Grabowski: Quasi-derivations and QD-algebroids. Rep. Math. Phys. 32, 445–451 (2003)

[2] A. A. Kirillov: Local Lie algebras (Russian). Uspekhi Mat. Nauk 31, 57–76 (1976); Englishversion: Russian Mathematical Surveys 31, 55–75 (1976)

[3] C.-M. Marle: Calculus on Lie algebroids, Lie groupoids and Poisson manifolds. Disserta-tiones Math. (Rozprawy Mat.) 457 (2008), 57 pp.

9

Spaces: A perspective view

Mariano Santander

Departamento de Fısica Teorica, Facultad de CienciasUniversidad de Valladolid, Spain

E-mail: msn@fta.uva.es

Abstract

‘Space’ is probably one of the most overburdened terms nowadays used in Mathematics andin Physics. From a viewpoint which considers the long-time evolution of ideas, this is acomparatively rather recent phenomenon.

In this minicourse I plan to provide a perspective view of this ‘Space to spaces’ devel-opment, discussing several aspects of the ‘spaces’ idea, which are the closest to the historicalmeaning of the Space term. In particular, I will put some emphasis in questions (like visual-ization) which are less frequently explored. I plan to start from a swift historical presentation,covering from the prehistory of the Space idea until the groundbreaking work by Riemannand Klein which actually set the starting point for the modern inteligence of the spaces.Then I will center attention in Symmetric Spaces, which provide a quite natural extensionto the basic and deep properties the Space was assumed to have during its previous history.Surprisingly, (or perhaps not so much) these properties have also turned up to be essentialin physics.

The following is a list of the topics which I plan to discuss in the course: interspersed therewill be examples and comments on the ubiquous presence of Symmetric spaces all-around inPhysics.• Basics on Symmetric Spaces: Metric and connection.• Rank of a Symmetric Space. Beyond rank-one: Plucker geometry and phase spaces as

rank-two spaces.• Classification of Symmetric Spaces: Cartan and Berger lists. Understanding the Cartan

and Berger lists: the generic real, complex and quaternionic spaces and the exceptionaloctonionic spaces.

• The exceptional spaces and the magic square.• Visualizing symmetric spaces: strategies and the example of CP 2

References

[1] B.A. Rosenfeld: A History of non-euclidean geometry, Springer (1988);

[2] B.A. Rosenfeld: The geometry of Lie groups, Kluwer (1997);

[3] M. Santander Matrix Models for (exceptional) simple Lie algebras: Two constructionsSymmetries in Gravity and Field Theory, V. Aldaya, Jose M. Cervero y Pilar Garcıa Eds.97-114, (2004).

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B. INVITED LECTURES

B-fields and Morita equivalence in Poisson geometry

and quantization

Henrique Bursztyna

a IMPA, BrazilE-mail: henrique@impa.br

Abstract

This talk will discuss B-field symmetries of Poisson structures and their quantum counter-part. Using the deformation quantization of Poisson manifolds resulting from Kontsevich’sformality theorem, we will show that, upon an integrality condition, B-field transformationsare the classical versions of Morita equivalence for star products.

13

Covariantizing Classical Field Theories

Mark J. Gotaya and Marco Castrillon Lopezb

a Pacific Institute for the Mathematical SciencesUniversity of British Columbia

Vancouver, BC V6T 1Z2 Canada E-mail: gotay@pims.math.ca

b Departamento de Geometrıa y TopologıaFacultad de Ciencias Matematicas

Universidad Complutense de Madrid28040 Madrid, Spain

E-mail: mcastri@mat.ucm.es

Abstract

We show how to enlarge the covariance groups of a wide variety of classical field theoriesin such a way that the resulting “covariantized” theories are ‘essentially equivalent’ to theoriginals. In particular, our technique will render many classical field theories generallycovariant, that is, the covariantized theories will be spacetime diffeomorphism-covariant andfree of absolute objects. Our results thus generalize the well-known parametrization techniqueof Dirac and Kuchar. Our constructions apply equally well to internal covariance groups, inwhich context they produce natural derivations of both the Utiyama minimal coupling andStuckelberg tricks.

14

Duality in String Theory

Yolanda Lozanoa

a University of OviedoE-mail: ylozano@uniovi.es

Abstract

Duality symmetries have played a key role in the most important recent developments inString Theory, such as the emergence of M-theory as the“Unifying Theory of Everything”[1], the discovery of D-branes [2], and the derivation thereof of the celebrated gauge/gravitycorrespondence [3]. After a review on the role played by duality in these developments wewill concentrate on recent achievements on the formulation of Abelian T-duality in terms ofgeneralized geometry [4] and on its possible extension to non-Abelian isometry groups [5].

References

[1] E. Witten, String theory dynamics in various dimensions, Nucl. Phys. B 443 (1995) 85[hep-th/9503124].

[2] J. Polchinski, Dirichlet Branes and Ramond-Ramond charges, Phys. Rev. Lett. 75 (1995)4724 [hep-th/9510017].

[3] J. M. Maldacena, The Large N limit of superconformal field theories and supergravity,Adv. Theor. Math. Phys. 2 (1998) 231 [hep-th/9711200]

[4] C. Hull and B. Zwiebach, Double Field Theory, JHEP 0909 (2009) 099 [arXiv:0904.4664[hep-th]].

[5] G. Itsios, Y. Lozano, E. OColgain and K. Sfetsos, Non-Abelian T-duality and consistenttruncations in type-II supergravity, arXiv:1205.2274 [hep-th].

15

Gauge fixing and dynamical r-matrices

in Chern-Simons theory

Catherine Meusburgera and Torsten Schonfeldb

a Department Mathematik, Friedrich-Alexander Universitat Erlangen-Nurnberg, Cauerstr.11, 91058 Erlangen, Germany

E-mail: catherine.meusburger@math.uni-erlangen.de

b Department Mathematik, Friedrich-Alexander Universitat Erlangen-Nurnberg, Cauerstr.11, 91058 Erlangen, Germany

E-mail: torsten.schoenfeld@math.uni-erlangen.de

Abstract

We apply Dirac’s gauge fixing procedure to Chern-Simons theory with the Poincare groupin three dimensions as a gauge group. For all gauge fixing conditions that satisfy certainstructural requirements, this yields an explicit description of the symplectic structure onthe moduli space of flat Poincare-connections in terms of classical dynamical r-matrices.We classify all dynamical r-matrices and Dirac brackets that arise from such gauge fixingprocedures and discuss the role of dynamical r-matrix symmetries in the theory. In particular,we show that these dynamical r-matrices have a direct physical interpretation in the Chern-Simons formulation of (2+1)-gravity.

References

[1] C. Meusburger, T. Schonfeld: Gauge fixing and classical dynamical r-matrices inISO(2,1)-Chern-Simons theory , arXiv:1203.5609.

[2] C. Meusburger, T. Schonfeld: Gauge fixing in (2+1)-gravity: Dirac bracket and spacetimegeometry . Class. Quant. Grav. 28, 125008 (2011)

16

Higher symplectic geometry

Chris Rogers

Mathematisches Institut, Georg-August-Universitiat Gottingen, Bunsenstr. 3-5, D-37073,Gottingen, DE

E-mail: crogers@uni-math.gwdg.de

Abstract

In higher symplectic geometry, we use techniques borrowed from higher category theoryand homotopical algebra to study manifolds equipped with a closed non-degenerate form ofdegree > 2. We will explain in this talk how the higher analogs of many important structuresfound in symplectic geometry naturally arise on these manifolds. For example, instead of aPoisson algebra of functions, a higher symplectic manifold has a strong homotopy Lie algebraof differential forms whose bracket only satisfies the Jacobi identity up to coherent chainhomotopy. In certain cases, these differential forms correspond to observables for classicalfield theories. We will also describe a quantization procedure for higher symplectic manifolds.This involves structures such as U(1)-gerbes, twisted vector bundles, and module categories;just as ordinary geometric quantization uses U(1)-bundles, global sections of line bundles,and Hilbert spaces. Throughout the talk, some comparisons will be made with closely relatedsubjects, such as Dirac geometry and multisymplectic geometry.

References

[1] C. Rogers, L∞-algebras from multisymplectic geometry, Lett. Math. Phys. 100, 29–50(2012). (arXiv:1005.2230).

[2] C. Rogers, 2-plectic geometry, Courant algebroids, and categorified prequantization. Toappear in J. Symp. Geom. (arXiv:1009.2975).

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C. CONTRIBUTED TALKS

Hypersymplectic structures on Courant algebroids

Paulo Antunesa and Joana M. Nunes da Costab

a CMUC, Department of Mathematics, University of Coimbra, 3001-454 Coimbra, PortugalE-mail: pantunes@mat.uc.pt

b CMUC, Department of Mathematics, University of Coimbra, 3001-454 Coimbra, PortugalE-mail: jmcosta@mat.uc.pt

Abstract

Hypersymplectic structures on manifolds were introduced by Xu in [3] where he proved thathypersymplectic structures are in 1-1 correspondence with (pseudo-)hyperKahler structures.In [1], we generalized the notion of hypersymplectic structures to Lie algebroids and showedthat the 1-1 correspondence is still valid and also that this framework is very rich in compatiblepairs of tensors. On Courant algebroids, hyperKahler structures were introduced, mimickingthe classical definition, by Bursztyn, Cavalcanti and Gualtieri in [2].

In the present work, we define hypersymplectic structures on Courant algebroids andwe show that, with this definition, we recover, as particular cases, all the classical resultsand examples already known on Lie algebroids. This framework is also very rich in alreadyknown compatible structures on Courant algebroids such as hypercomplex triples, generalizedKahler pairs and Nijenhuis pairs. As on Lie algebroids, we prove a 1-1 correspondence the-orem between hypersymplectic structures and (pseudo-)hyperKahler structures on Courantalgebroids.

References

[1] P. Antunes: Crochets de Poisson gradues et applications: structures compatibles etgeneralisations des structures hyperkahleriennes. These de doctorat de l’ Ecole Polytech-nique, (2010).

[2] H. Bursztyn, G. Cavalcanti and M. Gualtieri: Generalized Kahler and hyper-Kahler quo-tients. Poisson geometry in mathematics and physics, Contemp. Math., 450, 61–77, Amer.Math. Soc., Providence, RI, (2008).

[3] P. Xu: Hyper-Lie Poisson structures. Annales Scientifiques de lEcole Normale Superieure30, Issue 3, 279–302 (1997).

21

Dynamics of particles under the action of general force fields

A.M. Candelaa, A. Romerob and M. Sanchezb

a Dipartimento di Matematica, Universita degli Studi di Bari “A. Moro”,Via E. Orabona 4, 70125 Bari, Italy

E-mail: candela@dm.uniba.it

b Departamento de Geometrıa y Topologıa, Facultad de Ciencias,Universidad de Granada, 18071 Granada, Spain

E-mails: aromero@ugr.es, sanchezm@ugr.es

Abstract

The aim of this talk is analyzing the extendability of the trajectories which are solutions ofthe second order differential equation

Dγ

dt(t) = F(γ(t),t) γ(t) +X(γ(t),t) (E)

on a (connected, finite–dimensional) Riemannian manifold (M, g), where F is a (1,1) smoothtensor field and X is a smooth vector field along the natural projection π : M × R→M .

From a phisical point of view, (E) describes the dynamics of a classical particle under theaction of a force field F , which linearly depends on its velocity, and an external force fieldX, which is independent of the motion of the particle.

In the recent paper [1] we provide some general criteria for the completeness of the trajec-tories of (E) also including the general time–dependent anholonomic case. Furthermore, somerefinements for autonomous systems or forces derived from a potential are proved, extendingclassical results for Lagrangian and Hamiltonian problems (e.g., see [1]).

The optimality of these results is discussed by means of several examples; moreover, anapplication to the geodesic completeness of relativistic pp-waves is pointed out in [3].

References

[1] R. Abraham, J. Marsden: Foundations of Mechanics, 2nd Ed. (6th printing). Boston:Addison–Wesley Publishing Co., 1987.

[2] A. Candela, A. Romero, M. Sanchez: Completeness of the Trajectories of Particles Cou-pled to a General Force Field. Preprint (2011). http://arxiv.org/abs/1202.0523.

[3] A. Candela, A. Romero, M. Sanchez: Remarks on the completeness of plane waves andthe trajectories of accelerated particles in Riemannian manifolds. In: Proc. Int. Meetingon Differential Geometry (Cordoba, November 15-17, 2010).

22

On the variational discretization of optimal control problems

Leonardo Colombo,a Sebastian Ferraro b and David Martın de Diego c

a Instituto de Ciencias Matematicas (CSIC-UAM-UCM-UC3M), calle Nicolas Cabrera 15,Campus UAM, Cantoblanco, Madrid, Spain

E-mail: leo.colombo@icmat.es

b Departamento de Matematicas e Instituto de Matematicas, Universidad Nacional del Sur.Av. Alem 1253, 8000. Bahia Blanca, Argentina

E-mail: sferraro@uns.edu.ar

c Instituto de Ciencias Matematicas (CSIC-UAM-UCM-UC3M), calle Nicolas Cabrera 15,Campus UAM, Cantoblanco, Madrid, Spain

E-mail: david.martin@icmat.es

Abstract

Many continuous optimal control problems for mechanical systems can be modeled using asecond-order Lagrangian system. In this talk we will discuss the exact discretization of thistype of systems and the explicit construction of geometric integrators and their geometricinvariant properties.

References

[1] J. Marsden and M. West: Discrete mechanics and variational integrators. Acta Numer.,10, 357-514 (2001).

[2] S. Ober-Blobaum, O. Junge, and J. Marsden: Discrete mechanics and optimal control:an analysis.ESAIM Control Optim. Calc. Var., 17(2):322-352, 2011.

[3] G. Patrick: Lagrangian mechanics without ordinary differential equations. Rep. Math.Phys. 57, 437-443 (2006).

23

Mixed superposition rules: theory and applications

Javier de Lucasa and Janusz Grabowskib

a Institute of Mathematics, Polish Academy of Sciences, Warsaw, PolandE-mail: delucas@impan.pl

b Institute of Mathematics, Polish Academy of Sciences, Warsaw, PolandE-mail: j.grabowski@impan.pl

Abstract

Mixed superposition rules, i.e., functions describing the general solution of a system of first-order differential equations in terms of a generic family of particular solutions of certainfirst-order systems and some constants, are studied [1]. The main achievement is a general-ization of the celebrated Lie-Scheffers Theorem [2], characterizing systems admitting a mixedsuperposition rule. This somehow unexpected result says that such systems are exactly Liesystems, i.e., they admit a standard superposition rule. This provides a new and powerfultool for finding Lie systems, which is applied here to studying the Riccati hierarchy and toretrieving some known results in a more efficient and simpler way [1].

References

[1] J.F. Carinena and J. de Lucas: Lie systems: theory, generalisations, and applications.Dissertationes Math. 479, 1–162 (2011)

[2] J.F. Carinena, J. Grabowski and G. Marmo: Superposition rules, Lie theorem and partialdifferential equations. Rep. Math. Phys. 60, 237–258 (2007)

[3] J. Grabowski and J. de Lucas: Mixed superposition rules and the Riccati hierarchy.ArXiv:1203:0123

24

T-duality in the supermembrane

Maria Pilar Garcia del Moral

Departamento de Fısica, Universidad de OviedoE-mail: garciamormaria@uniovi.es

Abstract

T-duality is an important duality in the context of String Theory. It is proposed that M-theory as a theory candidate to unify all the 5 string theories as well as supergravity, shouldit realize as a proper symmetry. However a realization was so far unknown. Previously ithas been searched in two different contexts: Supermembrane and more recently the doublefield theory description of string theories in terms of torus bundles. In this talk I wouldexplain how the T-duality emerge as a natural symmetry when the precise global geometricdescription of the supermembrane is known. The supermembrane with central charges isdescribed in terms of a simplectic torus bundle with SL(2,Z) monodromy. These results arepart of a paper [1] done in collaboration with A. Restuccia and J.M.Pena.

References

[1] Supermembrane origin of type II gauged supergravities in 9D Authors: M. P. Garcia delMoral, J. M. Pena, A. Restuccia , arXiv:1203.2767

25

Reduction of time-dependent mechanical systems

and Poisson structures with magnetic terms

I. Lacirasellaa, J.C. Marrerob, and E. Padronc

a Dipartimento di Matematica, Universita degli Studi di Bari (Italy)E-mail: lacirasella@dm.uniba.it

b Departamento de Matematica Fundamental, Universidad de La Laguna (Spain)E-mail: jcmarrer@ull.es

c Departamento de Matematica Fundamental, Universidad de La Laguna (Spain)E-mail: mepadron@ull.es

Abstract

If a Lie group G acts on the cotangent bundle of a manifold by cotangent lifts, then one maycanonically construct a reduced symplectic manifold. In such a case, a magnetic term, i.e. adeformation given by a basic 2-form, may appear.In the time-dependent setting, we replace the cotangent bundle by a suitable symplecticprincipal R-bundle, where the phase space is given by a Poisson manifold of corank 1 (see[1]). We will develop an orbit bundle reduction procedure and, as a contravariant version ofthe orbit cotangent reduction, we will obtain some magnetic terms which deform the standardPoisson bivector. We will apply the procedure to the symplectic principal R-bundle associatedto the projection of a principal G-bundle with 1-dimensional base space and describe the exactnature of the Poisson structure on the ν-shape space, where ν ∈ g∗. Finally, we will showthat the procedure may be used for other interesting mechanical systems, as the dampedoscillator and the time-dependent heavy top.

References

[1] I. Lacirasella, J.C. Marrero, E. Padron: Reduction of symplectic principal R-bundles.Preprint: http://arxiv.org/abs/1201.4690 (2012).

[2] K. Grabowska, J. Grabowski, P. Urbanski, AV-differential geometry: Poisson and Jacobistructures. J. Geom. Phys. bf 52, no. 4, 398–446 (2004).

[3] J. Marsden, A. Weinstein, Reduction of symplectic manifolds with symmetry, Rep. Math-ematical Phys. 5, no. 1, 1974, Pages 121-130.

26

Multisymplectic BRST

Simon Pellinga and Alice Rogersb

a Department of Mathematics, King’s College London, Strand, London WC2R 2LSE-mail: simon.pelling@kcl.ac.uk

b Department of Mathematics, King’s College London, Strand, London WC2R 2LSE-mail: alice.rogers@kcl.ac.uk

Abstract

Developing the work of Hrabak [1] we investigate the construction of the supersymmetric two-dimensional sigma model [2] as the multisymplectic BRST quantization of a simple bosonicmodel.

The multisymplectic approach is appropriate because the symmetry of the model involvesconditions on the spacetime derivatives of the fields.

To implement this scheme recent ideas of Lopez-Osorio and one of the authors [3] onHamiltonian BRST techniques need development to include spacetime rather than simplytime dependence.

References

[1] S.P. Hrabak. On the multisymplectic origin of the nonabelian deformation algebra ofpseudoholomorphic embeddings into strictly almost Kahler ambient manifolds, and thecorresponding BRST algebra. Preprint, math-ph/9904026, 1999.

[2] E. Witten. Topological sigma models. Comm.Math.Phys. 118, 411–449 (1988)

[3] M. A. Lopez-Osorio and A. Rogers. BRST cohomology of systems with secondary con-straints

www.mth.kcl.ac.uk/staff/fa_rogers/Secondary_120518.pdf

27

Yang-Mills functional on Kahler manifolds

and generalized Bogomolov type inequalities

Carlos Tejero Prieto a

a Departamento de Matematicas andInstituto Universitario de Fısica Fundamental y Matematicas

Universidad de SalamancaE-mail: carlost@usal.es

Abstract

We study the Yang-Mills functional for principal fibre bundles with structure group a com-pact Lie group G over a Kahler manifold. In particular we analyze the minimizers for thisfunctional and prove that they are exactly the Einstein-Hermitian G-connections. By meansof the structure of the Yang-Mills functional at a minimum we prove that the characteristicclasses of a principal G-bundle which admits an Einstein-Hermitian connection satisfy twoinequalities. One of them is a generalization of the Bogomolov inequality [1, 2] whereas theother is a new inequality related to the center of the structure group. Therefore, this way weoffer a new and natural proof of Bogomolov inequality that helps understanding its origin.Finally, in view of the Hitchin-Kobayashi correspondence [3] we prove that every (poly)stableprincipal G-bundle has to satisfy these generalized Bogomolov type inequalities.

References

[1] F. A. Bogomolov: Holomorphic tensors and vector bundles on projective varieties. Math.USSR Izv. 13, 499–555 (1978)

[2] M. Lubke: Chernklassen von Hermite-Einstein Vektorbundeln. Math. Ann. 260, 133–141(1982)

[3] B. Anchouche, I. Biswas, Indranil: Einstein-Hermitian connections on polystable principalbundles over a compact Kahler manifold. Amer. J. Math. 1123 207228 (2001).

28

On the geometry of the Hamilton-Jacobi equation

for generating functions

S. Ferraroa, M. de Leonb, Juan C. Marreroc, David Martın de Diegod andMiguel Vaqueroe

a Universidad Nacional del Sursferraro@uns.edu.ar

bInstituto de Ciencias Matematicasmdeleon@icmat.es

c Universidad de La Lagunajcmarrrer@ull.es

dInstituto de Ciencias Matematicasdavid.martin@icmat.es

e Instituto de Ciencias Matematicasmiguel.vaquero@icmat.es

Abstract

It is well-known the usefulness of the Hamilton-Jacobi theory to integrate the equationsof motion of a hamiltonian system. In fact, a large enough number of solutions of theHamilton-Jacobi equation allow the construction of a canonical transformation that reducethe hamiltonian function to a form such that the Hamilton’s equations can be easily inte-grated. In this talk we give a geometric interpretation of this fact using some results aboutsymplectic groupoids, in particular, cotangent groupoids. Our methodology follows the am-bitious program proposed by A. Weinstein [3] in order to develop geometric formulationsof the dynamical behavior o Lagrangian and Hamiltonian systems on Lie algebreoids andLie groupoids. The complete solutions of the Hamilton-Jacobi equations are related withHamilton’s equations even in the case of reduction.

References

[1] R. Abraham, J.E. Marsden: Foundations of Mechanics. 2nd ed., Benjamin-Cummings,(1978).

[2] Z. Ge: Generating functions, Hamilton-Jacobi equations and symplectic groupoids onPoisson manifolds. Indiana Univ. Math. J., 39, 859–876 (1990).

[3] A. Weinstein: Lagrangian mechanics and groupoids. Mechanics Day, volume 7 of FiledsInst. Commun., 207–231 (1996).

29

Strongly Gauduchon metrics

M. Ceballosa, A. Otalb, L. Ugartec and R. Villacampab

a Universidad de Sevillamceballos@us.es

b Centro Universitario de la Defensa (Zaragoza)aotal@unizar.es, raquelvg@unizar.es

c Universidad de Zaragozaugarte@unizar.es

Abstract

Let (M,J) be a complex manifold of complex dimension n. Given any Hermitian metric,Gauduchon proved that in its conformal class there exists a metric g for which the Lee formis co-closed [2]. This condition is equivalent to ∂∂Fn−1 = 0, where F denotes the fundamentalform of (J, g), and a metric with this property is called standard or Gauduchon. Recently,Popovici has introduced and studied in [3] a more specific class of Hermitian metrics, knownas strongly Gauduchon, (sG), characterized by the ∂-exactness of the form ∂Fn−1. Anotherinteresting class of metrics is the class of balanced Hermitian structures, defined by thestronger condition dFn−1 = 0, which in dimension six plays an important role in heteroticstring compactifications (see [1] and the references therein). So, the sG metrics constitute aclass between the class of balanced metrics and the class of Gauduchon metrics.

In [3] the relation between the degeneration of the Frolicher spectral sequence at the firststep and the existence of sG metrics is studied, showing that these two notions are unrelated.We show the relations among the different classes of Hermitian metrics on 6-dimensionalnilmanifolds endowed with an invariant complex structure. We also prove that the existenceof a sG metric implies the degeneration of the Frolicher sequence at the second step, and thebehaviour under small deformation of the complex structure is shown.

These results can be found in arXiv:1111.5873v2 [math.DG].

References

[1] M. Fernandez, S. Ivanov, L. Ugarte, R. Villacampa: Non-Kaehler heterotic string com-pactifications with non-zero fluxes and constant dilaton, Commun. Math. Phys. 288(2009), 677–697.

[2] P. Gauduchon: La 1-forme de torsion d’une variete hermitienne compacte, Math. Ann.267 (1984), 495-518.

[3] D. Popovici: Deformation openness and closedness of various classes of compactcomplex manifolds; Examples, to appear in Ann. Scuola Norm. Sup. Pisa Cl. Sci.(arXiv:1102.1687v1 [math.AG]).

30

Invariants under deformation of Hamiltonian actions

Andres Vinaa

a Departamento de Fısica. Avda. Calvo Sotelo. 33007 Oviedo.E-mail: vina@uniovi.es

Abstract

Let ψ be a Hamiltonian action of a compact group K on a symplectic manifold M and let Tbe a maximal torus of K. We will define deformations of ψ, and we will prove the invarianceunder these deformations of:

(i) the values of the moment map over the fixed point set of T ,(ii) the product of the weights of the isotropy representation induced by T on each tangent

space to M at the fixed points, when these points form a finite set.We will generalize property (ii) to the case when the fixed point set is not discrete. We

will also extend these results to the action of a reductive group G, such that a real compactform K of G acts in a Hamiltonian fashion on M .

The ideas involved in the proofs are the following: On one hand, the coupling class [3] canbe obtained from the cohomology of the universal fibration of the Hamiltonian group, thus,it enjoys the corresponding invariance under deformations of the action. On the other hand,the coupling class can be considered as an equivariant cohomology class, hence, it is possibleto apply the localization formula [1] to integrals in which is involved this class. From thatapplication, we will derive the aforementioned results.

References

[1] N. Berline, E. Getzler, M. Vergne Heat Kernels and Dirac Operators. Heidelberg: Springer1992.

[2] V. Ginzburg, V. Guillemin, Y. Karshon Moment maps, cobordisms, and Hamiltoniangroup actions. Providence: AMS 2002.

[3] V. Guillemin, E. Lerman, S. Sternberg Symplectic Fibrations and Multiplicity Diagrams.Cambridge: Cambridge U P 1996.

31

Holonomy and singular foliations

Marco Zambona and Iakovos Androulidakisb

a Universidad Autonoma de Madrid (Departamento de Matematicas)and ICMAT(CSIC-UAM-UC3M-UCM),

Campus de Cantoblanco, 28049 - Madrid, SpainE-mail: marco.zambon@uam.es, marco.zambon@icmat.es

b National and Kapodistrian University of Athens, Department of Mathematics,Panepistimiopolis, GR-15784 Athens, Greece.

E-mail: iandroul@math.uoa.gr

Abstract

We study the geometry associated to singular foliations. A singular foliation F on M isdefined as a submodule of vector fields, and contains more information than the inducedpartition of M into (singular) leaves. For instance, any vector field on M gives rise to asingular foliation, and any Lie algebroid on M also does.

We will focus on the notion of holonomy [AnZa12]. Unlike the case of regular folia-tions, holonomy can not be defined in terms of paths in the leaves. Its definition will in-volve a groupoid H, which was ingeniously associated to (M,F) by Androulidakis-Skandalis[AnSk06]. We will present simple examples and show that this notion specializes to theclassical one for regular foliations.

As an application, we will describe the relation between the linearization of the foliationabout a leaf L and the notion of holonomy. When L satisfies certain regularity conditions,it turns out that the restriction H|L is an honest Lie groupoid [AnZa11]. From this we willdeduce some simple statements about the linearization problem (the singular version of theReeb stability theorem).

References

[AnSk06] I. Androulidakis and G. Skandalis: The holonomy groupoid of a singular foliation.J. Reine Angew. Math. 626 (2009), 1–37.

[AnZa11] I. Androulidakis and M. Zambon Smoothness of holonomy covers for singular foli-ations and essential isotropy. arXiv:1111.1327

[AnZa12] I. Androulidakis and M. Zambon Holonomy transformations for singular foliations.In progress.

32

D. POSTERS

Wave packet revivals in graphene nanostructures

I.G. Ayalaa, T. Garcıab, N.A. Corderoc and E. Romerad

a University of Burgos. Physics Department.E-mail: isagomez@ubu.es

b University of Granada. Department of Electronics and Computer Technology.E-mail: trinidad.gcia@gmail.com

c University of Burgos. Physics Department.E-mail: ncordero@ubu.es

c University of Granada. Dept Fısica Atomica Molecular y NuclearE-mail: eromera@ugr.es

Abstract

Graphene (a one-atom-thick gaphite monolayer) has attracted a lot of attention in recent yearsfrom both theoretical and experimental points of view due to its extraordinary mechanical,electronic and transport properties. These properties are due to the honeycomb-like geometryof the carbon atoms that, for instance, makes this material a zero-band-gap semiconductor.Besides, the dispersion relation at low energies for the electrons in this system does not obeythe typical parabolic band structure but a conical one that translates into electrons behavinglike massless Dirac fermions.

The quantum evolution of wave functions in confined systems exhibits some interestingproperties. One of them is the revival of the wave packet. Revivals appear when wavepackets return in their temporal evolution to a shape that is very similar to the initial shape.Assuming an initial wave packet that is a superposition of eigenstates sharply peaked aroundsome level, this wave packet evolves oscillating and spreading out until the delocalization isso big that the packet nearly collapses. But at a later time the packet starts to regenerateitself and finally reaches a state in which it recovers approximately its initial shape. The timeit takes for this phenomenon to occur depends on the energy eigenvalue spectrum.

We present examples of this kind of behaviour in small mono- and bi-layer graphene ringsand disks. We have used three different models (a continuum model, Density FunctionalTheory and Tight-Binding) to calculate the energy spectrum of these systems and studiedthe difference in revival times among these nanostructures

35

The anisotropic oscillator on the sphere

and the hyperbolic plane

Angel Ballesterosa, Francisco J. Herranza and Fabio Mussoa

a Physics Department, University of Burgos, 09001 Burgos (Spain)

Abstract

The aim of this contribution is to present an integrable generalization of the Hamiltonian

H =12

(p21 + p2

2) + δq21 + (δ + Ω)q22 +λ1

q21+λ2

q22,

on the 2D sphere S2 and on the hyperbolic plane H2, in the sense that the Hamiltonian Hwill be recovered when the Euclidean limit is considered. Such a curved Hamiltonian will beexplicitly shown to be integrable (albeit not superintegrable) for any values of the parametersδ,Ω, λ1, λ2, and it can be interpreted as an anisotropic generalization of the (isotropic andsuperintegrable) Higgs oscillator [1, 2].

In order to obtain such a –to the best of our knowledge– new integrable system, we shallmake use of the constant Gaussian curvature κ of these spaces as an explicit “deformation”parameter, and all the results here presented will hold simultaneously for S2 (κ > 0), H2

(κ < 0) and E2 (κ = 0). In this way, the “flat/linear” limit κ → 0 in the spherical andhyperbolic systems will be always well defined leading to the corresponding Euclidean Hamil-tonian. Conversely, the non-trivial dynamical nature of the transition from the ‘flat/linear’systems defined on E2 to the “curved/nonlinear” ones associated to S2 and H2 will be clearlyappreciated.

References

[1] P.W. Higgs: Dynamical symmetries in a spherical geometry I. J. Phys. A: Math. Gen. 12309–323 (1979).

[2] M.F. Ranada and M. Santander: Superintegrable systems on the two-dimensional sphereS2 and the hyperbolic plane H2. J. Math. Phys. 40 5026–5057 (1999).

[3] A. Ballesteros, F.J. Herranz and F. Musso, The anisotropic Higgs oscillator on the two-dimensional sphere and the hyperbolic plane, arXiv:1207.0071.

36

Hamiltonian formulation for the generalized ladder system

Alfonso Blascoa, Angel Ballesterosa and Fabio Mussoa

a Physics Department, University of Burgos, 09001 Burgos (Spain)

Abstract

The N -dimensional generalized ladder system (hereafter GLS) was introduced by Imai andHirata [1, 2]) and is a system of N homogeneous quadratic first order differential equations

xi = xi

N∑j=1

(ai − aj + a0)xj , i = 1, . . . , N, (1)

where ak (k = 0, . . . , N) are arbitrary real parameters. A particular relevant subcase of(1), the so-called “ladder” system (LS)[1], arises for the particular choice a0 = 1, ai = i,(i = 1, . . . , N) of the parameters, namely:

xi = xi

N∑j=1

(i− j + 1)xj . (2)

Imai and Hirata proved in [1] the integrability of the system (1) by means of Lie symmetries.Since the equations (1) are quadratic, the GLS can be also considered as a new class ofN -dimensional integrable Lotka-Volterra systems, which makes the GLS noteworthy.

To the best of our knowledge the Hamiltonian structure of (1) was still unknown. Inthis contribution we explicitly present such (generalized) Poisson structure and the Hamilto-nian functions for the GLS (1), obtaining also its first integrals as functionally independentCasimir functions for the associated Poisson algebra and thus proving the complete Liou-ville integrability of the GLS [3]. In order to obtain this result, two disjoint classes of GLSwill be considered: firstly, the non-degenerate systems with at least one pair of differentparameters al 6= am (l,m ∈ 1, . . . , N) and, secondly, the maximally degenerate GLS witha1 = a2 = · · · = aN and a0 6= 0.

References

[1] K. Imai and Y. Hirata: New Integrable Family in n-Dimensional Homogeneous Lotka-Volterra Systems with Abelian Lie Algebra. J. Phys. Soc. Jpn. 72 973–975 (2003).

[2] Y. Hirata and K. Imai: A Necessary Condition for Existence of Lie Symmetries in GeneralSystems of Ordinary Differential Equations. J. Phys. Soc. Jpn. 71 (2002) 2396–2400.

[3] A. Ballesteros, A. Blasco and F. Musso, J. Phys. Soc. Jpn. 81 (2012) 076001.

37

Subalgebras of the Witt-Virasoro algebra

and dynamical symmetries of integrable systems

R. Campoamor-Stursberga

a I.M.I.-U.C.M, Plaza de Ciencias 3, E-28040 MadridE-mail: rutwig@pdi.ucm.es

Abstract

It is proved that many of the well known (super)-integrable systems are related, by means ofthe Lagrangian formalism and generalized dynamical symmetries (inclusing those correspond-ing to Lie point, Cartan, Noether and non-Noether symmetries) to certain (finite dimensional)subalgebras of the Witt-Virasoro algebra. It is further analyzed under which conditions theknowledge of pure non-trivial point symmetries can be useful for the construction of newintegrable systems.

References

[1] M. Tsamparlis, A.Paliathanasis: Two-dimensional dynamical systems which admit Lieand Noether symmetries. J.Phys. A: Math. Theor. 44 175202 (2011)

[2] G. E. Prince, C. J. Eliezer: Symmetries of the time-dependent N-dimensional oscillator.J.Phys. A: Math. Gen. 13 815 (1981)

38

The modular class of Dirac structures

Raquel Caseiro

Universidade de Coimbra, PortugalE-mail: raquel@mat.uc.pt

Abstract

We explore some properties of the modular class of Dirac structures and Dirac maps. We seethat, in case of a simple null distribution, there is a natural relation between the modular classof the Dirac structure and the modular classes of the null distribution and the underlyingPoisson manifold. These kind of relation are also explored for the modular classes of Diracmaps.

39

Holomorphic maps of Grassmannian sigma models

Laurent Delislea, Veronique Hussinb and Wojtek J. Zakrzewskic

a Departement de Mathematiques et de Statistique, Universite de Montreal, C.P. 6128,Succ. Centre-ville, Montreal (Quebec) H3C 3J7, Canada.

E-mail: delisle@dms.umontreal.ca

b Centre de Recherches Mathematiques, Universite de Montreal, C.P. 6128,Succ. Centre-ville, Montreal (Quebec) H3C 3J7, Canada.

E-mail: hussin@dms.umontreal.cac Department of Mathematical Sciences, University of Durham, Durham DH1 3LE, United

Kingdom.E-mail:w.j.zakrzewski@durham.ac.uk

Abstract

We present a general formula for the Gaussian curvature of curved holomorphic 2-spheres inGrassmannian manifolds G(m,n). We show how to construct such solutions with constantcurvature. We also make some relevant conjectures for the admissible constant curvatures inG(m,n) and give a complete classification of holomorphic maps from S2 into G(2, 4).

References

[1] L. Delisle, V. Hussin and W. J. Zakrzewski: Constant curvature solutions of Grassman-nian sigma models: (1) Holomorphic solutions. Submitted to J. Geom. Phys (2012).

[2] J. Fei, X. Jiao and X. Xu: On conformal minimal 2-spheres in complex Grassmannmanifold G(2, n). Proc. Indian Acad. Sci. 121, 181–199 (2011)

[3] V. Hussin, I. Yurdusen and W. J. Zakrzewski: Canonical surfaces associated with projec-tors in Grassmannian sigma models. J. Math. Phys. 51 103509-1-15 (2010)

[4] W. J. Zakrzewski, Low dimensional sigma models Bristol: Adam Hilger (1989)

40

Topology of 3-quasi-Sasakian manifolds

Beniamino Cappelletti Montanoa, Antonio De Nicolab and Ivan Yudinc

a Universita degli Studi di Cagliari, Dipartimento di Matematica e Informatica, ViaOspedale 72, 09124 Cagliari, Italia

E-mail: b.cappellettimontano@gmail.com

b CMUC, Department of Mathematics, University of Coimbra, 3001-454 Coimbra, PortugalE-mail: antondenicola@gmail.com

c CMUC, Department of Mathematics, University of Coimbra, 3001-454 Coimbra, PortugalE-mail: yudin@mat.uc.pt

Abstract

I will present our preliminary results on the topology of the general class of compact 3-quasi-Sasakian manifolds. They include both 3-cosymplectic and 3-Sasakian manifolds as specialcases of minimum and maximum rank, respectively. Recently, there has been an increasinginterest toward these manifolds, especially the 3-Sasakian case, due to their applicationsto physics. We studied the topology of compact 3-cosymplectic manifolds in [2], while thetopology of 3-Sasakian manifolds was analyzed in [3].

Furthermore, it was proven in [1] that any 3-quasi-Sasakian manifold is locally the productof a Hyper-Kaehler manifold and a 3-Sasakian manifold. Nevertheless, we find examples ofcompact 3-quasi-Sasakian manifolds which are not the global product of a Hyper-Kaehlermanifold and a 3-Sasakian manifold.

References

[1] B. Cappelletti Montano, A. De Nicola, G. Dileo: The geometry of 3-quasi-Sasakian man-ifolds. Internat. J. Math. 20, (2009), 1081–1105.

[2] B. Cappelletti Montano, A. De Nicola, I. Yudin: Topology of 3-cosymplectic mani-folds. 2011, accepted for publication in The Quarterly Journal of Mathematics, 24 pp.DOI:10.1093/qmath/har038 [arXiv:1108.1810].

[3] K. Galicki, S. Salamon: Betti numbers of 3-Sasakian manifolds. Geom. Dedicata 63(1996), 45 - 68.

41

On Killing vector fields on tangent bundle

with g-natural metric

Stanislaw Ewert-Krzemieniewski

Address: West Pomeranian University of TechnologyAl. Piastow 17, 70-310 Szczecin, Poland

E-mail: ewert@zut.edu.pl

Abstract

We develope the methods of Tanno ([1]) to classify the Killing vector fields on tangent bundlewith g-natural metric ([2]).

References

[1] S. Tanno, Killing vector fields and geodesic flow vectors on tangent bundle, J. ReineAngew. Math., 238(1976), 162-171.

[2] O. Kowalski.; M. Sekizawa, Natural transformations of Riemannian metrics on manifoldto metrics on tangent bundle - a classification, Bull. Tokyo Gakugei Univ. (4) 40(1988),1-29.

42

Mathematical and physical description of resonances

O. Civitaresea and M. Gadellab

a Department of Physics, University of La Plata, c.c.67 (1900) La Plata, Argentina

b Departament of FTAO. Facultad de Ciencias. University of Valladolid,46071 Valladolid, Spain

Abstract

Starting from the definition of Gamow States, we discuss their properties and some applica-tions to scattering theory, nuclear structure models and reactions. The poster is devoted tothe description of specific features of the resonances in the framework of an exactly solvablemodel. We extend the standard Friedrichs model to include time-dependent interactions.The time-dependence of the poles of the reduced resolvent of the model is explicitly calcu-lated. It is found that these poles behave as analytical functions of the added time-dependentinteraction.

43

The Lie algebroid of the symplectic grupoid T ∗G

I. Alvaro Gutierrez1

1 Departamento de Matematicas,Universidad de Zaragoza,

50009 Zaragoza, SPAIN E-mail: alvaro@unizar.es

Abstract

I consider a integrble Lie algebroid E by a grupoid G.In [1] is defined the prolongation of Lie algebroid over the vector bundle proyection of

the dual bundle. It is denoted T EE∗. It is proved that this prolongation is a Lie symplecticLie algebroid and the Hamilton Equations, and Legendre trasformation are also defined. In[3], the Lagrange equations on a Lie algebroid are defined from the Lie grupoid T ∗G and thePoisson structure defined in the dual bundle E∗.

In [2], the structure of grupoids TG and T ∗G are studied as particular cases of VB−grupoidsand it’s proved that T ∗G is a Lie symplectic grupoid ([2] page 439) with respect the canonicalsymplectic structure.

I show that the Lie algebroid of grupoid T ∗G is just T EE∗ and the symplectic struc-ture defined in [1], is just the canonical symplectic structure on T ∗G when we consider thealgebroids elements as tangent vectors of the grupoid T ∗G.

References

[1] M. De Leon, J.C: Marrero, E. Martinez: Lagrangian submanifolds and dynamics on LieAlgebroids. J. Phis. A: Matt. Gen. 38 (2005), R241-R308, math. DG/0407528.

[2] K. Makenzie: General Theory of Lie Grupoids and Lie Algebroids. London MathematicalSociety Lecture Note Series, Vol. 213, Cambridge University Press, 2005

[3] A. Weinstein: Lagrangian Mechanic and grupoids. Fields Inst. Comm 7(1996), 207-23157 (1983)

44

Superintegrable Lorentzian metrics and Bertrand spacetimes

Angel Ballesterosa, Alberto Encisob, Francisco J. Herranza,Orlando Ragniscoc and Danilo Riglionid

a Departamento de Fısica, Universidad de Burgos, E-09001 Burgos, SpainE-mail: angelb@ubu.es fjherranz@ubu.es

b Instituto de Ciencias Matematicas (CSIC-UAM-UCM-UC3M), Nicolas Cabrera 14-16,E-28049 Madrid, SpainE-mail: aenciso@icmat.es

c Dipartimento di Fisica, Universita di Roma Tre and Instituto Nazionale di Fisica Nuclearesezione di Roma Tre, Via Vasca Navale 84, I-00146 Roma, Italy

E-mail: ragnisco@fis.uniroma3.it

dCRM, Montreal, CanadaE-mail: riglioni@CRM.UMontreal.ca

Abstract

Bertrand’s theorem asserts that any spherically symmetric natural Hamiltonian system H =12 |p|

2 + V (|q|) in R3 that has a stable circular trajectory and all whose bounded trajectoriesare closed is either a harmonic oscillator or a Kepler system. This statement finds a naturalextension through the definition of Bertrand spacetime introduced by Perlick [1]; this is aspherically symmetric and static spacetime whose timelike geodesics satisfy properties analo-gous to those of the trajectories of the harmonic oscillator or Kepler systems. Furthermore, ifone writes the Lorentzian metric as η = gij(q) dqi dqj − V −1(q) dt2, where g is a Riemannianmetric on a 3-manifold, the timelike geodesics in spacetime are related to the trajectories ofthe Hamiltonian system H = gij(q)pipj+V (q). Perlick’s classification of Bertrand spacetimesconsists of two multi-parametric families. We show that they correspond to either an intrinsicoscillator or an intrinsic Kepler system on a curved manifold [2] and they are superintegrable.Then superintegrability properties of H are translated into superintegrable Lorentzian met-rics and several properties, such a Kepler–oscillator duality, are also analyzed [3].

References

[1] V. Perlick: Bertrand spacetimes. Class. Quantum Grav. 9 1009 (1992).

[2] A. Ballesteros, A. Enciso, F.J. Herranz and O. Ragnisco: Bertrand spacetimes as Ke-pler/oscillator potentials. Class. Quant. Grav. 25 165005 (2008).

[3] A. Ballesteros, A. Enciso, F.J. Herranz, O. Ragnisco and D. Riglioni: SuperintegrableLorentzian metrics: Bertrand spacetimes, Hamiltonians and duality. Submitted (2012).

45

Jacobi vector fields on algebroids

Micha l Jozwikowskia

a IM PAN, Sniadeckich 8, 00-956 WarszawaE-mail: mjozwikowski@gmail.com

Abstract

We study the geometric nature of the Jacobi equiation for Lagrnagian systems on skew-symmetric algebroids. We prove that Jacobi vector fields along a solution of the Euler-Lagrange aquation for such a system are themselves solutions of the Euler-Lagrnage equationfor the tangent lift of the system. We also generalize some classical results about the relation ofJacobi vector fields and null subspaces of the bilinear form associated to the second variationof the Lagrangian action.

References

[1] K. Grabowska, J. Grabowski and P. Urbanski: Geometrical Mechanics on algebroids, Int.J. Geom. Meth. Mod. Phys. 3 (2006), 559–75.

[2] M. Jozwikowski Jacobi vector fields on algebrods - Lagrangian systems, in preparation.

[3] J. Milnor: Morse Theory, Princeton University Press 1963.

46

Strongly degenerate homogeneous pseudo-Kahler structures

of linear type and “complex” plane waves

M. Castrillon Lopeza and I. Lujanb

a ICMAT(CSIC-UAM-UC3M-UCM), Universidad Complutense de madridE-mail: mcastri@mat.ucm.es

b Universidad Complutense de MadridE-mail: ilujan@mat.ucm.es

Abstract

In [3], the author proves that a pseudo-Riemannian manifold admitting a degenerate homo-geneous structure of linear type is locally a singular scale-invariant homogeneous plane wave.Additionally, in [2], it is proved that a pseudo-Riemannian manifold admitting a degener-ate homogeneous structure in the class T1 + T3 has the underlying geometry of a singularhomogeneous plane wave.

The purpose of this work is the generalization of these ideas to pseudo-Kahler manifolds,that is, pseudo-Riemannian manifolds endowed with a compatible complex structure. Weprove that pseudo-Kahler manifolds admitting a strongly degenerate homogeneous structurein the class K2+K4 (according to the classification in [1]) locally has the form of a generalizedhomogeneous complex plane wave.

Finally, the same question is also tackled in the case of pseudo-hyper-Kahler and pseudo-quaternion Kahler manifolds.

References

[1] W. Batat, P. M. Gadea, Jose A. Oubina: Homogeneous pseudo-Riemannian structures oflinear type. J. Geom. Phys. 61, 745–764 (2011).

[2] P. Meessen: Homogeneous Lorentzian spaces admitting a homogeneous structure of typeT1 + T3. J. Geom. Phys. 56, 754–761 (2006).

[3] A. Montesinos Amilibia: Degenerate homogeneous structures of type S1 on pseudo-Riemannian manifolds. Rocky Mountain J. Math. 31, 561–579 (2001).

47

The hidden quantum groups symmetry of

super-renormalizable gravity

Stephon Alexandera, Antonino Marcianob and Leonardo Modestoc

a Department of Physics, The Koshland Integrated Natural Science Center, HaverfordCollege, Haverford, PA 19041 USA

Department of Physics, Princeton University, New Jersey 08544, USAInstitute for Gravitation and the Cosmos, Penn State University, University Park, PA

16802, USAE-mail: salexand@haverford.edu

b Department of Physics, The Koshland Integrated Natural Science Center, HaverfordCollege, Haverford, PA 19041, USA

Department of Physics, Princeton University, New Jersey 08544, USAE-mail: amarcian@haverford.edu

c Perimeter Institute for Theoretical Physics, 31 Caroline St., Waterloo, ON N2L 2Y5,Canada

E-mail: lmodesto@perimeterinstitute.ca

Abstract

I will present results from a work in collaboration with S. Alexander and L. Modesto, Ref. [1],in which we have considered the relation between the super-renormalizable theories of quan-tum gravity (SRQG) studied in [2, 3] and an underlying non-commutativity of spacetime.For one particular super-renormalizable theory, I will show that at linear level (quadratic inthe Lagrangian) the propagator of the theory is the same we obtain starting from a theory ofgravity endowed with θ-Poincare quantum groups of symmetry. Such a theory is over the socalled θ-Minkowski non-commuative spacetime. The study we have done will shed new lighton this link. In will then show that among the theories considered in [2, 3], there exist onlyone non-local and Lorentz invariant super-renormalizable theory of quantum gravity that canbe described in terms of a quantum group symmetry structure. I will also emphasize contactwith pre-existent works in the literature and discuss preservation of the equivalence principlein our framework.

References

[1] S. Alexander, A. Marciano and L. Modesto, accepted in Phys. Rev. D, arXiv:1202.1824[hep-th].

[2] T. Biswas, E. Gerwick, T. Koivisto, A. Mazumdar, accepted in Phys. Rev. Letters[arXiv:1110.5249v2].

[3] L. Modesto, accepted in Phys. Rev. D, arXiv:1107.2403 [hep-th]; L. Modesto, accepted inPhys. Rev. D, [arXiv:1202.0008 [hep-th]].

48

Integrable deformations of bihamiltonian systems

through Poisson-Lie groups

Fabio Musso

Universita degli Studi Roma TREE-mail: musso@fis.uniroma3.it

Abstract

We present a procedure to construct integrable deformations of bihamiltonian systems whenthe two Poisson tensors are of Lie–Poisson type. The main idea is to use a semi-algorithmicprocedure that we implemented in Maple [1] to construct Poisson-Lie groups associated witha given Lie bialgebra. By applying this procedure to the bihamiltonian Poisson pencil weobtain a deformed pencil whose Casimirs define the integrable deformations of the initialbihamiltonian system. We illustrate this procedure through an elementary example.

References

[1] A. Ballesteros, F. Musso, Quantum algebras as quantizations of dual Poisson-Lie groups,in preparation

49

Electromagnetic energy tensor for charged p-branes

Navarro, J.a and Sancho, J.B.b

a Department of Mathematics, University of ExtremaduraE-mail: navarrogarmendia@unex.es

b Department of Mathematics, University of ExtremaduraE-mail: jsancho@unex.es

Abstract

Let (X, g) be a relativistic spacetime of dimension 1 + n; that is, X is a smooth manifold ofdimension 1 + n and g is a Lorentzian metric of signature (+,−, n. . .,−).

An electromagnetic field on X is represented by a differential 2-form F , and its electro-magnetic energy tensor T is a 2-covariant tensor defined in a local chart by the formula:

Tab := −(Fa

iFbi −14F ijFijgab

). (3)

If particles are not punctual, but of dimension p ≤ n (hence called p-branes), therealso exists a generalized theory of electromagnetism, where the electromagnetic field F is adifferential form of order 2 + p ([1]).

In [3], we introduce the electromagnetic energy tensor for this theory of charged p-branes,whose role in the theory is analogous to that of (3) in the case of punctual particles. Thisgeneralized electromagnetic energy tensor happens to coincide with the super-energy tensorof a (2 + p)-form, previously introduced in [2] in a different setting.

Finally, we also characterize this electromagnetic energy tensor as the only natural tensorsatisfying certain properties, which are justified on physical grounds.

References

[1] Henneaux, M., Teitelboim, C. p-Form electrodynamics. Found. Phys., 16, 593–617 (1986)

[2] Senovilla, J.M.M.: Superenergy tensors. Class. Quant. Gravity, 17, 2799–2841 (2000)

[3] Navarro, J., Sancho, J.B. Energy and electromagnetism of a differential k-form.arXiv:1201.3504

50

Action-angle variables, ladder operators and coherent states

R. Campoamor-Stursberga, M. Gadellab, S. Kuruc and J. Negrob

aDep. de Geometrıa y Topologıa, U. Complutense de Madrid, 28040 Madrid, Spain.E-mail: rutwigcs@gmail.com

bDep. de Fısica Teorica, Atomica y Optica, U. de Valladolid 47071 Valladolid, SpainE-mail: manuelgadella1@gmail.com, jnegro@fta.uva.es

cDep. of Physics, Faculty of Science, Ankara University, 06100 Ankara, TurkeyE-mail: Sengul.Kuru@science.ankara.edu.tr

Abstract

This work is devoted to the building of coherent states from arguments based on classicalaction-angle variables. First, we show how these classical variables are associated to an al-gebraic structure in terms of Poisson brackets. In the quantum context these considerationsare implemented by ladder type operators and an structure known as spectrum generat-ing algebra. All this allows to generate coherent states and thereby the correspondence ofclassical-quantum properties by means of the aforementioned underlying structure. This ap-proach is illustrated with the example of the one-dimensional Poschl-Teller potential system.

References

[1] J.P. Gazeau and J. R. Klauder: Coherent states for systems with discrete and continuousspectrum. J. Phys. A 32 123 (1999) .

[2] A. M. Perelomov: Generalized Coherent States and Their Applications Berlin: SpringerVerlag, 1986.

[3] R. W. Robinett: Quantum wave packet revivals, Phys. Rep. 392 1 (2004).

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Nijenhuis and compatible tensors on exact Courant algebroids

Joana M. Nunes da Costaa and Paulo Antunesb

a CMUC, Department of Mathematics, University of Coimbra, 3001-454 Coimbra, PortugalE-mail: jmcosta@mat.uc.pt

b CMUC, Department of Mathematics, University of Coimbra, 3001-454 Coimbra, PortugalE-mail: pantunes@mat.uc.pt

Abstract

It is known that Nijenhuis tensors on Lie algebroids give rise to Nijenhuis tensors on exactCourant algebroids [2]. We show that Poisson bivector fields and presymplectic forms onLie algebroids can also be viewed as Nijenhuis tensors on exact Courant algebroids. Thisobservation allows us to characterize some well known structures on Lie algebroids [3], suchas complementary 2-forms, Poisson-Nijenhuis, ΩN and PΩ structures, as Nijenhuis tensorson exact Courant algebroids. We study compatibility of all these structures in terms of theirassociated tensors on Courant algebroids [1].

References

[1] P. Antunes, C. Laurent-Gengoux, J. M. Nunes da Costa: Hierarchies and compatibilityon Courant algebroids. arXiv:1111.0800

[2] J. Carinena, J. Grabowski, G. Marmo: Courant algebroid and Lie bialgebroid contractions.J. Phys. A 37, no. 19, 5189–5202 (2004)

[3] Y. Kosmann-Schwarzbach, V. Rubtsov: Compatible structures on Lie algebroids andMonge-Ampere operators. Acta Appl. Math. 109, no. 1, 101–135 (2010)

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Derivations of some groupoid algebra

Leszek Pysiaka and Wiesaw Sasinb and Joanna Tarkac

a Warsaw University of Technology, Department of Mathematics and Information Sciences,Koszykowa 75, 00-662 WarsawE-mail: lpysiak@mini.pw.edu.pl

b Warsaw University of Technology, Department of Mathematics and Information Sciences,Koszykowa 75, 00-662 WarsawE-mail: wsasin@mini.pw.edu.pl

c Warsaw University of Technology, Department of Mathematics and Information Sciences,Koszykowa 75, 00-662 Warsaw

E-mail: tarka.joanna@op.pl

Abstract

In earlier papers we presented a model unifying general relativity and quantum mechanics.The model was based on the (noncommutative) convolutive algebra A consisting of smoothcompactly supported functions on the transformation groupoid Γ = E×G, where E was thetotal space of the frame bundle over spacetime, and G the Lorentz group. The differentialgeometry, based on derivations of A was constructed and a generalized Einstein equation wasobtained.

Now we present more profound study of derivations of the algebra A. We define anextension A of the algebra by the space Z of smooth functions on spacetime manifold. Westudy Z−module of derivations of the extended algebra. This module is more natural todevelop geometry.

We illustrate the properties of derivations of the groupoid algebra in the case of the pairgroupoid G×G, where G is a Lie group.

References

[1] Heller, M., Pysiak, L. and Sasin, W.: Noncommutative unification of general relativityand quantum mechanics. Journal of Mathematical Physics 46, 122501. (2005)

[2] Heller, M., Pysiak, L., Sasin, W.: Conceptual unification of gravity and quanta. Int. J.Theor. Phys. 46, 2494-2512 (2007)

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Higher-order Noether symmetries in

k-symplectic field theories

N. Roman-Roya, M. Salgadob and S. Vilarinoc

a Departamento de Matemaatica Aplicada 4. Edificio C-3, Campus Norte UPC. C/ JordiGirona 1. 08034-Barcelona. Spain

E-mail: nrr@ma4.upc.edu

b Departamento de Xeometrıa e Topoloxıa, Facultade de Matematicas. Universidade deSantiago de Compostela. 15782-Santiago de Compostela. Spain

E-mail: modesto.salgado@usc.es

c Centro Universitario de La Defensa, Academia General Militar, Carretera de Huesca s/n,50090-Zaragoza, Spain

E-mail: silviavf@unizar.es

Abstract

For k-symplectic Hamiltonian field theories, we study infinitesimal transformations generatedby certain kinds of vector fields which are not Noether symmetries, but which allow us toobtain conservation laws by means of a suitable generalization of the Noether theorem.

References

[1] N. Roman-Roy, M. Salgado, S. Vilarino: Symmetries and conservation laws in the Gun-ther k-symplectic formalism of field theory, Rev. Math. Phys. 19(10), 1117–1147 (2007).

[2] W. Sarlet, F. Cantrijn: Higher-order Noether symmetries and constants of the motion, J.Phys. A: Math. Gen. 14 (1981) 479-492.

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Parabolicity of spacelike hypersurfaces in

generalized Robertson-Walker spacetimes.

Applications to uniqueness results

A. Romeroa, R.M. Rubiob and J.J. Salamancab

a Departamento de Geometrıa y Topologıa, Universidad de Granada, 18071 Granada, SpainE-mail: aromero@ugr.es

b Departamento de Matematicas, Campus de Rabanales, Universidad de Cordoba, 14071Cordoba, Spain

E-mails: rmrubio@uco.es, jjsalamanca@uco.es

Abstract

We study the non-compact complete spacelike hypersurfaces in Generalized Roberton-Walkerspacetimes of arbitrary dimension whose fiber is parabolic. Under boundedness assumptionson the warping function restricted on any spacelike hypersurface and on the hyperbolic angleof the hypersurface, we prove that a complete spacelike hypersurface is parabolic if theRiemannian universal covering of the fiber it is. As an application of this new technique,several uniqueness results on complete maximal spacelike hypersurfaces are obtained. Also,the corresponding Calabi-Bernstein problems are solved.

References

[1] M. Caballero, A. Romero and R.M. Rubio: Uniqueness of maximal surfaces in GeneralizedRobertson-Walker spacetimes and Calabi-Bernstein type problems. J. Geom. Phys. 60,394–402 (2010).

[2] A. Romero, R.M. Rubio, J.J. Salamanca: Parabolicity of n-dimensional complete spacelikehypersurfaces in certain spacetimes. Applications to uniqueness of maximal hypersurfaces.Preprint.

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Differential equations of smooth loops related to some

space-time models: integrability conditions and geometry

Larissa Sbitnevaa

a Universidad Autonoma del Estado de Morelos, Av. Universidad 1001, C.P. 62209, Col.Chamilpa, Cuernavaca, Morelos, Mexico

E-mail: larissa@uaem.mx

Abstract

The original approach of S. Lie for Lie groups on the basis of differential equations beingapplied to smooth loops has permitted the development of the infinitesimal theory of smoothloops generalizing Lie groups theory ([1, Ch. 4])

A loop with the identity of associativity is a group. It is well known that the systemof differential equations characterizing a smooth loop with the right Bol identity and theintegrability conditions leads to the binary-ternary algebra as a proper infinitesimal object,which terns out to be the Bol algebra (i.e. a Lie triple with an additional bilinear skew-symmetric operation). The corresponding geometry is related to homogeneous spaces similarto symmetric spaces and can be described in terms of the tensors of curvature and torsion([1, Ch. 5]). There exists the analogous consideration for Moufang loops.([1, Ch. 6])

We will consider the differential equations of smooth loops, generalizing smooth Bol loops,with the identities that are the characteristic identities for the algebraic description of somerelativistic space-time models [2]. Further examination of the integrability conditions for thedifferential equations allows to introduce proper infinitesimal objects for the class of loopsunder consideration [3]. This development leads to a Lie algebra g and a subalgebra h ⊂ g

with the decomposition g = h+m.The geometry of corresponding homogeneous spaces can be described in terms of tensors

of curvature and torsionThere is a relation to the notion of a left Bol loop action, since, in the smooth case, a left

Bol loop action coincides with the local triple Lie family of Nono.

References

[1] Lev V. Sabinin: Smooth Quasigroups and Loops. Kluwer Academic Publishers, Dordrecht,The Netherlands, 1999.

[2] A. Ungar: Thomas Precession: Its Underlying Gyrogroup Axioms and Their Use in Hy-perbolic Geometry and Relativistic Physics. Foundations of Physics. 27, 881–951 (1997)

[3] L. Sbitneva: M -loops and transsymmetric spaces. Aportaciones Matemticas. SMM. 27,77-86 (2007)

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