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Vincent Rodgers © 2006
Vincent Rodgers © Courant brackets are a framework for describing new string compactifications Goal is to construct a geometric action for the Courant Bracket (Dirac bundle) Provide clues to relations to symplectic structures and complex structures Geometric actions often related to anomalies Natural way to incorporate other algebras Collaborators: Leopoldo Pando-Zayas, Leo Rodriguez, Xiaolong Liu
Vincent Rodgers © Generalized Complex Structures: the origins of the Courant Bracket Extension of almost complex structures Early work by Duff and Tsetlyn Later developed by Hitchin and Gualtieri Motivated by T-duality
Vincent Rodgers © T-Duality Consider a string theory: Ingredients: tangent bundle, metric and Kalb-Ramond field
Vincent Rodgers © Alternatively we could have written: a) Integrating outgives back the actions since b) Integrating outgives a new sigma model Buscher rules
Vincent Rodgers © Theories are dual to each other and related by T symmetry T – symmetry T-Duals
Vincent Rodgers © Put dual elements on equal footing This space admits vectors and one forms
Vincent Rodgers © Generalize this to vector fields and p-forms T,X,Y, are vector fields , are p-forms
Vincent Rodgers © Courant Bracket Contains the Lie derivative of vector fields Canonical inner products: different “helicity” SO(d,d) symmetry
Vincent Rodgers © Jacobi Identity and Complex Structure Jacobiator measures violation of Jacobi identity Nijenhuis Operator
Vincent Rodgers © Generalized Complex Structure based on a Symplectic form Complex Symplectic
Vincent Rodgers © FOCUS ON DIRAC STRUCTURE = 0 This allows us to use coadjoint orbit techniques
Vincent Rodgers © Coadjoint Orbits Admit a natural symplectic structure Focus of Dirac Bundle where WZW models arise from affine Lie Algebras Two Dimensional Polyakov Gravity arises from Virasoro algebra Also related to bosonization of fermions
Vincent Rodgers © Introduce dual space of E Dual elements to E Element of E C(*,*) is a suitable pairing
Vincent Rodgers © Coadjoint Representation Invariant pairing By Leibnitz Defines Coadjoint representation
Vincent Rodgers © Natural Symplectic Two Form on Coadjoint Orbit Defines a Natural Action
Vincent Rodgers © 2006
Vincent Rodgers © 2006
Vincent Rodgers © Two Important Features ISOTROPY or stability algebra - those generators that do nothing to coadjoint elements, group generated by H COADJOINT ORBITS – space swept out by applying all elements of group generated by G to a fixed coadjoint vector G/H describes orbits
Vincent Rodgers © Example Semi-direct product of Virasoro and Affine Lie Algebras
Vincent Rodgers © Virasoro Algebra
Vincent Rodgers © MODE DECOMPOSITION OF LIE DERIVATIVES AND AFFINE LIE ALGEBRAS
Vincent Rodgers © Physical Interpretation
Vincent Rodgers © As transformation laws
Vincent Rodgers © THE GEOMETRIC ACTION B. Rai and V.G.J.R Nucl. Phys 1990
Vincent Rodgers © Courant Case p=0 Courant bracket for p=0 Suitable Pairing Variations
Vincent Rodgers © The Invariant Actions
Vincent Rodgers © Courant Bracket, p > 0
Vincent Rodgers © Future: Getting off the Orbits WZW vs. Yang Mills Example Space of orbits may have its on symplectic structure and constraints
Vincent Rodgers © Natural Foliation of E* Elements of E * live on one surface only. Each surface has its own natural Poisson bracket structure. Geometric Actions
Vincent Rodgers © Geometric Actions (A 1, D 1 ) Geometric Actions Fixed Coadjoint Orbit Here A and D are background Fields WZNW Models - Affine Lie Algebras 2D Polyakov Gravity – Virasoro Algebras W Algebras, etc symmetry is G/H
Vincent Rodgers © Transverse Actions (A 3, D 3 ) (A 2, D 2 ) (A 1, D 1 ) (A 4, D 4 ) (A 0, D 0 ) Transverse Actions Use isotropy to identify second symplectic structure that moves off the orbits. Transverse Actions
Vincent Rodgers © Geometric and Transverse Actions (A 3, D 3 ) (A 2, D 2 ) (A 1, D 1 ) (A 4, D 4 ) (A 0, D 0 ) Geometric Actions Transverse Actions Fixed Coadjoint Orbit Here A and D are background Fields Geometric Actions: action describes collective coordinates about fixed A and D Transverse Actions: Field equations yield constraints for A and D and dynamics
Vincent Rodgers © Conclusion Built a geometric action with natural symplectic structure Associate these orbits with complex structures How are these complex structures related to the original ones on E? Can we move off the orbits using the isotropy equations?
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1 A camera is modeled as a map from a space pt (X,Y,Z) to a pixel (u,v) by ‘homogeneous coordinates’ have been used to ‘treat’ translations ‘multiplicatively’
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