1. Vincent Rodgers © 2006 www.physics.uiowa.edu

2. Vincent Rodgers © 2006 www.physics.uiowa.edu 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

3. Vincent Rodgers © 2006 www.physics.uiowa.edu 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

4. Vincent Rodgers © 2006 www.physics.uiowa.edu T-Duality Consider a string theory: Ingredients: tangent bundle, metric and Kalb-Ramond field

5. Vincent Rodgers © 2006 www.physics.uiowa.edu Alternatively we could have written: a) Integrating outgives back the actions since b) Integrating outgives a new sigma model Buscher rules

6. Vincent Rodgers © 2006 www.physics.uiowa.edu Theories are dual to each other and related by T symmetry T – symmetry T-Duals

7. Vincent Rodgers © 2006 www.physics.uiowa.edu Put dual elements on equal footing This space admits vectors and one forms

8. Vincent Rodgers © 2006 www.physics.uiowa.edu Generalize this to vector fields and p-forms T,X,Y, are vector fields ,  are p-forms

9. Vincent Rodgers © 2006 www.physics.uiowa.edu Courant Bracket Contains the Lie derivative of vector fields Canonical inner products: different “helicity” SO(d,d) symmetry

10. Vincent Rodgers © 2006 www.physics.uiowa.edu Jacobi Identity and Complex Structure Jacobiator measures violation of Jacobi identity Nijenhuis Operator

11. Vincent Rodgers © 2006 www.physics.uiowa.edu Generalized Complex Structure based on a Symplectic form Complex Symplectic

12. Vincent Rodgers © 2006 www.physics.uiowa.edu FOCUS ON DIRAC STRUCTURE = 0 This allows us to use coadjoint orbit techniques

13. Vincent Rodgers © 2006 www.physics.uiowa.edu 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

14. Vincent Rodgers © 2006 www.physics.uiowa.edu Introduce dual space of E Dual elements to E Element of E C(*,*) is a suitable pairing

15. Vincent Rodgers © 2006 www.physics.uiowa.edu Coadjoint Representation Invariant pairing By Leibnitz Defines Coadjoint representation

16. Vincent Rodgers © 2006 www.physics.uiowa.edu Natural Symplectic Two Form on Coadjoint Orbit Defines a Natural Action

17. Vincent Rodgers © 2006 www.physics.uiowa.edu

18. Vincent Rodgers © 2006 www.physics.uiowa.edu

19. Vincent Rodgers © 2006 www.physics.uiowa.edu 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

20. Vincent Rodgers © 2006 www.physics.uiowa.edu Example Semi-direct product of Virasoro and Affine Lie Algebras

21. Vincent Rodgers © 2006 www.physics.uiowa.edu Virasoro Algebra

22. Vincent Rodgers © 2006 www.physics.uiowa.edu MODE DECOMPOSITION OF LIE DERIVATIVES AND AFFINE LIE ALGEBRAS

23. Vincent Rodgers © 2006 www.physics.uiowa.edu Physical Interpretation

24. Vincent Rodgers © 2006 www.physics.uiowa.edu As transformation laws

25. Vincent Rodgers © 2006 www.physics.uiowa.edu THE GEOMETRIC ACTION B. Rai and V.G.J.R Nucl. Phys 1990

26. Vincent Rodgers © 2006 www.physics.uiowa.edu Courant Case p=0 Courant bracket for p=0 Suitable Pairing Variations

27. Vincent Rodgers © 2006 www.physics.uiowa.edu The Invariant Actions

28. Vincent Rodgers © 2006 www.physics.uiowa.edu Courant Bracket, p > 0

29. Vincent Rodgers © 2006 www.physics.uiowa.edu Future: Getting off the Orbits WZW vs. Yang Mills Example Space of orbits may have its on symplectic structure and constraints

30. Vincent Rodgers © 2006 www.physics.uiowa.edu Natural Foliation of E* Elements of E * live on one surface only. Each surface has its own natural Poisson bracket structure. Geometric Actions

31. Vincent Rodgers © 2006 www.physics.uiowa.edu 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

32. Vincent Rodgers © 2006 www.physics.uiowa.edu 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

33. Vincent Rodgers © 2006 www.physics.uiowa.edu 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

34. Vincent Rodgers © 2006 www.physics.uiowa.edu 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?