1. Higher-Spin Geometry and String Theory Augusto SAGNOTTI Universita’ di Roma “Tor Vergata” QG05 – Cala Gonone, September, 2005 Based on: Francia, AS, hep-th/0207002,,0212185, 0507144 AS, Tsulaia, hep-th/0311257 AS, Sezgin, Sundell, hep-th/0501156 Also: D. Francia, Ph.D. Thesis, to appear Based on: Francia, AS, hep-th/0207002,,0212185, 0507144 AS, Tsulaia, hep-th/0311257 AS, Sezgin, Sundell, hep-th/0501156 Also: D. Francia, Ph.D. Thesis, to appear

2. QG05 - Cala Gonone, Sept. 20052 Plan  The (Fang-) Fronsdal equations  Non-local geometric equations  Local compensator forms  Off-shell extensions  Role in the Vasiliev equations

3. QG05 - Cala Gonone, Sept. 20053 The Fronsdal equations (Fronsdal, 1978) Originally from massive Singh-Hagen equations (Singh and Hagen, 1974) Unusual constraints: Gauge invariance for massless symmetric tensors:

4. QG05 - Cala Gonone, Sept. 20054 Bianchi identities Why the unusual constraints: 1. Gauge variation of F 2. Gauge invariance of the Lagrangian As in the spin-2 case, F not integrable As in the spin-2 case, F not integrable Bianchi identity: Bianchi identity:

5. QG05 - Cala Gonone, Sept. 20055 Constrained gauge invariance If in the variation of L one inserts: Are these constraints really necessary?

6. QG05 - Cala Gonone, Sept. 20056 The spin-3 case A fully gauge invariant (non-local) equation: Reduces to local Fronsdal form upon partial gauge fixing (Francia and AS, 2002)

7. QG05 - Cala Gonone, Sept. 20057 Spin 3: other non-local eqs Other equivalent forms: Lesson: full gauge invariance with non-local terms

8. QG05 - Cala Gonone, Sept. 20058 Kinetic operators Index-free notation: Now define: Then:

9. QG05 - Cala Gonone, Sept. 20059 Kinetic operators generic kinetic operator for higher spins when combined with traces: Defining:

10. QG05 - Cala Gonone, Sept. 200510 Kinetic operators Are gauge invariant for n > [(s-1)/2] Satisfy the Bianchi identities For n> [(s-1)/2] allow Einstein-like operators The F (n) :

11. QG05 - Cala Gonone, Sept. 200511 Geometric equations Christoffel connection: Generalizes to all symmetric tensors (De Wit and Freedman, 1980)

12. QG05 - Cala Gonone, Sept. 200512 Geometric equations 1.Odd spins (s=2n+1): 2.Evenspins (s=2n): 2.Even spins (s=2n): (Francia and AS, 2002)

13. QG05 - Cala Gonone, Sept. 200513 Bosonic string: BRST  The starting point is the Virasoro algebra:  In the tensionless limit, one is left with:  Virasoro contracts (no c. charge):

14. QG05 - Cala Gonone, Sept. 200514 String Field equation Higher-spin massive modes: massless for 1/  ’  0 Free dynamics can be encoded in: (Kato and Ogawa, 1982) (Witten, 1985) (Neveu, West et al, 1985) NO NO trace constraints on  or L

15. QG05 - Cala Gonone, Sept. 200515 Low-tension limit  Similar simplifications hold for the BRST charge:  With zero-modes manifest:

16. QG05 - Cala Gonone, Sept. 200516 Symmetric triplets (A. Bengtsson, 1986) (Henneaux,Teitelboim, 1987) (Pashnev, Tsulaia, 1998) (Francia, AS, 2002) (AS, Tdulaia, 2003)  Emerge from  The triplets are:

17. QG05 - Cala Gonone, Sept. 200517 (A)dS symmetric triplets  Directly, deforming flat-space triplets, or via BRST (no Aragone-Deser problem) Directly:  Directly: insist on relation between C and others BRST:  BRST: gauge non-linear constraint algebra  Basic commutator:

18. QG05 - Cala Gonone, Sept. 200518 Compensator Equations  In the triplet: compensator  spin-(s-3) compensator:  The second becomes:  The first becomes:  Combining them:  Finally (also Bianchi):

19. QG05 - Cala Gonone, Sept. 200519 (A)dS Compensator Eqs  Flat-space compensator equations can be extended to (A)dS: (no Aragone-Deser problem)  Gauge invariant under  First can be turned into second via (A)dS Bianchi

20. QG05 - Cala Gonone, Sept. 200520 Off-Shell Compensator Equations  Lagrangian form of compensator: BRST techniques Formulation due to Pashnev and Tsulaia (1997)  Formulation due to Pashnev and Tsulaia (1997)  Formulation involves a large number of fields (O(s))  Interesting BRST subtleties  For spin 3 the fields are: (AS and Tsulaia, 2003) Gauge fixing 

21. QG05 - Cala Gonone, Sept. 200521 Off-Shell Compensator Equations  “Minimal” Lagrangians can be built directly for all spins Only two extra fields,  (spin-(s-3)) and  (spin-(s-4))  Only two extra fields,  (spin-(s-3)) and  (spin-(s-4)) (Francia and AS, 2005)  Equation for   compensator equation Equation for   current conservation  Equation for   current conservation  Lagrange multiplier  :

22. QG05 - Cala Gonone, Sept. 200522 The Vasiliev equations (Vasiliev, 1991-2003;Sezgin,Sundell, 1998-2003) Integrablecurvature constraints  Integrable curvature constraints on one-forms and zero-forms  Cartan integrable systems Key new addition of Vasiliev:  Key new addition of Vasiliev: twisted-adjoint representation (D’Auria,Fre’, 1983) Minimal case (only symmetric tensors of even rank)Sp(2,R)  Minimal case (only symmetric tensors of even rank), Sp(2,R) zero-form  :  zero-form  : Weyl curvatures one-form A :  one-form A : gauge fields

23. QG05 - Cala Gonone, Sept. 200523 The Vasiliev equations  Curvature constraints:  [extra non comm. Coords] Gauge symmetry:  Gauge symmetry:

24. QG05 - Cala Gonone, Sept. 200524 The Vasiliev equations  “Off-shell”: Riemann-like curvatures Ricci-like = 0  “On-shell”: (Riemann-like = Weyl-like l)  Ricci-like = 0  What is the role of Sp(2,R) in this transition? (AS,Sezgin,Sundell, 2005)  Sp(2,R) generators:  Key on-shell constraint: NOT constrained  gauge fields NOT constrained Strong constraint:proper scalar masses  Strong constraint: proper scalar masses emerge regulate projector  At the interaction level  must regulate projector Gauge fields: extended (unconstrained) gauge symmetry  Gauge fields: extended (unconstrained) gauge symmetry  Alternatively: weak constraint, no extra symmetry (Vasiliev) (Dubois-Violette, Henneaux, 1999) (Bekaert, Boulanger, 2003)

25. QG05 - Cala Gonone, Sept. 200525 The spin-3 compensator (AS,Sezgin,Sundell, 2005) In the   0 limit the linearized Vasiliev equations become: Can be solved recursively for the W’s in terms of  : Since C is traceless, the k=2 equation implies: Explicitly: This implies: Last term (compensator): “exact” in sense of Dubois-Violette and Henneaux

26. QG05 - Cala Gonone, Sept. 200526 The Vasiliev equations  Non-linear corrections:  Non-linear corrections: from dependence on internal Z- coordinates Does the projection that “leaves” the compensators produce singular interactions? Vasiliev: Vasiliev: works with traceless conditions all over and feels it does My feeling: My feeling: eventually not, and we are seeing a glimpse of the off-shell form More work will tell us….

27. QG05 - Cala Gonone, Sept. 200527 The End

28. QG05 - Cala Gonone, Sept. 200528 Fermions Notice: Example: spin 3/2 (Rarita-Schwinger) (Francia and AS, 2002)

29. QG05 - Cala Gonone, Sept. 200529 Fermions One can again iterate: The relation to bosons generalizes to: The Bianchi identity generalizes to:

30. QG05 - Cala Gonone, Sept. 200530 Fermionic Triplets (Francia and AS, 2003)  Counterparts of bosonic triplets GSO:  GSO: not in 10D susy strings Yes:  Yes: mixed sym generalizations type-0 models  Directly in type-0 models all  Propagate s+1/2 and all lower ½-integer spins

31. QG05 - Cala Gonone, Sept. 200531 Fermionic Compensators  Recall:  Spin-(s-2) compensator:  Gauge transformations: First compensator equation  second via Bianchi (recently, also off shell  Buchbinder,Krykhtin,Pashnev, 2004)

32. QG05 - Cala Gonone, Sept. 200532 Fermionic Compensators could  We could extend the fermionic compensator eqs to (A)dS could not  We could not extend the fermionic triplets BRST:  BRST: operator extension does not define a closed algebra  First compensator equation  second via (A)dS Bianchi identity: (AS and Tsulaia, 2003)

33. QG05 - Cala Gonone, Sept. 200533 Compensator Equations (s=3)  Gauge transformations:  Field equations:  Gauge fixing:  Other extra fields: zero by field equations 