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Higher-Spin Geometry and String Theory

Higher-Spin Geometry and String Theory. Augusto SAGNOTTI Universita’ di Roma “Tor Vergata”. 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.

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Higher-Spin Geometry and String Theory

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  1. Higher-Spin Geometryand String Theory Augusto SAGNOTTI Universita’ di Roma “Tor Vergata” • 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 QG05 – Cala Gonone, September, 2005

  2. Plan • The (Fang-) Fronsdal equations • Non-local geometric equations • Local compensator forms • Off-shell extensions • Role in the Vasiliev equations QG05 - Cala Gonone, Sept. 2005

  3. The Fronsdal equations (Fronsdal, 1978) Gauge invariance for massless symmetric tensors: Originally from massive Singh-Hagen equations (Singh and Hagen, 1974) Unusual constraints: QG05 - Cala Gonone, Sept. 2005

  4. Bianchi identities Why the unusual constraints: • Gauge variation of F • Gauge invariance of the Lagrangian • As in the spin-2 case, F not integrable • Bianchi identity: QG05 - Cala Gonone, Sept. 2005

  5. Constrained gauge invariance If in the variation of L one inserts: Are these constraints really necessary? QG05 - Cala Gonone, Sept. 2005

  6. The spin-3 case (Francia and AS, 2002) A fully gauge invariant (non-local) equation: Reduces to local Fronsdal form upon partial gauge fixing QG05 - Cala Gonone, Sept. 2005

  7. Spin 3: other non-local eqs Other equivalent forms: Lesson: full gauge invariance with non-local terms QG05 - Cala Gonone, Sept. 2005

  8. Kinetic operators Index-free notation: Now define: Then: QG05 - Cala Gonone, Sept. 2005

  9. Kinetic operators Defining: • generic kinetic operator for higher spins • when combined with traces: QG05 - Cala Gonone, Sept. 2005

  10. Kinetic operators The F(n): • Are gauge invariant for n > [(s-1)/2] • Satisfy the Bianchi identities • For n> [(s-1)/2] allow Einstein-like operators QG05 - Cala Gonone, Sept. 2005

  11. Geometric equations Christoffel connection: Generalizes to all symmetric tensors (De Wit and Freedman, 1980) QG05 - Cala Gonone, Sept. 2005

  12. Geometric equations (Francia and AS, 2002) • Odd spins (s=2n+1): • Evenspins (s=2n): QG05 - Cala Gonone, Sept. 2005

  13. Bosonic string: BRST • The starting point is the Virasoro algebra: • In the tensionless limit, one is left with: • Virasoro contracts (no c. charge): QG05 - Cala Gonone, Sept. 2005

  14. String Field equation • Higher-spin massive modes: • massless for 1/a’  0 • Free dynamics can be encoded in: (Kato and Ogawa, 1982) (Witten, 1985) (Neveu, West et al, 1985) NO trace constraints on y or L QG05 - Cala Gonone, Sept. 2005

  15. Low-tension limit • Similar simplifications hold for the BRST charge: • With zero-modes manifest: QG05 - Cala Gonone, Sept. 2005

  16. Symmetric triplets (A. Bengtsson, 1986) (Henneaux,Teitelboim, 1987) (Pashnev, Tsulaia, 1998) (Francia, AS, 2002) (AS, Tdulaia, 2003) • Emerge from • The triplets are: QG05 - Cala Gonone, Sept. 2005

  17. (A)dS symmetric triplets • Directly, deforming flat-space triplets, or via BRST (no Aragone-Deser problem) • Directly: insist on relation between C and others • BRST: gauge non-linear constraint algebra • Basic commutator: QG05 - Cala Gonone, Sept. 2005

  18. Compensator Equations • In the triplet: • spin-(s-3) compensator: • The first becomes: • The second becomes: • Combining them: • Finally (also Bianchi): QG05 - Cala Gonone, Sept. 2005

  19. (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 QG05 - Cala Gonone, Sept. 2005

  20. Off-Shell Compensator Equations (AS and Tsulaia, 2003) • Lagrangian form of compensator: BRST techniques • 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: Gauge fixing  QG05 - Cala Gonone, Sept. 2005

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

  22. The Vasiliev equations (Vasiliev, 1991-2003;Sezgin,Sundell, 1998-2003) • Integrablecurvature constraints on one-forms and zero-forms • Cartan integrable systems • Key new addition of Vasiliev: twisted-adjoint representation (D’Auria,Fre’, 1983) • Minimal case (only symmetric tensors of even rank), Sp(2,R) • zero-form F : Weyl curvatures • one-form A : gauge fields QG05 - Cala Gonone, Sept. 2005

  23. The Vasiliev equations • Curvature constraints: • [extra non comm. Coords] • Gauge symmetry: QG05 - Cala Gonone, Sept. 2005

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

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

  26. The Vasiliev equations • Non-linear corrections: from dependence on internal Z- coordinates • Does the projection that “leaves” the compensators produce singular interactions? • Vasiliev: works with traceless conditions all over and feels it does • My feeling: eventually not, and we are seeing a glimpse of the off-shell form More work will tell us…. QG05 - Cala Gonone, Sept. 2005

  27. The End QG05 - Cala Gonone, Sept. 2005

  28. Fermions (Francia and AS, 2002) Notice: Example: spin 3/2 (Rarita-Schwinger) QG05 - Cala Gonone, Sept. 2005

  29. Fermions One can again iterate: The relation to bosons generalizes to: The Bianchi identity generalizes to: QG05 - Cala Gonone, Sept. 2005

  30. Fermionic Triplets (Francia and AS, 2003) • Counterparts of bosonic triplets • GSO: not in 10D susy strings • Yes: mixed sym generalizations • Directly in type-0 models • Propagate s+1/2 and all lower ½-integer spins QG05 - Cala Gonone, Sept. 2005

  31. Fermionic Compensators (recently, also off shell Buchbinder,Krykhtin,Pashnev, 2004) • Recall: • Spin-(s-2) compensator: • Gauge transformations: First compensator equation  second via Bianchi QG05 - Cala Gonone, Sept. 2005

  32. Fermionic Compensators (AS and Tsulaia, 2003) • We could extend the fermionic compensator eqs to (A)dS • We could not extend the fermionic triplets • BRST: operator extension does not define a closed algebra • First compensator equation  second via (A)dS Bianchi identity: QG05 - Cala Gonone, Sept. 2005

  33. Compensator Equations (s=3) • Gauge transformations: • Field equations: • Gauge fixing: • Other extra fields: zero by field equations  QG05 - Cala Gonone, Sept. 2005

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