Higher Spin Gauge Theories Lecture 1 1-
Introduction Main topic – HS gauge fields Generalization to higher tensor gauge fields of SPIN 1 Y-M gauge potential A n : SPIN 2 metric field g nm : SPIN 3 gravitino ψ nα : 2 Goal: non-Abelian HS gauge symmetries = nonlinear HS gauge interactions 2
Gauge symmetries guarantee consistency both for massless and massive theories like HS gauge theory and String Theory String theory via spontaneous breaking of HS gauge symmetries!? HS Theory evolves to a nonlocal theory with emergent concepts of space-time dimension, metric and local event Example: 4 d massless fields live on a delocalized 3-brane in ten dimen- sions 3
Some Reviews A.Sagnotti, D.Sorokin, P.Sundell, MV: to never appear A. Fotopoulos and M. Tsulaia, 0805.1346 X. Bekaert, S. Cnockaert, C. Iazeolla and MV, hep-th/0503128 MV, hep-th/0401177; 9910096; 9611024 A. Sagnotti, E. Sezgin and P. Sundell, hep-th/0501156 N. Bouatta, G. Compere and A. Sagnotti, hep-th/0409068 D. Sorokin, arXiv:hep-th/0405069 4
Plan Lecture I a Introduction: 1. Free symmetric fields 2. Structure of HS interactions Lecture I b 1. Gravity as a gauge theory 2. Frame-like formulation of massless HS fields 3. Free action 4. Central-On-Shell Theorem 5
Lecture II a 1. Weyl algebra 2. Star product 3. Simplest HS algebra 4. Properties of HS algebras 5. Singletons and AdS/CFT Lecture II b 1. Cubic HS action 2. Unfolded dynamics 3. Equations of motion in all orders 4. 4 d HS fields in ten-dimensional space-time 6
HS fields Symmetric massless HS fields - main subject of these lectures • m � = 0 symmetric fields of any spin: Singh-Hagen (1974) Traceless symmetric tensors φ n 1 ...n s , φ n 1 ...n s − 2 , φ n 1 ...n s − 3 , . . . , φ � �� � supplementary fields • m = 0 symmetric fields of any spin: Fronsdal (1978) φ n 1 ...n s , φ n 1 ...n s − 2 ∼ ϕ n 1 ...n s double traceless η n 1 n 2 η n 3 n 4 ϕ n 1 ...n s = 0 Mixed symmetry fields • m = 0 of any symmetry in flat space Labastida (1989), Skvortsov (2008), Campoleony, Francia, Mourad, Sagnotti (2008) • m = 0 of any symmetry in AdS Brink, Metsaev, MV (2000), Alkalaev, Shaynkman, MV (2003) , N.Boulanger,C.Iazeolla and P.Sundell (2008) , Skvortsov (2009) A lot of particular examples in the literature 7
String String Field Theory: Massive fields of all symmetry types � m s 1 n s 2 ψ m 1 ...m s 1 , n 1 ...n s 2 , ... a m 1 − 1 a n 1 | Ψ � = − 1 . . . a − 2 . . . a − 2 . . . | 0 � Q | Ψ � = 0 equations + constraints δ | ψ � = Q | ε � gauge symmetries: true+Stueckelberg Mass scale m 2 ∼ 1 /α ′ Tensionless limit α ′ → ∞ : All fields become massless High-energy symmetries?! A HS symmetric String Theory = HS gauge theory 8
Fronsdal theory ϕ n 1 ...n s - rank s double traceless symmetric tensor Gauge transformation: ε mmk 3 ...k s − 1 = 0 δϕ k 1 ...k s = ∂ ( k 1 ε k 2 ...k s ) , ( . . . )- symmetrization: A (( a 1 ...a n )) = A ( a 1 ...a n ) . ε k 1 ...k s − 1 is symmetric traceless Comment : δϕ nnmmk 5 ...k s = 0 Field equations G k 1 ...k s ( x ) = 0 , G k 1 ...k s ( x ) = � ϕ k 1 ...k s ( x ) − s∂ ( k 1 ∂ n ϕ k 2 ...k s n ) ( x ) + s ( s − 1) ∂ ( k 1 ∂ k 2 ϕ n k 3 ...k s n ) ( x ) 2 Problem 1.1. Check that G k 1 ...k s is gauge invariant. 9
Analysis of Fronsdal equations δϕ nnm 1 ...m s − 2 ∼ ∂ n ε nm 1 ...m s − 2 choose a partial gauge ϕ nnm 1 ...m s − 2 = 0 ∂ n ε nm 1 ...m s − 2 = 0 ∂ n ∂ m ϕ nm... = 0 By field equation: δ∂ n ϕ nm 1 ...m s − 1 = � ε m 1 ...m s − 1 Taking into account ∂ n ϕ nm 1 ...m s − 1 = 0 choose the gauge Leftover gauge symmetry parameter ε m 1 ...m s − 1 satisfies � ε m 1 ...m s − 1 = 0 ∂ m 1 ε m 1 ...m s − 1 = 0 ε nnm 1 ...m s − 3 = 0 . Field equations � ϕ m 1 ...m s = 0 ϕ nnm 1 ...m s − 2 = 0 ∂ n ϕ nm 1 ...m s − 1 = 0 . 10
Fronsdal action � � 1 � 2 ϕ m 1 ...m s G m 1 ...m s ( ϕ ) − 1 8 s ( s − 1) ϕ nn m 3 ...m s G pp m 3 ...m s ( ϕ ) S = M d Important property ∀ ϕ , δϕ : � � � δϕ m 1 ...m s G m 1 ...m s ( ϕ ) − 1 4 s ( s − 1) δϕ nn m 3 ...m s G pp m 3 ...m s ( ϕ ) δS = M d � � � ϕ m 1 ...m s G m 1 ...m s ( δϕ ) − 1 4 s ( s − 1) ϕ nn m 3 ...m s G pp m 3 ...m s ( δϕ = M d Problem 1.2. prove Gauge variation δS = 0 because δG nm = 0 . s = 0 ϕ scalar s = 1 ϕ n Maxwell potential s = 2 ϕ nm linearized metric 11
Various formulations of massless fields: frame-like, unrestricted, BRST, etc, differ by adding auxiliary fields that are expressed algebraically by their field equa- tions via derivatives of dynamical fields and/or Stueckelberg fields along with Stueckelberg shift gauge symme- tries. Interactions as the most crucial test: frame-like formulation 12
Yang-Mills theory A nij - elements of a Lie algebra l G nm = ∂ n A m − ∂ m A n + g [ A n , A m ] , ε ij ( x ) ∈ l. δA n = ∂ n ε + g [ A n , ε ] , δG nm = g [ G nm , ε ] , Yang-Mills Action � � � S = − 1 S = S Maxw + g A 4 , tr ( G mn G mn ) , A 2 ∂A + g 2 4 � δS = − 1 tr [ G mn G mn , ε ] = 0 . 4 g 13
• The coupling constants are fine tuned field spectra are distinguished: A ij − elements • of a Lie algebra: not any set of fields A n is allowed • interactions to other fields are restricted, requiring covariant deriva- tives ∂ n χ α → D n χ α = ∂ n χ i + A nαβ χ β χ α – some l - module • Cubic vertex contains one derivative 14
Gravity Spin 2: g nm – gauge field Riemann tensor R nm , kl transforms homogeneously under diffeomorphisms δg nm = ∂ n ( ξ k ( x )) g km + ∂ m ( ξ k ( x )) g kn + ξ k ( x ) ∂ k g nm for g nm = η nm + κϕ nm diffeomorphisms provide a nonlinear deformation of the Fronsdal transformation δϕ nm � √ g R S = − 1 Einstein action is a nonliner deformation of the 4 κ 2 Fronsdal action for spin two. Highly restricted field spectrum: only one spin-2 field. Two derivatives in interactions. Interactions via covariant derivatives. ∂ → D = ∂ − Γ − Christoffel connection 15
Goal To find a nonlinear HS theory such that (i) Fronsdal (or Labastuda) theory in the free field limit (ii) HS gauge symmetries related to HS parameters ε m 1 ...m s − 1 deform to non-Abelian These conditions were believed for a long time to admit no solution. S − matrix argument Coleman, Mandula (1967) If symmetry is larger than usual (super)symmetries in Minkowski space- time + inner symmetries the scattering is trivial: no interaction. 16
HS Problem HS-gravity interaction problem Aragone, Deser (1979) ∂ n → D n = ∂ n − Γ n [ D n D m ] = R nm . . . Riemann tensor R nm,kl � = 0 in a curved background. δϕ nm... → D n ε m... � R ... ( ε ... Dϕ ... ) � = 0 δS cov = ?! s ↑ Weyl tensor for s > 2 For s ≤ 2, δS cov contains only the Ricci tensor to be compensated by s the variation of the Einstein action, allowing nonlinear gravity and su- pergravity. For s > 2 , Weyl tensor contributes to δS cov : difficult to achieve HS gauge s symmetry at the nonlinear level. 17
Higher Derivatives in HS Interactions A.Bengtsson, I.Bengtsson, Brink (1983) Berends, Burgers, van Dam (1984) S = S 2 + S 3 + . . . � ( D p ϕ )( D q ϕ )( D r ϕ ) ρ p + q + r + 1 S 3 = 2 d − 3 p,q,r √ α ′ String: ρ ∼ HS Gauge Theories ( m = 0 ): Fradkin, M.V. (1987) AdS d : ( X 0 ) 2 + ( X d ) 2 − ( X 1 ) 2 − . . . − ( X d − 1 ) 2 = ρ 2 , ρ = λ − 1 [ D n , D m ] ∼ ρ − 2 = λ 2 The ρ → ∞ limit is ill-defined at the interaction level both in string theory and in HS gauge theory 18
HS Fields in AdS Background Anti-de Sitter space: R mn,kl = R mn,kl − λ 2 ( g mk g nl − g ml g nk ) R mn,kl = 0 , ρ = λ − 1 is AdS d radius. Symmetry: o (2 , d − 1) To preserve HS gauge symmetries of massless fields, mass-like terms have to be adjusted in terms of λ L flat = ∂ϕ∂ϕ → L AdS = DϕDϕ + λ 2 ϕϕ For general mixed symmetry fields it is impossible to keep all flat space HS gauge symmetries unbroken in AdS background Metsaev (1995), Brink, Metsaev, M.V. (2000) 19
Role of AdS Background in HS Theories Near AdS: expansion in powers of the shifted Riemann tensor R mn,kl (which is zero in the AdS space) rather than in powers of the Riemann tensor R [ D n , D m ] ∼ λ 2 ∼ O (1) + O ( R ) . The action is modified by cubic terms � � S int = λ − ( p + q ) D p ( ϕ ) D q ( ϕ ) R M 4 p,q which contain higher derivatives along with negative powers of λ . There exists such S int that its HS gauge variation compensates the nonzero gauge variation of the free covariantized action. ( Fradkin, M.V. (1987) ) For given spin, a highest order of derivatives in a vertex is finite increas- ing linearly with spin. 20
Spin 3 example MV ( ( ≤ 1986) ), unpublished , Zinoviev (2008) δϕ mnk = D ( m ǫ nk ) + . . . � ( DϕDϕ + λ 2 ϕ 2 ) S 33 = S 332 = λ − 2 S 2 332 + S 0 332 � � S 2 D 2 ( ϕϕR ) , S 0 332 = 332 = ϕϕR � � � δS 2 [ D, D ] D ( ϕǫR ) ∼ λ 2 D ( ϕǫR ) 332 = [ D n , D m ] ∼ λ 2 ∼ O (1) + O ( R ) λ − 2 δS 2 δS 33 + δS 0 332 compensates 332 For analogous analysis for s = 5 / 2 see Sorokin (2004) 21
Recommend
More recommend
