A Higher Derivative Extension of the Salam-Sezgin Model from Superconformal Methods

Transcription

1 A Higher Derivative Extension of the Salam-Sezgin Model from Superconformal Methods Frederik Coomans KU Leuven Workshop on Conformal Field Theories Beyond Two Dimensions 16/03/2012, Texas A&M Based on work with E. Bergshoeff, E. Sezgin and A. Van Proeyen Frederik Coomans (KU Leuven) 16/03/ / 36

2 Introduction Possible SUGRA theories for a given spacetime dimension and number of supersymmetry generators [Van Proeyen, Freedman, 2012] Frederik Coomans (KU Leuven) 16/03/ / 36

3 Introduction Method For D = 4, 5, 6 and #Q 16 (i.e when there are matter multiplets) matter-coupled SUGRA actions can be constructed via superconformal tensor calculus (SCTC) What? Use superconformal symmetry as a tool to construct SUGRA theories (which are only invariant under super Poincaré). Construct superconformal theory by coupling a compensating matter multiplet to the Weyl multiplet (gauge multiplet of the superconformal group). Compensator multiplet compensates for the redundant conformal symmetries. Frederik Coomans (KU Leuven) 16/03/ / 36

4 Introduction Method For D = 4, 5, 6 and #Q 16 (i.e when there are matter multiplets) matter-coupled SUGRA actions can be constructed via superconformal tensor calculus (SCTC) Why? (advantages over other methods like Noether method, superspace,...) Conformal symmetry severely restricts # couplings that can be written Extra symmetry gives inside in the structure of the theory Different compensators give rise to different formulations of the Poincaré theory Easy to construct off-shell actions Frederik Coomans (KU Leuven) 16/03/ / 36

5 Introduction Use of SCTC for minimal D = 6 SUGRA: [Bergshoeff, Sezgin, Van Proeyen, 1986]: Weyl & matter multiplets, action formulas [Van Proeyen, FC, 2011]: Complete off-shell pure SUGRA action [Bergshoeff, Rakowski, 1987 & Bergshoeff, Sezgin, Salam, 1987]: Off-shell supersymmetric Riem 2 action In this talk: [Bergshoeff, Sezgin, Van Proeyen, FC, 2012]: Gauge U(1) R-symmetry of pure theory Add Riem 2 action and study gauging procedure in presence of higher derivative terms Study solutions of gauged higher derivative action Frederik Coomans (KU Leuven) 16/03/ / 36

6 Introduction Why interest in higher-derivative terms? Appear as α corrections in effective action of string theory Corrections to black hole entropy (much progress in D = 4, 5 not yet in D = 6) Compactification to D = 3: make graviton a (massive) propagating mode (e.g. [Bergshoeff et al, 2010]) Frederik Coomans (KU Leuven) 16/03/ / 36

7 Outline 1 SUSY in D = 6 2 Construction of actions via superconformal tensor calculus -Construction of the pure action -Coupling to a vector multiplet and gauging of the R-symmetry -Connection to Salam-Sezgin -Construction of R 2 -action -The total Lagrangian 3 Vacuum solutions of the gauged R + R 2 Lagrangian 4 Conclusions and outlook Frederik Coomans (KU Leuven) 16/03/ / 36

8 1. SUSY in D = 6 Frederik Coomans (KU Leuven) 16/03/ / 36

9 Spinors in minimal D = 6 SUGRA Irreducible spinor has 8 real components Is Weyl spinor: λ = P L λ or λ = P R λ Symplectic Majorana condition: λ i = ɛ ij (λ j ) C Minimal SUSY algebra has 8 supercharges (like N = 2 in D = 4) Pair of SUSY generators Q i α = {Q 1 α, Q 2 α} of the same chirality, hence N = (1, 0) R-symmetry group is SU(2) Transformations between the SUSY parameters ɛ i = {ɛ 1, ɛ 2 } preserving the symplectic structure are SU(2) transformations Frederik Coomans (KU Leuven) 16/03/ / 36

10 p-form gauge fields in minimal D = 6 SUGRA S p = 1 2 F (p+1) F (p+1), F (p+1) = da (p) Reducible gauge symmetry δa (p) = dθ (p 1) Watch out when counting degrees of freedom! Degrees of freedom Off-shell: as antisymmetric tensor in SO(5) On-shell: as antisymmetric tensor in SO(4) p-forms are dual to (D p 2) = (4 p)-forms, hence 2-forms are selfdual Frederik Coomans (KU Leuven) 16/03/ / 36

11 Off-shell vs on-shell multiplets SUSY theories are built up from SUSY multiplets (i.e. field representations of the SUSY algebra) off-shell multiplets: SUSY-algebra closes on the fields of the multiplet; # off-shell bosonic d.o.f. = # off-shell fermionic d.o.f. on-shell multiplets: SUSY-algebra only closes on the fields of the multiplet modulo EOM; # on-shell bosonic d.o.f. = # on-shell fermionic d.o.f. Sum of two off-shell actions is again off-shell supersymmetric; no modification of the SUSY rules necessary! Frederik Coomans (KU Leuven) 16/03/ / 36

12 2. Construction of actions via superconformal tensor calculus Frederik Coomans (KU Leuven) 16/03/ / 36

13 Gravity as a conformal gauge theory P a M ab D K a ξ a λ ab λ D λ a K e a µ ω ab µ b µ f a µ Constraints determine two gauge fields R µν (P a ) = 0 = ω µ ab = ω µ ab (e, b) e ν br µν (M ab ) = 0 = f µ a = f µ a (e, b) Weyl multiplet : e µ a, b µ Frederik Coomans (KU Leuven) 16/03/ / 36

14 Gravity as a conformal gauge theory Use scalar field φ as compensator Conformal gravity: L C = gφ C φ = gφ φ 1 6 grφ Gauge fixing, δ(λ K )b µ = e µa λ a K = special conformal gauge fixing: b µ = 0, δ(λ D )φ = λ D φ = dilatational gauge fixing: φ = 3M P, leads to EH action: L = M2 P gr 2 Frederik Coomans (KU Leuven) 16/03/ / 36

15 Gravity as a conformal gauge theory Frederik Coomans (KU Leuven) 16/03/ / 36

16 Minimal D = 6 SUGRA P a M ab D K a SU(2) Q i S i ξ a λ ab λ D λ a K Λ ij ɛ i η i e µ a ω µ ab b µ f µ a V ij µ ψ µ i φ µ i Constraints determine ω µ ab, f µ a and φ µ i in terms of the others Weyl multiplet: e µ a, b µ, V ij µ, σ, B µν, ψ µ i, ψ i PS: there is also another choice of extra fields, i.e. another Weyl multiplet, but this one is chosen to obtain an invariant action Frederik Coomans (KU Leuven) 16/03/ / 36

17 Minimal D = 6 SUGRA Compensating multiplet: linear multiplet (off-shell, SC action known) Field Off-shell dof On-shell dof L ij 3 3 E µνρσ 5 1 ϕ i 8 4 We know: superconformal action for coupling of vector and linear multiplet plus embedding of linear into vector multiplet = We know: superconformal action for linear multiplet Gauge fixing: L ij = 1 2 δ ij, ϕ i = 0, b µ = 0 fixes D, SU(2)/U(1), K, S Frederik Coomans (KU Leuven) 16/03/ / 36

18 Minimal D = 6 SUGRA Weyl Linear Gauge fixing off-shell Poincaré e µ a (15) P a, M ab e µ a (15) P a, M ab 9 b µ (0) K a b µ = 0 K a V µ i j (15) SU(2) V µ i j (17) SO(2) σ (1) σ (1) 1 B µν (10) Λ µ B µν (10) Λ µ 6 dilatations ( 1) L ij (3) L ij = 1 δ ij 2 D, SU(2)/SO(2) E µνρσ (5) Λµνρ E µνρσ (5) Λµνρ ψ i µ (40) Q i ψ i µ (40) Q i 12 ψ i (8) ψ i (8) 4 S-susy ( 8) ϕ i (8) ϕ i = 0 S i Frederik Coomans (KU Leuven) 16/03/ / 36

19 Minimal D = 6 SUGRA e 1 L L=1 R = 1 2 R 1 2 σ 2 µσ µ σ 1 24 σ 2 F µνρ(b)f µνρ (B) + V µij V µij 1 4 E µ E µ + 1 E µ V µ ψ µγ µνρ D νψ ρ 2σ 2 ψγ µ D µψ +... [Van Proeyen, FC, 2011] PS: we split the gauge field V ij µ = V ij µ δij V µ into traceless plus trace and denote E µ as the dual of the 4-form field strength Frederik Coomans (KU Leuven) 16/03/ / 36

20 Gauging the theory To obtain R-symmetry gauging we add a vector multiplet: Field Off-shell dof On-shell dof W µ 5 4 Y ij 3 0 Ω i 8 4 -superconformal invariant action (includes also fields of the Weyl multiplet) e 1 L V = σ ( 1 ) 4 Fµν(W )F µν (W ) 2 Ωγ µ D µ(ω)ω + Y ij Y ij -coupling with linear multiplet 1 16 e 1 ε µνρσλτ B µνf ρσ(w )F λτ (W ) +... e 1 L VL = Y ij L ij + 2 Ωϕ L ij ψµi γ µ Ω j Wµ ( E µ ψ νγ νµ ϕ ) [Bergshoeff, Sezgin, Van Proeyen, 1986] Frederik Coomans (KU Leuven) 16/03/ / 36

21 Gauging the theory L 1 = (L R + L V + gl VL ) L=1 Contains terms σy ij Y ij + g 2 δ ij Y ij V = 1 4 g 2 σ 1 (after solving for Y ij ) Has U(1) R U(1) gauge symmetry E µνρσ field equation E µ = µ φ + 2V µ + gw µ V µ field equation E µ + (fermion bilinears) = 0 δ gauge φ = 2λ gη, hence fixing the scalar φ = φ 0 implies U(1)R U(1) U(1) diag R 2Vµ + gw µ = 0 (bosonically) Frederik Coomans (KU Leuven) 16/03/ / 36

22 To the Salam-Sezgin model... U(1) gauged Einstein-Maxwell SUGRA in D = 6, spontaneously compactifies on Mink 4 S 2 breaking half of the supersymmetries [Salam, Sezgin, 1984] e 1 L SS = 1 2 R 1 2 σ 2 aσ a σ 1 4 g 2 σ σ2 G µνρg µνρ 1 4 σfµν(w )F µν (W ) +... on-shell model U(1) vector appears inside the 2-form field strength: G (3) = d B (2) + dw (1) W (1) How does our model, described by L 1, relate to the SS model? Frederik Coomans (KU Leuven) 16/03/ / 36

23 To the Salam-Sezgin model... After elimination of auxiliary fields our model becomes e 1 L on-shell = 1 2 R 1 2 σ 2 µσ µ σ 1 4 g 2 σ σ 2 F µνρ(b)f µνρ (B) 1 4 σfµν(w )F µν (W ) e 1 ε µνρσλτ F µνρ(b)f λτ (W )W σ +... To obtain the SS-model we have to dualize the 2-form B µν. We can do this by adding a Lagrange multiplier 1 24 εµνρσλτ F µνρ (B) σ Bλτ solving for F µνρ (B) Frederik Coomans (KU Leuven) 16/03/ / 36

24 An alternative off-shell formulation Other gauge fixing, σ = 1, L ij = 1 2 δ ij L, ψ i = 0, b µ = 0 amounts to L 2 = (L R + L V + gl VL ) σ=1 Essentially: σ and ψ i replaced by L and ϕ i Frederik Coomans (KU Leuven) 16/03/ / 36

25 The Riem 2 invariant Trick from [Bergshoeff, Rakowski, 1987 & Bergshoeff, Sezgin, Salam, 1987]: Embed Weyl multiplet in non-abelian vector multiplet Denote ω µ ab = ω µ ab 1 2 F µ ab (B) (bosonic torsion) In the gauge σ = 1 : ( ) ) 2 ω ab µ, R abi (Q), 2 F abij (V) (W I µ, Ω I i, Y I ij Lagrangian for the non-abelian vector multiplet is known [Bergshoeff, Sezgin, Van Proeyen, 1986]: e 1 L YM σ=1 = 1 4 Fµν I (W )F µνi (W ) 2 Ω I γ µ D µ(ω)ω I + Y Iij Y I ij 1 16 e 1 ε µνρσλτ B µνf I ρσ(w )F I λτ (W ) FνρI Ω I γ µ γ νρ ψ µ Fµνρ(B) Ω I γ µνρ Ω I Frederik Coomans (KU Leuven) 16/03/ / 36

26 The Riem 2 invariant Making the substitution amounts to e 1 L R 2 σ=1 = R ab µν (ω )R µν ab(ω ) 2F ab (V)F ab (V) 4F abij (V)F abij (V) e 1 ε µνρσλτ B µνr ρσ ab (ω )R λτ ab (ω ) +2 R +ab (Q)γ µ D µ(ω, ω )R+ ab (Q) Rνρab (ω ) R +ab (Q)γ µ γ νρ ψ µ ( 8F µν ij (V) ψ µ i γ λ R+j λν (Q) + 1 ) 6 ψ µ i γ F (B)ψj ν 1 ab R + 12 (Q)γ F (B)R +ab(q) 1 [ D µ(ω, Γ +)R µρab (ω ) 2 2F µν ρ (B)R µνab (ω )] ψ aγ ρψ b Frederik Coomans (KU Leuven) 16/03/ / 36

27 The total Lagrangian L tot = (L R + L V + gl VL 1 8M 2 L R 2) σ=1 Off-shell: every term is seperately invariant! (no 1/M 2 corrections to the transformation rules) e 1 L tot = 1 2 LR glδ ij Y ij + Y ij Y ij L 1 µl µ L 1 24 LFµνρ(B)F µνρ (B) +LV a kl V a kl 1 4 L 1 E µe µ E µ( V µ gw µ ) 1 4 Fµν(W )F µν (W ) 1 16 e 1 ε µνρσλτ B µνf ρσ(w )F λτ (W ) 1 [ 8M 2 R ab µν (ω )R µν ab(ω ) 2F ab (V)F ab (V) 4F abij (V )F abij (V ) + 1 ] 4 e 1 ε µνρσλτ B µνr ab ρσ (ω )R λτ ab (ω ) +... Frederik Coomans (KU Leuven) 16/03/ / 36

28 3. Vacuum solutions Frederik Coomans (KU Leuven) 16/03/ / 36

29 Perturbative or Toy Model L R 2 contains kinetic terms for some auxiliaries! Can we still eliminate them? 1 Consider 1/M 2 as a small parameter and eliminate auxiliaries perturbatively SUSY only order by order in parameter 1/M 2 Open question: does this on-shell Lagrangian correspond to the one obtained by compactifying the effective heterotic string Lagrangian? 2 Consider 1/M 2 as an arbitrary (not necessarily small) parameter Propagating auxiliaries give rise to ghosts Consider theory as a toy model with exact SUSY We will take this approach when studying solutions of the theory Frederik Coomans (KU Leuven) 16/03/ / 36

30 Vacuum solutions Only consider bosonic field equations (background fermions vanish) Lagrangian without higher derivative terms Mink4 S 2, preserving half of the supersymmetries (Salam-Sezgin) No Mink 6 or (A)dS solution Lagrangian with higher derivative terms Elimination of Y ij, E µνρσ still possible Elimination of V ij µ, V µ no longer possible since they acquired kinetic terms Solutions without fluxes Solutions with 2-form flux or 3-form flux Frederik Coomans (KU Leuven) 16/03/ / 36

31 Vacuum solutions without fluxes Only non-vanishing fields are metric and L = L 0 Mink 6 only a solution if we switch off gauging g = 0, since R = g 2 L 0 For g 0 only solutions of the form M D M 6 D, i.e. consider Ansatz R µνρσ = n 1 g 2 L 0 (g µρ g νσ g µσ g νρ ), R pqrs = n 2 g 2 L 0 (g pr g qs g ps g qr ), L = L 0, M 2 = n 3 g 2 All these solutions are non-susy Spacetime n 1 n 2 n 3 Mink 4 S 2 0 1/2 1/2 ds 4 T 2 1/12 0 1/12 ds 4 S 2 1/14 1/14 1/14 Mink 3 S 3 0 1/6 1/6 ds 3 T 3 1/6 0 1/6 ds 3 S 3 1/12 1/12 1/12 Frederik Coomans (KU Leuven) 16/03/ / 36

32 Vacuum solutions with 2-form flux Consider solutions of the form M 4 M 2 Metric and L = L 0 non-vanishing and V µ, W µ have fluxes on M 2 Consider Ansatz R µν = 3a g µν, R rs = b g rs, L = L 0, F rs (W ) = c g 2 ε rs, F rs (V) = g 2 c g 2 ε rs, Mink 4 S 2 is still a solution, preserving half of the supersymmetries! Other solutions include AdS4 S 2, ds 4 S 2, ds 4 H 2 (all non-susy) Frederik Coomans (KU Leuven) 16/03/ / 36

33 Vacuum solutions with 3-form flux Consider solutions of the form M 3 M 3 Metric and L = L 0 non-vanishing and B µν has fluxes on both M 3 s Consider Ansatz R µν ρσ = 2a δ ρ [µ δσ ν], R pq rs = 2b δ r [p δs q], L = L 0, F µνρ (B) = 2c g 1 ε µνρ, F rst (B) = 2c g 2 ε rst c 2 = a = b: AdS 3 S 3, preserving full susy Other solutions include AdS3 S 3, ds 3 S 3, ds 3 H 3 (all non-susy) Frederik Coomans (KU Leuven) 16/03/ / 36

34 4. Conclusions and outlook Frederik Coomans (KU Leuven) 16/03/ / 36

35 Conclusions Use of superconformal calculus to construct minimal D = 6 R-symmetry gauged supergravity with higher derivative (Riem ) terms All parts of the action seperately off-shell Auxiliaries can be eliminated perturbatively; correspondence with compactified string Lagrangian? Potential is not modified by Riem 2 -terms (no couplings with Y ij in L R 2) Supersymmetric Mink 4 S 2 solution is still valid Also other solutions found Frederik Coomans (KU Leuven) 16/03/ / 36

36 Outlook D = 6 is highest dimension that allows off-shell formulation: worthwile to investigate further Adding matter couplings (Yang-Mills multiplets, hypermultiplets) Anomalies (Lorentz CS term is part of the Riem 2 -invariant) Computation of the spectrum in Mink 4 S 2 Existence of other higher curvature invariants in D = 6? Higher derivative terms contribute corrections to the BH entropy; are important for connection microscopic/macroscopic entropy In D = 4, N = 2 [Lopes, Cardoso, de Wit, Mohaupt, 2004] In D = 5, N = 2 [de Wit, Katmadas, 2011] Need to find BH solutions! so far only BH solutions for ungauged theory without higher derivative terms [Gibbons, Maeda, 1988] Frederik Coomans (KU Leuven) 16/03/ / 36