Recent Progress on Curvature Squared Supergravities in Five and Six Dimensions

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1 Recent Progress on Curvature Squared Supergravities in Five and Six Dimensions Mehmet Ozkan in collaboration with Yi Pang (Texas A&M University) hep-th/ April 24, 2013 Mehmet Ozkan () April 24, / 41

2 Contents 1 Motivation and Background 2 Superconformal Multiplets in 5D 3 Off-Shell Poincaré Supergravities 4 All Higher Derivative Extensions of 5D Model 5 Ricci Scalar Squared Extended 6D Model 6 Summary Mehmet Ozkan () April 24, / 41

3 Higher Order Curvature Terms in Supergravity String Theory At low energies, superstring theory reduce to supergravity Superstring actions originate higher curvature terms as well as Einstein-Hilbert supergravities Supergravity Inclusion of Lorentz-Chern-Simons 3-form for the cancellation of anomalies in YM and gravitational gauge currents in 10D SUGRA coupled to YM [GS] Modify the definition of H H = db + (AF A 3 ) (Rω ω 3 ) Mehmet Ozkan () April 24, / 41

4 Higher Order Curvature Terms in Supergravity Restore Supersymmetry Supersymmetric completion of LCS term is needed String Theory: String amplitues in tree level and one loop level Supergravity: Superspace methods and Noether methods Result: R µνρσ R µνρσ Problems Modified Lagrangian contains ghost particles [Siegel 84] Gauss - Bonnet combination is ghost free [Zwiebach 85] e 1 L = R µνρσ R µνρσ 4R µν R µν + R 2 Modifications to vacuum solutions? Mehmet Ozkan () April 24, / 41

5 Higher Order Curvature Terms in Supergravity Solution String theory is on-shell! Field Redefinition Higher curvature terms take the form of infinite series expansion as the on-shell supersymmetry works only order by order Only the coefficient of R µνρσ R µνρσ has a definite meaning A field redefinition is always possible g µν = g µν + ar µν + bg µν R Can shift the coefficients of R µν R µν and R 2 to arbitrary values Mehmet Ozkan () April 24, / 41

6 Higher Order Curvature Terms in Supergravity Exact Supersymmetric Models Off-shell formulations needed Full answer to exact supersymmetric models was achieved in D = 4, N = 1 Riem 2 action was constructed in 5 and 6 dimensions Weyl 2 action was constructed in 5 dimensions Mehmet Ozkan () April 24, / 41

7 Motivation More Motivation Higher order effects in AdS/CFT Corrections to black hole entropy Today s Objectives Off-shell Poincaré Supergavities in Five Dimensions All supersymmetric curvature squared invariants in Five Dimensions Off-shell R 2 invariant in Six Dimensions Brief discussion on supersymmetric f (R) theory Mehmet Ozkan () April 24, / 41

8 Current Status Status of D = 5, N = 2 Supergravity Off-Shell Poincaré Supergravity [Bergshoeff et. al., Fujita & Ohashi, Zucker, Coomans & Ozkan] Off-Shell Weyl Squared Invariant [Hanaki, Ohashi & Tachikawa] Off-Shell Riemann Squared Invariant [Bergshoeff, Rosseel & Sezgin] Status of D = 6, N = (1, 0) Supergravity Off-Shell Poincaré Supergravity [Bergshoeff, van Proeyen & Sezgin] Off-Shell Riemann Squared Invariant [Bergshoeff, Salam & Sezgin] Mehmet Ozkan () April 24, / 41

9 Superconformal Tensor Calculus Mehmet Ozkan () April 24, / 41

10 Toy Model: Proca Action from a U(1) Invariant Action Proca Action Proca Action e 1 L Proca = 1 4 G µνg µν 1 2 m2 C µ C µ Mass term forbids the local U(1) symmetry U(1) Invariant Action Introduce a compensating scalar Redefine the Proca field δ Λ σ = mλ C µ = A µ m 1 µ σ, δ Λ A µ = µ Λ Mehmet Ozkan () April 24, / 41

11 Toy Model: Proca Action from a U(1) Invariant Action U(1) Invariant Action U(1) invariant action e 1 L U(1) = 1 4 F µνf µν 1 2 D µσd µ σ, where D µ σ = µ σ ma µ Fix U(1) symmetry via σ = 0 Proca action Mehmet Ozkan () April 24, / 41

12 Superconformal Multiplets Mehmet Ozkan () April 24, / 41

13 Weyl Multiplets of N = 2 Supergravity in Five Dimensions N = 2, D = 5 Superconformal tensor calculus is based on the superconformal algebra F 2 (4) Generators and Gauge Fields P a e a µ Translations M ab ω ab µ Rotations D b µ Dilatations K a f µ a Special conformal Λ ij V ij µ SU(2) R-Symmetry Q i ψµ i Susy S i φ i µ S-Susy Mehmet Ozkan () April 24, / 41

14 Weyl Multiplets of N = 2 Supergravity in Five Dimensions Conventional Constraints R µν (P a ) = 0 e ν b R µν (M ab ) = 0 Identifies ω ab µ Identifies f a µ γ µ Ri µν (Q) = 0 Identifies φ i µ Independent Gauge Fields e µ a b µ ψ i µ V µ ij Independent gauge fields have 21(bosonic) + 24 (fermionic) degrees of freedom. Cannot represent a superconformal multiplet Mehmet Ozkan () April 24, / 41

15 Weyl Multiplets of N = 2 Supergravity in Five Dimensions Additional matter fields must be added to gauge fields in order to obtain an off-shell closed multiplet There are two choices [Bergshoeff et al] Standard Weyl Multiplet e µ a b µ ψ i µ V µ ij T µν (10) D(1) χ i (8) Dilaton Weyl Multiplet e µ a b µ ψ i µ V µ ij B µν (6) C µ (4) σ(1) ψ i (8) Construction of an off-shell Gauss - Bonnet combination requires Dilaton Weyl Multiplet Mehmet Ozkan () April 24, / 41

16 Weyl Multiplets of N = 2 Supergravity in Five Dimensions Map Between the Weyl Multiplet χ i = 1 8 iσ 1 /Dψ i iσ 2 /Dσψ i 1 32 σ 2 γ Gψ i + D = 1 32 R σ 2 µ σ µ σ 1 16 σ 2 G µν G µν +... T ab = 1 8 σ 1 G ab σ 2 ε abcde H cde +... Mehmet Ozkan () April 24, / 41

17 Compensating Multiplets: The Vector Multiplet Vector Multiplet Field Weyl Weight A µ 0 ρ 1 Y ij 2 λ i 3 2 Contains a gauge field A µ with gauge invariance µ Λ Scalar is S-invariant δ s ρ = 0 Mehmet Ozkan () April 24, / 41

18 Compensating Multiplets: The Linear Multiplet Linear Multiplet Field Weyl Weight L ij 3 ϕ i 7 2 E a 4 N 4 Contains a conserved current D a E a = 0 E a = 1 12 e µ a e 1 ε µνρσλ D ν E ρσλ Invariant Action Formula: e 1 L VL = Y ij L ij + ρn + A a E a Mehmet Ozkan () April 24, / 41

19 Intermezzo: How to construct off-shell actions? Vector - Linear Action e 1 L VL = Y ij L ij + ρn + A a P a Embedding Formulae Define the elements of Vector (or Linear) multiplet in terms of other multiplets Use these expressions in the Vector - Linear action (or any other superconformal action) to obtain a superconformal action Fix the redundant superconformal symmetries Mehmet Ozkan () April 24, / 41

20 Off-Shell Poincaré Supergravity Mehmet Ozkan () April 24, / 41

21 Construction of the Linear Multiplet Action Identify the elements of Vector Multiplet in terms of Linear Multiplet Step 1: Composite expressions for ρ, λ i, Y ij and F µν ρ = 2L 1 N + il 3 L ij ϕ i ϕ j λ i =... Y ij = L 1 C L ij D a L k(i D a L j)k L km L 3 N 2 L ij L 3 E µ E µ L ij L T 2 L 1 L ij + 4L 1 DL ij + 2E µ L k(i D µ L k j) L F µν = 4D [µ (L 1 E ν] ) + 2L 1 Rµν ij (V )L ij 2L 3 L l k D [µl kp D ν] L lp +... Mehmet Ozkan () April 24, / 41

22 Construction of the Linear Multiplet Action Construct a superconformal action Step 2: Use L VL to construct Linear Multiplet action e 1 L L = L 1 L ij C L ij L ij D a L k(i D a L j)k L km L 3 N 2 L 1 E µ E µ L LT 2 + 4DL 1 2 L 3 E µν L l k µl kp ν L pl +2E µν µ (L 1 E ν + V ij ν L ij L 1 ) L ij c L ij = L ij L ij L2 R D = 1 32 R +... Can describe a consistent Poincaré supergravity upon gauge fixing Mehmet Ozkan () April 24, / 41

23 Construction of the Poincaré Theory Gauge Fixing L ij = 1 2 δ ij L σ = 1 b µ = 0 ψ i = 0 Breaks SU(2) to U(1) R Dilatation Conformal boost S - Supersymmetry Off-Shell Poincaré Supergravity e 1 L LR σ=1 = 1 2 LR 1 4 LG µνg µν 1 6 LH µνρh µνρ 1 2 L 1 µ L µ L +L 1 N 2 L 1 E µ E µ + 2E µ V µ + LV ij µ V µ ij Mehmet Ozkan () April 24, / 41

24 Construction of the Poincaré Theory Remarks Decomposed V ij µ into its trace and traceless part V ij µ = V ij µ V µ δ ij, V ij µ δ ij = 0 U(1) R symmetry is gauged by the auxiliary V µ Corresponding on-shell theory is the Ungauged Maxwell - Einstein theory [Gunaydin - Sierra - Townsend] Mehmet Ozkan () April 24, / 41

25 Off-Shell Higher Derivative Extensions Riemann Squared Invariant [ Bergshoeff - Rosseel - Sezgin ] Weyl Tensor Squared Invariant [C µνρσ C µνρσ R2 ] Ricci Scalar Squared Invariant Mehmet Ozkan () April 24, / 41

26 Construction of the Riemann Squared Action Riemann Squared Action - Key Observation Based on a map between the Yang - Mills multiplet and Dilaton Weyl multiplet Riemann Squared Action - Construction Procedure Step 1: Establish a map between the Yang-Mills multiplet and Dilaton Weyl multiplet [Bergshoeff - Rosseel - Sezgin 11] ( ) ( A I µy ij I, λ i I, ρ I ω +µ, ab ij V ab, ψ ) ab i, Ĝab ω +µ ab is torsionful spin connection ω ab +µ = ω ab ab µ + Ĥµ Mehmet Ozkan () April 24, / 41

27 Construction of the Riemann Squared Action Step 2: Construction of the Yang-Mills Action Linear Multiplet Dilaton Weyl Multiplet Yang-Mills Multiplet L ij = σy ij +... ϕ i = 1 2 iσ /Dλ i +... E a = D b ( 1 2 σ F ab +...) 1 8 ɛabcde G bc F de N = 1 2 ρ c σ σ c ρ 4σρD +... Use L VL e 1 L YM = a IJ ( σy I ij Y ij J 1 4 σf I µν F µν J 1 2 ρi F J µν G µν 8σρ I F µνt µν ρi ρ J C σ σρi C ρ J ρi D aρ I D a σ 4σρ I ρ J (D T 2 ) +4ρ I ρ J G µνt µν 1 8 ɛµνρσλ F I µνf J ρσc λ ) Mehmet Ozkan () April 24, / 41

28 Construction of the Riemann Squared Action Step 3: Gauge Fix the Yang-Mills action and use the map Bosonic part of the Riemann tensor squared action e 1 L Riem 2 = 1 4 (R µνab (ω +) G µνg ab )(R µνab (ω +) G µν G ab) µ(ω+)g ab µ (ω +)G ab + V µν ij V µν ij 1 8 ɛµνρσλ( R µνab (ω +) G µνg ab )( R ρσab (ω +) G ρσg ab )C λ 1 2 ɛµνρσλ B ρσ ( R µνab (ω +) G µνg ab )( R ρσab (ω +) G ρσg ab ) λ (ω +)G ab V µν ij V µν ij - Kinetic term for the auxiliary V ij µ Massive particles in the spectrum Mehmet Ozkan () April 24, / 41

29 Intermezzo What do we know so far? Off-shell Poincaré supergravity in σ = 1 gauge fixing condition Riemann squared invariant in σ = 1 gauge fixing Riemann squared term generates massive graviton Auxiliary vector V µ becomes dynamical, and generates massive dynamical auxiliary vector Key Observation Massive graviton and massive auxiliary vector falls into same multiplet Mehmet Ozkan () April 24, / 41

30 Construction of Gauss-Bonnet Combination: The Idea Key Idea Construct a new higher derivative invariant Combine two off-shell higher derivative invariants Vanishing V ij µν Massive multiplet decouples Gauss - Bonnet combination Mehmet Ozkan () April 24, / 41

31 Construction of Gauss-Bonnet Combination Weyl Tensor Squared Invariant Weyl tensor square C µνρσ C µνρσ = R µνρσ R µνρσ 4 3 R µνr µν R2 Gauss-Bonnet requires C µνρσ C µνρσ R2 In superconformal language, Ĉ µνρσ Ĉ µνρσ D2 Claim: If the supersymmetrization of Weyl tensor square includes V µν ij then it has to include D2 Mehmet Ozkan () April 24, / 41

32 Construction of Gauss-Bonnet Combination Weyl Tensor Squared Invariant Linear Multiplet Dilaton Weyl Multiplet L ij = 1 4 i R(i ab(q) R j) ab (Q) i χ(i χ j) R ab ij (V )T ab ϕ i = 1 8 γ cd R i ab (Q)Ĉ abcd +... E a = 1 16 ɛ abcdeĉ bcfg Ĉ de fg +... N = 1 8ĈabcdĈabcd D2 1 3 V ab ij V ab ij +... Mehmet Ozkan () April 24, / 41

33 Construction of Gauss-Bonnet Combination Weyl Tensor Squared Invariant Use L VL e 1 L ρr 2 = ρc µνρσc µνρσ ρd2 1 3 ρv ab ij V ab ij ɛ abcdea a C bcfg C de fg +... Problem Riemann 2 is written purely in terms of Dilaton Weyl multiplet Weyl 2 uses both Dilaton Weyl multiplet and Vector multiplet Need a map: Vector multiplet Dilaton Weyl multiplet Mehmet Ozkan () April 24, / 41

34 Construction of Gauss-Bonnet Combination Vector multiplet Dilaton Weyl multiplet ( ) ( A µ, Y ij, λ i, ρ C µ, 1 ) 4 iσ 1 ψ i ψ j, ψ i, σ e 1 L σr 2 σ=1 = C µνρσc µνρσ D2 1 3 V ab ij V ab ij ɛ abcdec a C bcfg C de fg +... Mehmet Ozkan () April 24, / 41

35 Gauss-Bonnet Combination Off-Shell Theory e 1 (L LR + αl R 2 + βl W 2) = 1 2 LR L 1 µ L µ L 1 4 LG µνg µν + [ +α 1 ] 4 R abcdr abcd + V ij ab V ab ij +... [ 1 +β 8 R abcdr abcd 1 6 R abr ab R2 1 ] 3 V ab ij V ab ij +... Mehmet Ozkan () April 24, / 41

36 Gauss-Bonnet Combination Gauss-Bonnet Combination Set β = 3α to obtain Gauss - Bonnet combination e 1 (L LR + αl GB ) = 1 2 LR α(r abcdr abcd 4R ab R ab + R ) Remarks No kinetic term for the auxiliary V µ ij Curvature terms read the Gauss-Bonnet combination Corresponding on-shell theory is the Gauss-Bonnet extended Einstein-Maxwell theory Mehmet Ozkan () April 24, / 41

37 Ricci Scalar Squared Action and Off-Shell f (R) Theory Key Idea Define the elements of Vector (or Linear) multiplet in terms of other multiplets Use these expressions in the Vector - Linear action (or any other superconformal action) to obtain a superconformal action Fix the redundant superconformal symmetries Mehmet Ozkan () April 24, / 41

38 Ricci Scalar Squared Action and Off-Shell f (R) Theory Ricci Scalar Squared Action Use the composite linear multiplet fields in vector multiplet action Y ij = L 1 C L ij D a L k(i D a L j)k L km L 3 N 2 L ij L 3 E µ E µ L ij L T 2 L 1 L ij + 4L 1 DL ij + 2E µ L k(i D µ L j) k L e 1 L R 2 = 1 4 R2 + (H 2 ) 2 + (E 2 ) Mehmet Ozkan () April 24, / 41

39 Summary Summary Poincaré Riem 2 Weyl 2 + R 2 R 2 Rab 2 5D 6D In Progress Mehmet Ozkan () April 24, / 41

40 Outlook Outlook No AdS 5 solution Gauging Generalize to D = 6, N = (1, 0) theory, and reduction to D = 4, N = 2 theory Higher order effects in AdS/CFT correspondence Corrections to black hole entropies Mehmet Ozkan () April 24, / 41

41 Thank you! Mehmet Ozkan () April 24, / 41