The N = 2 Gauss-Bonnet invariant in and out of superspace

The N = 2 Gauss-Bonnet invariant in and out of superspace Daniel Butter NIKHEF College Station April 25, 2013 Based on work with B. de Wit, S. Kuzenko, and I. Lodato Daniel Butter (NIKHEF) Super GB 1 / 29

Motivation The Gauss-Bonnet is a very nice higher derivative invariant: L GB = 1 4 εmnpq R mn ab R pq cd ε abcd = C abcd C abcd 2R ab R ab + 2 3 R2. Although it is topological by itself, it often appears multiplied by a scalar function in specific applications (e.g. anomalies, 5D to 4D reductions, etc.). Its supersymmetric version will appear in corresponding situations. The manifestly supersymmetric 4D N = 1 GB is well-known. [Townsend and van Nieuwenhuizen; Ferrara and Villasante; Buchbinder and Kuzenko] This reflects the completeness of our understanding of 4D N = 1 higher derivative terms. We should better understand higher derivative terms in 4D N = 2! Daniel Butter (NIKHEF) Super GB 2 / 29

Outline 1 Superspace philosophy 2 The N = 1 Gauss-Bonnet in superspace and an N = 2 mystery 3 Superconformal tensor calculus in superspace 4 The Gauss-Bonnet invariant in and out of N = 2 superspace Daniel Butter (NIKHEF) Super GB 3 / 29

Quick review: N = 1 supersymmetry in spinor notation In a Weyl-basis, the γ-matrices are ( ) 1 0 γ 5 =, γ m = 0 1 ( 0 ) (σ m ) α α ( σ m ) αα 0 A Dirac fermion Ψ and its conjugate Ψ look like ( ) χα ( ) Ψ = ψ α, Ψ = ψ α χ α N = 1 supersymmetry in four dimensions: {Q α, Q α } = 2i (σ m ) α α m. Daniel Butter (NIKHEF) Super GB 5 / 29

Superspace makes supersymmetry manifest Superspace: add new Grassmann coordinates θ α and Interpret the supersymmetry generator Q α D α as Chiral multiplet is short D α = θ α + i(σm ) α α θ α m D α Φ = 0 = Φ = φ + θ α ψ α + θ 2 F + (x-derivative terms) The free massless chiral multiplet action is given by ( d 4 x d 2 θ d 2 θ ΦΦ = d 4 i x φ φ 2 ψ α ( σ m ) αα m ψ α + F F ) θ α If we want to gauge a U(1) symmetry, the standard lore is to add an explicit vector multiplet prepotential V, do Wess-Zumino gauge, etc. Daniel Butter (NIKHEF) Super GB 6 / 29

How to avoid prepotentials and Wess-Zumino gauges Encode gauge connection in superspace: A = dx m A m A = dz M A M. Build covariant derivative D A = (D α, D α, D a ) in superspace. This is supersymmetric minimal substitution. Keep the same action ΦΦ but replace the flat chiral constraint with a covariant chiral constraint D α Φ = 0 D α Φ = 0. We define the components of Φ not by a θ expansion but by φ Φ θ=0, ψ α = D α Φ θ=0, F = 1 4 Dα D α Φ θ=0. The supergravity story is essentially analogous. 1 Lift all connections to superconnections. 2 Replace flat derivatives with covariant derivatives. Daniel Butter (NIKHEF) Super GB 7 / 29

From superspace to tensor calculus and back again For every covariant field e.g. ψ α, there is some superfield, e.g. D α Φ with ψ α = D α Φ θ=0 Supersymmetry on the component field corresponds to covariant spinor derivative on the superfield: δ Q ψ α = ξ β Q β ψ α + ξ β Q βψ α δ ξ D α Φ θ=0 = ξ β D β D α Φ θ=0 + ξ β D βd α Φ θ=0 Given covariant superfields, one can evaluate all their components and derive supersymmetry transformations. The converse is also possible: one can often lift component results to superspace expressions. Properties/symmetries of the tensor calculus reflect those of the superspace. Poincaré tensor calculus Poincaré superspace Superconformal tensor calculus (Super)conformal superspace Daniel Butter (NIKHEF) Super GB 8 / 29

N = 1 Poincaré superspace Old minimal Poincaré supergravity involves the field content e a m, ψ α m, V }{{} m, }{{} M massive vector complex scalar Superspace geometry involves curvatures which are built out of the superfields W αβγ, G a, R. These contain (respectively) C abcd, R ab and R. General integrals in superspace look like S D = d 4 x e [L] D = d 4 x d 2 θ d 2 θ E L, L is unconstrained S F = d 4 x e [L c ] F = d 4 x d 2 θ E L c, D α L c = 0 A caveat: any D-term can be written as an F -term. Let s consider only honest F -terms. Daniel Butter (NIKHEF) Super GB 10 / 29

N = 1 actions and higher derivatives The Einstein-Hilbert Lagrangian is L EH = 3 2 d 4 x d 2 θ d 2 1 θ E 2 M P 2 d 4 x e R In N = 1, the only purely chiral (non-singular) invariants are L c = c 1 W αβγ W αβγ + c 2 3 (W αβγ W αβγ ) 2 + [L c ] F c 1 C abcd C abcd + c 2 3 (C ) 2 (Dψ) 2 + The other (non-singular) terms generically are D-term invariants with L = c 3 R R + c 4 G a G a + c 5 D α D α R + c 6 4 (D α G a ) 4 + c 7 3 (W αβγ ) 2 G 2 + [L D ] c 3 R 2 + c 4 R ab R ab + c 5 R + c 6 4 (R ab ) 4 + c 7 3 (C ) 2 (Dψ) 2 + Daniel Butter (NIKHEF) Super GB 11 / 29

The mystery in N = 2 For simplicity, let s stick with the minimal SU(2) multiplet: e m a, ψ mα i, V m i j, W m, = gauge connections T ab ij, χ i α, D, Y ij, V m = covariant fields Superspace geometry includes curvatures W αβ }{{}, S ij, G a, Y αβ chiral Weyl superfield The minimal multiplet action is the F-term integral of a constant S minimal = 3 ( 2 M P 2 d 4 x d 4 θ E = 2 d 4 x e 1 ) 2 R + 3D + The obvious higher derivative chiral invariant is L c = W αβ W αβ [L c ] F (C ) 2 D-term invariants lead to dimension 6 and higher: L = 1 ( ) 2 S ij S ij + Y αβ Y αβ +, [L] D 1 ( ) 2 R 3 + R R + Daniel Butter (NIKHEF) Super GB 12 / 29

Review: Conformal gravity Let s briefly review conformal gravity since the superconformal case is quite similar. Introduce covariant derivative so that c = a a. e m a a = m 1 2 ω m ab M ab b m D f m a K a We may consistently impose curvature constraints to determine ω m ab and f m a. Only independent fields are e m a and b m. To recover Poincaré gravity, use K-transformation to fix b m = 0. Fix dilatations by gauging some compensator field to a constant. Daniel Butter (NIKHEF) Super GB 14 / 29

N = 2 superconformal vs. Poincaré tensor calculus For N = 2 conformal supergravity, we have the fundamental connections e m a, ψ m αi, b m, A m, V m i j Constraints = composite ω m ab, f m a and φ m α i Choosing a certain compensator, we can reduce to a Poincaré tensor calculus real scalar multiplet (128+128) U(2) Poincaré tensor calculus U(2) superspace [Howe 81] chiral multiplet (16+16) SU(2) Poincaré tensor calculus SU(2) superspace [Grimm 80] vector multiplet (8+8) minimal SU(2) Poincaré tensor calculus reduced SU(2) superspace Poincaré supergravity multiplets still can describe superconformal theories. Then the additional compensator fields drop out of the action. But this can be complicated. Daniel Butter (NIKHEF) Super GB 15 / 29

N = 2 conformal superspace For a superconformal model, use conformal superspace (no compensator!) [DB 11] Introduce covariant derivative A = ( αi, αi, a ) with connections in the full superconformal algebra. Single curvature superfield W αβ contains all of the superconformal curvatures and the matter fields T ab ij, χ i α, D All constraints and curvatures of N = 2 superconformal tensor calculus are reproduced. This was worked out long ago at the linearized level in superspace. [Bergshoeff, de Roo, de Wit 81] Daniel Butter (NIKHEF) Super GB 16 / 29

Gauss-Bonnet invariant in conformal gravity c c ln φ is K-invariant for any choice of weight w for φ and equals ln φ + D a( 2 ) 3 RD a ln φ 2R ab D b ln φ + w 6 R w 2 Rab R ab + w 6 R2. We can get very close to the usual Gauss-Bonnet in conformal gravity: L GB = C abcd C abcd + 4 w c c ln φ = L GB + 2 3 R + 4 ) (D w Da D a ln φ + 2 3 RD a ln φ 2R ab D b ln φ This combination is known to physicists as the Riegert operator. [Riegert 84; Paneitz 08] Daniel Butter (NIKHEF) Super GB 18 / 29

Gauss-Bonnet invariant in conformal gravity c c ln φ is K-invariant for any choice of weight w for φ and equals ln φ + D a( 2 ) 3 RD a ln φ 2R ab D b ln φ + w 6 R w 2 Rab R ab + w 6 R2. We can get very close to the usual Gauss-Bonnet in conformal gravity: L GB = C abcd C abcd + 4 w c c ln φ = L GB + 2 3 R + 4 ) (D w Da a ln φ + 2 3 RD a ln φ 2R ab D b ln φ This combination is known to physicists as the Riegert operator. [Riegert 84; Paneitz 08] This allows you to see that under a finite Weyl transformation ( g L GB + 2 ) 3 R = g (L GB + 2 ) 3 R 4 gd a( D a σ + 2 ) 3 RD aσ 2R ab D b σ Daniel Butter (NIKHEF) Super GB 18 / 29

Supersymmetrizing the Gauss-Bonnet invariant Let us take the point of view that we wish to supersymmetrize L GB = C abcd C abcd + 4 w c c ln φ This can be done using superconformal methods. The first term should supersymmetrize easily to the superconformal Weyl-squared invariant. The second term can be supersymmetrized once we decide on the multiplet for φ. For both N = 1, 2, the natural choice is to take φ to be complex and the lowest component of a chiral multiplet. For both N = 1, 2 the natural quantity will be complex: L Γ = 1 8 (C ) 2 + 1 4w c c ln φ + Daniel Butter (NIKHEF) Super GB 19 / 29

N = 1 conformal superspace construction In N = 1 conformal superspace, only a single superspace curvature W αβγ. [DB 09] The only option is an F -term action with L c = W αβγ W αβγ [W αβγ W αβγ ] F = 1 16 C abcdc abcd 1 16 C abcd C abcd + additional terms. In flat space, it is easy to check that 1 64 we covariantize? Take chiral superfield Φ of weight w and introduce S(ln Φ) = 1 64 2 2 2 ln Φ. d 2 θ D 2 D 2 D2 ln Φ = ln φ. Can One can check that S is chiral and also S-invariant. This means we can use it to build chiral actions. Daniel Butter (NIKHEF) Super GB 20 / 29

N = 1 conformal superspace construction We choose the chiral F -term Lagrangian is Γ := W αβγ W αβγ + 1 S(ln Φ), 4w [Γ] F = 1 8 (C ) 2 + 1 4w c c ln φ + additional terms Rewriting in N = 1 Poincaré superspace... S(ln Φ) = 1 4 ( D ( 1 2 8R) 16 D2 D2 ln Φ + D α (G α α D α ln Φ) + 4wG a G a + 8wR R 1 ) 2 wd2 R We can see that we recover the topological N = 1 Gauss-Bonnet combination d 4 x d 2 θ E Γ = d 4 x d 2 θ E W αβγ W αβγ + d 4 x d 2 θ d 2 θ E (2R R + G a G a 1 ) 8 D2 R Daniel Butter (NIKHEF) Super GB 21 / 29

N = 2 conformal superspace construction Again: we want to supersymmetrize L Γ = 1 8 (C ) 2 + 1 4w c c ln φ The first term is easy. The second resembles the N = 2 kinetic multiplet. [de Wit, Katmadas, van Zalk 11] In flat N = 2 superspace, the second term is generated by d 4 θ D 4 ln Φ = D 4 D4 ln Φ θ=0 = ln φ In N = 2 conformal superspace, we can simply covariantize: T := 4 ln Φ. We call this the nonlinear kinetic multiplet. For this to be a valid chiral Lagrangian, we must check that it is chiral and S-invariant, αi T = S α i T = S i αt = 0. Then T is a proper conformally primary chiral multiplet, and we can write d 4 x d 4 θ E T = d 4 x e c c ln φ + Daniel Butter (NIKHEF) Super GB 22 / 29

Some details of chiral multiplets in superspace In N = 2 superspace, we need to know how to convert F -terms to component actions. ( ) d 4 x d 4 θ E L c = d 4 x e 4 L c + θ=0 The exact formula coincides with the usual chiral invariant density of N = 2 superconformal tensor calculus: [L c ] F C ε ij ψµi γ µ Λ j 1 8 ψ µi T ab jk γ ab γ µ Ψ l ε ij ε kl 1 16 A(T ab ijε ij ) 2 1 2 ψ µi γ µν ψ νj B kl ε ik ε jl + ε ij ψµi ψ νj (F µν 1 2 A T µν kl ε kl ) 1 2 εij ε kl e 1 ε µνρσ ψµi ψ νj ( ψ ρk γ σ Ψ l + ψ ρk ψ σj A) where A,, C are the components of the chiral multiplet L c. In our case, we need to know the chiral components corresponding to T := 4 ln Φ. Daniel Butter (NIKHEF) Super GB 23 / 29

Components of the nonlinear kinetic multiplet The components of the ln Φ multiplet ˆĀ := ln Φ θ=0, ˆ Ψ αi := αi ln Φ θ=0,, ˆ C : 4 ln Φ The components of the nonlinear kinetic multiplet T := 4 ln Φ A T = T θ=0, Ψ αi T = αi T θ=0,, C T 4 T θ=0 A straightforward (increasingly tedious) calculation, using αi ln Φ = 0, gives A T ˆ C,, C T 4 4 ln Φ θ=0 = c c ln φ + Daniel Butter (NIKHEF) Super GB 24 / 29

After the dust settles... We have a new invariant [T] F 4( c + 3 D) c log φ 1 2 D a( T ab ij ij T ) cb D c log φ ( + D a ε ij D a T bcij ˆF +bc + 4 ε ij T ab ij D c + ˆF cb T bc ij T ac ij D b log φ ) + ( 6 D b D 8iD a ) R(A) ab D b log φ wr(v) + i ab j R(V) + abj i + 8w R(D) + ab abr(d)+ wd c( D a T ab ij ) T cb ij wda T ab ij D c T cb ij +, The Weyl squared invariant is L W 2 C abcd C abcd C abcd Cabcd 4R(A) ab R(A) ab + R(V) abi jr(v) abj i + 6D 2 T ac ij D a D b T bc ij 1 128 T ab ij T ab kl T cd ijt cd kl Putting them together in the right combination we find (up to an explicit total derivative) C abcd C abcd 2R ab R ab + 2 3 R2 C abcd Cabcd + 2R(A) ab R(A)ab R(V) ab i j R(V) abj i Daniel Butter (NIKHEF) Super GB 25 / 29

So we have a new invariant, and it turns out to match the combination we needed to find from dimensional reduction from 5D... But what about the mystery of the minimal multiplet? Daniel Butter (NIKHEF) Super GB 26 / 29

A new chiral invariant in the minimal multiplet! Let s try to understand the nonlinear kinetic multiplet by writing it in SU(2) superspace: 4 ln Φ = ln Φ + wt 0 T 0 = 1 12 D ij Sij + 1 2 S ij Sij + 1 2Ȳ α βȳ α β. The operator is the SU(2) superspace chiral operator, generalizing D 4 of flat superspace. Under a full superspace integral, one can show that d 4 x d 4 θ E ln Φ = d 4 x d 4 θ d 4 θ E ln Φ = 0 so the dependence on Φ lies only within a total derivative. The remaining combination T 0 must be chiral (this can be checked explicitly), which is remarkable since none of its individual pieces is chiral! The minimal multiplet has an additional chiral invariant! Daniel Butter (NIKHEF) Super GB 27 / 29

Conclusions / Open questions (1/2) We have constructed a new chiral invariant based on N = 2 conformal supergravity coupled to a chiral multiplet. If the chiral multiplet is taken to be a vector multiplet and gauged to unity, it gives a new chiral invariant in the SU(2) minimal multiplet. It corresponds to certain actions which arise from reduction from 5D. The chiral multiplet can also be considered composite. Then in addition to D-term invariants d 4 x d 4 θ d 4 θ E H(X, X), X I H I = 0 we have intrinsic chiral invariants that look like d 4 x d 4 θ E Φ (X)T(ln Φ( X)), X I Φ I = 0, X J Φ J = w Φ In flat space, the second class corresponds to the first with the choice H = Φ ln Φ, but not in the curved case. In fact, a term of this form was exactly seen from dimensional reduction from 5D. Daniel Butter (NIKHEF) Super GB 28 / 29

Open questions (2/2) But there are several things we don t know. Does the new invariant class contribute to black hole entropy? but hopefully soon... The new invariant is peculiar in that it is (almost) independent of the compensator. Can we construct generic R 2 and (R ab ) 2 terms by introducing the second (tensor?) comensator? In principle, this should be so. Daniel Butter (NIKHEF) Super GB 29 / 29