Coset CFTs, high spin sectors and non-abelian T-duality

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1 Coset CFTs, high spin sectors and non-abelian T-duality Konstadinos Sfetsos Department of Engineering Sciences, University of Patras, GREECE GGI, Firenze, 30 September 2010 Work with A.P. Polychronakos (City University of New York) Nucl.Phys. B840 (2010) , arxiv: [hep-th]. Nucl.Phys. B (2010), arxiv: [hep-th].

2 Motivation Field equations in curved stringy backgrounds In string theory want to go beyond (super)gravity: exact theories realizing this, particularly, coset (G /H) CFTs. When groups are non-abelian there are no isometries (generic). Solving the field equations is an impossible task with traditional methods, i.e. separation of variables. In physical applications this is precisely what is needed, i.e. propagating fluctuations, etc. Understanding Non-abelian T-duality Unlike abelian T-duality: Not well understood. Not likely to be an exact symmetry. Yet, what is it good for? Maybe for some effective description?

3 Outline Gauged WZW models and Non-abelian T-duality: Gauged WZW models and their geometry. Non-abelian WZW T-duals and their geometry. Relating non-abelian T-duality to gauged WZW models. Solving field equations in coset CFT backgrounds. Example: SU(2) SU(2)/SU(2) The infinitely large spin limit and the effective non-abelian T-dual. Example: Non-abelian T-dual of SU(2) WZW. Concluding remarks. Towards Non-Abelian T-duality in RR-backgrounds.

4 Gauged WZW models and Non-abelian T-duality Gauged WZW models: Action Let a group G and a subgroup H G. Introduce g G and gauge fields A ± L(H). The gauged WZW action is WZW {}}{ S(g, A ± ) = k I 0 (g) + k [ dσ + dσ Tr A + gg 1 A + g 1 g π ] + A ga + g 1 A A +. Not minimally coupled gauged fields. Gauge invariance: For Λ(σ +, σ ) H g Λ 1 gλ, A ± Λ 1 (A ± ± )Λ.

5 Gauged WZW models: The geometry The gauge fields are non-dynamical and can be integrated out A a + = (M 1 ) ab ( + gg 1 ) b, A a = (g 1 g) b (M 1 ) ba, where (a, b H) M ab = Tr(t a gt b g 1 ) η ab. Gauge fix dim(h) parameters in g or, better, choose those that are H-singlets. That leaves dim(g /H) x µ s. One obtains the σ-model of the form (only NS fields) S = k (G µν + B µν ) + X µ X ν π and a dilaton Φ = 1 ln det(m). 2 Background solves beta-functions for conformal invariance. Isometries G L G R of the WZW are generically broken.

6 Non-abelian WZW T-duals and their geometry The starting action is S NonAb (g, v, A ± ) = S(g, A ± ) i k Tr(vF + ). π }{{} Lagrange mult. Gauge invariance: As before and in addition v Λ 1 vλ. Gauge fix dim(h) parameters in g and v, leaving dim(g ) variables X M. Integrate out the A ± s and get the σ-model. Properties: Isometries G L G R of the WZW generically broken. Even if G is compact, (some) variables of its non-abelian dual appear non-compact. Transformation is not invertible at the action level. Drastically different than abelian T-duality. Is it useful? What does it describe?

7 Relating non-abelian T-duality to gauged WZW models Start with the gauged WZW action for G k H l H k+l for two group elements g G, h H and gauged field in L(H). Expand infinitesimally around the identity ( ) k 1 h = I + i l v + O l 2 and take the limit l. We get the non-abelian T-duality action, i.e. classically [KS 94] Remarks: G k H l H k+l = dual of G k with respect to H. l In the limit some variables become non-compact. A well defined limit can be taken on the geometric background. Can this be the effective background describing a consistent sector of the parent theory?

8 Solving field equations in coset CFT backgrounds For the background corresponding to G k H l /H k+l, we would like to solve the scalar equation 1 e 2Φ G µe 2Φ GG µν ν Ψ = E Ψ. We will obtain its general solution from that of the scalar equation for the WZW model for G H. Start with Reps of G H. The eigenstates are R αβ (g) r µν (h), which are, relatively, easy to construct. The eigenvalues (semiclassically, for k, l 1) are E (R, r) = C 2(R) k where the C 2 s are the Casimirs. + C 2(r) l,

9 Under the vector H-transf they transform as (R r) ( R r) = (r 1 r 2 ) ( r 1 r 2 ). We decompose R r and its conjugate into Reps r i of H. We get a singlet from all products of the form r i r i. These singlets, representing the coset eigenstates, are ψ R,r;ri (g, h) = G H {}}{ Cαµ(R, a r; r i )Cβν a (R, r; r i ) R αβ (g)r µν (h) a;α,β,µ,ν }{{}}{{} Clebsch Gordan gauged fixed. Conclusion: The states in the G H/H coset theory are the H-singlet combinations of the states in the WZW model G H as this is dictated by group theory.

10 Remarks: The eigenvalues get shifted as E (R, r; r i ) = C 2(R) k + C 2(r) l C 2(r i ) k + l. This is in accordance to the algebraic coset construction [Goddard-Kent-Olive, 85]. The coset background fields receives 1/k corrections. They become simple in the semiclassical limit for k 1. Remarkably, the eigenstates do not depend on α 1/k, only the eigenvalues do (indicated expressions are for k 1).

11 Example: SU(2) SU(2)/SU(2) Parametrization We parametrize g 1 g 2 SU(2) SU(2) as ( ) ( α0 + iα g 1 = 3 α 2 + iα 1 β0 + i β, g α 2 + iα 1 α 0 iα 2 = 3 β 2 + i β 1 3 β 2 + i β 1 β 0 i β 3 ), where from unitarity α α2 = 1, β β 2 = 1. Under the diagonal SU(2) they transform as vectors δα i = ɛ ijk α j ɛ k, δβ i = ɛ ijk β j ɛ k, The SU(2)-singlets are α 0, β 0 and γ = α β, obeying 0 α 0, β 0 1, γ 1 α β 2 0. and represent the three compact geometrical coordinates.

12 The background fields Following the general procedure outlined above... The metrics is ds 2 = k 1 + k 2 (1 α 2 0 )(1 β2 0 ) γ2 ( αα dα ββd β γγdγ 2 +2 αβ dα 0 d β αγ dα 0 dγ + 2 βγ d β 0 dγ ). The s are functions of α 0, β 0, γ and of r = k 1 /k 2. The field B µν = 0 and the dilaton Φ = 1 2 ln ((1 α 2 0 )(1 β2 0 ) γ2). Background is a bit complicated, with no isometries.

13 Solution of the eigenvalue problem A general SU(2) spin j Rep has matrix elements Rm j 1,m 2 (a, b, c, d) = A j m 1,m 2,k aj m1 k d j+m2 k b k c k+m 1 m 2, k The general state is SU(2) SU(2) d functions Ψ j j 2 {}}{ j 1,j 2 = C j,m j 1,m m 2,j 2,m 2 C j,m j 1,m n 2,j 2,n 2 R j 1 m m2,m n 2 (g 1 )R j 2 m2,n 2 (g 2 ) m m 2,n 2 = j 2 }{{}}{{} Clebsch Gordan gauged fixed Examples: For j 2 = 0 and thus j 1 = j: Ψ j j,0 = Character {}}{ j Rm,m(g j 1 ) = m= j Chebyshev Pol. {}}{ U 2j (α 0 ) 2nd kind..

14 For (j 1, j 2 ) = (1, 1/2): Ψ 1/2 1,1/2 = (4α2 0 1)β 0 + 4α 0 γ, Ψ 3/2 1,1/2 = (4α2 0 1)b 0 2α 0 γ. For (j 1, j 2 ) = (1, 1): Ψ 0 1,1 = 4(α 0β 0 + γ) 2 1, Ψ 1 1,1 = 6α2 0 β α 0β 0 γ 2γ 2 2(α β2 0 ) + 1, Ψ 2 1,1 = 26α2 0 β2 0 20α 0β 0 γ + 2γ 2 6(α β2 0 ) + 1. Due to luck of isometries, there is no factorization. With increasing j 1,2 expressions become complicated. Impossible to obtain with other methods. What about the high spin limit, i.e. when j 1, j 1?

15 Large spins and the effective non-abelian T-dual High spin limit in the G k H l /H k+l theory? Let Reps in L(H) with highest weight (spin) j 1. The Reps in the tensor product with those in L(G ) have also large spin. We may expand as C 2 (r) = a(r)j 2 + b(r)j + O(1). Similarly for C 2 (r i ), with j replaced by j + n (n =finite). Keeping the eigenenergies finite requires the correlated limit l = k δ j, δ = positive real. The limit of the eigenfunction is delicate. It involves the limiting behaviour of the Clebsch Gordans. But, l is associated to the non-abelian T-dual of G k. Hence: Non-abelian T-duality provides an effective description of the high spin sector of the parent theory.

16 Example: Non-abelian T-dual of SU(2) WZW The background fields Non-abelian T-dual of the SU(2) WZW model w.r.t. SU(2) ds 2 = dψ 2 + cos2 ψ x3 2 dx1 2 + (x 3dx 3 + (sin ψ cos ψ + x 1 + ψ)dx 1 ) 2 x3 2 cos2 ψ, plus a dilaton Φ = 1 2 ln(x2 3 cos2 ψ). A bit complicated with no isometries. ψ is periodic and x 1, x 3 are non-compact. What do the eigenfunctions and eigenenergies look like? They should effectively describe the large spin sector of the SU(2) SU(2)/SU(2) coset.

17 Solution of the eigenvalue problem We will take the limit in the states and eigenvalues of the coset. Consider the high spin-level limit j 1 = j n, k 1 = k 2 δ j, j 2, n = finite, j 1. The energy eigenvalues E j remain finite j 1,j 2 = j 1(j 1 +1) k 1 + j 2(j 2 +1) E j2,n,δ = lim E j j j 1,j 2 = j 2(j 2 + 1) + δ 2n δ. k 2 k 2 In the high spin limit the Clebsch Gordan coefficients k 2 j(j+1) k 1 +k 2, lim C j,m j j n,m m 2,j 2,m 2 = d j 2 m2,n(ζ), cos ζ = m j. They get associated with an auxiliary SU(2) rotation. Expected for a classical body given extra angular momentum.

18 At the end we obtain the finite sum Ψ j2,n,δ(x 1, x 3, ψ) = lim Ψ j j 2 j j n,j 2 = Γ j2,m 2,n,δ(x 3 ) R j 2 m2,m 2 (g 2 ) m 2 = j }{{} 2 gauged fixed where Γ j2,m 2,n,δ(x 3 ) = π ( dζ sin ζ d j 2 2 m2,n(ζ)) e 2iδv 3 cos ζ. 0 Explicit expressions become complicated fast, as j 2 increases. Fair to say: Solution would have never been found without using this method.,

19 Examples of states: Define v 3 = (x 1 + ψ) 2 + x 2 3, β 0 = sin ψ, x β 1 = 3 cos ψ (x 1 + ψ) 2 + x3 2 For instance, for j 2 = 1 (and δ = 1):, β 3 = (x 1 + ψ) cos ψ (x 1 + ψ) 2 + x3 2. Ψ 1,±1 = β2 1 2β 3(β 3 2β 0 v 3 ) 2v 2 3 cos 2v 3 + 2β2 3 β β 0β 3 v 3 + 4(β 2 0 β2 3 )v 2 3 4v 3 3 sin 2v 3, Ψ 1,0 = 2β2 3 β2 1 v 2 3 cos 2v 3 + β2 1 2β (1 2β2 1 )v 2 3 2v 3 3 sin 2v 3.

20 Concluding remarks Using group theoretical methods one may solve field equations for general G /H, for which: Generically, there are no isometries. Conventional techniques are not applicable. Non-abelian duality generates solutions that: effectively describe high spin sectors. Taking the limit is a delicate procedure, but nevertheless the only way to solve field equation of the T-dual background. Method works in other occasions with non-abelian isometries. For instance, when the symmetry group acts from on side, i.e. in Principal Chiral Models. Use non-compact groups leading to Minkowski signature spacetimes, i.e. SL(2, IR) SL(2, IR)/SL(2, IR). Explore physical applications, i.e. in cosmology.

21 Towards Non-Abelian T-duality in RR-backgrounds with D. Thompson (Vrije Universiteit Brussels Solvay Institute) So far Non-abelian T-duality has been formulated only in pure NS-NS backgrounds. What about backgrounds with RR-fluxes? Abelian T-duality: From type-iia to type-iib and vice versa. Natural expectation: Non-abelian T-duality changes (remains in the same) type-ii theory if the dimension of the isometry group is odd (even). A natural formulation is within the pure spinor formalism allowing the description of the superstring in general curved backgrounds with non-trivial RR sectors [Berkovits 07].

22 Example: The near horizon of the D1-, D5-brane system. AdS 3 S 3, but with RR-fluxes (proportional to volume forms). NS-NS sector as in Principal Chiral Model for SL(2, R) SU(2). Some evidence that non-abelian T-duality w.r.t. SU(2) gives a solution of the massive IIA Romans theory.