Higher-Spin Black Holes and Generalised FZZ Duality

Transcription

1 Higher-Spin Black Holes and Generalised FZZ Duality Batsheva de Rothschild Seminar on Innovative Aspects of String Theory, Ein Bokek, Israel, 28 February 2006 Based on: Anindya Mukherjee, SM and Ari Pakman, in preparation

2 Outline Introduction 1 Introduction

3 Introduction We start with the bosonic c = 1 closed string background. This has one space ( Liouville ) and one time ( matter ) direction. It arises by coupling a timelike c = 1 matter field X (z, z) to worldsheet gravity. The Liouville mode φ(z, z) of the metric provides a spatial direction, and a linear dilaton and cosmological perturbation are generated: S c=1 = d 2 z ( X X + φ φ + 2ˆR(z, z)φ + 4πµ e 2φ) The central charge of this theory is 1 from X and 25 from φ, where the latter comes from the linear dilaton.

4 The string coupling in this background is g s e Φ = e Φ 0 e 2φ where Φ is the dilaton field and Φ 0 an arbitrary constant value. Thus the theory is weakly coupled at one end of the spatial direction and strongly coupled at the other: g s 0 as φ, g s as φ, The string loop expansion in this theory is an expansion in 1 µ 2.

5 For this talk, I will study the case of finite temperature. In this case, the time X is Euclidean and compactified at a radius 2πR. Thus, X (z, z) X (z, z) + 2πR The BRST procedure tells us the physical fields of this theory. One important class are the momentum modes or tachyons: T k R = e i k R X e (2 k R )φ, k integer with left and right conformal dimensions equal to: X + φ = 1 4( k R ) 2 +(1 1 2 k R )(1+ 1 k 2 R )=1

6 Another important class of observables are the winding modes: T kr = e i kr X e (2 kr)φ, k integer which are clearly also (1, 1) operators. The modes T k and T kr are dual to each other under R (timelike) T-duality: X = X L + X R X = X L X R, φ φ log R under which R 1 R, µ µr Note that T 0 = T 0 = e 2φ, the cosmological operator.

7 There are other modes of dimension (1, 1) called discrete states. These are easiest to write at self-dual radius R = 1: Y + j;m,m = P (j 2 m 2 ),(j 2 m 2 )( n X, n X ) e 2imX L e 2imX R e (2 2j)φ where j = 0, 1 2, 1..., and m, m = j, j 1,..., 1 j, j. Here P is a polynomial in derivatives of X with the given left (right) conformal dimensions. For m = m = ±j these are the momentum modes T ±2j while for m = m = ±j they are the winding modes T ±2j. All other modes are discrete states. The modes with m = m = 0 exist for all radius R, but of course only for integer j.

8 An interesting discrete state is: Y + 1,0,0 = X X which is the radius-changing operator. In the critical string this would have just been the zero-momentum mode of the graviton/dilaton. Here it is a remnant of those fields, being forced to have zero momentum. The other discrete states are similar remnants of excited tensor states of the string, with fixed momenta. We also see that at any radius R, the lowest winding mode T ±R = e ±ir(x L X R ) e (2 R)φ is a marginal operator in the matter sector if R < 2. This fact will be important later.

9 All the operators considered so far have a Liouville dependence e (2 p)φ, p > 0 Such operators are non-normalisable, since they peak at weak coupling. Their insertion creates a local deformation of the worldsheet. For every such operator there is a corresponding normalisable operator, that decays at weak coupling: e (2+p)φ, p > 0 that creates a non-local deformation. Thus, for the radius operator there is a non-local counterpart Y 1,0,0 = X X e 4φ which will play a role in what follows.

10 In the presence of the cosmological constant, the physical operators are really a combination of the two types of dressing. Upon Lorentzian continuation these correspond to incident and reflected waves on the Liouville wall.

11 Outline Introduction 1 Introduction

12 Let us consider deforming the c = 1 string background by a condensate of the lowest winding mode: S c=1 S c=1 + λ (T R + T R ) = S c=1 + λ e (2 R)φ cos R(X L X R ) This is called Sine-Liouville theory.

13 The Sine-Liouville term is unbounded below because of the cosine part. For R < 2, the Liouville part grows towards the strong coupling region. The theory has three marginal parameters: µ, R, λ.

14 Outline Introduction 1 Introduction

15 Noncritical string theory admits a black hole solution that can be written as an exact CFT (all orders in α ). This is an SL(2, R)/U(1) CFT whose Euclidean version, at large k, is: ( ds 2 = k (1 e 4φ ) dt dφ2) 1 e4φ Φ Φ 0 = 2φ, < φ < 0 By a change of coordinates, this can also be written: ds 2 = k ( dr 2 + tanh 2 r dθ 2) Φ Φ 0 = 2 log cosh r, < r < 0 Here k is the level of the CFT.

16 The Euclidean black hole is a cigar ending at r = 0. Its asymptotic radius as r is R = k The value of the dilaton at the tip, Φ 0, can be identified with the mass of the black hole: M e 2Φ 0 Note that conformal invariance of the worldsheet theory requires: c tot = 3k k 2 1=26 = k= 9 4 = R= 3 2 Therefore we are actually in the regime of small k, and the spacetime solution is not very reliable.

17 Note the absence of a cosmological perturbation µ in the black hole background. For large negative φ, the black hole metric can be written ds 2 = k ((1 e 4φ ) dt 2 + (1 + e 4φ )dφ 2) = k (dt 2 + dφ 2 (dt 2 dφ 2 )e 4φ) Thus, infinitesimally the black hole is generated by a perturbation S = ( X X φ φ)e 4φ The second term is a pure gauge in BRST cohomology. We recognise the first term as the normalisable version of the radius operator, Y 2,0,0.

18 This is misleading: the full perturbation is non-normalisable for k < 3. This can be shown directly from the SL(2, R) symmetry of the coset theory. Indeed, viewing the black hole as flat (linear dilaton) space perturbed by the above operator is misleading in many ways. For example the operator conserves winding number but the Euclidean black hole, being a cigar, does not.

19 So in what sense is the black hole operator useful at all? Conceptually, it is useful in that it seeds an exact CFT. It was shown (S. Mukherji, SM, A.Sen, Phys. Lett. 1991) that starting from such a perturbation of the c = 1 string, there is no obstruction to finding a classical solution of closed string field theory (CSFT) to all orders in α. The solution is unique. This closes the gap between the spacetime solution (valid only for large k) and the CFT (which lacks a direct spacetime interpretation).

20 But it also suggests a generalisation. We observed that: Y j;0,0 = P j 2,j 2( n X, n X ) e (2+2j)φ for j = 0, 1, 2,... defines an infinite family of normalisable operators, of which the first two (j = 0, 1) are the cosmological and black hole perturbations. And we showed that all these operators generate classical solutions of CSFT. They appear to be infinitely many higher-spin generalisations of the 2d black hole. To date their physical interpretation has not been understood any better.

21 Based on experience with the black hole, it is reasonable to conjecture each of these operators, despite being apparently normalisable perturbations, sum up into non-normalisable states. Thus they should describe genuinely new backgrounds (superselection sectors) of the c = 1 string.

22 Outline Introduction 1 Introduction

23 Returning to the 2d black hole, the FZZ conjecture is: 2d black hole Sine-Liouville theory This is a strong-weak duality on the worldsheet. It is motivated by comparing two and three-point functions. We saw that Sine-Liouville theory has three marginal parameters: µ, R, λ. However, the LHS of the above conjecture has neither µ nor R. The former makes no sense in the black hole, because spacetime terminates before strong coupling is reached. The latter is fixed by R = k = 3 2. Therefore in the above conjecture, the Sine-Liouville action has µ = 0 and R = 3 2.

24 Thus the Sine-Liouville action relevant for the FZZ conjecture is: ( ) S c=1 (µ = 0, R = 3) + λ e 1 2 φ 3 cos 2 2 (X L X R ) As we saw, this is nothing but the unit winding condensate in the theory with R = 3 2. So the c = 1 string with this condensate, of magnitude λ, describes a 2d black hole (of mass M = λ 8 ). Because the Sine-Liouville term is unbounded below, the limit µ 0 risks being unstable. A version of the FZZ conjecture was proved (as mirror symmetry ) for the N = 2 supersymmetric version of these theories. There, it relates the N = 2 black hole to N = 2 Liouville theory.

25 Though the FZZ duality holds only at R = 3 2, the Sine-Liouville theory can be defined in principle for all R. The Sine-Liouville term has a matter vertex operator of conformal dimension 1 4 R2. This is a relevant operator for R < 2. Above R = 2 it is irrelevant. Also, for such R the Liouville part no longer grows at strong coupling. Therefore the theory will not have a sensible limit as µ = 0. Below R = 1, it has been argued that the theory is unstable and decays to a c = 0 vacuum when µ 0. Thus the theory is sensible only for 1 R 2. The black hole value luckily lies inside this region. It is believed that the black hole has a continuation for other values of R between 1 and 2, but we do not know precisely what this is.

26 One nice way to think about the FZZ conjecture comes from an algebraic relation between Sine-Liouville and black-hole perturbations that we call the FZZ algebra. This consists of an OPE between the non-normalisable Sine-Liouville operator: T + R and the normalisable one: T R = cos R(X L X R ) e (2 R)φ = cos R(X L X R ) e (2+R)φ

27 It is easy to check that this OPE contains a unique (1, 1) operator on the RHS: T + R (z, z)t R (w, w) 1 z w 2 X X e 4φ + We see that perturbing a linear-dilaton theory by the Sine-Liouville operator (of both dressings) automatically generates the 2d black hole operator. It fits in with the general fact that in Liouville type theories, both dressings should be used as screening charges to get the correlation functions right. And it generalises nicely, as we will now see.

28 The infinitely many operators we found in CSFT suggest a generalised FZZ conjecture. Recall that the conformal dimension for the matter part of the generalised perturbations is: ( ) M Y j,0,0 = j 2, j = 1, 2, 3,... On the other hand, the winding modes T kr have matter conformal dimension: M (T kr ) = 1 4 k2 R 2, k = 1, 2, 3,... The two sets of dimensions can be put in one to one correspondence. And they are identical precisely at R = 2, the point beyond which all winding modes become irrelevant.

29 Therefore we conjecture: Y j,0,0 T jr for all integer j, at some fixed radius (not R = 2) which could depend on j. Recall that for j = 1, this radius is R = 3 2. Note that the conjecture is true for j = 1 (where it is FZZ) and for j = 0, where both sides are equal to e 2φ! For each T jr one needs to investigate the range of R values for which the Sine-Liouville makes sense. Also we would like to know the the analogue of the black hole radius on the other side. We believe the relevant radius for the jth perturbation is R = 2j + 1 j(j + 1)

30 Now we will find some support for the conjecture using a generalisation of the FZZ algebra. To be concrete, suppose that instead of the unit winding perturbation T R, we perturb the action by Sine-Liouville operators of winding number 2: T ± 2R = sin 2R(X L X R ) e (2 2R)φ The OPE between these again gives the black hole perturbation: T + 2R (z, z)t 2R (w, w) 1 z w 2 X X e 4φ + In fact, the same will be true for pairs of mutually conjugate operators of any winding number. In every case, the output of the OPE is the 2d black hole perturbation.

31 But suppose we perturb the theory simultaneously by operators of single and double windings. In this case, examining the OPE algebra, we find that the product of three suitable operators produces a new (1, 1) operator on the RHS: T + 2R (z 1, z 1 )T R (z 2, z 2 )T R (z 3, z 3 ) P 2,0 ( X ) P 2,0 ( X ) e 6φ Thus the second higher-spin black hole operator is produced by a product of the double-winding Sine-Liouville perturbation (of positive dressing) with two unit-winding perturbations (of negative dressing).

32 This result is quite general. For example, one can check that: T + nr (z 1, z 1 )T R (z 2, z 2 ) T R (z n+1, z n+1 ) P n,0 ( X ) P n,0 ( X ) e (2+2n)φ We see that the multiply wound Sine-Liouville operators are linked to higher-spin black holes. More precisely, perturbing by all Sine-Liouville operators of winding numbers 1, 2,, n gives rise to higher-spin black holes with all labels up to n (the spins realised in this way are 2k 2, k = 1, 2, n).

33 In Matrix Quantum Mechanics, multiple winding modes are equivalent to thermal Wilson-Polyakov loop operators. In turn, these are related to the nonsinglet sector involving various irreps of SU(N). These are important sectors of the matrix theory, of which the adjoint rep is only the simplest. Since the adjoint is related to the unit winding mode and thence the 2d black hole, all irreps and hence their corresponding multiply wound operators should have independent dualities to other discrete states. Our proposal precisely fills this gap.

34 Outline Introduction 1 Introduction

35 Higher-spin generalisations of 2d black holes exist, but are not understood. We conjectured a duality to higher winding modes. We know the geometrical picture of the 2d black hole is unreliable (due to α corrections) but we have an exact CFT description. What is the CFT for the higher-spin black holes? The 2d black hole at k = 9 4 is qualitatively just below the black-hole-string transition. Can the same be said for higher-spin black holes? Is there a corresponding transition for them? Do higher-spin black holes exist in type 0 strings, which also have RR fluxes? Can they be understood using matrix models?

36 Thank You