Holography and large-N Dualities Is String Theory Holographic? Lukas Hahn 1 Introduction1 2 Classical Strings and Black Holes2 3 The Strominger-Vafa Construction3 3.1 AdS/CFT for the D1/D5 System......................3 3.2 The Instanton Moduli Space.........................6 3.3 The Elliptic Genus.............................. 10 1 Introduction The holographic principle [1] is based on the idea that there is a limit on information content of spacetime regions. For a given volume V bounded by an area A, the state of maximal entropy corresponds to the largest black hole that can fit inside V . This entropy bound is specified by the Bekenstein-Hawking entropy A S ≤ S = (1.1) BH 4G and the goings-on in the relevant spacetime region are encoded on "holographic screens". The aim of these notes is to discuss one of the many aspects of the question in the title, namely: "Is this feature of the holographic principle realized in string theory (and if so, how)?". In order to adress this question we start with an heuristic account of how string like objects are related to black holes and how to compare their entropies. This second section is exclusively based on [2] and will lead to a key insight, the need to consider BPS states, which allows for a more precise treatment. The most fully understood example is 1 a bound state of D-branes that appeared in the original article on the topic [3]. The third section is an attempt to review this construction from a point of view that highlights the role of AdS/CFT [4,5]. We will focus on a version of the story which touches part of the subsequent mathematical physics literature related to the geometry of K3 surfaces [6,7,8,9, 10]. 2 Classical Strings and Black Holes We consider a heuristic model for a (classical bosonic) string made out of "string bits" 2.1. It can be thought of as arising from a random walk process in d-dimensional p Figure 2.1: We assume that the bits are of length l = α0 and the whole string has mass M, length L and number of bits N = L=l. spacetime. For such processes the average end-to-end distance can be calculated and is given by p p phD2i = lL ∼ M; (2.1) while its Schwarzschild radius is 2GM R = ∼ M: (2.2) S c2 By this analysis of the dependence on mass we deduce that sufficiently massive strings are indeed expected to form black holes. This can be rephrased as follows: E.g. in the case of open strings with zero momentum we have 1 N M 2 = E2 = (N − 1) ' if N 1: (2.3) α0 α0 Therefore a string at zero momentum and sufficiently high degree of excitation will look like a stationary Schwarzschild black hole for an outside observer. 2 The Bekenstein-Hawking entropy of this black hole is given by (c = ~ = 1) A 2 SBH = 2 = 4πGM ; (2.4) 4lp 2 with the Planck-length lp and proportional to the mass . What is the entropy of the random-walk string? We can estimate the number of microstates as p p L p 0 0 Ω ∼ d α0 ∼ dM α ∼ eM α log(d); (2.5) 1 by using M ∼ TL ∼ α0 L, where T is the tension. An approximation for the entropy is therefore p 0 Sstr = log Ω ∼ α M: (2.6) The result of this heuristic approach is not too far from the correct one [2] p 0 Sstr = 4π α M (2.7) and in any case the entropy is proportional to the mass1. The disagreement of both calculations for the entropy is of course expected. In the purely statistical derivation of Sstr the entropy will certainly be an extensive quantity and adding string bits of a given chunk of the mass M is expected to increase the entropy linearly. That this is different from the gravitational situation and the expression for SBH is a well-known feature of gravitational physics. There is however an obvious reason why this comparison is questionable: A non-vanishing black hole entropy requires interactions because G ∼ g2α0; (2.8) where g is the (closed) string coupling, while we treated the random-walk "string" as free. Although there is further heuristic evidence that a Schwarzschild black hole is the strong coupling version of a highly excited string [2], an exact computation is only feasible if it remains valid when changing from g = 0 to g 6= 0. Unfortunately the strong coupling regime in string theory is in general very hard to control. This means we have to consider a state which is BPS, where a computation at weak coupling allows for making statements about the strong coupling version. 3 The Strominger-Vafa Construction 3.1 AdS/CFT for the D1/D5 System The first and by now best understood example of such a BPS state is due to Strominger and Vafa and constructed from a bound state of D1- and D5-branes in type IIB string 3 theory [3], see also [11]. For a correct number of branes the system is BPS and we can try to deduce some information of the strong coupling situation from the weak coupling computation. Although the number of microstates will not be invariant there is a certain topological index which acts as a bound on the degeneracy of BPS microstates. Let X be a complex, projective1 and compact K3 surface and consider type IIB string 1;4 1 theory compactified to five dimensions on R ×X ×S with stacks of Q1 D1-branes and 4 Q5 D5-branes wrapping cycles according to 3.1. Let (2π) V be the volume of X and R be the radius of the S1. Using the superposition principle for supergravity solutions of Figure 3.1: The D5-branes wrap all of the compact space and the D1-branes only wrap the S1. A five dimensional physicist sees a worldline in R1;4. intersecting branes [12] the metric of the non-compact part of this spacetime looks like 2 −2=3 2 1=3 2 2 2 ds = − (H1H5 (1 + K)) dt + (H1H5 (1 + K)) dr + r dΩ3 (3.1) with r2 r2 r2 H (r) = 1 + 1 H (r) = 1 + 5 K(r) = m (3.2) 1 r2 5 r2 r2 and gQ l6 g2Nl8 r2 = 1 s r2 = gQ l2 r2 = s ; (3.3) 1 V 5 5 s m R2V where N is the momentum quantum number corresponding to motion along the S1. This supergravity solution is valid if gQ1; gQ5; gN 1 which we will enforce by assuming a very high number of D1- and D5-branes as well as a very high momentum quantum number. 1This assumption is only necessary for the viewpoint we want to take and will become clear later on. 4 If the system is BPS this solution describes an extremal, Reissner-Nordström black hole with Hawking temperature TH = 0. Its "macroscopic" entropy can be calculated by using the Bekenstein-Hawking formula (the horizon is at r = 0) A 1 2 3 3=6 p SBH = = π r (H1H5 (1 + K)) r=0 = 2π Q1Q5N: (3.4) 4G5 4G5 The basic motivation behind the construction of Strominger and Vafa is to determine a microscopic origin of this entropy at weak coupling. As it turns out, this is an applica- tion of AdS/CFT. To explain how the AdS/CFT correspondence enters into the story we consider a slightly modified version of the geometry in which we unwrap the S1 (or, equivalently, set N = 0)2. In this case the supergravity solution for the five dimensional non-compact part of the spacetime becomes 2 −1=2 −1=2 2 2 1=2 1=2 2 2 2 ds = H1 H5 −dt + dx + H1 H5 dr + r dΩ3 : (3.5) If we vary the coupling the system has two different descriptions 3.2, see [4,5]. AdS/CFT Figure 3.2: The black string case (left) is valid for gQ1; gQ5 1 and is related to a system of (proper) D-branes whose worldvolume theory is a gauge theory determined by open strings (right), valid for gQ1; gQ5 1, when we lower the coupling and keep fixed the number of branes. The system of D-branes has SO(1; 1)×SO(4) as symmetry group and preserves N = (4; 4) supersym- metry. The gauge theory lives on the intersection of all branes. We omitted the K3 surface on both sides. will be valid in a further low-energy limit (α0 ! 0), in which we keep all dimensionless parameters fixed, in this case r V g v := g := p : (3.6) α0 (2π)4(α0)2 6 v There are two interpretations for this low energy limit, depending on the value of the coupling. 2In this case the area of the horizon and the entropy shrinks to zero, but this will not interfere with the point that is made. 5 a) For weak coupling gQ 1, we have a decoupling of supergravity in the bulk and gauge theory on the branes. The gauge theory has a Coulomb branch, which corresponds to varying expectation values of gauge fields, and (unlike for example the N = 4 SYM on a D3) also a Higgs branch, corresponding to varying expec- tation values of matter fields that arise from strings that stretch between the two types of branes. There is an argument [5] how the Coulomb branch can be re- moved by symmetries such that we are interested in the Higgs branch of the (1+1) dimensional gauge theory with N = (4; 4) supersymmetry on the worldvolume.