String Theory and Quantum Field Theories in Three Dimensions A Thesis submitted to the Tata Institute of Fundamental Research, Mumbai for the degree of Doctor of Philosophy in Physics by Shiroman Prakash Department of Theoretical Physics Tata Institute of Fundamental Research Mumbai July 2012 Declaration This thesis is a presentation of my original research work. Wherever contributions of others are involved, every effort is made to indicate this clearly, with due reference to the literature, and acknowledgement of collaborative research and discussions. The work was done under the guidance of Professor Sandip P. Trivedi, at the Tata Institute of Fundamental Research, Mumbai. Shiroman Prakash In my capacity as supervisor of the candidate's thesis, I certify that the above statements are true to the best of my knowledge. Sandip P. Trivedi Date i Acknowledgment I would like to thank my advisor, Prof Sandip Trivedi, for all his help and encouragement during my time at TIFR and for sharing with me his knowledge, insight and invaluable phys- ical intuition. I would like to thank all my collaborators for their invaluable contributions to my understand- ing of theoretical physics; it was a privilege to work with them. I would also like to thank members of the string theory group, other members of the department and other members of the institute with whom I have interacted for their invaluable contribution to my education. I would like to thank all the former and present graduate students with whom I have shared an office for teaching me a great deal of physics. Finally, I would like to thank the people of India for their generous support for the basic sciences. ii Contents Synopsis iv List of Papers....................................... xviii 1 Introduction1 1.1 Holography of Dilaton Black Branes.......................1 1.2 Dyonic Dilaton Black Branes...........................3 1.3 A Chern-Simons Vector Model..........................5 1.4 Summary......................................9 2 Holography of Charged Dilaton Black Branes 11 2.1 Introduction..................................... 11 2.2 Gravity solution.................................. 14 2.3 Thermodynamics.................................. 18 2.4 Attractor Behavior................................. 20 2.5 Conductivity.................................... 23 2.6 Summary and Discussion............................. 29 3 Holography of Dyonic Dilaton Black Branes 30 3.1 Introduction..................................... 30 3.2 Review of Earlier Results............................. 31 3.3 The DC Conductivity............................... 33 3.4 Purely Magnetic Case............................... 37 3.5 The SL(2;R) Invariant Case........................... 41 3.6 Conductivity in the SL(2;R) Invariant Case................... 44 3.7 Attractor Behavior................................. 56 3.8 Discussion...................................... 61 4 A Chern-Simons Vector Model 67 4.1 Introduction..................................... 67 2 4.2 Free Energy on R at Finite Temperature.................... 68 4.3 Operator Spectrum................................. 91 4.4 Discussion...................................... 107 5 Conclusion 110 iii Synopsis Introduction Recent developments in string theory and M-theory have led to increased interest and new progress in the study of three-dimensional quantum field theories. One of the main reasons for this is that conformal field theories in three dimensions (CFT3's) involving U(N) gauge groups are expected to be dual to quantum theories of gravity in 4-dimensional anti-de Sitter space (AdS4), via the AdS=CF T (or holographic) duality [1]. The primary example of the AdS4=CF T3 correspondence is the ABJM theory discovered in 2008 [2] which is believed to describe the low-energy dynamics of multiple M2-branes in a Zk orbifold, and hence is conjectured to be dual to M-theory on AdS4 × S7=Zk. An important aspect of the AdS=CF T duality is that the classical limit of the four- dimensional quantum theory of gravity takes on a new role as a large-N saddle point to the three-dimensional conformal field theory in a particular strongly interacting limit. Various authors (e.g., [3,4,5,6,7]) have therefore been inspired to use gravitational systems as toy models for strongly interacting quantum systems near a quantum critical point. Motivated by these developments, in this thesis we study some new examples of strongly interacting conformal quantum field theories in three dimensions in the large-N limit, using both holographic (i.e., gravitational) and quantum field theory techniques. This thesis is based on work communicated in [8], [9] and [10]. We first consider holography of charged dilaton black branes in AdS4 in [8]. The most interesting feature of the solutions we study is that they have vanishing entropy at zero- temperature, in contrast with the Reissner-Nordstr¨ombranes studied earlier. We then con- sider dyonic charged dilaton black branes in [9]. We are able to calculate various transport properties of the strongly-coupled field theories dual to these gravitational theories at finite chemical potential and in the presence of a magnetic field. Our work here builds on previous studies aimed at modelling strongly interacting quantum systems using holography including [3,4,5, 11, 12,6, 13, 14,7], as well as previous work on non-supersymmetric attractors [15]. In addition to exploiting this duality to generate and study new examples of strongly interacting conformal field theories in three dimensions using gravity; it is also of interest to find models (particularly non-supersymmetric ones) that can be solved directly from the field theory side at strong coupling in the large N limit, with a view towards gaining some intuition into the mechanism behind the duality itself. For this purpose, we conclude with a study [10] of U(N) Chern-Simons theory coupled to fundamental fermions { a theory that turns out to iv be remarkably educational and (at least partially) exactly-solvable using traditional large N techniques. We briefly highlight the main results of these studies below. Holography of Charged Dilaton Black Branes The earliest charged black hole solutions to low-energy string theory were found by Garfinkle, Horowitz and Strominger [16] (and had appeared earlier as part of a family of solutions in [17]). Those authors studied the Einstein-Maxwell action with the gauge coupling controlled by a scalar dilaton φ. These black holes, and generalisations thereof, were found to have thermodynamic properties very different from traditional charged black holes [18, 19]. Hence, with a view towards generating holographic models with thermodynamic behaviour different from the usual Reissner-Nordstr¨omblack branes (which have the unpleasant feature of a large entropy at zero temperature) previously studied, it is natural to consider their AdS generalisations. Gravity solution We consider the following action, Z p S = d4x −g R − 2(rφ)2 − f(φ)F 2 − 2Λ : (1) which generalises the action studied in [16] to include a negative cosmological constant, and arbitrary dilaton coupling f(φ). We will focus on the specific case f(φ) = e2αφ, though in [8] we also briefly consider some other possibilities. The maximally symmetric vacuum solution 3 for this action is AdS space with AdS scale L determined by Λ = − L2 . We will set L = 1 below. We will look for electrically-charged black branes solutions of the above action, taking the metric to be of the form ds2 = −a(r)2 dt2 + a(r)−2 dr2 + b(r)2 (dx2 + dy2); (2) with gauge field Q e2αφF = dt ^ dr: (3) b(r)2 We first look for a near horizon solution to the above action by considering the following scaling ansatz: γ β a = C2(r − rh) ; b = C1(r − rh) ; φ = − K log(r − rh) + C3 ; (4) We find that an exact solution is obtained if the exponents take the values α ( α )2 γ = 1;K = 2 ; β = 2 : (5) α 2 1 + ( α )2 1 + ( 2 ) 2 v The constant C2 is given by 6 C2 = : (6) 2 (β + 1)(2β + 1) By rescaling (r − rh), t, x and y, one can set the constant C3 to zero and C1 to unity. Q is then determined in terms of α by 6 Q2 = : (7) (α2 + 2) In terms of w = r − rh, the exact near-horizon solution is: β a = C2w; b = w ; φ = − K log(w) : (8) Interestingly, this solution almost has a Lifshitz scaling symmetry under w ! λw, t ! λt, x ! λzx, and y ! λzy, with Lifshitz exponent z = 1/β; the symmetry is only broken by constant shifts in the dilaton φ. The metric component gtt has a second order zero at w = 0, and (8) is thus an extremal solution. A non-extremal generalisation is the following: 2 2 2 2 m 2 dw 2β 2 2 ds = C2 w (1 − 2β+1 )dt + 2 2 m + w (dx + dy ); (9) w C2 w (1 − w2β+1 ) which depends on the parameter m. A numerical asymptotically AdS4 solution −αφ In the extremal solution, the gauge coupling gU(1) ∼ e becomes weak at the horizon, as φ ! 1. When r ! 1 the gauge coupling becomes very strong; therefore, although exact, the solution (9) must be understood as a near-horizon geometry of a larger solution with different asymptotics. (The near-horizon solution appeared in a different coordinate system in the unpublished work [14], where it appeared unphysical due to this pathology.) In particular, we would like to find a generalisation of the extremal solution with a controlled and asymptotically constant dilaton as well as an asymptotically AdS4 geometry. The strategy for obtaining such a solution is to numerically integrate the equations of motion following from (1), starting near w = 0 with initial data taken from the near-horizon solution. However, the near-horizon solution is exact, so numerical integration using initial data drawn from it would simply reproduce the near-horizon solution unmodified. To nu- merically integrate to a solution that is asymptotically AdS4, we must also take into account possible subleading corrections to the near-horizon solution.