Double Field Theory and Closed String T-Duality Sean S. Cooper Queen Mary University of London, Mile End Road, London, E1 4NS Email: [email protected] Abstract. Double Field Theory is a formalisation of string theory in which T-duality, a uniquely stringy phenomenon, is promoted to a manifest symmetry. The theory of closed strings on toroidal backgrounds is reviewed, leading to the notion of dual coordinates. The Double Field Theory, invariant under the O(d; d; R) duality group, is then constructed and its local symmetries discussed. An O(d; d; R) invariant action is then presented which, upon imposing the section condition, is shown to reduce to the NS-NS sector supergravity action. Contents 1. Introduction1 1.1. General Introduction . .1 1.2. Closed Bosonic Strings . .2 1.3. Conventions . .3 2. T-Duality4 2.1. The Winding Modes of Closed Strings . .4 2.2. Momentum Quantisation on Closed Dimensions . .5 2.3. A Glimpse of a Hidden Symmetry . .6 2.4. The T-Duality Transformation . .9 2.5. The Generalised Metric . 12 2.6. Doubling the Dimensions . 14 3. The O(d; d; R) Duality Group 17 3.1. Properties of O(d; d)............................. 17 3.2. O(d; d) Fundamental Objects for DFT . 18 3.3. O(d; d) Transformations . 19 i 3.4. Constraints in DFT . 21 3.4.1. The Weak Constraint . 21 3.4.2. The Strong Constraint . 22 3.4.3. The Section Condition . 23 4. Local Transformations in DFT 24 4.1. The Lie Deriviative . 24 4.2. Kaluza-Klein Theory as an Analogy . 26 4.3. The Generalised Lie Derivative . 27 4.4. Generalised Diffeomorphisms in DFT . 28 4.5. Properties of the Generalised Lie Derivative . 30 4.5.1. A Bracket for Generalised Lie . 30 4.5.2. Closure . 33 4.5.3. Jacobi Identity . 35 5. An Action for DFT 39 5.1. The DFT Action . 39 5.2. Reduction of DFT to the Supergravity Action . 39 6. Conclusion 45 A. The Lie Derivative of an Arbitrary Tensor Field 47 B. The Automorphism of the Courant Bracket 49 Bibliography 51 Declaration I hereby certify that this project report, which is approximately eight and a half thousand words in length, has been written by me at the School of Physics and Astronomy, Queen Mary University of London, that all material in this dissertation which is not my own work has been properly acknowledged, and that it has not been submitted in any previous application for a degree. This document is presented as the written coursework requirement of the Physics Investigative Project (SPA7015U). ii 1. Introduction 1.1. General Introduction A duality between theories hints at a potential unification between them, and at the possibility that the theories in question are special cases of a more general one. Dual- ities are relations through which physical theories are equated. It is this concept with which this report will be concerned. More specifically, we will consider the T-duality relating different string theories. T-duality is a hidden symmetry arising in theories with backgrounds constrained by isometry. By doubling the dimensions of the space-time, a new theory, dubbed Double Field Theory, can be constructed. This approach promotes T-duality to a manifest symmetry. In doing so, the T-dual theories become special cases of the Double Field Theory. In string theory, the bosonic and fermionic state spaces result from choosing either Neveu-Schwarz (NS) or Ramond (R) boundary conditions. There are four closed string sectors, each constructing a spectrum of states from one left-moving sector and one right- moving sector.[1] The resulting four sectors are: NS-NS NS-R R-NS R-R In this report we will focus on the NS-NS sector of closed bosonic strings. The R-R sector, and the fermionic NS-R and R-NS sectors, will not be considered. We will begin by reviewing the T-duality of the closed bosonic string theory. After this the O(d; d; R) duality group, from which the Double Field Theory is constructed, is discussed. With this Double Field Theory in place, one can then begin to explore it's properties. The local symmetries of the theory are investigated, and are related to the local symmetries of the original theory by a reduction similar to that of Kaluza-Klein theory. After examining these local symmetries, and their gauge algebra, an action to describe the Double Field Theory is presented. It is then shown that, upon imposition of a suitable condition, this action can be reduced to the NS-NS sector supergravity action. 1 1.2. Closed Bosonic Strings In this section the basics of closed bosonic string theory are discussed. This is not a full review, instead it is a minimal presentation of the tools required for the remainder of the report. We will be considering the NS-NS sector; the massless fields of such a theory are: • A symmetric field G. This field is the metric of the background space-time. • The antisymmetric Kalb-Ramond gauge field B. • A scalar field Φ, known as the dilaton. Given these massless excitations of the string, one can write an action for the theory: −1 Z S = d2σ pjhjhµνG (@ Xi)(@ Xj) + µνB (@ Xi)(@ Xj) + pjhjα0R(2)Φ 4πα0 ij µ ν ij µ ν (1.1) This action is essentially a Polyakov action, with the inclusion of terms coupling the B field, and the dilaton, to the string. As the string propagates it sweeps out a two-dimensional surface. This surface is known as the string worldsheet, and is the string theory analogue of a point particle's worldline. This surface is parametrised by the worldsheet coordinates τ and σ. τ is a timelike coordinate with the range τ 2 [−∞; 1]. σ corresponds to the spacelike coordinate along the length of the string. Since the string is closed we find that σ is periodic in 2π, with the range σ 2 [0; 2π). The objects h and R(2) in the above action are the metric on the worldsheet, and the worldsheet's scalar curvature, respectively. The string is a one-dimensional object embedded in a background space-time known as the target space.[1] The string's position in this target space is then given by the string coordinates Xi(τ; σ). It is necessary to specify which point on the string we are referring to. As such, the string coordinates are in fact functions of the worldsheet coordinates. We will encounter derivatives of the string coordinates with respect to τ and σ. To aid readability, the following conventions are used: _ i i X := @τ X (1.2) 0i i X := @σX (1.3) The integration measure in the action, (1.1), is a shorthand notation with d2σ := dτ dσ. That is to say, the action is an integration over both of the worldsheet coordinates. Each string coordinate can be decomposed into a left-moving part and a right-moving i i i part as follows: X (τ; σ) = XL(τ + σ) + XR(τ − σ). As we will see in 2.3, these left- 2 i moving and right-moving oscillators can be expanded in terms of the modes αn and i α¯n respectively. After quantising the theory these modes correspond to creation and annihilation operators, of the type seen in the quantum harmonic oscillator. The constant α0 appearing in (1.1) is the slope parameter. It has dimensions of length- squared, and is related to the string tension by: 1 T = (1.4) 2πα0 1.3. Conventions Unless stated otherwise, the following conventions will be used in this report: • We will work in natural units; i.e., ~ = c = 1. • Greek characters (µ, ν, :::) will be used to label indices corresponding to the worldsheet coordinates, τ and σ. • Lowercase latin indices (i, j, :::) will be run of d coordinates. In the case of the Double Field Theory, these lowercase labels correspond to a d-dimensional subset of the total 2d coordinates. • Uppercase latin characters (M, N, :::) will be used to denote O(d; d) indices. Such indices correspond to the 2d-dimensional space-time. 3 2. T-Duality 2.1. The Winding Modes of Closed Strings The T-duality of closed strings is a phenomenon arising in theories with toroidal back- grounds; i.e, geometries with periodic isometry. In this section we will examine a simple example of such a geometry, namely a two-dimensional cylinder. The coordinates x and y will be used to describe this surface, with the x denoting the closed dimension, and y the direction along the length of the cylinder. One can then describe the periodicity of x by the identification:[1] (x; y) ∼ (x + 2πR; y) (2.1) Where, R is the radius of the circular dimension. Figure 2.1 provides a pictorial repre- sentation of such a geometry. Figure 2.1.: Left: a collection of closed strings living on the surface of a two-dimensional cylinder. Right: the same set of strings represented on the covering space of the cylinder. Strings with nontrivial winding numbers appear in the covering space as open strings. [1] The strings in this background exhibit a new behaviour, they can wrap around the 4 closed dimension. Since these strings are closed they cannot be unwound from the cylinder. Additionally, we see that the circumference of the circular dimension defines a minimum size for strings wrapped around it. We can define a winding number, m 2 Z, to count the number of times a string is wound around the cylinder. The aforementioned figure displays five examples of how strings could be wrapped around the cylinder, with arrows denoting the positive σ direction along each string.