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Master Thesis Superstring and M Theory on a Manifold with G2 Holonomy

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Master Thesis Superstring and M Theory on a Manifold with G2 Holonomy Master thesis Superstring and M Theory on a Manifold with G2 Holonomy Toshiaki Shimizu February 24, 2003 Contents 1 Introduction 3 1.1 Introduction .................................. 3 1.2 Basic facts ................................... 4 1.2.1 Why superstings require 10 dimensions? .............. 4 1.2.2 Emergence of the eleventh dimension ................ 6 1.2.3 The condition for N = 1 spacetime supersymmetry ........ 7 1.2.4 Calabi-Yau manifolds ......................... 9 1.2.5 Manifolds with G2 holonomy . ................... 12 2 Witten index 13 2.1 Witten index . .............................. 13 2.1.1 Definitions and physical implications ................ 13 2.1.2 Forbidden changes in parameters .................. 15 2.2 Some generalizations ............................. 16 2.3 A simple example . .............................. 19 2.4 Non-linear sigma models ........................... 20 3 M theory on G2 22 3.1 Flop ...................................... 22 3.1.1 Preliminaries ............................. 22 3.1.2 Calabi-Yau/Landau-Ginzburg correspondence . ........ 25 3.1.3 Symplectic quotient .......................... 27 3.1.4 A flop . .............................. 29 3.2 Topological closed strings .......................... 32 3.2.1 N = 2 superconformal field theory .................. 32 3.2.2 Topological twists ........................... 38 3.2.3 Topological sigma models . ................... 40 3.2.4 Topological strings; coupling to gravity ............... 50 3.2.5 Deriving all genus answers . ................... 55 3.3 Topological open strings ........................... 61 3.3.1 Boundary conditions ......................... 62 3.3.2 Coupling to gauge fields ....................... 62 3.3.3 Large t limit .............................. 64 1 3.3.4 Space-time physics .......................... 66 3.3.5 Relations to superstring ....................... 69 3.4 Duality of topological strings ......................... 70 3.4.1 Open/Closed duality ......................... 70 3.4.2 Proof of duality ............................ 71 3.5 Duality of superstrings ............................ 75 3.5.1 RR fluxes and topological string amplitudes ............ 75 3.5.2 Embedding the duality ........................ 77 3.6 M-theory flop and duality .......................... 80 3.6.1 Generalized flop ............................ 80 3.6.2 Gauge theoretical interpretation ................... 84 3.6.3 Type IIA description ......................... 84 4 Superstring on G2 86 4.1 Fundamental properties ........................... 86 4.1.1 G2 CFT................................ 86 4.1.2 Heighest weight states and a twist operator ............ 89 4.1.3 Geometry and G2 CFT........................ 92 4.1.4 Left-Right sector ........................... 94 4.1.5 Example of Joyce ........................... 95 4.1.6 Topological twist? ........................... 96 5 Conclusion and summary 99 2 Chapter 1 Introduction 1.1 Introduction String theory [1, 2] was first introduced as a model for explaining the exotic features of the strong interaction such as quark confinement, Regge slope, dual resonances, and so on. However, it was found that string theory is not successful and QCD is the correct description for strong interactions. Therefore, string theory seemed to die. However, after many turns and twists, string theory is now thought to be a strong candidate for unifying all four interactions. The main reasons are its self-consistency, existence of gravity, no UV divergences, and no free parameters. There are five string theories which have space-time supersymmetries, type I, type IIA, type IIB, heterotic string with SO(32) gauge group, and heterotic string with E8 ×E8 gauge group and they all live in ten dimensions. At the perturbative level, they are totally different from each other. Therefore, in the past, it was conjectured that one of five theories was the truely unified theory and others were not so interesting. And that one was the E8 ×E8 heterotic string. The reason was the similarity to the gauge group of GUT. And, as we will explain in the next section, heterotic string on Calabi-Yau manifolds provides us with N = 1 supersymmetry in four dimensions. Therefore, heterotic strings on Calabi-Yau manifolds lead to N = 1 supersymmetric theories with realistic gauge groups and correct dimension. For these reasons, they have been studied vigorously. However, this point of view was changed dramatically by the emergence of D-branes. D-branes are the solitons of the string theory and inform us of non-perturbative aspects of string theory. And it was found that there is an unique eleven dimensional theory called “M-theory” which is the truly unified theory. From this perspective, string theories are thought to describe the physics around some particular vacuum. Under these circumstances, manifolds with G2 holonomy have received much attention. The main reason for this is that if we compactify M-theory on such a manifold, we can get a four dimensional theory with 4 supercharges, which is of phenomenological importance. They also play fundamental roles in realizing the N = 1 supersymmetric gauge theories. In fact, Atiyah, Maldacena and Vafa proposed a new duality by using the M theory on G2 manifolds [43]. This work is based on the previous work by Vafa and his coraborators [40, 41, 42]. These are the origin of the recent work by Dijkgraaf and Vafa 3 [56] and their work is the realization of the possibility that the non-perturbative physics can be understood by the perturbative calculations, which was first pointed out in [27]. Therefore, manifolds with G2 holonomy are important from both phenomenological and theoretical point of view. With this in mind, we review the fundamental properties of superstring and M-theory on G2 manifolds. The organization is as follows. We will first recall some of the basic facts about string theory and geometry in chapter 1. Then, we will discuss the Witten’s index in chapter 2. And we will go to the main chapter, M-theory on G2 manifolds. In this chapter, we will review the work of Atiyah, Maldacena and Vafa and the fundamentals for understanding their work. Or more concretely, we will review conifold transitions, N = 2 superconformal field theories, open and closed topological strings, relations to superstring, and their duality properties. In chapter 4, we will discuss CFT description of G2 manifolds. Chapter 5 is conclusion and summary. 1.2 Basic facts 1.2.1 Why superstings require 10 dimensions? Let us first review briefly what determines the dimensions. This chapter only contains basic facts which we assume the reader to be familiar with. To avoid technical difficulties, we first consider bosonic string. The action is 1 1 − − 2 ab µ S = dσdτ( γ) γ ∂aX ∂bXµ (1.1) 4πα M where γab is a world sheet metric (Lorentz signature(−, +)) and γ is determinant of it. µ runs 0 to D − 1. (D is the dimension of spacetime.) This action has the following symmetries. 1. D-dimensional Poincar´e invariance µ µ ν µ X (τ,σ)=Λ ν X (τ,σ)+a (1.2) γab(τ,σ)=γab(τ,σ) (1.3) 2. Two dimensional diff invariance Xµ(τ ,σ)=Xµ(τ,σ) (1.4) ∂σc ∂σd γ (τ ,σ)=γ (τ,σ) (1.5) ∂σa ∂σb cd ab 3. Two dimensional Weyl invariance Xµ(τ,σ)=Xµ(τ,σ) (1.6) γab(τ,σ) = exp(2ω(τ,σ))γab(τ,σ) (1.7) 4 This action of course describes a string propagating in D dimensional spacetime. But, we can also regard this as a two dimensional field theory. If we see from this point of view, Xµ are two dimensional bosons, interacting with two dimensional gravitons γab. And this theory has a large gauge symmetry which we must fix; i.e. local diff×Weyl symmetry. Let us fix this gauge symmetry. We first replace the world sheet metric with a Euclidean one and denote this new metric as gab. If we forget the global structures of the world sheet, it is easy to show that we can make this metric δab by combining the diff and Weyl gauge symmetry. Therefore, we can fix the gauge by restricting to the flat metric. By using a well-known Fadeev- Popov method, the action becomes 1 S = dzdz¯ ∂Xµ∂X¯ + b∂c¯ + ¯b∂c¯ (1.8) 2πα µ after the gauge fixing. Here we change the world sheet coordinates to z = σ1 + iσ2 = σ + i(iτ) and its complex cojugatez ¯. If we change coordinates in such a way that z is a holomorphic function of z z = f(z) (1.9) and combine this with a Weyl transformation, the metric becomes 2 −2 ds = exp(2ω)|∂zf| dz dz¯. (1.10) If we take ω =ln|∂zf| here, we have the same metric as before. This means that locally, we have an extra gauge symmetry (this is a local conformal symmetry) even after fixing the metric. Therefore, after fixing the gauge, we have two dimensional conformal field theories which consist of matter CFT (D free bonsons Xµ) and ghost CFT. As we know, a free bonson has central charge 1 and the ghost has central charge −26. For the theory to be consistent, the diff×Weyl invariance must not be amomalous. It can be shown that the theory can be regularized in a diff invariant way. So, we can concentrate only on the Weyl anomaly. Recall that the Weyl invariance makes the energy-momentum tensor a a traceless; i.e. T a = 0. The only possible form of the anomaly is known to be T a = a1R, where a1 is some constant and R is a scalar curvature of the world sheet. This can be rewritten in complex coordinates of the form a T = 1 g R. (1.11) zz¯ 2 zz¯ Let us calculate the constant a1. Taking the covariant derivative of both sides, we have a ∇z¯T = 1 ∂ R. (1.12) zz¯ 2 z By using the conservation of Tab, this can be written as a ∇zT = −∇z¯T = − 1 ∂ R. (1.13) zz zz¯ 2 z 5 The Weyl transformation of the right hand side is ∇2 ≈ 2 a1∂z δω 4a1∂z ∂z¯δω (1.14) where we expand around a flat world sheet.
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