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The Sixth Superstring

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The Sixth Superstring The Sixth Superstring Bachelor thesis in Physics Author: Pelle Werkman Supervisor: Diederik Roest June 10 2013 Abstract The objective of this thesis is to discuss the possible new superstring proposed by Savdeep Sethi in April 2013. In order to do this, we give an introduction to string theory pretty much from the ground up - starting with the 26 dimensional bosonic string and then on to the five different flavours of ten dimensional superstring. Along the way we will discuss some of the dualities and transformations that relate these strings to each other. Savdeep Sethi's superstring will arise from just such a transformation: an orientifold of the type IIB superstring. Eventually we will start to hone in on the modern picture of string theory, which is that the superstrings are all perturbative limits of an 11-dimensional theory called 'M-theory'. We will discuss how Savdeep Sethi's superstring may fit into this web of theories. By construction, the new string looks very similar to the Type I superstring. We will have to find a way to distinguish the two from each other. By comparing the Kaluza-Klein towers that result from their M-theory descriptions, we will find a sharp distinction between Type I and the new superstring. However, we will be left with questions about the consistency of the new string and about its place in the M-theory web. Contents 1 Introduction 3 2 The bosonic string 6 2.1 Constructing the action . 6 2.1.1 The point particle action . 6 2.1.2 The string action . 6 2.1.3 Choosing a parametrization . 7 2.1.4 The Polyakov action . 8 2.2 Symmetries of the Polyakov action and the conformal gauge . 8 2.2.1 Poincar´etransformations . 8 2.2.2 Reparametrization . 8 2.2.3 Weyl rescaling . 9 2.2.4 Gauge fixing and residual symmetry: the light cone gauge . 9 2.3 Boundary conditions: open and closed strings . 10 2.4 Mode expansion . 10 2.4.1 The closed string . 11 2.4.2 Virasoro constraints . 11 2.4.3 The open string . 12 2.5 Quantization . 12 2.5.1 Covariant quantization . 12 2.5.2 Dealing with the ghosts . 13 2.5.3 The mass operators . 13 2.5.4 Light-cone gauge quantization . 14 2.6 The spectrum . 15 2.6.1 The open string . 15 2.6.2 The closed string . 15 3 The supersymmetric string 17 3.1 The RNS formalism . 17 3.1.1 The action and equations of motion . 17 3.1.2 World-sheet supersymmetry . 18 3.1.3 The super-Virasoro constraints . 18 3.1.4 Mode expansion . 19 3.1.5 Covariant quantization . 20 3.1.6 Light-cone gauge quantization . 22 3.1.7 GSO projection . 24 3.1.8 Closed string spectrum . 24 3.2 The GS formalism . 25 3.2.1 Point particle action . 26 3.2.2 Superstring action . 26 3.2.3 Quantization . 27 3.2.4 The massless spectrum . 28 3.3 A look at supergravity . 29 3.3.1 11-dimensional supergravity . 29 3.3.2 Type IIB supergravity . 30 3.4 Perturbative symmetries of Type IIB . 31 3.5 Anomaly cancellation . 32 1 3.6 The heterotic strings . 32 3.7 The web of dualities . 32 4 Compactification 33 4.1 T-duality and D-branes in the bosonic theory . 33 4.1.1 Closed strings . 33 4.1.2 Open strings . 35 4.1.3 D-branes and gauge fields . 36 4.1.4 Wilson lines . 36 4.1.5 String charge . 39 4.2 T-duality and D-branes in superstring theory . 39 4.2.1 D-brane charges . 39 4.2.2 Stability of D-branes . 40 4.2.3 Type II T-duality . 41 4.3 Kaluza-Klein towers and 11-dimensional supergravity . 42 4.3.1 A note on BPS states . 42 4.3.2 M-theory - Type II duality . 42 4.4 The web of dualities: part II . 44 5 Orbifolds and Orientifolds 45 5.1 General features of orbifolds . 45 5.2 Type IIA as an orbifold . 45 5.3 General features of orientifolds . 46 5.4 Type I as an orientifold . 46 5.4.1 Type I' . 47 5.5 Heterotic-Type I S-duality . 47 5.6 The web of dualities: part III . 48 6 Savdeep Sethi's Superstring 49 6.1 A different orientifold . 49 6.2 M-theory BPS spectrum . 49 6.3 The strong coupling limit . 51 Conclusion 52 Appendices 53 A Review of spinors 53 A.1 Gamma matrices in d dimensions . 53 A.2 Spinors of SO(1; d − 1)............................................. 53 A.3 Spin transformations . 54 A.4 Fierz identity . ..
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