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Hyperbolic Geometry of Superstring Perturbation Theory

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Hyperbolic Geometry of Superstring Perturbation Theory Prepared for submission to JHEP Hyperbolic Geometry of Superstring Perturbation Theory Seyed Faroogh Moosavian, Roji Pius Perimeter Institute for Theoretical Physics, Waterloo, ON N2L 2Y5, Canada E-mail: [email protected] [email protected] Abstract: The hyperbolic structure of the RNS formulation of perturbative su- perstring theory is explored, with the goal of providing a systematic method for explicitly evaluating both the on-shell and the off-shell amplitudes in closed super- string theory with arbitrary number of external states and loops. A set of local coordinates around the punctures and a distribution of picture-changing operators on the worldsheet satisfying the off-shell factorization requirement needed for defining the off-shell amplitudes are specified using the hyperbolic metric on the worldsheet with constant curvature −1. The superstring amplitudes are expressed as sum of real integrals over certain covering spaces of the corresponding moduli space. These covering spaces are identified as simple domains inside the Teichm¨ullerspace of the worldsheet parameterized using the Fenchel-Nielsen coordinates. arXiv:1703.10563v2 [hep-th] 3 Feb 2019 Contents 1 Introduction1 1.1 Objectives1 1.1.1 Bosonic String Amplitudes and Moduli Space of Riemann Sur- faces2 1.1.2 Superstring Amplitudes and Picture-Changing Operators4 1.1.3 Off-Shell Superstring Amplitudes and 1PI Approach6 1.2 Summary of the Results9 1.2.1 Bosonic-String Measure and the Fenchel-Nielsen Coordinates 10 1.2.2 Gluing-Compatible Local Coordinates and Hyperbolic Metric 11 1.2.3 Integration of the Bosonic-String Measure 13 1.2.4 Superstring Amplitudes and the PCOs 14 1.3 Organization of the Paper 15 2 The Off-Shell Bosonic-String Amplitudes 15 2.1 The Worldsheet Theory 16 2.2 Off-Shell Bosonic-String Amplitudes 16 2.2.1 Tangent Vectors of Pg;n 17 2.2.2 Differential Forms on Pbg;n 19 2.2.3 Differential Form on Sections of Pbg;n 20 2.2.4 Gluing-Compatible Integration Cycle 21 3 Hyperbolic Geometry and Off-Shell Bosonic-String Amplitudes 22 3.1 A Gluing-Compatible Section of Pbg;n 22 3.1.1 Plumbing Fixture and Hyperbolic Riemann Surfaces 23 3.1.2 A Proposal for a Gluing-Compatible Section of Pbg;n 28 3.1.3 The Hyperbolic Metric on the Plumbing Family 29 3.1.4 A Gluing-Compatible Choice of Local Coordinates 34 3.2 Integration Measure on the Section of Pbg;n 37 3.3 Integration over the Moduli Space 39 3.3.1 The Covering Space of the Moduli Space 40 3.3.2 Integration of MCG-Invariant Functions over the Moduli Space 43 3.3.3 The Mirzakhani-McShane Identity 45 3.3.4 Computing the Volume of Moduli Spaces 47 4 The Off-Shell Superstring Amplitudes 50 4.1 The Worldsheet Theory 50 4.2 The Off-Shell Superstring Measure 51 4.3 A Gluing-Compatible Integration Cycle 52 { i { 4.4 Regulating the Infrared Divergences 54 4.5 Singularities Associated with PCOs 54 4.6 A Prescription for Avoiding the Spurious Poles: The Vertical Integration 55 4.6.1 A Warm-Up Example 56 4.6.2 The Vertical Integration Prescription 59 4.7 The Integration Measure 63 5 Hyperbolic Geometry and Off-Shell Superstring Amplitudes 64 5.1 Gluing-Compatibility and the Mirzakhani-McShane Decomposition 64 5.1.1 A Property of the Mirzakhani-McShane Decomposition 65 5.1.2 The Maximal Decomposition of the Off-Shell Superstring Am- plitude 66 5.1.3 A Gluing-Compatible Distribution of PCOs 69 5.1.4 A Gluing-Compatible Choice of Local Coordinates 70 5.2 The Infrared Regulator and the Maximal Decomposition 70 5.3 An Integration Cycle Avoiding the Spurious Poles 71 6 Discussion and Future Directions 71 A Hyperbolic Riemann Surfaces and their Teichm¨ullerspace 72 A.1 The Hyperbolic Pair of Pants 74 A.2 The Complex Structure of a Pair of Pants 75 A.3 The Teichm¨ullerSpace of a Hyperbolic Riemann Surfaces 76 A.4 The Teichm¨ullerSpace of a hyperbolic Pair of Pants 77 A.5 The Fenchel-Nielsen Coordinates 78 A.6 The Presentation for the Mapping Class Group 79 B The Matrices for the Fenchel-Nielsen Coordinates 82 B.1 The Fenchel-Nielsen System 82 B.1.1 The Ordering for the Boundary Components of the Pairs of Pants 83 B.1.2 Directing the Coordinate and the Boundary Geodesics 84 B.2 From the F-N System to the Generators of the Fuchsian Group 84 B.2.1 The Normalization 84 B.2.2 The Basic Building Blocks 85 B.2.3 Sphere with Holes 85 B.2.4 Closing the Handles 87 B.3 The Explicit Matrices 88 B.3.1 Reduction to Primitive Conjugators 88 B.3.2 The Normalized Pants Groups 90 B.3.3 Closing a Handle on a Pair of Pants 91 { ii { B.3.4 Matrices for the Universal Twist Map, Interchange and Ele- mentary Conjugators 92 B.4 The Algorithm 93 1 Introduction One of the main powers of quantum field theory is that it comes with a set of rela- tively simple rules for calculating measurable quantities. For instance, it allows us to compute the quantities of interest, such as the scattering amplitudes, as a perturba- tive expansion in the coupling parameters of the theory by evaluating the Feynman diagrams using the Feynman rules that are derived from the Lagrangian of the the- ory. The Feynman rules expresses the amplitudes with arbitrary external states and loops as well-defined integrals. Therefore, we can consider Feynman prescription as a calculable formulation of the perturbative quantum field theory. Similarly, in the conventional formulation of string theory, scattering amplitudes are obtained by evaluating the string diagrams which are the string theory analogue of the Feynman diagrams. Unlike in quantum field theory, the rules for evaluating the string diagrams are not derived from a Lagrangian. Instead the rules are given without referring to an underlying Lagrangian [1{3]. By exploring the conformal structure of string theory, powerful covariant methods to compute amplitudes in perturbative string theory were introduced in [3]. Some aspects of the formalism that remained slightly opaque also got cleared due to the recent studies and the revival of old results [9, 11{28]. Therefore, we can safely say that perturbative string theory provides a well-defined formal procedure for computing the amplitudes as an expansion in the string coupling. However, we can consider this as a calculable formulation of the string theory, only if it comes with a set of well defined practical rules for evaluating the amplitudes with any number of arbitrary external states and loops. Unfortunately, such a set of practical rules for computing the amplitudes is not readily available. In this paper, we describe a practical procedure for performing computations in perturbative string theory by using hyperbolic geometry. Below, we spell out the main objectives of this paper followed by a brief summary of the results. 1.1 Objectives The path integral formulation of string theory requires summing over all possible worldsheets traced by the relativistic strings in spacetime with contribution from each worldsheet weighted by a unimodular complex number whose phase is the area of the corresponding worldsheet. In his seminal papers [1,2], Polyakov showed { 1 { that this summation over the two dimensional surfaces reduces to two dimensional exactly solvable conformal field theory on Riemann surfaces. This connection was established by introducing a metric on each of the worldsheets. The net effect of this is to introduce an unphysical degree of freedom, Liouville field, into the problem that disappears from the worldsheet action due to its Weyl invariance. Demanding the Weyl invariance at the quantum level makes sure that the Liouville field will not reappear in the action due to the quantum correction. Moreover, if we assume that the external states involved in the scattering process satisfy the classical on- shell condition, then the amplitudes are also independent of the specific choice of the metric on the world-sheet. Since the classical on-shell condition is assumed for the external states, we shall call the amplitude computed this way as the on-shell string amplitudes. 1.1.1 Bosonic String Amplitudes and Moduli Space of Riemann Surfaces The conventional formulation of the bosonic string theory defines the g-loop con- tribution to the amplitude for the scattering of n asymptotic closed string states as a path integral over the embedding Xµ(z; z¯) of the string worldsheet, which is a Riemann surfaces with n boundaries, in spacetime and all inequivalent metrics on such Riemann surfaces. Here z denotes the complex coordinate on the worldsheet. Using the state-operator correspondence, it is possible to replace the boundary con- ditions with vertex operators, in an appropriate conformal field theory on the genus g Riemann surface. Let us denote the vertex operator representing a state carrying µ quantum numbers ai and momentum ki by Vai;ki (zi; z¯i). Then the momentum space expression for the g-loop string amplitude for n closed-string states having quantum µ numbers ai and momentum ki with i running from 1 to n is given by, n Z h i µ Z (n) µ −Sp[gab;X ] Y 2 p Ag (a1; k1; :::; an; kn) = dgab dX e d zi −g(zi) Vai;ki (zi); g;n | {z } i=1 On genus g Riemann surface with n punctures where Sp is the Polyakov action for a free relativistic string propagating through the space time. The path integral 1.1 can be made well-defined by carrying out the gauge fixing by introducing the Fadeev-Popov ghosts.
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