Superstrings
In the first part of this chapter we compute the one-loop partition function of the closed fermionic string. We will do this in light cone gauge. The possibility to assign to the world-sheet fermions periodic or anti-periodic boundary conditions leads to
- PDF / 411,975 Bytes
- 39 Pages / 439.36 x 666.15 pts Page_size
- 16 Downloads / 160 Views
Superstrings
Abstract In the first part of this chapter we compute the one-loop partition function of the closed fermionic string. We will do this in light cone gauge. The possibility to assign to the world-sheet fermions periodic or anti-periodic boundary conditions leads to the concept of spin structures. The requirement of modular invariance is then shown to result in the GSO projection. We also generalize some of the results of Chap. 6 to the case of fermions. We then consider open superstrings, i.e. we extend the formalism of conformal field theories with boundaries to include free fermionic fields. This gives rise to D-branes in superstring theories. We also discuss non-oriented superstrings, which result form performing a quotient of the type IIB superstring by the world-sheet parity transformation. We show that oneloop diagrams are divergent unless D-branes are present in the model. This defines the type I superstring, whose construction we discuss in some detail.
9.1 Spin Structures and Superstring Partition Function The RNS formulation of the superstring, discussed in the previous two chapters, possesses world-sheet spinor variables. Given a topological manifold we can always endow it with a Riemannian structure, i.e. we can always define a metric on it but there are topological conditions which have to be met in order to define spinors. On a d -dimensional orientable manifold, the transition functions of the frame bundle are elements of SO.d /. SO.d / has no spinor representations, but its double cover Spin.d / does. Spinors can be defined if the transition functions of the frame bundle can be consistently lifted to Spin.d /. The set of transition functions defines a spin structure, which does not have to be unique. Manifolds which admit a spin structure are called spin manifolds. On an orientable manifold the necessary and
R. Blumenhagen et al., Basic Concepts of String Theory, Theoretical and Mathematical Physics, DOI 10.1007/978-3-642-29497-6 9, © Springer-Verlag Berlin Heidelberg 2013
223
224
9 Superstrings
sufficient condition to be spin is that the second Stiefel-Whitney class of the tangent bundle vanishes.1 In string theory the question of spin structures poses itself at two levels: first at the level of the world-sheet and second at the level of the background spacetime through which the string propagates. The latter occurs when we discuss the low-energy effective action and solutions to its equations of motion. In this chapter we will only be concerned with world-sheet properties. (The space-time is always Minkowski space which certainly does allow spinors.) Except for a few comments at the end, the discussion in this section will be concerned with the closed string whose (Euclidean) world-sheet is a genus g Riemann surface ˙g . Riemann surfaces are spin and, as we will now discuss, except for g D 0, they admit more than one spin-structure. Let us begin by characterizing spin structures on ˙g . We know from Chap. 6 that there are two noncontractible loops associated with each of the g h
