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Abstract
We use the superspace formulation of supergravity in eleven and ten dimensions to compute fermion couplings on the M2-brane and on Dp-branes. In this formulation fermionic couplings arise naturally from the θ-expansion of the superfields from which the brane actions are constructed. The techniques we use and develop can in principle be applied to determine the fermionic couplings to general background fields up to arbitrary order. Starting with the superspace formulation of 11-dimensional supergravity, we use a geometric technique known as the ‘normal coordinate’ method to obtain the θ-expansion of the M2-brane action. We then present a method which allows us to translate the knowledge of fermionic couplings on the M2-brane to knowledge of such couplings on the D2-brane, and then to any Dp-brane. This method is based on superspace generalizations of both the compactification taking 11-dimensional supergravity to type IIA supergravity and the T-duality rules connecting the type IIA and type IIB supergravities.
Highlights
String Theory concerns itself with objects extended in a single dimension, the eponymous strings, and with objects extended in many dimensions, namely branes
Supersymmetry breaking is still not completely understood in String Theory proposals, but Dp-branes are good candidates to provide ways to achieve it without spoiling the solution to the Hierarchy Problem since their fermionic degrees of freedom can realize supersymmetry non-linearly [37,38,39,40,41]
We show that the usefulness of the approach can be limited because the size of the formulae grows quickly as one computes terms of higher θ-order in the superfields. This is the main obstacle we find in our computations, and it will bring us to the conclusion that unless one succeeds in combing terms obtained with NORCOR together into simple and manageable formulae, it remains extremely challenging to extract information valuable for physics
Summary
String Theory concerns itself with objects extended in a single dimension, the eponymous strings, and with objects extended in many dimensions, namely branes. We use this to determine the D2-brane action to quartic order in fermions. We demonstrate how this can be used in principle to move from the action for the D2-brane at a given order in fermions to that for any Dp-brane at the same order, and give explicit examples at second order. In this paper we are interested in the action describing these localized branes, with a particular interest in fermionic modes living on them and their couplings in the brane worldvolumes For this purpose it is convenient to approach branes from a slightly different perspective, that of the superspace formulation of the supergravity theories. Our strategy will be to use the ‘normal coordinate’ method to obtain the θ-expansion of the M2-brane action superfield, and pursue the results for Dp-branes using the superspace generalization of the duality web in figure 1
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