The supergravity Hamiltonian as a quadratic form
Introduction
Eleven-dimensional supergravity [1] on reduction to four dimensions yields the maximally supersymmetric theory (, ). Supergravity is similar in many ways to the other maximally supersymmetric theory in four dimensions, Yang–Mills. In Ref. [2], it was shown that the light-cone Hamiltonian of Yang–Mills could be expressed as a quadratic form. The key to this rewriting was the maximal supersymmetry present in the theory. Since this is true in as well, we conjecture that a similar rewriting must be possible. To start with, we show this explicitly to order κ. We explain why this quadratic-form structure is unique to maximally supersymmetric theories and simply does not apply to other cases.
A light-cone superspace formulation of supergravity was first achieved in [3], [4] wherein the authors constructed the action to order κ. The formulation has not been extended since. In this paper, we extend the action to order by constructing the quartic interaction vertex. We will then see that the Hamiltonian is a quadratic form at order as well.
When formulating a maximally supersymmetric theory (like supergravity) in light-cone superspace, it suffices to show that the superspace action correctly reproduces the component action for any one component field (our focus will be on the graviton). Once this is done, the remaining component terms in the action will follow from supersymmetry transformations. We will therefore construct the quartic interaction vertex in supergravity by requiring that it correctly reproduce pure gravity in light-cone gauge.
The long-term aim of this program of research [2], [5] is aimed at understanding the divergent behavior of supergravity. Curtright [6] conjectured that any divergences occurring in the theory, were attributable to the incomplete cancellation of Dynkin indices of representations. The light-cone offers an ideal framework to study this conjecture since it highlights the role of the spacetime little group. In addition, working on the light-cone ensures that we deal exclusively with the theory's physical degrees of freedom.
Section snippets
supergravity in light-cone superspace
With the space–time metric , the light-cone coordinates and their derivatives are The degrees of freedom of supergravity theory may be captured in a single superfield [3]. In terms of Grassmann variables , which transform as the 8 of (), we define1
The Hamiltonian as a quadratic form
Maximally supersymmetric theories (like Yang–Mills and supergravity) are special for various reasons. Specifically, the superfield governing these theories satisfies the inside-out constraint (Eq. (7)). This constraint allows us to express the Hamiltonian of the theory as a quadratic form. In this section, we illustrate this at lowest order and then at order κ.
The Hamiltonian to order : lessons from the quadratic form
The quadratic form studied so far will not immediately tell us the Hamiltonian to order . This is because the dynamical supersymmetry (and hence ) is known only to order κ. However the quadratic form still offers a lot of insight into possible forms the quartic interaction may take.
Our plan then is as follows. In this section, we will collect information from the quadratic form, dimensional analysis and helicity considerations. Based on these pointers, the general structure of the quartic
The supergravity action to order
By explicit comparison with the gravity vertex, we find that the supergravity quartic interaction vertex is where the explicit form of X is given in Appendix D. The final answer (as explained in the appendix) confirms our conjecture regarding the quadratic form, to order .
The answer does not seem to simplify very much based on the usual tricks (partial integrations, inside-out constraints and so on). Our result is surprisingly
Summary
The quartic interaction governing supergravity has been constructed. In addition, we have shown (to order ) that the Hamiltonian is expressible as a quadratic form. We believe this quadratic structure of the Hamiltonian will hold to higher orders as well. We expect to return to this issue and explore its consequences further.
The general structure of the four-point interaction term is quite simple. The complication stems from the many ways the various derivatives enter. However, when
Acknowledgements
We are grateful to Pierre Ramond for many valuable discussions. We thank Stefano Kovacs, Hermann Nicolai, G. Rajasekaran and Hidehiko Shimada for helpful comments. H.G.S. is supported by a Humboldt fellowship.
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