Abstract
A manifestly diffeomorphism invariant exact renormalization group requires extra diffeomorphism invariant ultraviolet regularisation at some effective cutoff scale Λ. This motivates construction of a ‘Parisi-Sourlas’ supergravity, in analogy with the gauge theory case, where the superpartner fields have the wrong spin-statistics such that they can become Pauli–Villars regulator fields after spontaneous symmetry breaking. We show that in contrast to gauge theory, the free theory around flat space is already non-trivial and in a sense already displays some spontaneous symmetry breaking. We show that the fluctuating fields form multiplets whose mass matrices imply that the fields propagate into each other not only with the expected 1/p 2 but also through propagators with improved ultraviolet properties, namely 1/p 4 and 1/p 6, despite the fact that the action contains a maximum of two space-time derivatives.
Highlights
In developments over a period of years this problem was solved for gauge field theory [18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44], where it was proved to work to all orders in perturbation theory. (For a short summary see ref. [15], and for reviews and further advances see refs. [45,46,47,48].2) In gauge theory, this extra regularisation is provided by generalising the gauge group from SU (N ) to SU (N |N ) and spontaneously breaking the fermionic gauge fields at the effective cutoff scale Λ
At high energies these degrees of freedom cancel each other, as happens with Parisi-Sourlas supersymmetry [49], at least sufficiently that, together with appropriately chosen covariant cutoff functions, the theory is regularised to all orders in perturbation theory [26,27,28]
As reviewed in sec. 1, the Parisi-Sourlas regularisation works in gauge theory by adding to the original gauge field A1μ, a complex pair of fermionic gauge fields Bμ, Bμ and a ghost copy, A2μ
Summary
We collect together here the basic material we will need to formulate Parisi-Sourlas supergravity. Note from (2.10), the difference in index-shifting conventions between up and down indices We wish to work in a Riemannian supermanifold and have to define a metric This is a real, c-type, non-degenerate supersymmetric (0,2) tensor g. We take the base manifold to be flat RD and discard boundary contributions, and assume a trivial bundle in the fermionic directions This will be required for a Wilsonian RG analysis, for example fixed points, since the manifold must remain invariant under Kadanoff blocking [7], but this is the obvious choice for determining the propagating degrees of freedom, as we do in the remainder of the paper.
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