Discussion and open questions

A general theme of AdS/CFT is comparison of field theory computations with supergravity computations. The analysis of the present paper is incomplete, and potentially leaves a mismatch between field theory and supergravity. Indeed, on the field theory side we use the formalism of Section Renormalization of the deformations of the action, Symmetries of undeformed theory act on deformations, Geometrical abstraction. While on the supergravity side, we use the formalism of Sections Geometrical abstraction, Normal form of the action of , which is similar but different.

Is it true that symmetries of QFT naturally act on the space of its deformations?

It is essential for our reasoning, that there is a natural action of the symmetries of QFT on the space of its deformations, Section Symmetries of undeformed theory act on deformations. This action should be natural, i.e. should not depend on how we describe deformations. Strictly speaking, our reasoning in Section Symmetries of undeformed theory act on deformations used a particular way of thinking about the deformations. Therefore, we are in danger of using an unnatural definition.

Gauge group is not an invariant

Renormgroup invariants in QFT match certain cohomological invariants of the action of the group of gauge transformations of supergravity, as described in Section Normal form of the action of

. But gauge group is not actually an invariant of the theory

Gauge transformations characterize the redundancy of the given Lagrangian desciption of the theory. A different Lagrangian descriptions of the same theory can have slightly different gauge symmetries.

. Therefore, we are in danger of being non-invariant. However, the invariants which we describe in Section Normal form of the action of

actually depend, in some sense, on the cohomology of the gauge transformations. Our construction uses certain “flabbyness” of the algebra of gauge transformations, essentially allowing to reduce the cohomologies to those of

(using the Shapiro’s lemma). We hope that the cohomologies are invariant.

Maybe there is no mismatch

If our conjectures in Section Normal form of the action of

hold, namely:

Exists and
Solution of MC Equation does not depend on the choice of

then the invariants of Section Normal form of the action of

just match the invariants on the QFT side, so there is no discrepancy. We think that this is, most likely, what actually happens.

Computation of the MC invariants

As we already mentioned in Section Outline of computation, an important open question is to develop the homological perturbation theory of [Bedoya:2010qz,Benitez:2018xnh] and to actually compute the Maurer-Cartan invariants which we defined.

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