Introduction

Renormalization of the deformations of the action

Consider a quantum field theory with an action invariant under some Lie algebra of symmetries . Let us study its infinitesimal deformations of the theory, corresponding to the deformations of the action:

where is an infinitesimal parameter, are some space-time-dependent coupling constants and is some set of local operators, closed under in the sense that the expressions on the RHS of Eq. (1) form a linear representations of . We call the linear space of this representation

In principle, we can take the set of all local operators of the theory. But there could be smaller -invariant subspaces.

We can study the effects on the correlation functions, or perhaps on the -matrix, of the deformation of the form (1), to the linear order in . We can also study the effects of the deformation (1) beyond linear order in , but this requires taking care of the definitions. To define (1), we expand in powers of , bringing down from the exponential expressions like:

This has to be regularized, because of singularities due to collisions of . Suppose that the set is big enough in the sense that all the required counterterms are linear combinations of . The counterterms are not unique, because we can always add a finite expression. Suppose that we have choosen some rule to fix this ambiguity. Then, we have a map, parameterized by a small parameter :

Symmetries of undeformed theory act on deformations

The space of finite deformations is not, in any useful sense, a linear space. It is a “nonlinear infinite-dimensional manifold”. But it naturally comes with an action of . Indeed, the regularized expression (2) is, in particular, a (non-local) operator in the original theory. As the symmetry group of the undeformed theory acts on operators, it therefore acts on deformations, bringing one deformation to another.

Because we had some freedom in the choice of regularization, the map does not necessarily commute with the action of . Can we choose regularization with some care, so the resulting does commute with ? Of course, we can not, there are obstacles.

In this paper we will introduce a geometrical framework for describing these obstacles and to what they correspond in the strong coupling limit via AdS/CFT duality.

Holographic renormalization

AdS/CFT correspondence relates the deformations of CFT to the classical solutions of SUGRA deforming AdS. As main example, consider Type IIB SUGRA in and SYM on the boundary . Deformations of the SYM action of the form (1) are mapped by AdS/CFT to the classical SUGRA solutions, deformations of . Linearized SUGRA solutions correspond to linearized deformations.

Renormalization of the deformations of QFT (Section Renormalization of the deformations of the action) should correspond to something on the AdS side. Most of the work on holographic renormalization was done along the lines of [Bianchi:2001kw,Skenderis:2002wp], and was based on the study of the bulk supergravity action.

On the other hand, the computation of the renormgroup flow of the beta-deformation done in Aharony:2002hx seems to use a different method. In particular, the authors of Aharony:2002hx did not need to know the action of the bulk theory. This, in particular, may allow to apply their method to the cases where the action is not known and maybe even does not exists, such as higher spin theories [Vasiliev:2004qz,Sharapov:2017yde].

Geometrical abstraction Suppose that a Lie algebra acts on a manifold , preserving a point . Then it acts in the tangent space to at . The question is, can we find a formal map:

parametrized by (“formal” means power series in ) from the tangent space to to , commuting with the action of ? There are, generally speaking, obstructions to the existence of such a map — see Section Obstacles to linearization of symmetry. We want to classify these obstructions. This is, essentially, equivalent to studying the normal form of the action of in the vicinity of the fixed point.

Tangent vectors as equivalence classes of trajectories Maps participate in the “usual” definition of the tangent space ( e.g. Arnold). The tangent space is defined as the space of equivalence classes of paths (maps from to ) such that . The equivalence relation is that two paths and are equivalent when in a coordinate patch. Giving a function as in Eq. (4) is same as giving a prescription of how to pick, for each tangent vector , one path from the corresponding equivalence class. That path is:

Of course, there are many such prescriptions. The question is, can we find some, which would be consistent with the action of ?

The space of formal paths such that can be denoted — similar to the space of -based loops in , but we only need a formal power series in at , not the whole loop. To summarize, we investigate the existence of a map:

commuting with the action of . We find that there are obstacles to the existence of such a map, and classify them. These obstacles are, roughly speaking, some cohomology groups. More precisely, they are solutions of a Maurer-Cartan equation, modulo gauge transformations (a nonlinear analogue of cohomology groups) — see Section Obstacles to linearization of symmetry.

The role of supergeometry and infinite-dimensional geometry

In our main application (AdS/CFT): — the superconformal algebra. Its even (bosonic) part is . If were a finite-dimensional “usual” (not super) manifold then there would be no obstacle in linearizing the action, because the relevant cohomology groups are zero. This makes our picture somewhat counter-unintuitive geometrically.

The relevant cohomology group is . In classical geometry, we would have nontrivial invariants if were or contained as a subalgebra. For example, a classical mechanical system can have a free limit and have in that limit periodic trajectories, but away from that free limit the trajectories are not periodic.

There are two reasons for having nontrivial invariants. The first reason is that is actually infinite-dimensional. But there is also the second reason: even when we can find some finite-dimensional “subsectors” (submanifolds in ), they are actually super-manifolds. This can make the cohomolgy nontrivial even in finite-dimensional case.

Summary of this paper

Cohomological framework for holographic renormalization

Here we will develop a formalism for computations along the lines of Aharony:2002hx, which makes them geometrically transparent. We interpret Aharony:2002hx as computing certain invariants of supergravity equations in the vicinity of AdS, namely the solution of some Maurer-Cartan equation modulo gauge transformations. We give the definition of these invariants in Section Geometrical abstraction. This is broadly similar to the obstructions to the existence of the -invariant map of Section Geometrical abstraction. The details, however, are more subtle, because we are dealing with gravity. The symmetry algebra of is actually a part of the larger infinite-dimensional symmetry, the gauge symmetries of the supergravity theory. This makes the analysis on the AdS side more interesting.

Finite-dimensional representations have nontrivial cohomology

The cohomological obstacles for linearization of the symmetry are usually rather complicated, because they involve the cohomology with coefficients in infinite-dimensional representations (see Section Infinite-dimensional extension of finite-dimensional ). But when the symmetry is super-symmetry, even finite-dimensional representations have nontrivial cohomology (Section Lifting of to superalgebra). We formulate some conjecture

The idea is, therefore, to study the space of all perturbative solutions of supergravity (instead of particular solutions) and describe its invariants, as a invariants of a supermanifold.

Plan In Section Obstacles to linearization of symmetry we develop geometrical formalism for studying the obstructions to the existence of the map (4) commuting with . We explain that the obstacle is a solution of the Maurer-Cartan (MC) equation with values in vector fields. In Section Relation to tree level Feynman diagrams we explain how to apply this formalism to the space of perturbative solutions of a classical field theory. We show that there is a natural operation of “amputation of the last leg” which converts Feynman diagrams into a solution of the MC equation. In particular, in Section Example: classical CFT on we consider the case of classical CFT in . In Section Supergravity in AdS space we discuss deformations of and holographic renormalization. In Section Beta-deformation and its generalizations we study the particular case of beta-deformation Leigh:1995ep. Finally, in Section Discussion and open questions we discuss some open questions and potential problems.