Relation to tree level Feynman diagrams

Here we will apply the formalism of Section Obstacles to linearization of symmetry to the case when is the space of solutions of some classical nonlinear field equations, constructed as perturbation of some zero solution . This is different from the context of deformations of QFT , but the AdS/CFT correspondence establishes a relation between these two contexts (Section Supergravity in AdS space).

Perturbation theory as a map

Let us take to be the space of perturbative solutions of nonlinear equations of the form:

where is some linear differential operator, and is a nonlinear function describing the interaction. We assume that is a polynomial starting with the terms of quadratic or higher order.

The point will be the zero solution . Then can be identified with the space of solutions of the linearized equation:

parameterized by a small parameter . As explained in Section Geometrical abstraction, it can be also understood as a map .

We will embed into the space of all field configurations, not necessarily satisfying equations of motion (subindex “os” means “off-shell“). We assume that is a linear space. We consider as a function . It can be described as a sum of tree level Feynman diagrams. Every incoming leg corresponds to a solution of the linearized equation (21). Every internal leg and the outgoing leg each correspond to a propagator . There is a recursion relation

where satisfies:

The definitions of the operator has an ambiguity (because one can add a solution of the free equation). Suppose that we made some choice of . The dependence on the choice of is controlled by Lemma 27 below.

As we already explained, we need an embedding of into the linear space of off-shell field configurations , just because we want to add Feynman diagrams. Obvously, the space of free solutions is also embedded into . Let us assume that the action of on agrees with this embedding. This is not really important, but we make this assumption for this Section. For example, suppose contains time translation . We assume that it acts as , both on and on .

Let us define as follows:

(the dependence of on the RHS comes from ).

Lemma : This is the same as defined in Eq. (12):

Proof We have to show that for any . In other words, for any :

Every time hits , we get :

— this gives the term on the LHS of Eq. (25). And when hits one of the , we get .

Lemma : An infinitesimal variation of :

where satisfies , corresponds to an infinitesimal gauge transformation of (see Eq. (15)) where:

Amputation of the last leg

We will now present a slightly different point of view on the construction. Suppose that for every linearized solution we constructed a nonlinear solutions (depending on a small parameter ). What should we do with , to obtain ? Remember that is a (nonlinear) vector field on the space of linearized solutions. Obviously, we have to somehow “project” to a linearized solution. According to Eq. (24) we should remove the last leg, and replace it with :

Remember that satisfies Eq. (23):

Let us define the “amputator” as the composition:

(Notice that is a projector to .) It satisfies

If is our perturbative solution ( i.e. ), then:

This leads to the following interpretation. The “projector” can be interpreted as a map (Section Action of a symmetry in local coordinates), the inverse of . Then, again, .

Trivial example

Consider a vector field . Suppose that vanishes at , and the derivative of also vanishes at :

In notations of Section Relation to tree level Feynman diagrams, be the space of all solutions of Eq. (28) and . This equation is invariant under translations of , generating . The generator of is .

For any functions , we can expand it in Taylor series around . We define as follows:

The map of Eq. (22) is:

According to Eq. (24), to construct we have to take and hit it with , which, according to Eq. (29) amounts to put . Therefore, in this case:

where is another vector field. Therefore, in this case the MC invariant computes the normal form of the vector field ( i.e. modulo nonlinear changes of coordinates).

In this example we had , and the structure of the first cohomology group was rather tautological: was the space of -invariants in . In the next example we will consider the non-abelian , with much more interesting (smaller!) cohomologies. In particular, can be nonzero only for infinite-dimensional .

Example: classical CFT on

Consider a conformally invariant classical theory on the Lorentzian , for example the theory, .

Realization of as the base of the lightcone

Consider the light cone in ( cp Eq. (76)):

A convenient model for the conformal is the projectivization of the light cone, which is parametrized by satisfying (30) modulo the equivalence relation:

A density of the weight is a function satisfying:

modulo functions divisible by . Let denote the space of such densities. The conformally invariant d’Alambert acts as follows:

This operator is only well-defined with this value of , because for other values of it would not annihilate modulo those functions which are divisible by .

The elements of the kernel of , i.e. the solutions of free field equations, are real sums of positive and negative frequency waves:

Conformal symmetry Besides the rotations of , there are also the following conformal transformations:

Amputator Introduce the Lie algebra cocycle :

As we explained in Section Amputation of the last leg, given a perturbative solution , the corresponding solution of the MC Eq. (13) is:

Consider elements of periodic in global time. Any such element can be written as:

where the summation is over a pair of integers and is a harmonic polynomial of of the following degree:

We define as follows:

These formulas partially define a cohomology class of Eq. (35). The definition is only partial, because Eq. (36) does not describe all elements of , but only those periodic which are periodic in the global time . Generic elements are linear combinations of:

To completely specify , we have to define on elements containing powers of , and compute for them the commutators, as in Eqs. (38), (39), (40). We will not do it here.

Relation to renormgroup

Our discussion of the classical field theory solutions in this section is a warm-up. However, it is related to renormalization. Given a set of operators , e.g. , and a set of infinitesimal coefficients , let us define the coherent state, schematically:

Comments on the structure of

is simpler than perturbative classical solutions Let us continue with the example of the previous section. Generally speaking, a perturbative solution is a sum of expressions of the form:

However, after the replacement of the last leg with of eq. (35), the resulting expression does not contain “bare” ( i.e. only contains via its exponentials). Indeed, is a solution of the free field equations. Solutions of the free field equations do not contain powers of . They only involve expressions of the form Eq. (31). No powers of . In this sense, the amputated is much simpler than full perturbative solution.

As we mentioned at the end of Section Amputator, we did not actually compute the amputator on the field configurations containing powers of (Eq. (41)). But we know in advance that the resulting expression will not contain any powers of .

Moreover, we know that satisfies a constraint: the Maurer-Cartan Eq. (13). In some situations, this might allow for some partial bootstrap, see Section Can be bootstrapped?.

There is a price to pay: the definition of contains an ambiguity. We could have choosen a different . This corresponds to the gauge transformation of Eq. (14). Moreover, the condition of -covariance is complicated:

The condition of -covariance is complicated

( cp Section Symmetries)

Under the false impression that all non-covariance is due to the “resonant” factors , one might conjecture that is -covariant in the sense that:

This, however, is not the case. At least when is a semisimple Lie algebra, Eq. (44) is incompatible with the MC equation:

because takes values in vector fields of degree 1 and higher. A semisimple Lie algebra cannot be represented by the vector fields of degree 1 and higher. In fact, it follows immediately from Eqs. (38), (39) and (40) that with chosen as in Eq. (37) is zero when is an infinitesimal rotation of . This already contradicts Eq. (44).

Instead of simple but wrong Eq. (44) we have more complicated Eq. (19).

Can be bootstrapped? Consider an infinitesimal -preserving deformation of the action which is a monomial of the order in the elementary fields. Then the corresponding cocycle, representing a class of , is given by the expression:

Could it be that all is exhausted by the expressions of this form for various -preserving deformations? This is certainly not true for SUGRA on . But in the situations when this is true, Eq. (13) allows to recursively compute , modulo gauge equivalence described in Section Gauge transformations, starting from the terms of the lowest order in given by Eq. (45).