Master Actions quadratic-linear in antifields

Suppose that the BV phase space is an odd cotangent bundle, i.e. is of the form for some supermanifold (the “field space”). If are coordinates on , then are coordinates on , and “” means that the statistics of is opposite to the statistics of . There is an odd Poisson bracket (the “BV bracket”):

This bracket is geometrically well-defined, in a sense that the bracket of two functions is actually independent of how the coordinates on are choosen. Equivalently, there is an odd symplectic form (which, as any differential form, can be considered a function on ):

(As a slight overuse of Einstein notations, we will omit the summation sign in such cases.) Suppose that the Master Action is of the form:

(writing rather than simplifies some signs later).

We will assume that satisfies the classical Master Equation:

If is purely even, we can think of functions on as polyvector fields on . For example, corresponds to the vector field , and corresponds to a Poisson bivector . The odd Poisson bracket corresponds to the Schouten bracket of polyvector fields.

If is a super-manifold, then this polyvector picture does not seem to be very illuminating. However, one can still apply the intuition of Hamiltonian mechanics. The linear function still defines a vector field; the derivative of a function along it is: . The quadratic function still defines a map from functions on to vector fields on :

The Master Equation (5) implies, order by order in expansion in :

It follows from Eq. (8) that vector fields of the form , , form a closed subalgebra in the algebra of vector fields. They are all tangent to a family of submanifolds of which can be called “symplectic leaves of ”. As a slight abuse of notations, the letter will denote both the BRST transformation and the function on . Eq. (6) says that generally speaking the BRST operator is only nilpotent on-shell Berkovits:2007rj.

We will show that under some conditions, this theory can be reduced to a simpler theory which has BRST operator nilpotent off-shell (and therefore its Master Action has no quadratic terms ).

The case when is non-degenerate Let us first consider the case when the Poisson bivector is nondegenerate. Eq. (7) implies that an odd function locally exists, such that . Suppose that is also defined globally. Let us consider the canonical transformation of the Darboux coordinates generated by :

More geometrically: and (functions on ) are pullbacks of and by the flux of the Hamiltonian vector field by the time . (The flux integrates to Eqs. (9) because only depends on , and therefore the velocity of is -independent.)

The -linear term is gone! The Master Equation implies that . Since we assumed that is nondegenerate, this implies:

The case of degenerate

We are actually interested in the case when is degenerate. Let be the distribution tangent to symplectic leaves of :

This distribution is integrable because satisfies the Jacobi identity. We also assume that is transverse to :

Let us also consider the distribution which is generated by elements of and by . Eqs. (7) and (8) imply that is also integrable. Let us assume the existence of a 2-form

on each integrable surface

of and a function which satisfy:

where and are defined as follows:

Existence of satisfying Eq. (14) locally follows from Eqs. (11) and (13), because they imply . But we also require this to be a globally well-defined function on . Contracting with we find that:

Eq. (13) implies that is an integrable distribution inside an integral surface of . Therefore Eq. (15) implies that is proportional to , i.e. there exists a function such that: . In fact , since and . We conclude:

Let us consider the canonical transformation (9) of Darboux coordinates generated by . With these new Darboux coordinates:

is automatically constant on symplectic leaves of . Also, it follows that consistently defines an odd nilpotent vector field on the moduli space of symplectic leaves of . These facts follow from . To summarize:

where is coordinates on the space of symplectic leaves of . We therefore constructed a new, simpler theory, on the space of symplectic leaves of .

This theory can be interpreted as the result of integrating out some antifields. More precisely, let us define a submanifold by picking one point from each symplectic leaf. Fibers of the odd conormal bundle

are isotropic submanifolds in , and we can integrate them out as described in Mikhailov:2016rkp. In this paper the coordinates in these fibers will be called (and integrated out).

Oversimplified example

We will now illustrate the relation by a toy sigma-model (we will actually run the procedure “in reverse”). Let be a two-dimensional worldsheet. Let us start with:

where does not not depend neither on the fermionic field nor on the bosonic field . (It depends on some other fields .) We postulate the odd symplectic form so that our fields are Darboux coordinates Mikhailov:2016rkp, as in Eq.~(3):

This action is highly degenerate; the path integral is undefined (infinity from integrating over times zero from integrating over ). To regularize , let us introduce a new field-antifield pair , where is a bosonic 1-form on the worldsheet and is a fermionic 1-form on the worldsheet:

(where is the field space differential, not the worldsheet differential). Let us add to the BV action:

(Notice that this does not involve the worldsheet metric.) This corresponds to:

(again, is the field space differential, not the worldsheet differential). In this case is the subspace of the tangent space generated by , and is generated by . Then, shift the Lagrangian submanifold by a gauge fermion:

Here we have run the procedure of Section Master Actions quadratic-linear in antifields “in reverse”. That is, Eq. (21) is an example of the of Eq. (4), and Eq. (19) is an example of the “split” Eq. (10). Notice that is degenerate, as it does not involve and . Because of that, the of Eq. (19) is not constant as in Eq. (18), but just independent of . The vector field is the -part of , as in Eq.~(15).

This is, still, not a quantizable action (the kinetic term for is a total derivative). One particular way of choosing a Lagrangian submanifold leading to quantizable action is to treat and asymmetrically (pick a worldsheet complex structure), see section on A-model in AKSZAlexandrov:1995kv and Section Constraint surface and its conormal bundle of this paper. This requires more than one flavour of .