1 Excision of disk

We use the fact that the odd symplectic form is given by a single integral over the worldsheet: .

This allows us to discuss the choice of Lagrangian submanifold locally on the worldsheet. We can agree to only change the Lagrangian submanifold (for example, metric), inside some limited compact region on the Riemann surface:

As we vary the metric, the state on the boundary (marked red) remains the same state in the same theory. This is possible because we do not change the theory in the region of insertion.

If we limit the variations of the Lagrangian submanifold in this way, then we can construct the form (and its base analogue) in the same way as we did on the closed Riemann surface. In this case the form is not closed:

If we restrict ourselves with the insertions of only physical operators , then the form is closed.

We will prove Eq. (1) in Proof of Eq. (1).

2 Variation of wave function

Let us study the quantum theory on a region with the boundary .

Let us bring our theory to some first order formalism, so that the Lagrangian is of the form . Fix the Lagrangian submanifold . Fix some polarization in the restriction of the fields to , for example the standard polarization where the leaves have constant . Let us consider the wave function:

— the integration is over the field configurations inside , and enters through boundary conditions. More generally, let denote the wave function obtained by the path integration with the insertion of some operator with compact support, not touching the boundary:

Theorem 1 Suppose that satisfies the Master Equation.

Any functional on the odd phase space, with compact support, determines the operator insertion, just by its restriction on the Lagrangian submanifold. We will denote it . Let us assume that is such that is well-defined (as discussed here and here). Then we have:

where is the restriction of to and is the Lie derivative of the half-density along the Hamiltonian vector field .

Proof: Let us introduce some Darboux coordinates near , so that is given by the equation . Let us expand in powers of . Let us first assume that only a constant in antifields term is present:

In this case the Lie derivative is equivalent to inserting into the path integral, giving Eq. (4).

If also depends on antifields, then we have to be careful restricting ourselves to such that the Lie derivative is well-defined, because otherwise Eq. (4) does not make sense. This assumption must include the vanishing of the integration by parts:

Theorem 2 Eq. (4) actually holds even without assuming that satisfies the Master Equation. In this case we define using the expansion of in Darboux coordinates: .

Proof: nothing in the proof of Theorem 1 requires the use of Master Equation.

Definition 1 Suppose that satisfies the Master Equation sufficiently close to the boundary, in the sense that has compact support which does not touch the boundary. Then defined as in Theorem 2 becomes a symmetry of sufficiently close to the boundary. Let us define via the insertion of the BRST current near the boundary:

3 Proof of Eq. (1)

3.1 Old form

Now we are ready to prove Eq. (1). Let us first study the usual “old” (not equivariant) form with the boundary. The derivation parallels the case with no boundary; we denote :

We should choose ; in this case .

3.2 Interpretation of as an intertwiner in the presence of a boundary

The equivariant is a particular case of when solves:

We interpret the path integral in the theory on , with a boundary state, as the path integral over the whole with insertions (inside ) determining this boundary state. Then our form is defined by the path integral in the theory on the whole compact Riemann surface :

But we only allow the variations of which do not change the theory inside the disk . In other words, we restrict to of compact support inside . We also assume that also has compact support localized inside .

As we explained, intertwines with :

Since the support of is in , and the support of is in , we have:

The first term should be interpreted as a nilpotent operator acting on the inserted state:

This implies that the presence of a boundary modifies the intertwiner property of . This is a particular case of the following general construction. Let be a representation of of the wavy Lie superalgebra. Then any complex defines a new representation of :

In our case is the Hilbert space of states on the boundary, with .

4 Base form

At the same time, is still horizontal and invariant. Horizontality follows from the construction, as we obtain our base form from the equivariant form by horizontal projection. Invariance follows from the fact that is also horizontal.

For our constructions presented in this part, it is important that we restrict to only those variations of the complex structure which are zero at the boundary. Otherwise, the variation would change the Hilbert space of states. In such case it would be nontrivial to even define , as we would need a connection on the bundle of Hilbert spaces.