Algebraisation of the problem

In principle, the higher order vertices could be constructed in a similar way. The only requirement is that the fibre sitting over a single point in M_0,5 has vanish-ing homotopy groups. If the bosonic cover of the moduli space requires elementary higher vertices, the geometric construction becomes more cumbersome as we would also have to take into account lower codimension boundaries of the elementary ver-tices. It is not clear that the correction terms are contact terms anymore. Another drawback is that this field theory is not entirely constructive. In order to be able to do meaningful calculations, we need to construct σ_k, k ≥ 4 explicitly, which seems not to be feasible. Alternatively, the procedure indicates that we may solve the A_∞-relations algebraically and take them as a replacement for the construction of σ_k. The only drawback is that we need to show that the correct S-matrix is indeed reproduced by (2.83) and their higher analogues. In chapter 3 we complete the con-struction and show indeed that all higher order vertices can be chosen as contact vertices if we start with Witten’s vertex.

The whole procedure can be reinterpreted within classical BV theory. After the suspension, the Poisson bracket is given by equation (2.74). Let us introduce a generating functionS_int,

S_int =^X

k≥3

1 k

Z

σ^∗_kΩ^k−3|k−2_0,k φ^k. (2.89)

S_int is a cyclic functional of degree 0. As the productsM_kconstitute anA∞algebra, we have

X

k≥3

1 k

Z

σ_k^∗Ω^k−3|k−2_0,k Qφ^k+1

2(S_int, S_int) = 0. (2.90) If the Poisson bracket (·,·) comes from a symplectic structureω, we can also express the first term in terms of the Poisson bracket,

S = 1

2hω|φ⊗Qφi+S_int, (2.91a)

0 = 1

2(S, S). (2.91b)

Invoking the results from section 2.3, we conclude that finding the maps induced by the integrals ofσ_kis equivalent to finding a symplectic structure inducing the Poisson bracket (2.74), solving the cyclic master equation (2.91) and showing that the S-matrix calculated from the master actionSindeed reproduces the usual perturbative string S-matrix. For the open superstring these problems are tackled in chapters 3, 5, 6 and 7.

With these results in mind, it is straightforward to generalise the construction problem to include Ramond fields and also to closed strings. If Ramond fields are present, the BV bracket does not come from a naive symplectic form. Giving up cyclicity, we can still solve (2.90) or equivalently theL∞- or A∞-relations. Formally, one may still consider the S-matrix given by (2.83) and show that it coincides with the traditional perturbative S-matrix. Thus the same reasoning can be applied to open superstrings based on a decomposition of the bosonic moduli space based on Witten’s star product with stubs, to heterotic strings and to closed type II super-strings. Due to some problems with the invertibility of the Poisson bracket when including Ramond fields, the generalisation is easiest when restricting to pure NS fields. Without the P structure it is still possible to construct a Q-manifold struc-ture giving rise to gauge-invariant equations of motion. The results are discussed in chapter 4 for the NS subsectors and in chapter 5 for the Q-structure for all fields, chapter 6 evaluates the S-matrix.

Chapter 3

Resolving Witten’s open superstring field theory

Classical open string field theories are determined by cyclic A_∞ algebras. In this chapter we construct such an algebra for the NS sector of open superstring the-ory. The construction starts with Witten’s singular open superstring field theory and regulates it by replacing the picture-changing insertion at the midpoint with a contour integral of picture changing insertions over the half-string overlaps of the cubic vertex. The resulting product between string fields is non-associative, but we provide a solution to the A_∞ relations defining all higher vertices. The result is an explicit covariant superstring field theory which by construction satisfies the classical BV master equation.

This chapter is based on the paperResolving Witten’s open superstring field theory by T. Erler, the author and I. Sachs [52].

3.1 Introduction

For the bosonic string, the construction of covariant string field theories is more or less well understood. We know how to construct an action, quantise it, and prove that the vertices and propagators cover the the moduli space of Riemann surfaces relevant for computing amplitudes. For the superstring this kind of understanding is largely absent. A canonical formulation of open superstring field theory was provided by Berkovits [41, 42], but it utilises the large Hilbert space which obscures the relation to supermoduli space. Moreover, quantization of the Berkovits theory is not completely understood [111–114]. Motivated by this problem, we seek a different formulation of open superstring field theory satisfying three criteria:

(1) The kinetic term is diagonal in mode number.

(2) Gauge invariance follows from the same algebraic structures which ensure gauge invariance in open bosonic string field theory.

(3) The vertices do not require integration over bosonic moduli.

We assume (1) since we want the theory to have a simple propagator. We assume (2) since we want to be able to quantise the theory in a straightforward manner, following the work of Thorn [115], Zwiebach [17] and others for the bosonic string.

Finally we assume (3) for simplicity, but also because we would like to know whether open string field theory can describe closed string physics through its quantum corrections. Once we add stubs to the open string vertices, the nature of the minimal area problem changes and requires separate degrees of freedom for closed strings at the quantum level [26].

Condition (1) rules out the modified cubic theory and its variants [38,39,116–119], and (2) rules out the Berkovits theory. This leaves the original proposal for open superstring field theory at picture −1, described by Witten [34]. The problem is that this theory is singular and incomplete. A picture changing operator in the cubic term leads to a divergence in the four point amplitude which requires subtraction against a divergent quartic vertex [37]. Likely an infinite number of divergent higher vertices are needed to ensure gauge invariance, but have never been constructed.¹

In this chapter we would like to complete the construction of Witten’s open su-perstring field theory in the NS sector. We achieve this by resolving the singularity in the cubic vertex by spreading the picture changing insertion away from the mid-point. As a result the product is non-associative. But we know how to formulate a gauge invariant action with a non-associative product [25]. The action takes the form

S = 1

2ω(Ψ, QΨ) +1

3ω(Ψ, M₂(Ψ,Ψ)) + 1

4ω(Ψ, M₃(Ψ,Ψ,Ψ)) +. . . , (3.1) whereωis the symplectic bilinear form andQ, M₂, M₃, . . .are multi-string products which satisfy the relations of an A∞ algebra. The fact that one can in principle construct a regularisation of Witten’s theory along these lines is well-known. The new ingredient we provide is an exact solution of theA∞ relations, giving an explicit and computable definition of the vertices to all orders.

The resulting theory is quite simple. However, its explicit form depends on a choice of non-local, BPZ even operator built from the picture changing operator

X =

I dz

2πif(z)X(z), (3.2)

which tells us how to spread the picture changing insertion in the cubic vertex away from the midpoint. As far as we know, there is no canonical way to make this choice. This suggests the result of a partial gauge fixing. In fact, a gauge fixed

1There have been some attempts to fix the problems with Witten’s theory by changing the nature of the midpoint insertions in the action. These include the modified cubic theory [38, 39] and the theory described in [120].

version of Berkovits’ theory resembling our approach has been explored in [51, 121].

Our regularisation of the cubic vertex is inspired by this work.

This chapter is organised as follows. In section 3.2 we describe our regularisation of the cubic vertex. The cubic vertex gives rise to a non-associative 2-product and we find the 3-product by requiring that the resulting action satisfies a master equation.

The main observation is that the 3-product is Q-exact and leads to the recursive construction of a full solution to the master equation described in section 3.3. As a cross check for our construction we calculate the four-point amplitude in field theory and show that it is identical to the first quantised amplitude in section 3.4.

We conclude the chapter with some discussion.