The String Coffee Table

(0) Introduction

The phenomenon that supersymmetric field theories look like Dirac-Kaehler systems when formulated in Schroedinger representation is well studied in the special limits where only a finite number of degrees of freedom are retained, such as the semi-classical quantization of solitons in field theory (see for instance [1] for a brief introduction and further references). That this phenomenon is rooted in the general structure of supersymmetric field theory has been noted long ago in [2]. A way to exploit this fact for the construction of covariant target space Hamiltonians applicable to the computation of curvature corrections of string spectra in nontrivial backgrounds has been proposed in [3].

In the following this technique is looked at more closely. The key idea is that the form of the (classical) superconformal algebra is preserved under the deformation

when $\mathbf{W}$ is an even graded operator that satisfies a certain consistency condition. Here $\mathbf{d}_K$ is the $K$-deformed exterior derivative on loop space and $K$ is the Killing vector field on loop space which induces reparametrizations.

The canonical (functional) form of the superconformal generators for all massless NS-NS backgrounds can neatly be expressed this way by deformation operators $\mathbf{W}$ that are bilinear in the fermions. It turns out that there is one further bilinear in the fermions which induces a background that is shown to be related to the R-R $C_2$ field.

It is straightforward to find further deformation operators and hence further backgrounds. Even though at the classical level no equations of motion for the backgrounds are obtained, there is a classical consistency condition which constrains the admissible deformation operators. A few further selected cases are studied here.

This approach for obtaining new superconformal algebras from existing ones by applying deformations is similar in spirit, but rather complementary, to an approach studied by Giannakis and Evans [4]. There, the superconformal generators of one chirality are deformed to lowest order as

Requiring the deformed generators to satisfy the desired algebra to first order shows that $\delta T$ and $\delta G$ can be any fields of conformal weight 1 and $1/2$, respectively. An adaption of this procedure to deformations of the BRST charge itself is discussed in [6].

The advantage of this method over the one discussed in the following is that it operates at the level of quantum SCFTs and has powerful CFT tools at its disposal, such as normal ordering and operator product expansion. The disadvantage is that it only applies perturbatively to the first order in the background fields.

On the other hand, the deformations induced by $\mathbf{d}\to e^{-W}\mathbf{d} e^{W} \sim e^{-W}(G + i\tilde G)e^{W}$ preserve the superconformal algebra for arbitrarily large perturbations $\mathbf{W}$. The drawback is that normal ordering is non-trivially affected, too, and without further work the resulting superconformal algebra is only available in classical form, i.e. on the level of Poisson brackets.

(1) Loop space

In this section the technical setup is briefly established, the main purpose being to emphasize the fact that the 0-mode $\mathbf{d}_K$ of the sum of the left- and the rightmoving supercurrents is indeed the $K$-deformed exterior derivative on loop space. Weak nilpotency of this $K$-deformed operator (namely nilpotency up to reparametrizations) is the essential property which implies that the modes of $\mathbf{d}_K$ and its adjoint generate a superconformal algebra. In this sense the loop space perspective on superstrings highlights a special aspect of the super Virasoro constraint algebra which turns out to be pivotal for the construction of classical deformations of that algebra.

The kinematical configuration space of the closed bosonic string is loop space $\mathcal{LM}$, the space of parameterized loops in target space $\mathcal{M}$. As discussed in section 2.1 of [3] the kinematical configuration space of the closed superstring is therefore the superspace over $\mathcal{LM}$, which can be identified with the 1-form bundle $\Omega^1\left(\mathcal{LM}\right)$. Superstring states in Schroedinger representation are super-functionals on $\Omega^1\left(\mathcal{LM}\right)$ and hence section of the form bundle $\Omega(\mathcal{LM})$ over loop space. This is perfectly analogous to the general situation for supersymmetric quantum mechanical systems, i.e. those with a finite-dimensional configuration space. In the following we will make ample use of this formal analogy to finite dimensional systems in order to develop superstring quantum dynamics as a relativistic supersymmetric quantum mechanical system in loop space.

The main technical consequence of the infinite dimensionality are the well known divergencies of certain objects, such as the Ricci-Tensor and the Laplace-Beltrami operator, which inhibit the naive implementation of quantum mechanics on $\mathcal{LM}$. But of course these are just the well known infinities that arise, when working in the Heisenberg (CFT) instead of in the Schroedinger picture, from operator ordering effects, and which are removed by imposing normal ordering. Since the choice of Schroedinger or Heisenberg picture is just one of language, the same normal ordering (now expressed in terms of functional operators instead of Fock space operators) takes care of infinities in loop space. We will therefore not have much more to say about this issue here. The main result of the following sections are various (deformed) representations of the super-Virasoro algebra on loop space (corresponding to different spacetime backgrounds), and will be derived in their classical (Poisson-bracket) form without considering normal ordering effects. The quantization of these classical algebras, which will not be discussed here, would yield anomalies and equations of motion for the background fields in order to cancel these.

A mathematical discussion of aspects of loop space can for instance be found in [6] [7]. A rigorous treatment of some of the objects discussed below is also given in [8].

(1.1) Definitions

Let $\left(\mathcal{M},g\right)$ be a pseudo-Riemannian manifold, the target space, with metric $g$, and let $\mathcal{LM}$ be its loop space consisting of smooth embeddings of the circle into $\mathcal{M}$:

The tangent space $T_X\mathcal{LM}$ of $\mathcal{LM}$ at a loop $X : S^1 \to \mathcal{M}$ is the space of vector fields along that loop. The metric on $\mathcal{M}$ induces a metric on $T_X\mathcal{LM}$: Let $g(p) = g_{\mu\nu}(p)dx^\mu \otimes dx^\nu$ be the metric tensor on $\mathcal{M}$. Then we choose for the metric on $\mathcal{LM}$ at a point $X$ the mapping

(For the intended applications $T\mathcal{LM}$ is actually too small, since there will be need to deal with distributional vector fields on loop space. Therefore one really considers $\bar T\mathcal{LM}$, the completion of $T\mathcal{LM}$ at each point $X$ with respect to the norm induced by the inner product.)

For brevity, whenever we refer to “loop space” in the following, we mean $\mathcal{LM}$ equipped with the induced metric described above.

To abbreviate the notation, formal multi-indices $(\mu,\sigma)$ may be introduced and we write equivalently

for a vector $U\in T_X\mathcal{LM}$, and similarly for higher-rank tensors on loop space.

Extending physicist’s Ricci-calculus to the infinite-dimensional setting in the obvious way, we also write:

For this to make sense we need to know how to “shift” the continuous index $\sigma$. Because of

it makes sense to write the metric tensor on loop space as

Therefore

and

Consequently, it is natural to write

A (holonomic) basis for $T_X\mathcal{LM}$ may now be written as

where the expression on the right denotes the functional derivative, so that

By analogy, many concepts known from finite dimensional geometry carry over to the infinite dimensional case of loop spaces. Problems arise when traces over the continuous “index” $\sigma$ are taken, like for contractions of the Riemann tensor, which leads to undefined diverging expressions. These are taken care of by the usual normal-ordering of quantum field theory.

(1.2) Differential geometry on loop space

With the metric on loop space in hand the usual objects of differential geometry can be derived for loop space. Simple calculations yield the Levi-Civita connection as well as the Riemann curvature, which will be frequently needed later on. The exterior algebra over loop space is introduced and the exterior derivative and its adjoint, which play the central role in the construction of the superconformal algebra in section 2, are constructed in terms of operators on the exterior bundle $\Omega(\mathcal{LM})$. Furthermore isometries on loop space are considered, both the one coming from reparametrization of loops as well as those induced from target space. The former leads to the reparametrization constraint on strings, while the latter is crucial for the the Hamiltonian evolution on loop space along the lines of [3].

(1.2.1) Basic geometric data The basic geometric data on loop space is obtained straightforwardly by the usual computations. For instance the inverse metric is obviously

A vielbein field $e^a = e^a{}_\mu d x^\mu$ on $\mathcal{M}$ gives rise to a vielbein field $E^{(a,\sigma)}$ on loop space:

which satisfies

For the Levi-Civita connection one finds:

Similarly one finds the Riemann tensor on loop space as

(1.2.2)Exterior and Clifford algebra over loop space

The anticommuting fields $\mathcal{E}^{+ (\mu,\sigma)}$, $\mathcal{E}_{(\mu,\sigma)}$, satisfying the CAR

are assumed to exist over loop space, in analogy with the form creators and form annihilators $c^{+\mu}$, $c_\mu$ on the exterior bundle in finite dimensions (in the notation of appendix A of [3]). (For a mathematically rigorous treatment of the continuous CAR compare [7] and references given there.) From them the Clifford fields

are obtained, which satisfy

The above operators will frequently be needed with respect to some orthonormal frame $E^{(a,\sigma)}$:

Just like in the finite dimensional case, the following derivative operators can now be defined:

The covariant derivative operator (cf. A.2 in [3]) on the exterior bundle over loop space is

or alternatively

One should note well the difference between the functional derivative $\partial^c_{(\mu,\sigma)}$ which commutes with the coordinate frame forms ($[{\partial^c_{(\mu,\sigma)}},{\mathcal{E}^{+ \nu}}] = 0$) and the functional derivative $\partial_{(\mu,\sigma)}$ which instead commutes with the ONB frame forms ($[{\partial_{(\mu,\sigma)}},{\mathcal{E}^{+ a}}] = 0$). See (A.29) of [3] for more details.

In terms of these operators the exterior derivative and coderivative on loop space read, respectively (A.39)

These operators are the basis for the construction of the superconformal algebras in section 2.

(1.2.3) Isometries

Regardless of the symmetries of the metric $g$ on $\mathcal{M}$, loop space $(\mathcal{LM},G)$ has an isometry generated by the reparametrization flow vector field $K$, which is defined by:

Here $T$ is just a constant, the string tension, which we include for later convenience.

The flow generated by this vector field obviously rotates the loops around. Since the loop space metric is diagonal in the parameter $\sigma$, this leaves the geometry of loop space invariant, and the vector field $K$ satisfies Killing’s equation

as is readily checked.

The operator inducing the Lie-derivative along $K$ on differential forms is (see section A.4 of [3])

This operator will be seen to be an essential ingredient of the super-Virasoro algebra in section 2.

Apart from this generic isometry, every symmetry of the target space manifold $\mathcal{M}$ gives rise to a family of symmetries on $\mathcal{LM}$: Let $v$ be any Killing vector on target space,

then every vector $V$ on loop space of the form

(where $\xi^\sigma = \xi(\sigma)$ is some differentiable function $S^1\to \C$), is a Killing vector on loop space. For the commutators of these Killing fields one finds

The reparametrization Killing vector $K$ will be used to deform the exterior derivative on loop space as discussed in section 2.1.1 of [3], and a target space induced Killing vector $V_\xi$ will serve as a generator of parameter evolution on loop space along the lines of section 2.2 of [3]. There it was found in equation (88) that the condition

needs to be satisfied for this to work. Due to the above commutators this means that one needs to choose $\xi = \mathrm{const}$, i.e. use the integral lines of $V_{\xi =1 }$ as the “time”-parameter on loop space. This is only natural: It means that every point on the loop is evolved equally along the Killing vector field $v$ on target space.

(2) Classical superconformal generators for various backgrounds

(2.1) Purely gravitational background

In this section it is described how to obtain representations of the classical superconformal algebra on loop space. For a trivial background the construction itself is relatively trivial and, possibly in different notation, well known. The point that shall be emphasized here is that the identification of super-Virasoro generators with modes of the deformed exterior(co-)derivative on loop space allows a convenient treatment of curved backgrounds as well as more general non-trivial background fields.

As was reviewed in section 2.1.1 of [3], (which is based on [2]) one obtains from the exterior derivative and its adjoint on a manifold the generators of a global $D=2$, $N=1$ superalgebra by deforming with a Killing vector. The generic Killing vector field on loop space is the reparametrization vector $K$. Using this to deform the exterior derivative and its adjoint as in equation (19) of [3] yields the operators

The corresponding modes are

Their auto-anticommutator gives

where $\mathcal{L}$ is the operator inducing the Lie derivative on differential forms over loop space.

Any field $A(\sigma)$ is said to have reparametrization weight $w(A)$ if

In particular

The weight is additive under operator multiplication, so that $w(d_K) = 3/2$ and hence

and

Similar equations hold for $\mathbf{d}_K^+$.

This yields part of the sought-after algebra. In order to find the rest, a very simple and apparently unproblematic, but rather crucial step is to now define the modes of the deformed Laplace-Beltrami operator as the right hand side of

For this definition to make sense one needs to check that the consistency condition

is satisfied. This is trivial in the present case, but becomes a crucial condition for more general backgrounds.

The remaining supercommutators all follow from the Jacobi identity, and one can check that the resulting algebra is indeed the superconformal one.