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We generalize the DeWitt-Virasoro (DWV) construction of arXiv:0912.3987 [hep-th] to tensor representations of higher ranks. A rank-

With the motivation of understanding exact conformal invariance of string worldsheet theory in certain pp-wave background in Hamiltonian framework [

The construction of [

In this work, we will generalize the above construction to higher rank tensor representations in a certain sense. Before we delve into the details of this generalization we first try to motivate and explain what we intend to achieve. String theory contains infinite number of higher rank tensor states. Around flat background such states are constructed by applying suitable creation operators on the ground state (which can be solved exactly). State-operator mapping of this sort should hold for any CFT representing an exact classical background in string theory and the set of all possible excited states, with suitable restriction, should correspond to the fluctuation of the tensor fields around that particular background (background independence restricted to conformal backgrounds has been studied in the context of classical closed string field theory (CSFT) [

Before we attempt to answer these questions, in this work, we would like to understand certain computations that are entirely guided by the tensorial property of the wavefunctions and therefore can be carried out without knowing the explicit construction of those. More precisely, our goal is to understand how to compute coordinate invariant matrix elements involving arbitrary tensor wavefunctions. To this end, we describe a framework where the tensor wavefunctions are encoded into the states in a special way. This is done by introducing the position eigenstates which themselves are higher rank tensors. The resulting analysis can be viewed as a tensor generalization of the work in [

The DWV algebra was computed in [

The rest of the paper is organized as follows. The tensor representation has been constructed in Section

As mentioned before, the analysis of [

We first discuss the kind of questions that are relevant for our present purpose. We define a tensor state to be one whose wavefunction is a tensor in the infinite-dimensional sense. More precisely, the latter can be obtained either by multiplying a tensor field with a scalar wavefunction or by applying covariant derivatives on it. Given all possible tensor states, one would next like to understand how to compute the general coordinate invariant matrix elements of the DWV generators between such states. In position space such a matrix element should be given by an invariant integral where the integrand is constructed out of the tensor wavefunctions defined above.

Recall that in flat space the vacuum state can be explicitly solved and the tensor states are obtained by applying suitable creation operators on the vacuum state (Notice that the Schrödinger wavefunctions [

Using the existing framework of [

Given the above discussion, here we introduce a new framework where a tensor state is not created by applying operators on a scalar state, rather it is related to the corresponding wavefunction in a special way. The resulting construction will enable us to compute invariant matrix elements in tensor representations.

In this framework a rank-

Certain comments regarding (

Since we are using a complex basis, the Hermitian conjugate of a basis state is given by

Here we will discuss the rank-

Notice that in the first terms of both the expressions in (

The right and left moving DWV generators

In [

The DWV algebra, as operator equations, should be valid independent of the representation considered. Therefore, the result in (

Notice that we have found the tensor representations in the previous section from first principle. This construction does not

All the analysis in this paper has been done using the infinite-dimensional/particle language introduced in [

A shift in the infinite-dimensional index is defined to be

To quantize the theory we first define the conjugate momentum

In [

The general coordinate transformation (GCT) is understood in the quantum theory in the following way. The eigenvalue

The DWV generators are defined in (

Here we will define and discuss the properties of the parallel displacement bivector

Let us consider two nearby points

For our calculations we will use the explicit representation of

The simplest way to justify the representation (

We now generalize the above discussion to a string. We consider two nearby string-embeddings

Here we will derive the tensor representation of

We will verify the second commutator in (

Next we will consider the matrix element of the second commutator in (

Using the results in (

It should be possible to understand GCT, as introduced below (

The rank-

To answer the above question, we first write the infinitesimal version of (

To prove (

Here we will reproduce the results in (

Using (

We begin with the

Let us now consider the linear term

Using (

In the first steps of (

This is the most important contribution as the anomaly term arises here. After a somewhat lengthy calculation we arrive at the following expressions:

The rest of the commutators are straightforward to calculate and do not introduce anymore complications than in the case of spin-zero representation studied in [

The author is thankful to G. Date, T. R. Govindarajan, and R. K. Kaul for illuminating discussions.