# DeWitt-Virasoro Construction in Tensor Representations

Authors: Partha Mukhopadhyay
Journal: Advances in High Energy Physics
 (2012)

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## Abstract

We generalize the DeWitt-Virasoro (DWV) construction of arXiv:0912.3987 [hep-th] to tensor representations of higher ranks. A rank-n tensor state, which is by itself coordinate invariant, is expanded in terms of position eigenstates that transform as tensors of the same rank. The representation of the momentum operator in these basis states is then obtained by generalizing DeWitt's argument (1952). Such a representation is written in terms of certain bivector of parallel displacement and its covariant derivatives. With this machinery at hand we find tensor representations of the DWV generators defined in the previous work. The results differ from those in spin-zero representation by additional terms involving the spin connection. However, we show that the DWV algebra found earlier as a scalar expectation value remains the same, as required by consistency, as all the additional contributions conspire to cancel in various ways. In particular, vanishing of the anomaly term requires the same condition of Ricci-flatness for the background.


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## Body

## 1. Introduction and Summary

With the motivation of understanding exact conformal invariance of string worldsheet theory in certain pp-wave background in Hamiltonian framework [1–3], a particular approach to study the latter was considered in [4, 5]. The simplest case of a bosonic string moving in an arbitrary metric-background was considered. The basic idea was to find a way to describe the string quantum mechanics in a coordinate independent way following the same line of argument that was used by DeWitt [6, 7] for particle (two-dimensional nonlinear sigma model has been studied in lagrangian framework in [8–18]. See [19–30] for earlier studies using hamiltonian framework. DeWitt’s argument was used earlier in string theory in [31]).The construction of [4, 5], hereafter called the DeWitt-Virasoro (DWV) construction, goes as follows. We first rewrite the worldsheet theory in terms of the Fourier modes in such a way that it can be interpreted as a single particle moving in an infinite-dimensional curved background subject to certain potential. Since the chosen degrees of freedom are nonlocal on the worldsheet the classical conformal invariance is not directly visible. However, the description possesses some more structure which look unusual from the point of view of a particle dynamics but captures the underlying worldsheet. This enables us to write down the classical Virasoro generators in terms of the phase-space variables of the infinite-dimensional/particle description. Then implementing DeWitt’s argument [6, 7] naturally leads to a quantum background independent definition for such generators, hereafter called DeWitt-Virasoro generators. The machinery available through [6] enables one to calculate the matrix element of an arbitrary operator constructed out of the DWV generators between any two scalar states. The result written in position space is manifestly generally covariant. Using this tool we computed the DWV algebra in spin-zero representation. The result is given by the Witt algebra with certain additional anomalous terms that vanish for Ricci-flat backgrounds.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) [32] in [33–36]. Construction of CSFT for a nonconformal background was discussed in [37]). However, as we will argue, state-operator mapping, at least in this form, is not applicable in our generally covariant description. What appears to be more relevant in this case are the Schrödinger wavefunctions of the tensor states in position representation. In an arbitrary background these wavefunctions should correspond to the fluctuations of the tensor fields in spacetime. One would like to properly formulate this connection in a precise way. For example, one would like to understand how such wavefunctions should be explicitly constructed in order to be able to define the string Hilbert space.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 [6]. In that work the momentum operator in position space was obtained by demanding that the canonical commutation relations and the hermiticity conditions hold true as expectation values between two position eigenstates of spin zero. The basic ingredient that goes into such an analysis is the orthonormality and completeness conditions for such basis states. A natural generalization of the orthonormality condition for higher rank basis states is given in terms of bivector of parallel displacement. A bivector is a bilocal object which carries a vector index associated with each of the two points in the argument. Given two such points in spacetime the parallel displacement bivector, contracted with another vector at one of the points, gives the same vector parallel transported to the other point along a given path. An example is the bivector of geodetic parallel transport as discussed in [38] where the path is the unique geodesic passing through the two points. Our analysis is sensitive to the expansion of the bivector only to first-order in separation and therefore is independent of the choice of path. It turns out that the desired representation of the momentum operator can be written in terms of such bivectors and their covariant derivatives. We use this to find the representation of the DWV generators as defined in our previous work. These expressions contain all the terms that are present in the scalar representation. Moreover, they receive additional contributions involving the spin connection. These are calculated by explicitly writing the parallel displacement bivector in terms of the vielbeins.The DWV algebra was computed in [4, 5] in spin-zero representation. Given that the calculation was done between two arbitrary scalar states, one expects the result to be true as operator equations and therefore is independent of the representation considered (I thank Romesh K. Kaul for emphasizing this point). Since in this work we have found the tensor representation from first principle, it is a good idea to check explicitly if this is true. We do this following the same basic method as described in [4, 5]. As expected, the computation gets more complicated than the previous one because of the additional contributions. There arise three kinds of additional terms which depend on Riemann tensor, covariant derivative of spin connection, and higher power of spin connection. It turns out that these terms are organized in such a way that they cancel each other due to various identities in general relativity reproducing indeed the same result for the DWV algebra.The rest of the paper is organized as follows. The tensor representation has been constructed in Section2. We discuss in what sense the existing framework in [4–6] is inadequate for the present purpose in Section 2.1. The new framework is described in Section 2.2. Tensor representation of the momentum operator has been discussed in Section 2.3. We use the results of Section 2 to calculate the DWV generators and algebra in Section 3. Various details of our technical derivations have been given in several appendices.


## 2. Construction of Tensor Representations

### 2.1. Scope of the Existing Framework

As mentioned before, the analysis of [4, 5] was done using an infinite-dimensional language. We have summarized the relevant details of it in Appendix A. All our discussion in this work will be done using the same language. Below we will establish that the scope of the existing framework in [4–6] is as follows
(2.1)Nontrivialinvariantmatrixelementscanbeconstructedonlyinscalarrepresentation.
Therefore, a new framework needs to be developed in order to construct the desired tensor representations.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 [39] of all such states are of the type that we are considering here. As mentioned in Section 1, the aim of this work is not to find a background independent generalization, if it exists, of this procedure. Rather here we want to understand certain computations that are entirely guided by the tensorial properties of the wavefunctions). This is expected from the point of view of ordinary quantum mechanics. However, as we will demonstrate with an example below, this procedure is not suitable for our generally covariant construction.Using the existing framework of [4–6], one can calculate invariant matrix elements of the following form:
(2.2)〈χ|𝒪^|ψ〉,
where |χ〉 and |ψ〉 are two arbitrary scalar states and 𝒪^ is a scalar operator which can in general depend on momentum. For example, the scalar expectation value of the DWV algebra as computed in [4, 5] is of the above form. Let us now consider the following two vector wavefunctions:
(2.3)χi(x)=ai(x)χ(x),ψi(x)=ai(x)ψ(x),
where χ(x) and ψ(x) are the wavefunctions (see Appendix A for our notations) of the states in (2.2), respectively. Usually the corresponding states will be defined as follows:
(2.4)|χi〉=ai(x^)|χ〉=∫‍dwχi(x)|x〉,|ψi〉=ai(x^)|ψ〉=∫‍dwψi(x)|x〉,
respectively. However, notice that the matrix element 〈χi|𝒪^|ψi〉 is not coordinate invariant unless 𝒪^ is a scalar field independent of momentum. This is simply because under a general coordinate transformation (GCT) this matrix element transforms as (GCT has been introduced in our quantum theory in Appendix A below (A.17). In Appendix D we have discussed how GCT is realized as a unitary transformation in tensor representations)
(2.5)〈χi|𝒪^|ψi〉→〈χj|λij(x^)𝒪^λki(x^)|ψk〉.
Since λij(x^) does not commute with 𝒪^ if the latter depends on momentum, the above matrix element is not invariant in general. Proceeding in the similar way one can argue that the only nontrivial (i.e., involving momentum operator) invariant matrix elements that can be constructed are of the type (2.2). This establishes (2.1).


### 2.2. A New Framework

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-n tensor state |ψ(n)〉 is by itself GCT invariant. It is expanded in position space basis in the following way:
(2.6)|ψ(n)〉=∫‍dwψi1i2⋯in(x)|i1i2⋯in;x〉,
where ψi1i2⋯in(x) is the corresponding rank-n tensor wavefunction, which is constructed following the general definition given in the previous section. We have introduced the rank-n position eigenstates |i1⋯in;x〉 satisfying the same eigenvalue equation as in (A.15). Under a GCT ψi1i2⋯in(x) and |i1i2⋯in;x〉 transform as a contravariant and a covariant tensor of rank n, respectively (see Appendix D). The orthonormality and completeness conditions of the basis states are given by
(2.7)〈i1⋯in;x|j1⋯jn~;x~〉=δn,n~Δi1⋯inj1⋯jn(x,x~)δ(x,x~),(2.8)∫‍dw|i1⋯in;x〉gi1j1(x)⋯ginjn(x)〈j1⋯jn;x|=1,
where,
(2.9)Δi1⋯inj1⋯jn(x,x~)=Δi1j1(x,x~)⋯Δinjn(x,x~).
Equations (2.7), (2.8) are the tensor generalizations of (A.16). Before discussing what Δij(x,x~) is, we will first introduce a bitensor in the sense of [38] in our notations. A bitensor ti1⋯inj1⋯jn~(x,x~) is a tensorial object with two sets of indices {i1,…,in} and {j1,…,jn~} associated with the points x and x~, respectively such that it transforms as a tensor of rank n and n~, respectively at the two points separately. Consider
(2.10)ti1⋯inj1⋯jn~(x,x~)→ti1⋯inj1⋯jn~′(x′,x~′)=λi1k1(x)⋯λinkn(x)λj1l1(x~)⋯λjnln~(x~)tk1⋯knl1⋯ln~(x,x~).
In this sense δ(x,x~) is a biscalar and Δij(x,x~) is a bivector. The latter is called a bivector of parallel displacement which we have defined and explained the properties of in Appendix B.Certain comments regarding (2.7) are in order. From the discussion of Appendix B it is clear that in the coincidence limit the bivector approaches the metric
(2.11)limx→x~Δij(x,x~)→gij(x~).
Therefore, it might appear unnecessary to use it in (2.7) due to the presence of the delta function. The reason for this is as follows. First of all, notice that the expression in (2.7) makes it manifest that the relevant matrix element is a bitensor. Secondly, the tensor representation of the momentum operator discussed in the next section will involve derivatives. As we will argue, such a quantity will be sensitive to the expansion of the bivector up to first-order in separation.Since we are using a complex basis, the Hermitian conjugate of a basis state is given by(2.12)(|i1,…in;x〉)†=〈i¯1,⋯i¯n;x|,
and that of any operator 𝒪 is defined to be (For example, for 𝒪=xi, using (A.12) on the left hand side of (2.14) one gets, xi¯δn,n~Δi1⋯inj1⋯jn(x,x~)δ(x,x~). Then using the reality property in (A.15) and,
(2.13)Δij(x,x~)*=Δi¯j-(x,x~),
one gets the same result from the right hand side of (2.14)),
(2.14)〈i1⋯in;x|𝒪†|j1⋯jn~;x~〉=〈j¯1⋯j¯n~;x~|𝒪|i¯1⋯i¯n;x〉*.
This completes the description of the present framework using which all the calculations of this paper will be done.


### 2.3. Tensor Representation of Momentum Operator

Here we will discuss the rank-n tensor representation of the momentum operator in position space. We will actually directly work with the shifted momentum operators as defined in (A.21) [40]. The derivation of the representation of π^k has been given in Appendix C. The result is given by (in ℏ=α′=1 unit)
(2.15)〈i1⋯in;x|π^k|j1⋯jn~;x~〉=-iδn,n~[Δi1⋯inj1⋯jn(x,x~)∂kδ(x,x~)+Fi1⋯inj1⋯jnk(x,x~)δ(x,x~)],=-iδn,n~[-Δi1⋯inj1⋯jn(x,x~)(∂~k+γk(x))δ(x,x~)+Fi1⋯inj1⋯jnk(x,x~)δ(x,x~)(∂~k+γk(x))],
where
(2.16)Fi1⋯inj1⋯jnk(x,x~)=-∇kΔi1⋯inj1⋯jn(x,x~),
where ∇k is the covariant derivative. We first notice that the above expressions display the expected tensorial properties of the relevant matrix element. The two equations in (2.15) are related by the second identity in (C.4). Because of the presence of delta functions in (2.15) the complete functional dependence of Δi1⋯inj1⋯jn(x,x~) and Fi1⋯inj1⋯jnk(x,x~) are not needed. The following expressions, which can be obtained (see Appendix C) by expanding these quantities to the desired order, will be used for all practical purposes. Consider the following
(2.17)〈i1⋯in;x|π^k|j1⋯jn~;x~〉=-iδn,n~[gi1⋯inj1⋯jn(x~)∂kδ(x,x~)-(ωi1⋯inj1⋯jnk(x)+γi1⋯inj1⋯jnk(x))δ(x,x~)],=-iδn,n~[-gi1⋯inj1⋯jn(x)(∂~k+γk(x))δ(x,x~)-(ωi1⋯inj1⋯jn,k(x)-γj1⋯jni1⋯ink(x))δ(x,x~)(∂~k+γk(x))].
Various tensors appearing in the above equations are given by
(2.18)gi1⋯inj1⋯jn=gi1j1⋯ginjn,ωi1⋯inj1⋯jnk=ωki1j1gi2j2⋯ginjn+gi1j1ωki2j2gi3j3⋯ginjn+⋯,ωkij=∇keii^ηi^j^ejj^=ωki^j^eii^ejj^,=-ωkji,(2.19)γi1⋯inj1⋯jnk=γki1j1gi2j2⋯ginjn+gi1j1γki2j2gi3j3⋯ginjn+⋯,γkij=γkii′gi′j,
where eii^, ηi^j^, and ωki^j^ are the infinite-dimensional analogues of the vielbein, the Minkowski metric, and the spin connection see (B.10) and (B.12).Notice that in the first terms of both the expressions in (2.17) the argument of gi1⋯inj1⋯jn and the variable with respect to which the derivative applies on the delta function are different. This is how we choose the argument to expand Δi1⋯inj1⋯jn(x,x~) around. Because of this choice the derivative can pass through the factor gi1⋯inj1⋯jn freely. This form is useful for performing various manipulations that will be used extensively in our calculation of the DWV algebra.


## 2.1. Scope of the Existing Framework

As mentioned before, the analysis of [4, 5] was done using an infinite-dimensional language. We have summarized the relevant details of it in Appendix A. All our discussion in this work will be done using the same language. Below we will establish that the scope of the existing framework in [4–6] is as follows
(2.1)Nontrivialinvariantmatrixelementscanbeconstructedonlyinscalarrepresentation.
Therefore, a new framework needs to be developed in order to construct the desired tensor representations.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 [39] of all such states are of the type that we are considering here. As mentioned in Section 1, the aim of this work is not to find a background independent generalization, if it exists, of this procedure. Rather here we want to understand certain computations that are entirely guided by the tensorial properties of the wavefunctions). This is expected from the point of view of ordinary quantum mechanics. However, as we will demonstrate with an example below, this procedure is not suitable for our generally covariant construction.Using the existing framework of [4–6], one can calculate invariant matrix elements of the following form:
(2.2)〈χ|𝒪^|ψ〉,
where |χ〉 and |ψ〉 are two arbitrary scalar states and 𝒪^ is a scalar operator which can in general depend on momentum. For example, the scalar expectation value of the DWV algebra as computed in [4, 5] is of the above form. Let us now consider the following two vector wavefunctions:
(2.3)χi(x)=ai(x)χ(x),ψi(x)=ai(x)ψ(x),
where χ(x) and ψ(x) are the wavefunctions (see Appendix A for our notations) of the states in (2.2), respectively. Usually the corresponding states will be defined as follows:
(2.4)|χi〉=ai(x^)|χ〉=∫‍dwχi(x)|x〉,|ψi〉=ai(x^)|ψ〉=∫‍dwψi(x)|x〉,
respectively. However, notice that the matrix element 〈χi|𝒪^|ψi〉 is not coordinate invariant unless 𝒪^ is a scalar field independent of momentum. This is simply because under a general coordinate transformation (GCT) this matrix element transforms as (GCT has been introduced in our quantum theory in Appendix A below (A.17). In Appendix D we have discussed how GCT is realized as a unitary transformation in tensor representations)
(2.5)〈χi|𝒪^|ψi〉→〈χj|λij(x^)𝒪^λki(x^)|ψk〉.
Since λij(x^) does not commute with 𝒪^ if the latter depends on momentum, the above matrix element is not invariant in general. Proceeding in the similar way one can argue that the only nontrivial (i.e., involving momentum operator) invariant matrix elements that can be constructed are of the type (2.2). This establishes (2.1).


## 2.2. A New Framework

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-n tensor state |ψ(n)〉 is by itself GCT invariant. It is expanded in position space basis in the following way:
(2.6)|ψ(n)〉=∫‍dwψi1i2⋯in(x)|i1i2⋯in;x〉,
where ψi1i2⋯in(x) is the corresponding rank-n tensor wavefunction, which is constructed following the general definition given in the previous section. We have introduced the rank-n position eigenstates |i1⋯in;x〉 satisfying the same eigenvalue equation as in (A.15). Under a GCT ψi1i2⋯in(x) and |i1i2⋯in;x〉 transform as a contravariant and a covariant tensor of rank n, respectively (see Appendix D). The orthonormality and completeness conditions of the basis states are given by
(2.7)〈i1⋯in;x|j1⋯jn~;x~〉=δn,n~Δi1⋯inj1⋯jn(x,x~)δ(x,x~),(2.8)∫‍dw|i1⋯in;x〉gi1j1(x)⋯ginjn(x)〈j1⋯jn;x|=1,
where,
(2.9)Δi1⋯inj1⋯jn(x,x~)=Δi1j1(x,x~)⋯Δinjn(x,x~).
Equations (2.7), (2.8) are the tensor generalizations of (A.16). Before discussing what Δij(x,x~) is, we will first introduce a bitensor in the sense of [38] in our notations. A bitensor ti1⋯inj1⋯jn~(x,x~) is a tensorial object with two sets of indices {i1,…,in} and {j1,…,jn~} associated with the points x and x~, respectively such that it transforms as a tensor of rank n and n~, respectively at the two points separately. Consider
(2.10)ti1⋯inj1⋯jn~(x,x~)→ti1⋯inj1⋯jn~′(x′,x~′)=λi1k1(x)⋯λinkn(x)λj1l1(x~)⋯λjnln~(x~)tk1⋯knl1⋯ln~(x,x~).
In this sense δ(x,x~) is a biscalar and Δij(x,x~) is a bivector. The latter is called a bivector of parallel displacement which we have defined and explained the properties of in Appendix B.Certain comments regarding (2.7) are in order. From the discussion of Appendix B it is clear that in the coincidence limit the bivector approaches the metric
(2.11)limx→x~Δij(x,x~)→gij(x~).
Therefore, it might appear unnecessary to use it in (2.7) due to the presence of the delta function. The reason for this is as follows. First of all, notice that the expression in (2.7) makes it manifest that the relevant matrix element is a bitensor. Secondly, the tensor representation of the momentum operator discussed in the next section will involve derivatives. As we will argue, such a quantity will be sensitive to the expansion of the bivector up to first-order in separation.Since we are using a complex basis, the Hermitian conjugate of a basis state is given by(2.12)(|i1,…in;x〉)†=〈i¯1,⋯i¯n;x|,
and that of any operator 𝒪 is defined to be (For example, for 𝒪=xi, using (A.12) on the left hand side of (2.14) one gets, xi¯δn,n~Δi1⋯inj1⋯jn(x,x~)δ(x,x~). Then using the reality property in (A.15) and,
(2.13)Δij(x,x~)*=Δi¯j-(x,x~),
one gets the same result from the right hand side of (2.14)),
(2.14)〈i1⋯in;x|𝒪†|j1⋯jn~;x~〉=〈j¯1⋯j¯n~;x~|𝒪|i¯1⋯i¯n;x〉*.
This completes the description of the present framework using which all the calculations of this paper will be done.


## 2.3. Tensor Representation of Momentum Operator

Here we will discuss the rank-n tensor representation of the momentum operator in position space. We will actually directly work with the shifted momentum operators as defined in (A.21) [40]. The derivation of the representation of π^k has been given in Appendix C. The result is given by (in ℏ=α′=1 unit)
(2.15)〈i1⋯in;x|π^k|j1⋯jn~;x~〉=-iδn,n~[Δi1⋯inj1⋯jn(x,x~)∂kδ(x,x~)+Fi1⋯inj1⋯jnk(x,x~)δ(x,x~)],=-iδn,n~[-Δi1⋯inj1⋯jn(x,x~)(∂~k+γk(x))δ(x,x~)+Fi1⋯inj1⋯jnk(x,x~)δ(x,x~)(∂~k+γk(x))],
where
(2.16)Fi1⋯inj1⋯jnk(x,x~)=-∇kΔi1⋯inj1⋯jn(x,x~),
where ∇k is the covariant derivative. We first notice that the above expressions display the expected tensorial properties of the relevant matrix element. The two equations in (2.15) are related by the second identity in (C.4). Because of the presence of delta functions in (2.15) the complete functional dependence of Δi1⋯inj1⋯jn(x,x~) and Fi1⋯inj1⋯jnk(x,x~) are not needed. The following expressions, which can be obtained (see Appendix C) by expanding these quantities to the desired order, will be used for all practical purposes. Consider the following
(2.17)〈i1⋯in;x|π^k|j1⋯jn~;x~〉=-iδn,n~[gi1⋯inj1⋯jn(x~)∂kδ(x,x~)-(ωi1⋯inj1⋯jnk(x)+γi1⋯inj1⋯jnk(x))δ(x,x~)],=-iδn,n~[-gi1⋯inj1⋯jn(x)(∂~k+γk(x))δ(x,x~)-(ωi1⋯inj1⋯jn,k(x)-γj1⋯jni1⋯ink(x))δ(x,x~)(∂~k+γk(x))].
Various tensors appearing in the above equations are given by
(2.18)gi1⋯inj1⋯jn=gi1j1⋯ginjn,ωi1⋯inj1⋯jnk=ωki1j1gi2j2⋯ginjn+gi1j1ωki2j2gi3j3⋯ginjn+⋯,ωkij=∇keii^ηi^j^ejj^=ωki^j^eii^ejj^,=-ωkji,(2.19)γi1⋯inj1⋯jnk=γki1j1gi2j2⋯ginjn+gi1j1γki2j2gi3j3⋯ginjn+⋯,γkij=γkii′gi′j,
where eii^, ηi^j^, and ωki^j^ are the infinite-dimensional analogues of the vielbein, the Minkowski metric, and the spin connection see (B.10) and (B.12).Notice that in the first terms of both the expressions in (2.17) the argument of gi1⋯inj1⋯jn and the variable with respect to which the derivative applies on the delta function are different. This is how we choose the argument to expand Δi1⋯inj1⋯jn(x,x~) around. Because of this choice the derivative can pass through the factor gi1⋯inj1⋯jn freely. This form is useful for performing various manipulations that will be used extensively in our calculation of the DWV algebra.


## 3. DWV Construction

The right and left moving DWV generatorsL^(i), and L~^(i) respectively are defined in the same way as done in [4, 5],
(3.1)4L^(i)=K^(i)-Z^(i)+V^(i),4L~^(i)=K^(i¯)+Z^(i¯)+V^(i¯),
where
(3.2)K^(i)=π^k⋆gkl+i(x^)π^l,Z^(i)=Z^(i)L+Z^(i)R,V^(i)=gkl(x^)ak(x^)al+i(x^),
where
(3.3)Z^(i)L=π^k⋆ak+i(x^),Z^(i)R=ak+i(x^)π^k.
Notice that both Z^(i)L, and Z^(i)R are coordinate invariant and both have the right classical limit. The relative coefficient between these two terms in the second equation of (3.2) is fixed by the hermiticity property of the DWV generators,
(3.4)L^(i)†=L^(i¯),L~^(i)†=L~^(i¯).
Given (2.17), one can compute the matrix elements of K^(i), Z^(i)L, and Z^(i)R between two arbitrary tensor states. The manipulations involved in such calculations have been demonstrated in deriving (C.15). The final results are given by (notice that according to (2.15) πk connects tensor states of the same rank only)
(3.5)〈χ(n)|K^(i)|ψ(n)〉=(-i)2∫‍dw[χ¯i1⋯in∇(i)2ψi1⋯in+∇k+iχ¯i1⋯inωi1⋯inj1⋯jnkψj1⋯jn-χ¯i1⋯inωi1⋯inj1⋯jnk∇k+iψj1⋯jn+χ¯i1⋯inωi1⋯ink1⋯knkωk1⋯knj1⋯jnk+iψj1⋯jn],(3.6)〈χ(n)|Z^(i)L|ψ(n)〉=i∫‍dw(∇kχ¯i1⋯in+χ¯j1⋯jnωj1⋯jni1⋯ink)ak+iψi1⋯in,(3.7)〈χ(n)|Z^(i)R|ψ(n)〉=-i∫‍dwχ¯i1⋯inak+i(∇kψi1⋯in-ωi1⋯inj1⋯jnkψj1⋯jn),
where χ¯i1⋯in=χ*i¯1⋯i¯n and ∇(i)2=∇k∇k+i is the shifted Laplace operator [4, 5]. Notice that only the first terms in the above expressions were found in [4, 5] for scalar representation. The spin connection terms, which originate from the F-term in (2.15), are the additional contributions due to tensor generalization.In [4, 5] the DWV algebra was calculated as expectation value between two spin-zero states. The result, written in operator form, is given by
(3.8)[L^(i),L^(j)]=(i-j)L^(i+j)+A^(i)(j)R,[L~^(i),L~^(j)]=(i-j)L~^(i+j)+A^(i)(j)L,[L^(i),L~^(j)]=A^(i)(j).
The anomaly terms are given by
(3.9)A^(i)(j)R=0,A^(i)(j)L=0,A^(i)(j)=18(π^⋆k+irkl(x^)al+j¯(x^)-ak+i(x^)rkl(x^)π^l+j¯),
where rij(x) is the Ricci tensor in the infinite-dimensional spacetime, which, according to the general map (A.5), is related to the same in physical spacetime, namely, Rμν(X) in the following way:
(3.10)rij(x)~2πδ(0)∮‍dσ2πRμν(X(σ))ei(m+n)σ.The DWV algebra, as operator equations, should be valid independent of the representation considered. Therefore, the result in (3.8) is correct up to possible additional operator terms whose expectation value vanishes in the calculation of [4, 5]. However, since the calculation was done between two arbitrary states, it is expected that such terms are not present.Notice that we have found the tensor representations in the previous section from first principle. This construction does nota priori know about the DWV algebra. Consistency would therefore require the results in (3.8), (3.9) to hold true as expectation value between two arbitrary tensor states of any rank. Following the basic method of [4, 5], we have checked this consistency condition to affirmative in Appendix E.



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*Source: 415634-2012-11-06.xml*