DeWitt-Virasoro Construction in Tensor Representations

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.

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 Section 2. 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 Nontrivialinvariantmatrixelementscanbeconstructedonlyinscalarrepresentation. (2.1) 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: 𝜒𝒪𝜓, (2.2) 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: 𝜒^𝑖(𝑥)=𝑎^𝑖(𝑥)𝜒(𝑥),𝜓^𝑖(𝑥)=𝑎^𝑖(𝑥)𝜓(𝑥), (2.3) 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: 𝜒^𝑖=𝑎^𝑖(𝑥)𝜒=𝑑𝑤𝜒^𝑖(𝑥)|𝑥⟩,𝜓^𝑖=𝑎^𝑖(𝑥)𝜓=𝑑𝑤𝜓^𝑖(𝑥)|𝑥⟩, (2.4) respectively. However, notice that the matrix element ⟨𝜒_𝑖|𝒪|𝜓^𝑖⟩ 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) 𝜒_𝑖𝒪𝜓^𝑖→𝜒_𝑗𝜆^𝑗_𝑖(𝑥)𝒪𝜆^𝑖_𝑘(𝑥)𝜓^𝑘. (2.5) Since 𝜆^𝑗_𝑖(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: 𝜓_(𝑛)=𝑑𝑤𝜓^{𝑖1𝑖2⋯𝑖𝑛}(𝑥)_{𝑖1𝑖2⋯𝑖𝑛};𝑥, (2.6) where 𝜓^{𝑖1𝑖2⋯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 𝜓^{𝑖1𝑖2⋯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 _{𝑖1⋯𝑖𝑛};𝑥_{𝑗1⋯𝑗𝑛};𝑥=𝛿_𝑛,𝑛Δ_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛}(𝑥,𝑥)𝛿(𝑥,𝑥),𝑑𝑤_{𝑖1⋯𝑖𝑛};𝑥𝑔^𝑖1𝑗1(𝑥)⋯𝑔^{𝑖𝑛𝑗𝑛}(𝑥)_{𝑗1⋯𝑗𝑛};𝑥=1, (2.7)(2.8) where, Δ_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛}(𝑥,𝑥)=Δ_𝑖1𝑗1(𝑥,𝑥)⋯Δ_{𝑖𝑛𝑗𝑛}(𝑥,𝑥). (2.9) Equations (2.7), (2.8) are the tensor generalizations of (A.16). Before discussing what Δ_𝑖𝑗(𝑥,𝑥) is, we will first introduce a bitensor in the sense of [38] in our notations. A bitensor 𝑡_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛}(𝑥,𝑥) is a tensorial object with two sets of indices {𝑖₁,…,𝑖_𝑛} and {𝑗₁,…,𝑗_𝑛} 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 𝑡_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛}(𝑥,𝑥)→𝑡^′_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛}𝑥^′,𝑥^′=𝜆^𝑘1_𝑖1(𝑥)⋯𝜆^𝑘𝑛_𝑖𝑛(𝑥)𝜆^𝑙1_𝑗1(𝑥)⋯𝜆^𝑙𝑛_𝑗𝑛(𝑥)𝑡_{𝑘1⋯𝑘𝑛𝑙1⋯𝑙𝑛}(𝑥,𝑥). (2.10) In this sense 𝛿(𝑥,𝑥) is a biscalar and Δ_𝑖𝑗(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 lim_𝑥→𝑥Δ_𝑖𝑗(𝑥,𝑥)→𝑔_𝑖𝑗(𝑥). (2.11) 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 _{𝑖1,…𝑖𝑛};𝑥^†=_{𝑖1,⋯𝑖𝑛};𝑥, (2.12) and that of any operator 𝒪 is defined to be (For example, for 𝒪=𝑥^𝑖, using (A.12) on the left hand side of (2.14) one gets, 𝑥^𝑖𝛿_𝑛,𝑛Δ_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛}(𝑥,𝑥)𝛿(𝑥,𝑥). Then using the reality property in (A.15) and, Δ_𝑖𝑗(𝑥,𝑥)^∗=Δ_𝑖𝑗(𝑥,𝑥), (2.13) one gets the same result from the right hand side of (2.14)), ;𝑥𝒪_{𝑖1⋯𝑖𝑛}^†_{𝑗1⋯𝑗𝑛};𝑥=_{𝑗1⋯𝑗𝑛};𝑥|𝒪|_{𝑖1⋯𝑖𝑛};𝑥^∗. (2.14) 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) ;𝑥𝜋_{𝑖1⋯𝑖𝑛𝑘𝑗1⋯𝑗𝑛};𝑥=−𝑖𝛿_𝑛,𝑛Δ_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛}(𝑥,𝑥)𝜕_𝑘𝛿(𝑥,𝑥)+𝐹_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛𝑘}(𝑥,𝑥)𝛿(𝑥,𝑥),=−𝑖𝛿_𝑛,𝑛−Δ_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛}(𝑥,𝑥)𝜕_𝑘+𝛾_𝑘(𝑥)𝛿(𝑥,𝑥)+𝐹_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛𝑘}(𝑥,𝑥)𝛿(𝑥,𝑥), (2.15) where 𝐹_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛𝑘}(𝑥,𝑥)=−∇_𝑘Δ_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛}(𝑥,𝑥), (2.16) 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 Δ_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛}(𝑥,𝑥) and 𝐹_{𝑖1⋯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 ;𝑥𝜋_{𝑖1⋯𝑖𝑛𝑘𝑗1⋯𝑗𝑛};𝑥=−𝑖𝛿_𝑛,𝑛𝑔_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛}(𝑥)𝜕_𝑘𝛿(𝑥)−𝜔𝑥,_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛𝑘}(𝑥)+𝛾_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛𝑘}(𝑥)𝛿(𝑥,𝑥),=−𝑖𝛿_𝑛,𝑛−𝑔_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛}(𝑥)𝜕_𝑘+𝛾_𝑘(𝑥)𝛿(𝑥)−𝜔𝑥,_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛,𝑘}(𝑥)−𝛾_{𝑗1⋯𝑗𝑛𝑖1⋯𝑖𝑛𝑘}(𝑥)𝛿(𝑥,𝑥). (2.17) Various tensors appearing in the above equations are given by 𝑔_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛}=𝑔_𝑖1𝑗1⋯𝑔_{𝑖𝑛𝑗𝑛},𝜔_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛𝑘}=𝜔_{𝑘𝑖1𝑗1}𝑔_𝑖2𝑗2⋯𝑔_{𝑖𝑛𝑗𝑛}+𝑔_𝑖1𝑗1𝜔_{𝑘𝑖2𝑗2}𝑔_𝑖3𝑗3⋯𝑔_{𝑖𝑛𝑗𝑛}𝜔+⋯,_𝑘𝑖𝑗=∇_𝑘𝑒_𝑖𝑖̂𝜂̂𝑖𝑗̂𝑒𝑗_𝑗̂=𝜔_𝑘̂𝑖𝑗̂𝑒𝑖_𝑖̂𝑒𝑗_𝑗̂,=−𝜔_𝑘𝑗𝑖,𝛾_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛𝑘}=𝛾_{𝑘𝑖1𝑗1}𝑔_𝑖2𝑗2⋯𝑔_{𝑖𝑛𝑗𝑛}+𝑔_𝑖1𝑗1𝛾_{𝑘𝑖2𝑗2}𝑔_𝑖3𝑗3⋯𝑔_{𝑖𝑛𝑗𝑛}𝛾+⋯,_𝑘𝑖𝑗=𝛾^𝑖′_𝑘𝑖𝑔_𝑖′𝑗, (2.18)(2.19) where 𝑒𝑖_𝑖̂, 𝜂̂𝑖𝑗̂, and 𝜔_𝑘̂𝑖𝑗̂ 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 𝑔_{𝑖1⋯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 Δ_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛}(𝑥,𝑥) around. Because of this choice the derivative can pass through the factor 𝑔_{𝑖1⋯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 generators 𝐿_(𝑖), and 𝐿_(𝑖) respectively are defined in the same way as done in [4, 5], 4𝐿_(𝑖)=𝐾_(𝑖)−𝑍_(𝑖)+𝑉_(𝑖),4𝐿_(𝑖)=𝐾_(𝑖)+𝑍_(𝑖)+𝑉_(𝑖), (3.1) where 𝐾_(𝑖)=𝜋^⋆_𝑘𝑔^{𝑘𝑙+𝑖}(𝑥)𝜋_𝑙,𝑍_(𝑖)=𝑍^𝐿_(𝑖)+𝑍^𝑅_(𝑖),𝑉_(𝑖)=𝑔_𝑘𝑙(𝑥)𝑎^𝑘(𝑥)𝑎^𝑙+𝑖(𝑥), (3.2) where 𝑍^𝐿_(𝑖)=𝜋^⋆_𝑘𝑎^𝑘+𝑖(𝑥),𝑍^𝑅_(𝑖)=𝑎^𝑘+𝑖(𝑥)𝜋_𝑘. (3.3) Notice that both 𝑍^𝐿_(𝑖), and 𝑍^𝑅_(𝑖) 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, 𝐿^†_(𝑖)=𝐿_(𝑖),𝐿^†_(𝑖)=𝐿_(𝑖). (3.4) Given (2.17), one can compute the matrix elements of 𝐾_(𝑖), 𝑍^𝐿_(𝑖), and 𝑍^𝑅_(𝑖) 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) 𝜋_𝑘 connects tensor states of the same rank only) 𝜒_(𝑛)𝐾_(𝑖)𝜓_(𝑛)=(−𝑖)²𝑑𝑤𝜒^{𝑖1⋯𝑖𝑛}∇²_(𝑖)𝜓_{𝑖1⋯𝑖𝑛}+∇^𝑘+𝑖𝜒^{𝑖1⋯𝑖𝑛}𝜔_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛𝑘}𝜓^{𝑗1⋯𝑗𝑛}−𝜒^{𝑖1⋯𝑖𝑛}𝜔_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛𝑘}∇^𝑘+𝑖𝜓^{𝑗1⋯𝑗𝑛}+𝜒^{𝑖1⋯𝑖𝑛}𝜔_{𝑖1⋯𝑖𝑛𝑘1⋯𝑘𝑛𝑘}𝜔^{𝑘1⋯𝑘𝑛}_{𝑗1⋯𝑗𝑛}^𝑘+𝑖𝜓^{𝑗1⋯𝑗𝑛},𝜒_(𝑛)𝑍^𝐿_(𝑖)𝜓_(𝑛)=𝑖𝑑𝑤∇_𝑘𝜒_{𝑖1⋯𝑖𝑛}+𝜒^{𝑗1⋯𝑗𝑛}𝜔_{𝑗1⋯𝑗𝑛𝑖1⋯𝑖𝑛𝑘}𝑎^𝑘+𝑖𝜓^{𝑖1⋯𝑖𝑛}𝜒_(𝑛),𝑍^𝑅_(𝑖)𝜓_(𝑛)=−𝑖𝑑𝑤𝜒^{𝑖1⋯𝑖𝑛}𝑎^𝑘+𝑖∇_𝑘𝜓_{𝑖1⋯𝑖𝑛}−𝜔_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛𝑘}𝜓^{𝑗1⋯𝑗𝑛}, (3.5)(3.6)(3.7) where 𝜒^{𝑖1⋯𝑖𝑛}=𝜒^{∗𝑖1⋯𝑖𝑛} and ∇²_(𝑖)=∇_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 𝐿_(𝑖),𝐿_(𝑗)=𝑖−𝑗𝐿_(𝑖+𝑗)+𝐴^𝑅_(𝑖𝑗),𝐿_(𝑖),𝐿_(𝑗)=𝑖−𝑗𝐿_(𝑖+𝑗)+𝐴^𝐿_(𝑖)𝑗,𝐿_(𝑖),𝐿_(𝑗)=𝐴_(𝑖)(𝑗). (3.8) The anomaly terms are given by 𝐴^𝑅_(𝑖𝑗)𝐴=0,_{𝐿(𝑖)𝑗}𝐴=0,_(𝑖)(𝑗)=18𝜋^⋆𝑘+𝑖𝑟_𝑘𝑙(𝑥)𝑎^𝑙+𝑗(𝑥)−𝑎^𝑘+𝑖(𝑥)𝑟_𝑘𝑙(𝑥)𝜋^𝑙+𝑗, (3.9) where 𝑟_𝑖𝑗(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, 𝑅_𝜇𝜈(𝑋) in the following way: 𝑟_𝑖𝑗(𝑥)∼2𝜋𝛿(0)𝑑𝜎𝑅2𝜋^𝜇𝜈(𝑋(𝜎))𝑒^{𝑖(𝑚+𝑛)𝜎}. (3.10)

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 not a 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.

Appendices

A. Summary of Previous Work

All the analysis in this paper has been done using the infinite-dimensional/particle language introduced in [4, 5]. Here, we will briefly review the basic results of that work. The matter part of the conformally gauge fixed worldsheet lagrangian takes the following form in this language: 𝐿(𝑥̇)=1𝑥,2𝑔_𝑖𝑗(𝑥)𝑥̇^𝑖𝑥̇^𝑗−𝑎^𝑖(𝑥)𝑎^𝑗(𝑥), (A.1) where the infinite-dimensional spacetime index is give by 𝑖=𝜇,𝑚, (A.2)𝜇=0⋯𝐷−1 and 𝑚∈𝑍 being the physical spacetime index and the string-mode-number, respectively (throughout the paper, we make the following type of index identifications: 𝑖={𝜇,𝑚}, 𝑗={𝜈,𝑛}, 𝑘={𝜅,𝑞}). The coordinate 𝑥^𝑖 is implicitly dependent on the worldsheet time coordinate 𝜏. A dot indicates derivative with respect to 𝜏. The map between the infinite-dimensional and worldsheet languages is given by 𝑥^𝑖=𝑑𝜎𝑋2𝜋^𝜇(𝜎)𝑒^{−𝑖𝑚𝜎},𝑔_𝑖𝑗(𝑥)=𝑑𝜎𝐺2𝜋^𝜇𝜈(𝑋(𝜎))𝑒^{𝑖(𝑚+𝑛)𝜎},𝑎^𝑖(𝑥)=𝑑𝜎2𝜋𝜕𝑋^𝜇(𝜎)𝑒^{−𝑖𝑚𝜎}, (A.3) where 𝐺_𝜇𝜈 is the metric in physical spacetime, ∮≡∫₀^2𝜋 and 𝜕≡𝜕_𝜎. Using the above map an arbitrary field in the infinite-dimensional spacetime can be mapped to a nonlocal worldsheet operator. A class of examples, which includes the classical Virasoro generators and their quantum version, is given by a multi-indexed object 𝑢^{𝑖1𝑗1⋯}_{𝑖2𝑗2⋯}(𝑥) constructed out of the metric, its inverse, their derivatives, and 𝑎^𝑖(𝑥) (but not its derivatives) such that 𝑢^{𝑖1𝑗1⋯}_{𝑖2𝑗2⋯}(𝑥) cannot be factored into pieces which are not contracted with each other. In this case one can construct a local worldsheet operator 𝑈^{𝜇1𝜈1⋯}_{𝜇2𝜈2⋯}(𝑋(𝜎)) simply by performing the following replacements in the expression of 𝑢^{𝑖1𝑗1⋯}_{𝑖2𝑗2⋯}(𝑥): 𝑔_𝑖𝑗(𝑥)→𝐺_𝜇𝜈(𝑋(𝜎)),𝑔^𝑖𝑗(𝑥)→𝐺^𝜇𝜈(𝑋(𝜎)),𝜕_𝑖→𝜕_𝜇,𝑎^𝑖(𝑥)→𝜕𝑋^𝜇(𝜎). (A.4) The two objects 𝑢^{𝑖1𝑗1⋯}_{𝑖2𝑗2⋯}(𝑥) and 𝑈^{𝜇1𝜈1⋯}_{𝜇2𝜈2⋯}(𝑋(𝜎)) are related to each other by the following general rule: 𝑢^{𝑖1𝑗1⋯}_{𝑖2𝑗2⋯}(𝑥)∼[2𝜋𝛿(0)]^𝑁𝑑𝜎𝑈2𝜋^{𝜇1𝜈1⋯𝜇2𝜈2⋯}(𝑋(𝜎))𝑒^{𝑖(𝑚2+𝑛2+⋯)𝜎−𝑖(𝑚1+𝑛1+⋯)𝜎}, (A.5) where 𝑁 is the number of traces in 𝑢 (e.g., 𝛾_𝑘 as defined in (A.22) has a single trace and therefore is divergent: 𝛾_𝑘(𝑥)=2𝜋𝛿(0)𝑑𝜎Γ2𝜋^𝜅(𝑋(𝜎))𝑒^𝑖𝑞𝜎, (A.6) where the expression for Γ_𝜅(𝑋) can be read out from (A.22) using the replacement (A.4). The Ricci tensor is another example of a single-trace field (see (3.10))), and the argument of the Dirac delta function 𝛿(0) appearing on the right hand side is the worldsheet space direction 𝛿(0)=lim_{𝜎→𝜎′}𝛿𝜎−𝜎^′=lim_{𝜎→𝜎′}12𝜋_𝑛∈𝑍𝑒^{𝑖𝑛(𝜎−𝜎′)}. (A.7) See [4, 5] for further discussion on the appearance of such 𝛿(0)-factors. In the infinite-dimensional language the problem at hand possesses certain shift properties which can be written as 𝑢^{𝑖1+𝑖𝑖2⋯}_{𝑗1𝑗2⋯}=𝑢^𝑖1𝑖2_𝑗^+𝑖⋯_1𝑗2⋯=𝑢^{𝑖1𝑖2⋯}_{𝑗1−𝑖𝑗2⋯}=𝑢^{𝑖1𝑖2⋯}_{𝑗1𝑗2−𝑖⋯}=⋯. (A.8)

A shift in the infinite-dimensional index is defined to be 𝑖+𝑗={𝜇,𝑚+𝑛} (notice that we choose to denote the physical spacetime index corresponding to 𝑖+𝑗 by the one associated with the first index (i.e., 𝑖) appearing in the shift. We will follow this convention in all our expressions). It is obvious that (A.8) is a direct consequence of (A.5). Last equation in (A.3) implies 𝜕_𝑗𝑎^𝑘=𝑖𝑗𝛿^𝑘_𝑗, (A.9) where given the spacetime index 𝑖 as in (throughout the paper we make the following type of index identifications: 𝑖={𝜇,𝑚}, 𝑗={𝜈,𝑛}, 𝑘={𝜅,𝑞}), we have defined (𝑖)=𝑚. An important identity which was proved in [4, 5] and will be used crucially in deriving the results in Appendix E is given by 𝑎^𝑘+𝑖𝜕_𝑘𝑢^{𝑖1𝑗1⋯}_{𝑖2𝑗2⋯}=𝑖𝑖+𝑖₁+𝑗₁+⋯−𝑖₂+𝑗₂+⋯𝑢^{𝑖1+𝑖𝑗1⋯}_{𝑖2𝑗2⋯}. (A.10)

To quantize the theory we first define the conjugate momentum 𝑝_𝑖=𝜕𝐿/𝜕𝑥̇^𝑖 and then impose (in ℏ=𝛼^′=1 unit), 𝑥^𝑖,𝑝_𝑗=𝑖𝛿^𝑖_𝑗, (A.11) where we have used a hat to denote a quantum operator. The hermiticity properties are given by 𝑥^𝑖†=𝑥^𝑖,𝑝^†_𝑖=𝑝_𝑖, (A.12) where given (A.2) we have defined 𝑖=𝜇,−𝑚. (A.13)

In [6, 7] it was shown how to construct the position space representation of this quantum theory such that general covariance is manifest. An arbitrary scalar state |𝜓⟩ is expanded in position basis in the following way (throughout this work we will suppress the 𝜏 dependence of the Schrödinger states to reduce clutter): 𝜓=𝑑𝑤𝜓(𝑥)|𝑥⟩, (A.14) where 𝑑𝑤=𝑑𝑥√𝑔 (𝑔=|det𝑔_𝑖𝑗|) is the invariant measure. The position eigenstate |𝑥⟩ satisfies 𝑥^𝑖|𝑥⟩=𝑥^𝑖|𝑥⟩,𝑥^𝑖∗=𝑥^𝑖 (A.15) and the following orthonormality and completeness conditions, ⟨𝑥∣𝑥⟩=𝛿(𝑥,𝑥),𝑑𝑤|𝑥⟩⟨𝑥|=1, (A.16) where 𝛿(𝑥,𝑥)=𝑔(𝑥)𝑔(𝑥)^−(1/4)𝛿(𝑥−𝑥), (A.17)𝛿(𝑥−𝑥) being the Dirac delta function.

The general coordinate transformation (GCT) is understood in the quantum theory in the following way. The eigenvalue 𝑥^𝑖 of the position operator is the coordinate of a point 𝒫 on the manifold ℳ, where the particle-dynamics is taking place. A GCT is given by a set of functions 𝑓={𝑓^𝑖(𝑥)∣𝑓^𝑖(𝑥)^∗=𝑓^𝑖(𝑥)} such that 𝑥^𝑖→𝑥^′𝑖=𝑓^𝑖(𝑥),𝑝_𝑖→𝑝^′_𝑖=12𝜆^𝑗_𝑖(𝑥)𝑝_𝑗+𝑝_𝑗𝜆^𝑗_𝑖(𝑥,𝑥_𝑖)→𝑥^′𝑖=𝑓^𝑖(𝑥), (A.18) where 𝜆^𝑖_𝑗(𝑥)=𝜕𝑥^′𝑖𝜕𝑥^𝑗,𝜆^𝑗_𝑖(𝑥)=𝜕𝑥^𝑗𝜕𝑥^′𝑖 (A.19) are the Jacobian matrix for the transformation and its inverse, respectively. The original system (𝑥^𝑖, 𝑝_𝑖, 𝑥^𝑖) and the transformed one (𝑥^′𝑖, 𝑝^′_𝑖, 𝑥^′𝑖) give equivalent descriptions of the quantum theory such that 𝑥^𝑖 and 𝑥^′𝑖 are associated to the same point 𝒫 in ℳ. Both the wavefunction and the position eigenstate in (A.14) are scalar under GCT, 𝜓(𝑥)→𝜓^′𝑥^′=𝜓(𝑥),|𝑥⟩→𝑥^′′=|𝑥⟩. (A.20)

The DWV generators are defined in (3.1), (3.2), (3.3) in terms of certain shifted momentum operators, 𝜋_𝑘=𝑝_𝑘+𝑖2𝛾_𝑘(𝑥),𝜋^⋆_𝑘=𝑝_𝑘−𝑖2𝛾_𝑘(𝑥), (A.21) where 𝛾_𝑘 are the contracted Christoffel symbols, 𝛾_𝑘=𝛾^𝑖_𝑘𝑖,𝛾^𝑖_𝑗𝑘=12𝑔^𝑖𝑙𝜕_𝑗𝑔_𝑙𝑘+𝜕_𝑘𝑔_𝑙𝑗−𝜕_𝑙𝑔_𝑗𝑘. (A.22) Position space representation of 𝜋_𝑘 is found to be 𝑥𝜋_𝑘𝑥=−𝑖𝜕_𝑘𝛿(𝑥,𝑥). (A.23) Using either the definitions in (A.21) or the above representation, one can establish the following transformations under GCT, 𝜋_𝑘→𝜋^′_𝑘=𝜆^𝑙_𝑘(𝑥)𝜋_𝑙,𝜋^⋆_𝑘→𝜋^⋆′_𝑘=𝜋^⋆_𝑙𝜆^𝑙_𝑘(𝑥), (A.24) which show that the DWV generators are scalar operators. Using this basic ingredient one calculates the DWV algebra in spin-zero representation. The result is as given in (3.8), (3.9).

B. Bivector of Parallel Displacement

Here we will define and discuss the properties of the parallel displacement bivector Δ_𝑖𝑗(𝑥,𝑥). As explained in Section 1, such a bivector contracted with another vector at one of the points, gives the same vector parallel transported to the other point along a given path. However, as remarked below (2.11), our analysis is sensitive to the expansion of Δ_𝑖𝑗(𝑥,𝑥) only to first-order in (𝑥−𝑥)^𝑖. Such contributions are independent of the path chosen. To make the discussion valid for arbitrary separation and for definiteness we choose to discuss the bivector of geodetic parallel displacement [38] where the aforementioned path is a geodesic. We will first define this object (following [38]) for a particle in the physical spacetime using our notation and elaborate, to some extent, on the explicit representation in terms of the vielbeins. Then we will consider the string-case and describe how it is constructed in the infinite-dimensional language.

Let us consider two nearby points 𝑋^𝜇 and 𝑋^𝜇 in the physical spacetime such that a unique geodesic passes through them. The world-function Σ(𝑋,𝑋), which is a biscalar, is given by [41] (the proportionality constant is usually taken as ±(1/2) with the ± sign corresponding to space-like and time-like separations respectively. However, for our purpose such details are not needed) 𝑋,𝑋∝𝑆𝑋,𝑋², (B.1) where 𝑆(𝑋,𝑋)=𝑆(𝑋,𝑋) is the biscalar of geodetic interval. The bivector of geodetic parallel displacement 𝐷_𝜇𝜈(𝑋,𝑋) satisfies the following defining equations [38]: ∇^𝜌Σ𝑋,𝑋∇_𝜌𝐷_𝜇𝜈𝑋∇𝑋,=0,_𝜌Σ𝑋,𝑋∇_𝜌𝐷_𝜇𝜈𝑋,𝑋=0,lim_𝑋→𝑋𝐷_𝜇𝜈𝑋,𝑋=𝐺_𝜇𝜈𝑋, (B.2) where ∇_𝜇 and ∇_𝜇 denote the covariant derivatives with respect to 𝑋^𝜇 and 𝑋^𝜇 respectively. It turns out that ∇^𝜌Σ(𝑋,𝑋) and ∇^𝜌Σ(𝑋,𝑋) are directed along (or opposite to) the tangent vectors to the geodesic at 𝑋 and 𝑋, respectively [41]. From this one infers the following geometrical interpretation of 𝐷_𝜇𝜈(𝑋,𝑋): 𝐷^𝜈_𝜇𝑋,𝑋𝑉_𝜈𝑋=𝑉_𝜇𝑋→𝑋,𝐷^𝜇_𝜈𝑋,𝑋𝑉_𝜇(𝑋)=𝑉_𝜈𝑋→𝑋, (B.3) where 𝑉_𝜇(𝑋) is an arbitrary vector and 𝑉_𝜇(𝑋→𝑋) denotes the vector 𝑉_𝜇(𝑋) parallel transported from 𝑋 to 𝑋. Therefore, 𝐷_𝜇𝜈(𝑋,𝑋) parallel displaces a vector along the geodesic. As argued in [38], 𝐷_𝜇𝜈(𝑋,𝑋) is unique. Therefore, (B.3) can be taken as an alternative definition. The following properties follow from this geometrical interpretation: 𝐷_𝜇𝜈𝑋,𝑋=𝐷_𝜈𝜇,𝐷𝑋,𝑋_𝜇𝜌𝑋,𝑋𝐷^𝜌_𝜈𝑋,𝑋=𝐺_𝜇𝜈(𝑋),𝐷_𝜌𝜇𝑋,𝑋𝐷^𝜌_𝜈𝑋,𝑋=𝐺_𝜇𝜈𝑋. (B.4)

For our calculations we will use the explicit representation of 𝐷_𝜇𝜈(𝑋,𝑋) in terms of the vielbeins 𝐸𝜇_𝜇(𝑋), where 𝜇 is the local Lorentz index. The representation is given by 𝐷_𝜇𝜈𝑋𝜇𝑋,=𝐸_𝜇𝑋→𝑋𝜂𝜇𝜈𝐸𝜈_𝜈𝑋,𝜇=𝐸_𝜇(𝑋)𝜂𝜇𝜈𝐸𝜈_𝜈𝑋→𝑋, (B.5) where the parallel transport of the vielbeins is defined by vanishing of the following covariant derivative: ∇_𝜇𝐸𝜇_𝜈≡𝜕_𝜇𝐸𝜇_𝜈−Γ^𝜅_𝜇𝜈𝐸𝜇_𝜅=−Ω_𝜇𝜇𝜈𝐸𝜈_𝜈, (B.6) along the geodesic, such that the Lorentz index remains unaffected. In the last step above we have written the result in terms of the spin connection Ω_𝜇, using the fact that the total covariant derivative of the vielbein vanishes.

The simplest way to justify the representation (B.5), is to show that it leads to the same geometrical interpretation in (B.3). According to (B.6) the Lorentz component 𝑉𝜇𝜇=𝐸_𝜇𝑉^𝜇 of a vector 𝑉^𝜇 remains constant under parallel transport. Therefore, to parallel transport a vector it is sufficient to transport only the vielbein, 𝑉^𝜇𝑋→𝑋=𝐸𝜇_𝜇𝜂𝜇𝜈𝑉𝜈𝑋,𝜇𝑋→𝑋=𝐸^𝜇𝑋→𝑋𝜂𝜇𝜈𝐸𝜈_𝜈𝑋𝑉^𝜈𝑋. (B.7) Combining the last equation above and the first equation in (B.3) one identifies the first equation in (B.5). The second equation in (B.3) can also be obtained in a similar way. These two equations make the first property in (B.4) manifestly true. It is also easy to prove the other two properties in (B.4) by recalling the fact that a vielbein remains so under a parallel transport [41]. So we must have 𝐸𝜇_𝜇𝑋→𝑋𝐸𝜇_𝜈𝑋→𝑋=𝐺_𝜇𝜈(𝑋). (B.8) Notice that the representations in (B.5) is unique up to a single Lorentz transformation which may be taken to be the one relating the objects, say, at 𝑋 in the two expressions. But according to the definition of covariant derivative in (B.6), a Lorentz transformation matrix is invariant under parallel transport and therefore the same transformation relates the objects at 𝑋. However, when 𝐷_𝜇𝜈(𝑋,𝑋) acts on a vector this ambiguity gets canceled by the same in defining the Lorentz component of that vector.

We now generalize the above discussion to a string. We consider two nearby string-embeddings 𝑋(𝜎) and 𝑋(𝜎) in physical spacetime, both parametrized by 𝜎, such that the pair of points on the two strings at the same value of 𝜎 are connected by a unique geodesic. This leads to the definition of the geodetic parallel displacement bivector 𝐷_𝜇𝜈(𝑋(𝜎),𝑋(𝜎)) for the string. The infinite-dimensional analogue of the same is given by the general map (A.5), Δ_𝑖𝑗(𝑥,𝑥)=𝑑𝜎𝐷2𝜋^𝜇𝜈𝑋(𝜎),𝑋(𝜎)𝑒^{𝑖(𝑚+𝑛)𝜎},𝑖̂=𝑒_𝑖()𝜂̂𝑖𝑗̂𝑒𝑗̂𝑥→𝑥_𝑗(𝑥),𝑖̂=𝑒_𝑖(𝑥)𝜂̂𝑖𝑗̂𝑒𝑗_𝑗̂(𝑥→𝑥), (B.9) where the infinite-dimensional analogues of the vielbeins and Minkowski metric are given by 𝑒𝑖_𝑖̂(𝑥)=𝑑𝜎𝐸𝜇2𝜋^𝜇(𝑋(𝜎))𝑒^𝑖(𝑚−𝑚^)𝜎,𝜂̂𝑖𝑗̂=𝑑𝜎𝜂𝜇𝜈𝑒2𝜋^{𝑖(𝑚+𝑛)𝜎}𝜇𝜈=𝜂𝛿_𝑚+𝑛,0, (B.10) respectively. This way all the relevant equations that we discussed in the context of physical spacetime have infinite-dimensional analogues. For example, the analogue of (B.6), which we use extensively in our computations, is given by ∇_𝑖𝑒𝑖_𝑗̂=𝜕_𝑖𝑒𝑖_𝑗̂−𝛾^𝑘_𝑖𝑗𝑒𝑖_𝑘̂𝑖̂=−𝜔_𝑖̂𝑗𝑒𝑗_𝑗̂, (B.11) where, 𝑤_𝑖̂𝑖𝑗̂=𝑑𝜎Ω2𝜋^𝜇𝜇𝜈(𝑋(𝜎))𝑒^𝑖(𝑚+𝑚⁺𝑛⁾. (B.12)

C. Tensor Representation of 𝜋_𝑘

Here we will derive the tensor representation of 𝜋_𝑘 as given in (2.17). We follow the basic line of argument in [6]. In particular, we demand that the following operator equations hold as matrix elements, 𝑥^𝑙,𝜋_𝑘=𝑖𝛿^𝑙_𝑘,𝜋_𝑘,𝜋_𝑙𝜋=0,_†𝑘=𝜋^⋆_𝑘. (C.1)(C.2) Using the first commutator in (C.1) one gets the following condition: (𝑥−𝑥)^𝑙_{𝑖1⋯𝑖𝑛};𝑥𝜋_{𝑘𝑗1⋯𝑗𝑛};𝑥=𝑖𝛿^𝑙_𝑘𝛿_𝑛,𝑛Δ_𝑖1𝑗1(𝑥,𝑥)⋯Δ_{𝑖𝑛𝑗𝑛}(𝑥,𝑥)𝛿(𝑥,𝑥). (C.3) Using the identities, (𝑥−𝑥)^𝑙𝜕_𝑘𝛿(𝑥,𝑥)=−𝛿^𝑙_𝑘𝛿(𝑥),𝜕𝑥,_𝑘𝛿(𝑥,𝑥)=−𝜕_𝑘+𝛾_𝑘(𝑥)𝛿(𝑥,𝑥), (C.4) we write down the general solution in the form as given in (2.15) with Δ_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛}(𝑥,𝑥) as given in (2.7). The hermiticity condition in (C.2) is easily verified by using the expression of 𝐹_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛𝑘}(𝑥,𝑥) as given in (2.16). Consider the following: ;𝑥𝜋_{𝑖1⋯𝑖𝑛}^†_{𝑘𝑗1⋯𝑗𝑛};𝑥=_{𝑗1⋯𝑗𝑛};𝑥𝜋_{𝑘𝑖1⋯𝑖𝑛};𝑥^∗,=𝑖𝛿_𝑛𝑛Δ_{𝑗1⋯𝑗𝑛𝑖1⋯𝑖𝑛}(𝑥,𝑥)𝜕_𝑘𝛿(𝑥,𝑥)+𝐹^∗_{𝑗1⋯𝑗𝑛𝑖1⋯𝑖𝑛𝑘}(𝑥,𝑥)𝛿(𝑥,𝑥),=−𝑖𝛿_𝑛𝑛Δ_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛}(𝑥,𝑥)𝜕_𝑘+𝛾_𝑘(𝑥)𝛿(𝑥,𝑥)+𝐹_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛𝑘}(𝑥)𝛿(𝑥,=𝑥,𝑥,)_{𝑖1⋯𝑖𝑛};𝑥𝜋^⋆_{𝑘𝑗1⋯𝑗𝑛};𝑥, (C.5) where in the second step we have used the reality property (2.13). In the third step we have used Δ_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛}(𝑥,𝑥)=Δ_{𝑗1⋯𝑗𝑛𝑖1⋯𝑖𝑛}(𝑥,𝑥), (C.6) the second equation in (C.4), and 𝐹^∗_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛𝑘}(𝑥,𝑥)=𝐹_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛𝑘}(𝑥),𝐹𝑥,_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛𝑘}(𝑥,𝑥)𝛿(𝑥,𝑥)=−𝐹_{𝑗1⋯𝑗𝑛𝑖1⋯𝑖𝑛𝑘}(𝑥,𝑥)𝛿(𝑥,𝑥). (C.7)

We will verify the second commutator in (C.1) between two arbitrary invariant tensor states using the expressions in (2.17) instead of the position eigenstates. This will make the expressions look more transparent. To this end we first show how one gets the results in (2.17) from (2.15). We first expand Δ_𝑖𝑗(𝑥,𝑥) about 𝑥, Δ_𝑖𝑗(𝑥,𝑥)=𝑔_𝑖𝑗(𝑥)+(𝑥−𝑥)^𝑘𝛾^𝑙_𝑘𝑖(𝑥)𝑔_𝑙𝑗(𝑥)+⋯. (C.8) This is obtained by expanding the vielbein 𝑒𝑖_𝑖̂(𝑥→𝑥) in the second equation of (B.9) up to first-order. Using this one finds Δ_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛}(𝑥,𝑥)𝜕_𝑘𝛿(𝑥,𝑥)=𝑔_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛}(𝑥)𝜕_𝑘𝛿(𝑥,𝑥)−𝛾_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛𝑘}(𝑥)𝛿(𝑥,𝑥), (C.9) where 𝑔_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛} and 𝛾_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛𝑘} are as given in (2.18) and (2.19), respectively. Interchanging 𝑥↔𝑥 and 𝑖↔𝑗 in (C.9) and using (C.6) and 𝑔_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛}(𝑥)=𝑔_{𝑗1⋯𝑗𝑛𝑖1⋯𝑖𝑛}(𝑥), (C.10) one establishes Δ_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛}(𝑥,𝑥)𝜕_𝑘𝛿(𝑥,𝑥)=𝑔_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛}(𝑥)𝜕_𝑘𝛿(𝑥,𝑥)−𝛾_{𝑗1⋯𝑗𝑛𝑖1⋯𝑖𝑛𝑘}(𝑥)𝛿(𝑥,𝑥). (C.11) Finally, it is straightforward to show 𝐹_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛𝑘}(𝑥,𝑥)𝛿(𝑥,𝑥)=𝜔_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛𝑘}(𝑥)𝛿(𝑥,𝑥). (C.12) Substituting the results (C.9), (C.11), and (C.12) in (2.15) one finds the results (2.17).

Next we will consider the matrix element of the second commutator in (C.1) between two arbitrary tensor states |𝜒_(𝑛)⟩ and |𝜓_(𝑛)⟩ of rank 𝑛. In order to do such calculations we need the following results: 𝜒_(𝑛)𝜋_{𝑘𝑗1⋯𝑗𝑛};𝑥=𝑖∇_𝑘+𝛾_𝑘(𝑥)𝜒_{𝑗1⋯𝑗𝑛}(𝑥)+𝜒^{𝑖1⋯𝑖𝑛}(𝑥)𝜔_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛𝑘}(𝑥)_{𝑖1⋯𝑖𝑛},;𝑥𝜋_𝑘𝜓_(𝑛)=−𝑖∇_𝑘𝜓_{𝑖1⋯𝑖𝑛}(𝑥)−𝜔_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛𝑘}(𝑥)𝜓^{𝑗1⋯𝑗𝑛}(𝑥). (C.13) Such results are derived using the expressions in (2.17) and the following relations, 𝜕_𝑘𝑔_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛}=𝛾_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛𝑘}+𝛾_{𝑗1⋯𝑗𝑛𝑖1⋯𝑖𝑛𝑘},∇_𝑘𝜓_{𝑖1⋯𝑖𝑛}=𝜕_𝑘𝜓_{𝑖1⋯𝑖𝑛}−𝛾_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛𝑘}𝜓^{𝑗1⋯𝑗𝑛}. (C.14) We will now calculate ⟨𝜒|𝜋_𝑘𝜋_𝑙|𝜓⟩ by first inserting the completeness relation (2.8) between the two 𝜋's and then using the results (C.13). To get to the final expression we need to go through certain manipulations that are generally involved in the computations of the matrix elements carried out in this paper. We give the details of the present calculation in the following series of steps: 𝜒_(𝑛)𝜋_𝑘𝜋_𝑙𝜓_(𝑛)=𝑑𝑤𝜒_(𝑛)𝜋_{𝑘𝑘1⋯𝑘𝑛};𝑥𝑔^𝑘1𝑙1(𝑥)⋯𝑔^{𝑘𝑛𝑙𝑛}(𝑥)_{𝑙1⋯𝑙𝑛};𝑥𝜋_𝑙𝜓_(𝑛),=𝑑𝑤∇_𝑘+𝛾_𝑘𝜒_{𝑘1⋯𝑘𝑛}+𝜒^{𝑖1⋯𝑖𝑛}𝜔_{𝑖1⋯𝑖𝑛𝑘1⋯𝑘𝑛𝑘}∇_𝑙𝜓^{𝑘1⋯𝑘𝑛}−𝜔^{𝑘1⋯𝑘𝑛}_{𝑙1⋯𝑙𝑛𝑙}𝜓^{𝑙1⋯𝑙𝑛},=𝑑𝑤𝜕_𝑘+𝛾_𝑘𝜒_{𝑘1⋯𝑘𝑛}−𝛾^𝑘′_𝑘𝑘1𝜒_{𝑘′𝑘2⋯𝑘𝑛}+𝛾^𝑘′_𝑘𝑘2𝜒_{𝑘1𝑘′𝑘3⋯𝑘𝑛}++⋯𝜒^{𝑖1⋯𝑖𝑛}𝜔_{𝑖1⋯𝑖𝑛𝑘1⋯𝑘𝑛𝑘}∇_𝑙𝜓^{𝑘1⋯𝑘𝑛}−𝜔^{𝑘1⋯𝑘𝑛}_{𝑙1⋯𝑙𝑛𝑙}𝜓^{𝑙1⋯𝑙𝑛},=−𝑑𝑤𝜒_{𝑘1⋯𝑘𝑛}𝜕_𝑘∇_𝑙𝜓^{𝑘1⋯𝑘𝑛}−𝜔^{𝑘1⋯𝑘𝑛}_{𝑙1⋯𝑙𝑛𝑙}𝜓^{𝑙1⋯𝑙𝑛}+𝜒_{𝑘1⋯𝑘𝑛}𝛾^𝑘1_𝑘𝑘′∇_𝑙𝜓^{𝑘′𝑘2⋯𝑘𝑛}−𝜔^{𝑘′𝑘2⋯𝑘𝑛}_{𝑙1⋯𝑙𝑛𝑙}𝜓^{𝑙1⋯𝑙𝑛}+𝛾^𝑘2_𝑘𝑘′∇_𝑙𝜓^{𝑘1𝑘′𝑘2⋯𝑘𝑛}−𝜔^{𝑘1𝑘′𝑘2⋯𝑘𝑛}_{𝑙1⋯𝑙𝑛𝑙}𝜓^{𝑙1⋯𝑙𝑛}⋯−𝜒^{𝑖1⋯𝑖𝑛}𝜔_{𝑖1⋯𝑖𝑛𝑘1⋯𝑘𝑛𝑘}∇_𝑙𝜓^{𝑘1⋯𝑘𝑛}−𝜔^{𝑘1⋯𝑘𝑛}_{𝑙1⋯𝑙𝑛𝑙}𝜓^{𝑙1⋯𝑙𝑛},=−𝑑𝑤𝜒_{𝑘1⋯𝑘𝑛}∇_𝑘∇_𝑙𝜓^{𝑘1⋯𝑘𝑛}−𝜔^{𝑘1⋯𝑘𝑛}_{𝑙1⋯𝑙𝑛𝑙}𝜓^{𝑙1⋯𝑙𝑛}+𝜒_{𝑘1⋯𝑘𝑛}𝛾^𝑙′_𝑘𝑙∇_𝑙′𝜓^{𝑘1⋯𝑘𝑛}−𝜔^{𝑘1⋯𝑘𝑛}_{𝑙1⋯𝑙𝑛𝑙′}𝜓^{𝑙1⋯𝑙𝑛}−𝜒^{𝑖1⋯𝑖𝑛}𝜔_{𝑖1⋯𝑖𝑛𝑘1⋯𝑘𝑛𝑘}∇_𝑙𝜓^{𝑘1⋯𝑘𝑛}−𝜔^{𝑘1⋯𝑘𝑛}_{𝑙1⋯𝑙𝑛𝑙}𝜓^{𝑙1⋯𝑙𝑛},=−𝑑𝑤𝜒_{𝑘1⋯𝑘𝑛}∇_𝑘∇_𝑙𝜓^{𝑘1⋯𝑘𝑛}−𝜒_{𝑘1⋯𝑘𝑛}∇_𝑘𝜔^{𝑘1⋯𝑘𝑛}_{𝑙1⋯𝑙𝑛𝑙}𝜓^{𝑙1⋯𝑙𝑛}+𝜒^{𝑖1⋯𝑖𝑛}𝜔_{𝑖1⋯𝑖𝑛𝑘1⋯𝑘𝑛𝑘}𝜔^{𝑘1⋯𝑘𝑛}_{𝑙1⋯𝑙𝑛𝑙}𝜓^{𝑙1⋯𝑙𝑛}+𝜒_{𝑘1⋯𝑘𝑛}𝛾^𝑙′_𝑘𝑙∇_𝑙′𝜓^{𝑘1⋯𝑘𝑛}−𝜔^{𝑘1⋯𝑘𝑛}_{𝑙1⋯𝑙𝑛𝑙′}𝜓^{𝑙1⋯𝑙𝑛}−𝜒_{𝑘1⋯𝑘𝑛}𝜔^{𝑘1⋯𝑘𝑛}_{𝑙1⋯𝑙𝑛𝑙}∇_𝑘+𝑘⟷𝑙𝜓^{𝑗1⋯𝑗𝑛}. (C.15) The last two terms in the above expression are symmetric in 𝑘 and 𝑙. Therefore, they do not contribute to the commutator which is given by 𝜒_(𝑛)𝜋_𝑘,𝜋_𝑙𝜓_(𝑛)=−𝑑𝑤𝑉, (C.16) where 𝑉=𝜒_{𝑘1⋯𝑘𝑛}∇_𝑘,∇_𝑙𝜓^{𝑘1⋯𝑘𝑛}−𝜒_{𝑘1⋯𝑘𝑛}∇_𝑘𝜔^{𝑘1⋯𝑘𝑛}_{𝑙1⋯𝑙𝑛𝑙}−𝑘⟷𝑙𝜓^{𝑙1⋯𝑙𝑛}+𝜒^{𝑖1⋯𝑖𝑛}𝜔_{𝑖1⋯𝑖𝑛𝑘1⋯𝑘𝑛𝑘}𝜔^{𝑘1⋯𝑘𝑛}_{𝑗1⋯𝑗𝑛𝑙}−𝑘⟷𝑙𝜓^{𝑗1⋯𝑗𝑛}. (C.17) Below we will show that 𝑉 vanishes. Using ∇_𝑘,∇_𝑙𝜓^{𝑘1⋯𝑘𝑛}=𝑟^𝑘1_{𝑘′𝑘𝑙}𝜓^{𝑘′𝑘2⋯𝑘𝑛}+𝑟^𝑘2_{𝑘′𝑘𝑙}𝜓^{𝑘1𝑘′𝑘3⋯𝑘𝑛}+⋯, (C.18)𝑟^𝑖_𝑗𝑘𝑙 being the Riemann tensor, one may write the first term in (C.17) as 𝜒_{𝑘1⋯𝑘𝑛}∇_𝑘,∇_𝑙𝜓^{𝑘1⋯𝑘𝑛}=𝜒^{𝑘1⋯𝑘𝑛}𝑟_{𝑘1⋯𝑘𝑛𝑙1⋯𝑙𝑛𝑘𝑙}𝜓^{𝑙1⋯𝑙𝑛}, (C.19) where 𝑟_{𝑘1⋯𝑘𝑛𝑙1⋯𝑙𝑛𝑘𝑙}=𝑟_{𝑘1𝑙1𝑘𝑙}𝑔_𝑘2𝑙2⋯𝑔_{𝑘𝑛𝑙𝑛}+𝑔_𝑘1𝑙1𝑟_{𝑘2𝑙2𝑘𝑙}𝑔_𝑘3𝑙3⋯𝑔_{𝑘𝑛𝑙𝑛}+⋯. (C.20) To calculate the second term in (C.17) one first derives, following the standard argument [42], ∇_𝑘𝑤_𝑙𝑖𝑗−𝑘⟷𝑙=𝑟_{𝑖𝑗𝑘𝑙}+𝑑_{𝑖𝑗𝑘𝑙},𝑑_{𝑖𝑗𝑘𝑙}=𝑤_{𝑘𝑖𝑖′}𝑤^𝑖′_𝑙𝑗−𝑘⟷𝑙, (C.21) using which one may write, ∇_𝑘𝜔_{𝑘1⋯𝑘𝑛𝑙1⋯𝑙𝑛𝑙}−𝑘⟷𝑙=𝑟_{𝑘1⋯𝑘𝑛𝑙1⋯𝑙𝑛𝑘𝑙}+𝑑_{𝑘1⋯𝑘𝑛𝑙1⋯𝑙𝑛𝑘𝑙}, (C.22) where we have defined, 𝑑_{𝑖1⋯𝑖𝑛,𝑗1⋯𝑗𝑛,𝑘𝑙}≡𝑑_{𝑖1𝑗1𝑘𝑙}𝑔_𝑖2𝑗2⋯𝑔_{𝑖𝑛𝑗𝑛}+𝑔_𝑖1𝑗1𝑑_{𝑖2𝑗2𝑘𝑙}𝑔_𝑖3𝑗3⋯𝑔_{𝑖𝑛𝑗𝑛}+⋯. (C.23) We note down the following antisymmetry properties for future use: 𝑟_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛𝑘𝑙}=−𝑟_{𝑗1⋯𝑗𝑛𝑖1⋯𝑖𝑛𝑘𝑙}=−𝑟_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛𝑙𝑘},𝑑_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛𝑘𝑙}=−𝑑_{𝑗1⋯𝑗𝑛𝑖1⋯𝑖𝑛𝑘𝑙}=−𝑑_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛𝑙𝑘}. (C.24) The last term in (C.17) is calculated in the following way: 𝜔_{𝑖1⋯𝑖𝑛𝑘1⋯𝑘𝑛𝑘}𝜔^{𝑘1⋯𝑘𝑛}_{𝑗1⋯𝑗𝑛𝑙}−𝑘⟷𝑙=𝜔_{𝑘𝑖1𝑘1}𝑔_𝑖2𝑘2⋯𝑔_{𝑖𝑛𝑘𝑛}+𝑔_𝑖1𝑘1𝜔_{𝑘𝑖2𝑘2}𝑔_𝑖3𝑘3⋯𝑔_{𝑖𝑛𝑘𝑛}𝜔+⋯_{𝑘1𝑙𝑗1}𝑔^𝑘2_𝑗2⋯𝑔^𝑘𝑛_𝑗𝑛+𝑔^𝑘1_𝑗1𝜔_𝑙^𝑘2_𝑗2𝑔^𝑘3_𝑗3⋯𝑔^𝑘𝑛_𝑗𝑛+⋯−𝑘⟷𝑙,=𝒟_{𝑖1⋯𝑖𝑛,𝑗1⋯𝑗𝑛,𝑘𝑙}+𝒞_{𝑖1⋯𝑖𝑛,𝑗1⋯𝑗𝑛,𝑘𝑙}−𝑘⟷𝑙, (C.25) where the diagonal and the cross terms are 𝒟_{𝑖1⋯𝑖𝑛,𝑗1⋯𝑗𝑛,𝑘𝑙}=𝜔_{𝑘𝑖1𝑘1}𝜔_𝑙^𝑘1_𝑗1𝑔_𝑖2𝑗2⋯𝑔_{𝑖𝑛𝑗𝑛}+𝑔_𝑖1𝑗1𝜔_{𝑘𝑖2𝑘2}𝜔_𝑙^𝑘2_𝑗2𝑔_𝑖2𝑗2⋯𝑔_{𝑖𝑛𝑗𝑛}𝒞+⋯,_{𝑖1⋯𝑖𝑛,𝑗1⋯𝑗𝑛,𝑘𝑙}=𝜔_{𝑘𝑖1𝑗1}𝜔_{𝑙𝑖2𝑗2}𝑔_𝑖3𝑗3⋯𝑔_{𝑖𝑛𝑗𝑛}+𝜔+𝑘⟷𝑙_{𝑘𝑖1𝑗1}𝑔_𝑖2𝑗2𝜔_{𝑙𝑖3𝑗3}𝑔_𝑖4𝑗4⋯𝑔_{𝑖𝑛𝑗𝑛}+𝑘⟷𝑙+⋯, (C.26)(C.27) respectively. The cross-term, being symmetric under 𝑘↔𝑙, does not contribute to (C.25). Therefore, 𝜔_{𝑖1⋯𝑖𝑛𝑘1⋯𝑘𝑛𝑘}𝜔^{𝑘1⋯𝑘𝑛}_{𝑗1⋯𝑗𝑛𝑙}−𝑘⟷𝑙=𝑑_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛𝑘𝑙}. (C.28)

Using the results in (C.19), (C.22), and (C.28) in (C.17) one shows that 𝑉 vanishes implying 𝜒_(𝑛)𝜋_𝑘,𝜋_𝑙𝜓_(𝑛)=0. (C.29) The above calculations show why the 𝐹-term in (2.15) or the spin connection term in (2.17) is essential. This is needed to cancel the Riemann tensor contributions which will otherwise survive in the calculation of the above commutator.

D. GCT as a Unitary Transformation

It should be possible to understand GCT, as introduced below (A.17), as a usual unitary transformation in quantum mechanics. This has been explained for spin-zero representation in [7]. Here we will generalize the argument to incorporate the tensor representations.

The rank-𝑛 tensor wavefunction and the position eigenstate in (2.6) transform under GCT in the following way: 𝜓^{𝑖1⋯𝑖𝑛}(𝑥)→𝜓^{′𝑖1⋯𝑖𝑛}𝑥^′=𝜆^𝑖1_𝑗1(𝑥)⋯𝜆^𝑖𝑛_𝑗𝑛(𝑥)𝜓^{𝑗1⋯𝑗𝑛}(𝑥),_{𝑖1𝑖2⋯𝑖𝑛};𝑥→_{𝑖1𝑖2⋯𝑖𝑛};𝑥^′′=𝜆^𝑗1_𝑖1(𝑥)𝜆^𝑗2_𝑖2(𝑥)⋯𝜆^𝑗𝑛_𝑖𝑛(𝑥)_{𝑗1𝑗2⋯𝑖𝑛};𝑥. (D.1) Using (A.18) and (D.1) one finds, as expected, 𝑥^′𝑖_{𝑖1⋯𝑖𝑛};𝑥^′′=𝑥^′𝑖_{𝑖1⋯𝑖𝑛};𝑥^′′. (D.2) Let us now consider the following unitary operator: 𝑈=𝑒^(𝑖/2)𝑉,𝑉=𝜋^⋆_𝑖𝑣^𝑖(𝑥)+𝑣^𝑖(𝑥)𝜋_𝑖, (D.3) where the 𝜋-operators are defined in (A.21) and 𝑣^𝑖(𝑥) is a vector field such that 𝑣^𝑖(𝑥)^∗=𝑣^𝑖(𝑥). (D.4) The above definition implies that 𝑉 is both Hermitian and GCT invariant. The latter can be checked easily by using the transformation property in (A.24). At the infinitesimal level 𝑈 produces the following coordinate transformation: 𝑥^′𝑖(𝑥)=𝑈𝑥^𝑖𝑈^†=𝑥^𝑖+𝑣^𝑖(𝑥). (D.5) The question that one would like to ask is how the state 𝑈|_{𝑖1⋯𝑖𝑛};𝑥⟩ is related to |_{𝑖1⋯𝑖𝑛};𝑥⟩^′. Notice that both of these states have the same eigenvalue for 𝑥^′𝑖, namely 𝑥^𝑖.

To answer the above question, we first write the infinitesimal version of (D.1) in the following way: ;𝑥−𝑣(_{𝑖1⋯𝑖𝑛}𝑥)=𝛿^𝑗1_𝑖1𝛿^𝑗2_𝑖2⋯𝛿^𝑗𝑛_𝑖𝑛+∇_𝑖1𝑣^𝑗1𝛿^𝑗2_𝑖2⋯𝛿^𝑗𝑛_𝑖𝑛+𝛿^𝑗1_𝑖1∇_𝑖2𝑣^𝑗2𝛿^𝑗3_𝑖3⋯𝛿^𝑗𝑛_𝑖𝑛+⋯_{𝑗1⋯𝑗𝑛};𝑥^′, (D.6) where the state appearing on the left hand side has a covariant Taylor expansion in the following sense: ;𝑥−𝑣(_{𝑖1⋯𝑖𝑛}𝑥)∣𝜓_(𝑛)=𝜓_{𝑖1⋯𝑖𝑛}(𝑥)−𝑣^𝑘(𝑥)∇_𝑘𝜓_{𝑖1⋯𝑖𝑛}(𝑥), (D.7) where |𝜓_(𝑛)⟩ is an arbitrary rank-𝑛 tensor state as in (2.6). Below we will show that the following relation is true, 𝑈_{𝑖1⋯𝑖𝑛};𝑥=𝑁^{𝑗1⋯𝑗𝑛}_{𝑖1⋯𝑖𝑛}(𝑥)_{𝑗1⋯𝑗𝑛};𝑥−𝑣(𝑥), (D.8) where the normalization is given by: 𝑁_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛}=𝑔_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛}+𝛿𝑁_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛},𝛿𝑁_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛}1=−2∇_𝑘𝑣^𝑘𝑔_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛}+𝜔_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛𝑘}𝑣^𝑘. (D.9)

To prove (D.8), (D.9) we will first calculate ⟨_{𝑖1⋯𝑖𝑛};𝑥|𝑈^†|𝜓_(𝑛)⟩ independently by using (D.8) and (D.3) and then compare the results. Using (D.8) one gets ;𝑥𝑈_{𝑖1⋯𝑖𝑛}^†𝜓_(𝑛)=𝑁^∗_{𝑖1⋯𝑖𝑛}^{𝑗1⋯𝑗𝑛}(𝑥)𝜓_{𝑗1⋯𝑗𝑛}(𝑥)−𝑣^𝑘(𝑥)∇_𝑘𝜓_{𝑗1⋯𝑗𝑛}(𝑥),=1−𝑣^𝑘(𝑥)∇_𝑘𝜓_{𝑖1⋯𝑖𝑛}(𝑥)+𝛿𝑁^∗_{𝑖1⋯𝑖𝑛}^{𝑗1⋯𝑗𝑛}𝜓_{𝑗1⋯𝑗𝑛}(𝑥), (D.10) where in the second step we have used the first equation in (D.9). On the other hand using (D.3) one finds ;𝑥𝑈_{𝑖1⋯𝑖𝑛}^†𝜓_(𝑛)=11−2∇_𝑘𝑣^𝑘(𝑥)−𝑣^𝑘(𝑥)∇_𝑘𝜓_{𝑖1⋯𝑖𝑛}(𝑥)+𝜔_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛𝑘}(𝑥)𝑣^𝑘(𝑥)𝜓^{𝑗1⋯𝑗𝑛}(𝑥), (D.11) where we have used the representation (2.17). Finally, comparing (D.10) and (D.11), we arrive at the second equation in (D.9).

E. Derivation of DWV Algebra

Here we will reproduce the results in (3.8), (3.9) in tensor representation. This will be done by showing that the results hold as matrix elements between two arbitrary tensor states|𝜒⟩ and |𝜓⟩ of rank 𝑛 (To simplify our notations we remove the subscript (𝑛) from the states in this appendix.) We begin by quoting certain results that will be needed as we go along. 𝜒𝐾_{(𝑖)𝑖1⋯𝑖𝑛};𝑥=(−𝑖)²∇²_(𝑖)𝜒_{𝑖1⋯𝑖𝑛}+2∇^𝑘+𝑖𝜒^{𝑗1⋯𝑗𝑛}𝜔_{𝑗1⋯𝑗𝑛𝑖1⋯𝑖𝑛𝑘}+𝜒^{𝑗1⋯𝑗𝑛}∇^𝑘+𝑖𝜔_{𝑗1⋯𝑗𝑛𝑖1⋯𝑖𝑛𝑘}+𝜒^{𝑗1⋯𝑗𝑛}𝜔_{𝑗1⋯𝑗𝑛𝑘1⋯𝑘𝑛𝑘}𝜔^{𝑘1⋯𝑘𝑛}_{𝑖1⋯𝑖𝑛}^𝑘+𝑖,_{𝑖1⋯𝑖𝑛};𝑥𝐾_(𝑖)𝜓=(−𝑖)²∇²_(𝑖)𝜓_{𝑖1⋯𝑖𝑛}−∇^𝑘+𝑖𝜔_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛𝑘}𝜓^{𝑗1⋯𝑗𝑛}−2𝜔_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛𝑘}∇^𝑘+𝑖𝜓^{𝑗1⋯𝑗𝑛}+𝜔_{𝑖1⋯𝑖𝑛𝑘1⋯𝑘𝑛𝑘}𝜔^{𝑘1⋯𝑘𝑛}_{𝑗1⋯𝑗𝑛}^𝑘+𝑖𝜓^{𝑗1⋯𝑗𝑛},𝜒𝑍^𝐿_{(𝑖)𝑖1⋯𝑖𝑛};𝑥=−𝑖−∇_𝑘𝜒_{𝑖1⋯𝑖𝑛}−𝜒^{𝑗1⋯𝑗𝑛}𝜔_{𝑗1⋯𝑗𝑛𝑖1⋯𝑖𝑛𝑘}𝑎^𝑘+𝑖,_{𝑖1⋯𝑖𝑛};𝑥𝑍^𝐿_(𝑖)𝜓=−𝑖∇_𝑘𝑎^𝑘+𝑖𝜓_{𝑖1⋯𝑖𝑛}−𝜔_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛𝑘}𝑎^𝑘+𝑖𝜓^{𝑗1⋯𝑗𝑛},𝜒𝑍^𝑅_{(𝑖)𝑖1⋯𝑖𝑛};𝑥=−𝑖−∇_𝑘𝜒_{𝑖1⋯𝑖𝑛}𝑎^𝑘+𝑖−𝜒^{𝑗1⋯𝑗𝑛}𝜔_{𝑗1⋯𝑗𝑛𝑖1⋯𝑖𝑛𝑘}𝑎^𝑘+𝑖,_{𝑖1⋯𝑖𝑛};𝑥𝑍^𝑅_(𝑖)𝜓=−𝑖𝑎^𝑘+𝑖∇_𝑘𝜓_{𝑖1⋯𝑖𝑛}−𝜔_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛𝑘}𝜓^{𝑗1⋯𝑗𝑛}. (E.1)(E.2)(E.3) To proceed with the computation of DWV algebra, we first introduce certain notations. We break each of the relevant commutators into several terms in the following way: _𝜒𝐿_(𝑖),𝐿_(𝑗)𝜓=1𝑇16^{𝐾(𝑖)𝐾(𝑗)}+𝑇^{𝑍(𝑖)𝑍(𝑗)}−𝑇^{𝐾(𝑖)𝑍(𝑗)}+𝑇−𝑖⟷𝑗_{𝐾(𝑖)𝑉(𝑗)}−𝑖⟷𝑗−𝑇^{𝑍(𝑖)𝑉(𝑗)}−𝑖⟷𝑗_𝜒,𝐿_(𝑖),𝐿_(𝑗)𝜓=1𝑇16^{𝐾(𝑖)𝐾(𝑗)}+𝑇^{𝑍(𝑖)𝑍(𝑗)}+𝑇^{𝐾(𝑖)𝑍(𝑗)}−𝑖⟷𝑗+𝑇^{𝐾(𝑖)𝑉(𝑗)}−𝑖⟷𝑗+𝑇^{𝑍(𝑖)𝑉(𝑗)}−𝑖⟷𝑗,_𝜒𝐿_(𝑖),𝐿_(𝑗)𝜓=1𝑇16^{𝐾(𝑖)𝐾(𝑗)}−𝑇^{𝑍(𝑖)𝑍(𝑗)}+𝑇^{𝐾(𝑖)𝑍(𝑗)}+𝑖⟷𝑗+𝑇^{𝐾(𝑖)𝑉(𝑗)}−𝑖⟷𝑗−𝑇^{𝑍(𝑖)𝑉(𝑗)}+𝑖⟷𝑗, (E.4) where _𝜒⟨⋯⟩_𝜓=⟨𝜒|⋯|𝜓⟩ and 𝑇^𝐴𝐵=_𝜒𝐴,𝐵_𝜓. (E.5) Terms involving the operator 𝑍_(𝑖) are further divided into components in the following way: 𝑇^{𝑍(𝑖)𝑍(𝑗)}=𝑇^{𝑍𝐿(𝑖)𝑍𝐿(𝑗)}+𝑇^{𝑍𝑅(𝑖)𝑍𝑅(𝑗)}+𝑇^{𝑍𝐿(𝑖)𝑍𝑅(𝑗)},𝑇−𝑖⟷𝑗_{𝐾(𝑖)𝑍(𝑗)}=𝑇^{𝐾(𝑖)𝑍𝐿(𝑗)}+𝑇^{𝐾(𝑖)𝑍𝑅(𝑗)},𝑇^{𝑍(𝑖)𝑉(𝑗)}=𝑇^{𝑍𝐿(𝑖)𝑉(𝑗)}+𝑇^{𝑍𝑅(𝑗)𝑉(𝑗)}. (E.6) Each of the commutator terms receives contributions at different orders in 𝜔_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛𝑘}, 𝑇^𝐴𝐵=(−𝑖)^𝑟𝑂^𝐴𝐵_𝜒𝜓+𝐿^𝐴𝐵_𝜒𝜓+𝑄^𝐴𝐵_𝜒𝜓+𝐶^𝐴𝐵_𝜒𝜓, (E.7) where 𝑟 is the number of 𝜋 operator involved in the computation and 𝑂, 𝐿, 𝑄, and 𝐶 denote the terms which are zeroth, first, second, and third order in 𝜔, respectively. We have made ordering of 𝜒 and 𝜓 explicit in the notations of such contributions as we will exploit certain symmetry under 𝜒↔𝜓 to compute a particular set of terms. Although for other terms it is not needed, we use a uniform notation. Similar contributions for a matrix element of product of two operators will be denoted by the same symbols, but without the bar (The quantity _𝜒⟨𝐾_(𝑖)𝐾_(𝑗)⟩_𝜓 receives contribution at the quartic order, but we will not have to calculate it explicitly.) Consider _𝜒𝐴𝐵_𝜓=(−𝑖)^𝑟𝑂^𝐴𝐵_𝜒𝜓+𝐿^𝐴𝐵_𝜒𝜓+𝑄^𝐴𝐵_𝜒𝜓+𝐶^𝐴𝐵_𝜒𝜓. (E.8) Below we will compute each of the terms appearing in (E.4), (E.6) separately.

𝑇^{𝐾(𝑖)𝐾(𝑗)}
Using (E.1), one can explicitly compute all the terms in _𝜒⟨𝐾_(𝑖)𝐾_(𝑗)⟩_𝜓. In [4, 5] it was shown that in spin-zero representation such terms are symmetric under 𝑖↔𝑗 by using the shift property in (A.8). In the present case the wavefunctions themselves carry tensor indices and a direct application of such an argument would require the wavefunctions also to possess similar shift property. According to our general definition of tensor wavefunctions which has been explained below (2.1), these objects may in general involve derivatives of 𝑎^𝑖(𝑥). Since the general form of 𝑢^{𝑖1𝑗1⋯}_{𝑖2𝑗2⋯} (see discussion above (A.4)), for which the shift property in (A.8) holds true, does not allow such factors, we do not want to assume the shift property for the wavefunctions. However, as will be discussed below, a slightly modified argument may be used to show that all the terms in _𝜒⟨𝐾_(𝑖)𝐾_(𝑗)⟩_𝜓 are indeed symmetric in 𝑖𝑗 without assuming any special property for the wavefunctions. This would therefore imply 𝑇^{𝐾(𝑖)𝐾(𝑗)}=0. (E.9)
We begin with the 𝒪(𝜔⁰)-term. By using integration by parts we write it in the following form: 𝑂^{𝐾(𝑖)𝐾(𝑗)}_𝜒𝜓=𝑑𝑤𝜒^{𝑖1⋯𝑖𝑛}∇²_(𝑖),∇²_𝑗𝜓_{𝑖1⋯𝑖𝑛}. (E.10) We then expand the commutator in the following way: ∇²_(𝑖),∇²_𝑗=∇_𝑘∇^𝑘+𝑖,∇^𝑙+𝑗∇_𝑙+∇_𝑘∇_𝑙∇^𝑘+𝑖,∇^𝑙+𝑗+∇^𝑘+𝑖,∇^𝑙+𝑗∇_𝑙∇_𝑘+∇_𝑙∇^𝑘+𝑖,∇^𝑙+𝑗∇_𝑘. (E.11) Notice that the terms have been written in such a way that the shifts always appear within the commutators. This can be done as we may write ∇^𝑘+𝑖∇_𝑘=𝑔^{𝑘+𝑖𝑙}∇_𝑙∇_𝑘=𝑔^{𝑙+𝑖𝑘}∇_𝑙∇_𝑘=∇_𝑘∇^𝑘+𝑖, (E.12) without assuming any shift property for a tensor field on which the derivatives are acting. Then we recognize that the following is true for an arbitrary tensor 𝑡^{𝑖1⋯𝑖𝑛}, which does not necessarily possess the shift property, ∇^𝑘+𝑖,∇^𝑙+𝑗𝑡^{𝑖1⋯𝑖𝑛}=𝑟^𝑖1_𝑖′^{𝑘+𝑖𝑙+𝑗}𝑡^{𝑖′𝑖2⋯𝑖𝑛}+𝑟^𝑖2_𝑖′^{𝑘+𝑖𝑙+𝑗}𝑡^{𝑖1𝑖′𝑖3⋯𝑖𝑛}+⋯,=𝑟^𝑖1_𝑖^+𝑖+𝑗_′^𝑘𝑙𝑡^{𝑖′𝑖2⋯𝑖𝑛}+𝑟^𝑖2_𝑖^+𝑖+𝑗_′^𝑘𝑙𝑡^{𝑖1𝑖′𝑖3⋯𝑖𝑛}+⋯, (E.13) where in the second step we have used the shift property (A.8) for the Riemann tensor. This shows that the right hand side of (E.11) is symmetric under 𝑖↔𝑗. However, the left hand side is manifestly antisymmetric. Therefore, it must vanish.
Let us now consider the linear term 𝐿^{𝐾(𝑖)𝐾(𝑗)}_𝜒𝜓. Following similar argument as above many of the terms can be shown to vanish by using integration by parts and the shift property of 𝜔_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛𝑘} and its covariant derivatives. The terms that are problematic are as follows: 𝐿^{𝐾(𝑖)𝐾(𝑗)}_𝜒𝜓=𝑑𝑤∇^𝑘+𝑖𝜒^{𝑖1⋯𝑖𝑛}∇^𝑙+𝑗𝜔_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛𝑙}∇_𝑘𝜓^{𝑗1⋯𝑗𝑛}+2∇^𝑘+𝑖𝜒^{𝑖1⋯𝑖𝑛}𝜔_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛𝑙}∇_𝑘∇^𝑙+𝑗𝜓^{𝑗1⋯𝑗𝑛}−∇_𝑙𝜒^{𝑖1⋯𝑖𝑛}∇^𝑘+𝑖𝜔_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛𝑘}∇^𝑙+𝑗𝜓^{𝑗1⋯𝑗𝑛}−2∇_𝑙∇^𝑘+𝑖𝜒^{𝑖1⋅𝑖𝑛}𝜔_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛𝑘}∇^𝑙+𝑗𝜓^{𝑗1⋯𝑗𝑛}−𝑖⟷𝑗. (E.14) Notice that each of the above terms is such that the shift 𝑖 cannot be moved to the position of the shift 𝑗 and vice versa without using the shift property of the wavefunctions. Therefore, such terms are not individually symmetric under 𝑖↔𝑗. However, by using integrations by parts the above terms can be rearranged to give the following form: 𝐿^{𝐾(𝑖)𝐾(𝑗)}_𝜒𝜓=𝑑𝑤−∇^𝑙+𝑗∇^𝑘+𝑖𝜒^{𝑖1⋯𝑖𝑛}𝜔_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛𝑙}∇_𝑘+𝑘⟷𝑙𝜓^{𝑗1⋯𝑗𝑛}+∇_𝑘𝜒^{𝑖1⋯𝑖𝑛}𝜔_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛𝑙}+𝑘⟷𝑙∇^𝑘+𝑖∇^𝑙+𝑗𝜓^{𝑗1⋯𝑗𝑛}+∇^𝑘+𝑖,∇^𝑙+𝑗𝜒^{𝑖1⋯𝑖𝑛}𝜔_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛𝑘}∇_𝑙𝜓^{𝑗1⋯𝑗𝑛}+∇_𝑘𝜒^{𝑖1⋅𝑖𝑛}𝜔_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛𝑙}∇^𝑘+𝑖,∇^𝑙+𝑗𝜓^{𝑗1⋯𝑗𝑛}−𝑖⟷𝑗. (E.15) The last two terms are symmetric in 𝑖 and 𝑗 because of the reason discussed earlier. Therefore, they drop out in the final result when we antisymmetrize. Each of the first two terms has a factor symmetric under 𝑘↔𝑙. Because of this such a term, combined with the counterpart with 𝑖 and 𝑗 interchanged, finally contains the commutator of the shifted covariant derivatives [∇^𝑘+𝑖,∇^𝑙+𝑗]. Therefore these terms also vanish following the previous argument. Similar arguments can be used to show that all the other relevant terms vanish leading to the result (E.9).

𝑇^{𝑍(𝑖)𝑍(𝑗)}
Using (E.2) and manipulating various terms we first write the contributions to _𝜒⟨𝑍^𝐿_(𝑖)𝑍^𝐿_(𝑗)⟩_𝜓 in the following forms: 𝑂^{𝑍𝐿(𝑖)𝑍𝐿(𝑗)}_𝜒𝜓=𝑑𝑤∇_𝑙∇_𝑘𝜒^{𝑖1⋯𝑖𝑛}𝑎^𝑘+𝑖𝑎^𝑙+𝑗𝜓_{𝑖1⋯𝑖𝑛}+∇_𝑘𝜒^{𝑖1⋯𝑖𝑛}∇_𝑙𝑎^𝑘+𝑖𝑎^𝑙+𝑗𝜓_{𝑖1⋯𝑖𝑛},𝐿^{𝑍𝐿(𝑖)𝑍𝐿(𝑗)}_𝜒𝜓=𝑑𝑤∇_𝑘𝜒^{𝑖1⋯𝑖𝑛}𝑎^𝑘+𝑖𝜔_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛𝑙}𝑎^𝑙+𝑗𝜓^{𝑗1⋯𝑗𝑛}++𝑖⟷𝑗𝜒^{𝑖1⋯𝑖𝑛}∇_𝑙𝜔_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛𝑘}𝑎^𝑘+𝑖𝑎^𝑙+𝑗𝜓^{𝑗1⋯𝑗𝑛}+𝜒^{𝑖1⋯𝑖𝑛}𝜔_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛𝑘}∇_𝑙𝑎^𝑘+𝑖𝑎^𝑙+𝑗𝜓^{𝑗1⋯𝑗𝑛},𝑄^{𝑍𝐿(𝑖)𝑍𝐿(𝑗)}_𝜒𝜓=𝑑𝑤𝜒^{𝑖1⋯𝑖𝑛}𝜔_{𝑖1⋯𝑖𝑛𝑘1⋯𝑘𝑛𝑘}𝜔^{𝑘1⋯𝑘𝑛}_{𝑗1⋯𝑗𝑛𝑙}𝑎^𝑘+𝑖𝑎^𝑙+𝑗𝜓^{𝑗1⋯𝑗𝑛}. (E.16)(E.17)(E.18) Antisymmetrizing the above results in 𝑖 and 𝑗 leads to the following contributions to the commutator: 𝑂^{𝑍𝐿(𝑖)𝑍𝐿(𝑗)}_𝜒𝜓=𝑑𝑤∇_𝑙,∇_𝑘𝜒^{𝑖1⋯𝑖𝑛}𝑎^𝑘+𝑖𝑎^𝑙+𝑗𝜓_{𝑖1⋯𝑖𝑛}+𝑖_𝑘(𝑘+𝑖)∇_𝑘𝜒^{𝑖1⋯𝑖𝑛}𝑎^{𝑘+𝑖+𝑗}𝜓_{𝑖1⋯𝑖𝑛},=−𝑖⟷𝑗𝑑𝑤𝜒^{𝑖1⋯𝑖𝑛}𝑟_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛𝑘𝑙}𝑎^𝑘+𝑖𝑎^𝑙+𝑗𝜓^{𝑗1⋯𝑗𝑛}+𝑖𝑖−𝑗∇_𝑘𝜒^{𝑖1⋯𝑖𝑛}𝑎^{𝑘+𝑖+𝑗}𝜓_{𝑖1⋯𝑖𝑛},𝐿^{𝑍𝐿(𝑖)𝑍𝐿(𝑗)}_𝜒𝜓=𝑑𝑤𝜒^{𝑖1⋯𝑖𝑛}∇_𝑙𝜔_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛𝑘}−∇_𝑘𝜔_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛𝑙}𝑎^𝑘+𝑖𝑎^𝑙+𝑗𝜓^{𝑗1⋯𝑗𝑛}+𝑖_𝑘(𝑘+𝑖)𝜒^{𝑖1⋯𝑖𝑛}𝜔_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛𝑘}𝑎^{𝑘+𝑖+𝑗}𝜓^{𝑗1⋯𝑗𝑛},=−−𝑖⟷𝑗𝑑𝑤𝜒^{𝑖1⋯𝑖𝑛}𝑟_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛𝑘𝑙}+𝑑_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛𝑘𝑙}𝑎^𝑘+𝑖𝑎^𝑙+𝑗𝜓^{𝑗1⋯𝑗𝑛}+𝑖𝑖−𝑗𝜒^{𝑖1⋯𝑖𝑛}𝜔_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛𝑘}𝑎^{𝑘+𝑖+𝑗}𝜓^{𝑗1⋯𝑗𝑛},𝑄^{𝑍𝐿(𝑖)𝑍𝐿(𝑗)}_𝜒𝜓=𝑑𝑤𝜒^{𝑖1⋯𝑖𝑛}𝜔_{𝑖1⋯𝑖𝑛𝑘1⋯𝑘𝑛𝑘}𝜔^{𝑘1⋯𝑘𝑛}_{𝑗1⋯𝑗𝑛𝑙}−𝑙⟷𝑘𝑎^𝑘+𝑖𝑎^𝑙+𝑗𝜓^{𝑗1⋯𝑗𝑛},=𝑑𝑤𝜒^{𝑖1⋯𝑖𝑛}𝑑_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛𝑘𝑙}𝑎^𝑘+𝑖𝑎^𝑙+𝑗𝜓^{𝑗1⋯𝑗𝑛}. (E.19)(E.20)(E.21)
In the first steps of (E.19) and (E.20), we have evaluated the covariant derivative ∇_𝑙𝑎^𝑘+𝑖 appearing in (E.16) using (A.9). The connection terms coming from these covariant derivatives do not contribute because of the antisymmetrization in 𝑖 and 𝑗. The first terms in (E.19) and (E.20) and the term in (E.21) are evaluated using (C.19), (C.22), and (C.28), respectively. Each of these terms vanishes individually because of the shift property. However, an interesting feature of these results is the following. Even if we do not use the shift property, such terms cancel each other when we add all the above contributions to get the relevant commutator. We will see as we proceed that this sort of cancellations will be very crucial in computing certain other terms of the DWV algebra. Using (3.6) the final result is written in the following form: 𝑇^{𝑍𝐿(𝑖)𝑍𝐿(𝑗)}=−𝑖−𝑗_𝜒𝑍^𝐿_{𝑖+𝑗𝜓}. (E.22) A similar calculation shows, 𝑇^{𝑍𝑅(𝑖)𝑍𝑅(𝑗)}=−𝑖−𝑗_𝜒𝑍^𝑅_{𝑖+𝑗𝜓}. (E.23) To calculate the last term in (E.6), we manipulate the terms in _𝜒⟨𝑍^𝐿_(𝑖)𝑍^𝑅_(𝑗)⟩_𝜓 and _𝜒⟨𝑍^𝑅_(𝑗)𝑍^𝐿_(𝑖)⟩_𝜓 to arrive at the following results for the terms in 𝑇^{𝑍𝐿(𝑖)𝑍𝑅(𝑗)} at various orders: 𝑂^{𝑍𝐿(𝑖)𝑍𝑅(𝑗)}_𝜒𝜓=𝑑𝑤𝜒^{𝑖1⋯𝑖𝑛}𝑎^𝑙+𝑗∇_𝑘,∇_𝑙𝑎^𝑘+𝑖𝜓_{𝑖1⋯𝑖𝑛}+∇_𝑘𝜒^{𝑖1⋯𝑖𝑛}𝑎^𝑙+𝑗∇_𝑙𝑎^𝑘+𝑖𝜓_{𝑖1⋯𝑖𝑛}+𝜒^{𝑖1⋯𝑖𝑛}∇_𝑘𝑎^𝑙+𝑗𝑎^𝑘+𝑖∇_𝑙𝜓_{𝑖1⋯𝑖𝑛}+𝜒^{𝑖1⋯𝑖𝑛}∇_𝑘𝑎^𝑙+𝑗∇_𝑙𝑎^𝑘+𝑖𝜓_{𝑖1⋯𝑖𝑛},=𝑑𝑤𝜒^{𝑖1⋯𝑖𝑛}𝑎^𝑘+𝑖𝑟_𝑘𝑙𝑎^𝑙+𝑗𝜓_{𝑖1⋯𝑖𝑛}+𝜒^{𝑖1⋯𝑖𝑛}𝑎^𝑘+𝑖𝑟_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛𝑘𝑙}𝑎^𝑙+𝑗𝜓^{𝑗1⋯𝑗𝑛}+𝜒^{𝑖1⋯𝑖𝑛}∇_𝑘𝑎^𝑙+𝑗∇_𝑙𝑎^𝑘+𝑖𝜓_{𝑖1⋯𝑖𝑛}+∇_𝑘𝜒^{𝑖1⋯𝑖𝑛}𝑎^𝑙+𝑗∇_𝑙𝑎^𝑘+𝑖𝜓_{𝑖1⋯𝑖𝑛}+𝜒^{𝑖1⋯𝑖𝑛}∇_𝑘𝑎^𝑙+𝑗𝑎^𝑘+𝑖∇_𝑙𝜓_{𝑖1⋯𝑖𝑛},𝐿^{𝑍𝐿(𝑖)𝑍𝑅(𝑗)}_𝜒𝜓=−𝑑𝑤𝜒^{𝑖1⋯𝑖𝑛}𝑎^𝑘+𝑖𝑎^𝑙+𝑗∇_𝑘𝜔_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛𝑙}+𝑎−𝑘⟷𝑙_𝑘+𝑖∇_𝑘𝑎^𝑙+𝑗𝜔_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛𝑙}−𝑖⟷𝑗𝜓^{𝑗1⋯𝑗𝑛},=𝑑𝑤𝜒^{𝑖1⋯𝑖𝑛}−𝑎^𝑘+𝑖𝑎^𝑙+𝑗𝑟_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛𝑘𝑙}+𝑑_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛𝑘𝑙}+𝑖𝑖−𝑗𝑎^{𝑘+𝑖+𝑗}𝜔_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛𝑘}𝜓^{𝑗1⋯𝑗𝑛},𝑄^{𝑍𝐿(𝑖)𝑍𝑅(𝑗)}_𝜒𝜓=𝑑𝑤𝜒^{𝑖1⋯𝑖𝑛}𝜔_{𝑖1⋯𝑖𝑛𝑘1…𝑘𝑛𝑘}𝜔^{𝑘1⋯𝑘𝑛}_{𝑗1⋯𝑗𝑛𝑙}−𝑘⟷𝑙𝑎^𝑘+𝑖𝑎^𝑙+𝑗𝜓^{𝑗1⋯𝑗𝑛},=𝑑𝑤𝜒^{𝑖1⋯𝑖𝑛}𝑑_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛𝑘𝑙}𝑎^𝑘+𝑖𝑎^𝑙+𝑗𝜓^{𝑗1⋯𝑗𝑛}. (E.24) In this case also the terms involving the Riemann tensor and 𝑑_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛𝑘𝑙} vanish individually due to the shift property. They also cancel each other when we add all the above contributions. Finally, the relevant commutator is given by 𝑇^{𝑍𝐿(𝑖)𝑍𝑅(𝑗)}=(−𝑖)²𝑑𝑤𝜒^{𝑖1⋯𝑖𝑛}𝑎^𝑘+𝑖𝑟_𝑘𝑙𝑎^𝑙+𝑗𝜓_{𝑖1⋯𝑖𝑛}+𝜒^{𝑖1⋯𝑖𝑛}∇_𝑘𝑎^𝑙+𝑗∇_𝑙𝑎^𝑘+𝑖𝜓_{𝑖1⋯𝑖𝑛}+∇_𝑘𝜒^{𝑖1⋯𝑖𝑛}𝑎^𝑙+𝑗∇_𝑙𝑎^𝑘+𝑖𝜓_{𝑖1⋯𝑖𝑛}+𝜒^{𝑖1⋯𝑖𝑛}∇_𝑘𝑎^𝑙+𝑗𝑎^𝑘+𝑖∇_𝑙𝜓_{𝑖1⋯𝑖𝑛}+𝑖𝑖−𝑗𝜒^{𝑖1⋯𝑖𝑛}𝑎^{𝑘+𝑖+𝑗}𝜔_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛𝑘}𝜓^{𝑗1⋯𝑗𝑛}. (E.25) As indicated in (E.6), we need to antisymmetrize the above quantity in 𝑖 and 𝑗. The first two terms drop out as they are symmetric. The final result is given by 𝑇^{𝑍𝐿(𝑖)𝑍𝑅(𝑗)}−𝑖⟷𝑗=−𝑖−𝑗_𝜒𝑍_{(𝑖+𝑗)𝜓}. (E.26) Substituting (E.22), (E.23) and (E.25) into (E.6) one gets 𝑇^{𝑍(𝑖)𝑍(𝑗)}=−2𝑖−𝑗_𝜒𝑍_{(𝑖+𝑗)𝜓}. (E.27)

𝑇^{𝐾(𝑖)𝑍(𝑗)}
This is the most important contribution as the anomaly term arises here. After a somewhat lengthy calculation we arrive at the following expressions: 𝑂^{𝐾(𝑖)𝑍𝐿(𝑗)}_𝜒𝜓=𝑑𝑤∇^𝑘+𝑖𝜒^{𝑖1⋯𝑖𝑛}𝑟_𝑘𝑙𝑎^𝑙+𝑗𝜓_{𝑖1⋯𝑖𝑛}+∇^𝑙+𝑖∇_𝑘𝜒^{𝑖1⋯𝑖𝑛}∇_𝑙𝑎^𝑘+𝑗𝜓_{𝑖1⋯𝑖𝑛}−∇_𝑘𝜒^{𝑖1⋯𝑖𝑛}∇^𝑙+𝑖𝑎^𝑘+𝑗∇_𝑙𝜓_{𝑖1⋯𝑖𝑛}−∇^𝑘+𝑖𝜒^{𝑖1⋯𝑖𝑛}𝑟_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛𝑘𝑙}𝑎^𝑙+𝑗𝜓^{𝑗1⋯𝑗𝑛}++𝜒⟷𝜓𝜒^{𝑖1⋯𝑖𝑛}𝑟_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛𝑘𝑙}∇^𝑘+𝑖𝑎^𝑙+𝑗𝜓^{𝑗1⋯𝑗𝑛},𝐿^{𝐾(𝑖)𝑍𝐿(𝑗)}_𝜒𝜓=−𝑑𝑤𝜒^{𝑖1⋯𝑖𝑛}𝜔_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛𝑙}∇_𝑘𝜓^{𝑗1⋯𝑗𝑛}+𝜒⟷𝜓∇^𝑘+𝑖𝑎^𝑙+𝑗−∇+𝑘⟷𝑙_𝑘+𝑖𝜒^{𝑖1⋯𝑖𝑛}𝑟_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛𝑘𝑙}+𝑑_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛𝑘𝑙}𝑎^𝑙+𝑗𝜓^{𝑗1⋯𝑗𝑛}+𝜒⟷𝜓+2𝜒^{𝑖1⋯𝑖𝑛}𝑟_{𝑖1⋯𝑖𝑛𝑘1⋯𝑘𝑛𝑘𝑙}𝜔^{𝑘1⋯𝑘𝑛}_{𝑗1⋯𝑗𝑛}^𝑘+𝑖𝑎^𝑙+𝑗𝜓^{𝑗1⋯𝑗𝑛}+𝜒^{𝑖1⋯𝑖𝑛}∇_𝑙𝜔_𝑖^𝑘+𝑖_{1⋯𝑖𝑛𝑗1⋯𝑗𝑛}∇_𝑘𝑎^𝑙+𝑗−∇_𝑙𝜔_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛𝑘}∇^𝑘+𝑖𝑎^𝑙+𝑗−∇_𝑘,∇_𝑙𝜔_𝑖^𝑘+𝑖_{1⋯𝑖𝑛𝑗1⋯𝑗𝑛}𝑎^𝑙+𝑗𝜓^{𝑗1⋯𝑗𝑛},𝑄^{𝐾(𝑖)𝑍𝐿(𝑗)}_𝜒𝜓=𝑑𝑤−∇^𝑘+𝑖𝜒^{𝑖1⋯𝑖𝑛}𝑑_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛𝑘𝑙}𝑎^𝑙+𝑗𝜓^{𝑗1⋯𝑗𝑛}+𝜒^{𝑖1⋯𝑖𝑛}𝑟_{𝑖1⋯𝑖𝑛𝑘1⋯𝑘𝑛𝑘𝑙}+𝑑_{𝑖1⋯𝑖𝑛𝑘1⋯𝑘𝑛𝑘𝑙}𝜔^{𝑘1⋯𝑘𝑛}_{𝑗1⋯𝑗𝑛}^𝑘+𝑖𝑎^𝑙+𝑗𝜓^{𝑗1⋯𝑗𝑛}+𝜒^{𝑖1⋯𝑖𝑛}𝜔_{𝑖1⋯𝑖𝑛𝑘1⋯𝑘𝑛𝑘}𝜔^{𝑘1⋯𝑘𝑛}_{𝑗1⋯𝑗𝑛𝑙}∇^𝑘+𝑖𝑎^𝑙+𝑗𝜓^{𝑗1⋯𝑗𝑛}+𝜒⟷𝜓,𝐶^{𝐾(𝑖)𝑍𝐿(𝑗)}_𝜒𝜓=−𝑑𝑤𝜒^{𝑖1⋯𝑖𝑛}𝜔_{𝑖1⋯𝑖𝑛𝑘1⋯𝑘𝑛𝑘}𝜔^{𝑘1⋯𝑘𝑛}_{𝑙1⋯𝑙𝑛}^𝑘+𝑖𝜔^{𝑙1⋯𝑙𝑛}_{𝑗1⋯𝑗𝑛𝑙}𝑎^𝑙+𝑗𝜓^{𝑗1⋯𝑗𝑛}+𝜒⟷𝜓,=−𝑑𝑤𝜒^{𝑖1⋯𝑖𝑛}𝑑_{𝑖1⋯𝑖𝑛𝑘1⋯𝑘𝑛𝑘𝑙}𝜔^{𝑘1⋯𝑘𝑛}_{𝑗1⋯𝑗𝑛}^𝑘+𝑖𝑎^𝑙+𝑗𝜓^{𝑗1⋯𝑗𝑛}+𝜒⟷𝜓. (E.28)(E.29)(E.30)(E.31) To get to the second step of (E.31) we have used (C.28) twice to push 𝜔^{𝑙1⋯𝑙𝑛}_{𝑗1⋯𝑗𝑛𝑙} to the left. In order to derive the expected result for 𝑇^{𝐾(𝑖)𝑍(𝑗)} we need cancellation of many terms to take place. This happens in two steps—some terms get canceled when we add all the above contributions to find 𝑇^{𝐾(𝑖)𝑍𝐿(𝑗)}. Another set of cancellations takes place when we add the latter to 𝑇^{𝐾(𝑖)𝑍𝑅(𝑗)} to get the final result for 𝑇^{𝐾(𝑖)𝑍(𝑗)}. To arrive at our final results, however, it will be easier for us to go in the reverse order. One can be easily convinced that various contributions to 𝑇^{𝐾(𝑖)𝑍𝑅(𝑗)} can be obtained by interchanging 𝜒 and 𝜓 in the expressions for the corresponding contributions to 𝑇^{𝐾(𝑖)𝑍𝐿(𝑗)}. Therefore to get the terms in 𝑇^{𝐾(𝑖)𝑍(𝑗)} at various orders we simply symmetrize the expressions in (E.28), (E.29), (E.30), and (E.31) under 𝜒↔𝜓, 𝑂^{𝐾(𝑖)𝑍(𝑗)}_𝜒𝜓=𝑑𝑤∇^𝑘+𝑖𝜒^{𝑖1⋯𝑖𝑛}𝑟_𝑘𝑙𝑎^𝑙+𝑗𝜓_{𝑖1⋯𝑖𝑛}+∇^𝑙+𝑖∇_𝑘𝜒^{𝑖1⋯𝑖𝑛}∇_𝑙𝑎^𝑘+𝑗𝜓_{𝑖1⋯𝑖𝑛}−∇_𝑘𝜒^{𝑖1⋯𝑖𝑛}∇^𝑙+𝑖𝑎^𝑘+𝑗∇_𝑙𝜓_{𝑖1⋯𝑖𝑛}−2∇^𝑘+𝑖𝜒^{𝑖1⋯𝑖𝑛}𝑟_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛𝑘𝑙}𝑎^𝑙+𝑗𝜓^{𝑗1⋯𝑗𝑛}+𝜒⟷𝜓,𝐿^{𝐾(𝑖)𝑍(𝑗)}_𝜒𝜓=𝑑𝑤−2𝜒^{𝑖1⋯𝑖𝑛}𝜔_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛𝑙}∇_𝑘𝜓^{𝑗1⋯𝑗𝑛}∇^𝑘+𝑖𝑎^𝑙+𝑗+𝑘⟷𝑙+2∇^𝑘+𝑖𝜒^{𝑖1⋯𝑖𝑛}𝑟_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛𝑘𝑙}+𝑑_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛𝑘𝑙}𝑎^𝑙+𝑗𝜓^{𝑗1⋯𝑗𝑛}−2𝜒^{𝑖1⋯𝑖𝑛}𝑟_{𝑖1⋯𝑖𝑛𝑘1⋯𝑘𝑛𝑘𝑙}𝜔^{𝑘1⋯𝑘𝑛}_{𝑗1⋯𝑗𝑛}^𝑘+𝑖𝑎^𝑙+𝑗𝜓^{𝑗1⋯𝑗𝑛}+𝜒⟷𝜓,𝑄^{𝐾(𝑖)𝑍(𝑗)}_𝜒𝜓=2𝑄^{𝐾(𝑖)𝑍𝐿(𝑗)}_𝜒𝜓,𝐶^{𝐾(𝑖)𝑍(𝑗)}_𝜒𝜓=2𝐶^{𝐾(𝑖)𝑍𝐿(𝑗)}_𝜒𝜓. (E.32) Notice that in the process of symmetrization the last terms of (E.28) and (E.29) dropped out. We now add all the above contributions to get 𝑇^{𝐾(𝑖)𝑍(𝑗)}, 𝑇^{𝐾(𝑖)𝑍(𝑗)}=(−𝑖)³𝑑𝑤∇^𝑘+𝑖𝜒^{𝑖1⋯𝑖𝑛}𝑟_𝑘𝑙𝑎^𝑙+𝑗𝜓_{𝑖1⋯𝑖𝑛}+∇^𝑙+𝑖∇_𝑘𝜒^{𝑖1⋯𝑖𝑛}∇_𝑙𝑎^𝑘+𝑗𝜓_{𝑖1⋯𝑖𝑛}−∇_𝑘𝜒^{𝑖1⋯𝑖𝑛}∇^𝑙+𝑖𝑎^𝑘+𝑗∇_𝑙𝜓_{𝑖1⋯𝑖𝑛}−2𝜒^{𝑖1⋯𝑖𝑛}𝜔_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛𝑙}∇_𝑘𝜓^{𝑗1⋯𝑗𝑛}×∇^𝑘+𝑖𝑎^𝑙+𝑗+𝑘⟷𝑙+2𝜒^{𝑖1⋯𝑖𝑛}𝜔_{𝑖1⋯𝑖𝑛𝑘1⋯𝑘𝑛𝑘}𝜔^{𝑘1⋯𝑘𝑛}_{𝑗1⋯𝑗𝑛𝑙}∇^𝑘+𝑖𝑎^𝑙+𝑗𝜓^{𝑗1⋯𝑗𝑛}+,=()𝜒⟷𝜓−𝑖³𝑑𝑤∇^𝑘+𝑖𝜒^{𝑖1⋯𝑖𝑛}𝑟_𝑘𝑙𝑎^𝑙+𝑗𝜓_{𝑖1⋯𝑖𝑛}+𝜒^{𝑖1⋯𝑖𝑛}𝑎^𝑘+𝑗𝑟_𝑘𝑙∇^𝑙+𝑖𝜓_{𝑖1⋯𝑖𝑛}+∇^𝑙+𝑖∇_𝑘𝜒^{𝑖1⋯𝑖𝑛}∇_𝑙𝑎^𝑘+𝑗𝜓_{𝑖1⋯𝑖𝑛}+−∇𝜒⟷𝜓_𝑘𝜒^{𝑖1⋯𝑖𝑛}∇_𝑙𝜓_{𝑖1⋯𝑖𝑛}+2𝜒^{𝑖1⋯𝑖𝑛}𝜔_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛𝑙}∇_𝑘𝜓^{𝑗1⋯𝑗𝑛}−∇_𝑘𝜒^{𝑖1⋯𝑖𝑛}𝜔_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛𝑙}𝜓^{𝑗1⋯𝑗𝑛}−2𝜒^{𝑖1⋯𝑖𝑛}𝜔_{𝑖1⋯𝑖𝑛𝑘1⋯𝑘𝑛𝑘}𝜔^{𝑘1⋯𝑘𝑛}_{𝑗1⋯𝑗𝑛𝑙}𝜓^{𝑗1⋯𝑗𝑛}∇^𝑘+𝑖𝑎^𝑙+𝑗+𝑘⟷𝑙, (E.33) where in the last step we have simply rearranged the terms in a different way. We will now manipulate the terms in the last three lines in certain ways. Let us first consider the terms in the following second line: ∇^𝑙+𝑖∇_𝑘𝜒^{𝑖1⋯𝑖𝑛}∇_𝑙𝑎^𝑘+𝑗𝜓_{𝑖1⋯𝑖𝑛}=𝑖_𝑘𝑘+𝑗∇^{𝑘+𝑖+𝑗}∇_𝑘𝜒^{𝑖1⋯𝑖𝑛}𝜓_{𝑖1⋯𝑖𝑛}+𝑎^𝑙′𝛾_𝑙^𝑘+𝑗_′𝑙∇^𝑙+𝑖∇_𝑘𝜒^{𝑖1⋯𝑖𝑛}𝜓_{𝑖1⋯𝑖𝑛},=𝑖_𝑘𝑘+𝑗∇^{𝑘+𝑖+𝑗}∇_𝑘𝜒^{𝑖1⋯𝑖𝑛}𝜓_{𝑖1⋯𝑖𝑛}+12𝑎^𝑙′𝛾_𝑙^{𝑘+𝑖+𝑗}_′𝑘𝑔𝑘_𝑙∇_𝑙,∇_𝑘+∇_𝑙,∇_𝑘𝜒^{𝑖1⋯𝑖𝑛}𝜓_{𝑖1⋯𝑖𝑛},=𝑖_𝑘𝑘+𝑗∇^{𝑘+𝑖+𝑗}∇_𝑘𝜒^{𝑖1⋯𝑖𝑛}𝜓_{𝑖1⋯𝑖𝑛}+12𝑎^𝑙′𝛾_𝑙^{𝑘+𝑖+𝑗}_′𝑘𝑔𝑘_𝑙+𝛾_𝑙^{𝑙+𝑖+𝑗}_′𝑘𝑔𝑘_𝑘∇_𝑙∇_𝑘𝜒^{𝑖1⋯𝑖𝑛}𝜓_{𝑖1⋯𝑖𝑛}+12𝑎^𝑙′𝛾_𝑙^{𝑘+𝑖+𝑗}_′𝑘𝑔𝑘_𝑙𝜒^{𝑖1⋯𝑖𝑛}𝑟_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛𝑘𝑙}𝜓^{𝑗1⋯𝑗𝑛}. (E.34) Following [4, 5] we identify the term inside the round bracket in the second line as −𝜕_𝑙′𝑔^{𝑘+𝑖+𝑗} and then use the identity (A.10). This gives for the whole term in the second line, −𝑖2_𝑘,𝑙𝑘+𝑙+𝑖+𝑗𝑔^{𝑘+𝑖+𝑗𝑙}∇_𝑙∇_𝑘𝜒^{𝑖1⋯𝑖𝑛}𝜓_{𝑖1⋯𝑖𝑛}𝑖=−2𝑖+𝑗∇²_𝑖+𝑗𝜒^{𝑖1⋯𝑖𝑛}𝜓_{𝑖1⋯𝑖𝑛}−𝑖2_𝑘(𝑘)∇^{𝑘+𝑖+𝑗}∇_𝑘+∇_𝑘∇^{𝑘+𝑖+𝑗}𝜒^{𝑖1⋯𝑖𝑛}𝜓_{𝑖1⋯𝑖𝑛}. (E.35) The sum over 𝑘 in the second term is not a tensor contraction because of the presence of the factor (𝑘). For the same reason the two terms inside the square bracket are not same. However, because of the antisymmetry property of 𝑟_{𝑖1⋯𝑖𝑛𝑗1⋯𝑗𝑛𝑘𝑙} in (C.24) such terms give the same result once we symmetrize between 𝜒 and 𝜓, ∇^{𝑘+𝑖+𝑗}∇_𝑘𝜒^{𝑖1⋯𝑖𝑛}𝜓_{𝑖1⋯𝑖𝑛}+𝜒⟷𝜓=∇_𝑘∇^{𝑘+𝑖+𝑗}𝜒^{𝑖1⋯𝑖𝑛}𝜓_{𝑖1⋯𝑖𝑛}+𝜒⟷𝜓, (E.36) where the index 𝑘 is not summed over. Because of the same reason the last term in (E.34) drops out when we symmetrize between 𝜒 and 𝜓. The final result for the second line in (E.33) is given by ∇^𝑙+𝑖∇_𝑘𝜒^{𝑖1⋯𝑖𝑛}∇_𝑙𝑎^𝑘+𝑗𝜓_{𝑖1⋯𝑖𝑛}+𝑖𝜒⟷𝜓=2𝑗−𝑖∇²_𝑖+𝑗𝜒^{𝑖1⋯𝑖𝑛}𝜓_{𝑖1⋯𝑖𝑛}+𝜒^{𝑖1⋯𝑖𝑛}∇²_𝑖+𝑗𝜓_{𝑖1⋯𝑖𝑛}. (E.37) To evaluate the terms in the last two lines of (E.33) we first derive ∇^𝑘+𝑖𝑎^𝑙+𝑗+𝑘⟷𝑙=𝑖𝑗−𝑖𝑔^{𝑘+𝑖+𝑗𝑙}, (E.38) using the same trick as described below (E.34). Using (E.37) and (E.38) in (E.33) one finally finds 𝑇^{𝐾(𝑖)𝑍(𝑗)}=−2𝑖−𝑗_𝜒𝐾_{(𝑖+𝑗)𝜓}+_𝜒𝜋^⋆𝑘+𝑖𝑟_𝑘𝑙(𝑥)𝑎^𝑙+𝑗(𝑥)−𝑎^𝑘+𝑗(𝑥)𝑟_𝑘𝑙(𝑥)𝜋^𝑙+𝑖_𝜓, (E.39) where we have used (3.5), (C.13).

The Rest
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 [4, 5]. The relevant results are as follows: 𝑇^{𝐾(𝑖)𝑉(𝑗)}−𝑖⟷𝑗=−2𝑖−𝑗_𝜒𝑍_{(𝑖+𝑗)𝜓},𝑇^{𝑍𝐿(𝑖)𝑉(𝑗)}=𝑇^{𝑍𝑅(𝑖)𝑉(𝑗)}=−𝑖−𝑗_𝜒𝑉_{(𝑖+𝑗)𝜓}. (E.40) Finally, using the results (E.9), (E.27), (E.39), (E.40) in (E.6), and (E.4) suitably one finds the algebra as shown in (3.8), (3.9).

Acknowledgment

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