Infinite-Dimensional Symmetries of Two-Dimensional Coset Models Coupled to Gravity

We take as our starting point a symmetric-space non-linear sigma model defined on the coset manifold ${\bf K}={\bf G}/{\bf H}$, and coupled to gravity in three spacetime dimensions. The commutation relations for the corresponding generators of the Lie algebra ${\cal G}$ take the form

$$[{\cal H},{\cal H}]={\cal H}\,,\qquad[{\cal H},{\cal K}]={\cal K}\,,\qquad[{\cal K},{\cal K}]={\cal H}\,.$$

(2.1)

The condition that ${\bf K}$ is a symmetric space is reflected in the absence of ${\cal K}$ generators on the right-hand side of the last commutation relation. The symmetric-space algebra implies that there is an involution $\sharp$ under which

$${\cal K}^{\sharp}={\cal K}\,,\qquad{\cal H}^{\sharp}=-{\cal H}\,.$$

(2.2)

In many cases, such as when ${\bf G}=SL(n,{\mathbbm{R}})/O(n)$, the involution map is given by Hermitean conjugation,

$${\cal K}^{\dagger}={\cal K}\,,\qquad{\cal H}^{\dagger}=-{\cal H}\,.$$

(2.3)

In some cases, such as ${\bf G}=E_{(8,8)}$, ${\cal H}=O(16)$, the involution $\sharp$ is more involved.

The fields of the sigma model ${\bf G}/{\bf H}$ may be parameterised by a coset representative ${\cal V}$, in terms of which we may define

$$M={\cal V}^{\sharp}\,{\cal V}\,,\qquad A=M^{-1}dM\,.$$

(2.4)

Under transformations

$${\cal V}\longrightarrow h{\cal V}g\,,$$

(2.5)

where $g$ is a global element in the group ${\bf G}$ and $h$ is a local element in the denominator subgroup ${\bf H}$, we have shall have

$$M\longrightarrow g^{\sharp}Mg\,,\qquad A\longrightarrow g^{-1}Ag\,,$$

(2.6)

since it follows from ${\cal H}^{\sharp}=-{\cal H}$ that $h^{\sharp}=h^{-1}$. Henceforth, we shall consider for simplicity cases where the involution $\sharp$ is just Hermitean conjugation. For the general case, all occurrences of $\dagger$ should be replaced by $\sharp$.

The Cartan-Maurer equation $d(M^{-1}dM)=-(M^{-1}dM)\wedge(M^{-1}dM)$ implies that the field strength for $A$ vanishes:

$$F\equiv dA+A\wedge A=0\,.$$

(2.7)

The Lagrangian for the three-dimensional model may be written as

$${\cal L}_{3}=\sqrt{-\hat{g}}\,\big{(}\hat{R}-{\textstyle{\frac{\scriptstyle 1}{\scriptstyle 4}}}\hat{g}^{MN}\,{\rm tr}(A_{M}A_{N})\big{)}\,,$$

(2.8)

where $\hat{g}_{MN}$ is the three-dimensional spacetime metric tensor. The sigma-model equations of motion are therefore given by

$$\hat{\nabla}^{M}A_{M}=0\,.$$

(2.9)

We now assume that the metric and sigma-model fields are all independent of one of the coordinates, which we shall denote by $z$. We may reduce the metric according to the Kaluza-Klein ansatz

$$d\hat{s}_{3}^{2}=e^{\psi}ds_{2}^{2}+\rho^{2}\,dz^{2}\,.$$

(2.10)

Note that the two-dimensional field $\psi$ is redundant, in the sense that it could be absorbed into a conformal rescaling of the two-dimensional metric $ds_{2}^{2}$. However, it it useful to retain it since it will allow us later to take $ds_{2}^{2}$ to be just the Minkowski metric. It is straightforward to see that the three-dimensional Lagrangian (2.8) reduces to give

$${\cal L}_{2}=\sqrt{-g}\,\rho\,\big{(}R-{\textstyle{\frac{\scriptstyle 1}{\scriptstyle 4}}}g^{\mu\nu}{\rm tr}(A_{\mu}A_{\nu})+\rho^{-1}\,g^{\mu\nu}{\partial}_{\mu}\rho\,{\partial}_{\nu}\psi\big{)}\,,$$

(2.11)

(after an integration by parts).

The equations of motion that follow from varying ${\cal V}$, $\psi$  $\rho$ and $g_{\mu\nu}$ are, respectively,

	$\displaystyle\nabla^{\mu}(\rho\,A_{\mu})$	$\displaystyle=$	$\displaystyle 0\,,$		(2.12)
	$\displaystyle\square\rho$	$\displaystyle=$	$\displaystyle 0\,,$		(2.13)
	$\displaystyle\square\psi$	$\displaystyle=$	$\displaystyle R-{\textstyle{\frac{\scriptstyle 1}{\scriptstyle 4}}}g^{\mu\nu}\,{\rm tr}(A_{\mu}A_{\nu})\,,$		(2.14)
	$\displaystyle\ 0$	$\displaystyle=$	$\displaystyle{\partial}_{\mu}\rho\,{\partial}_{\nu}\psi+{\partial}_{\nu}\rho\,{\partial}_{\mu}\psi-g_{\mu\nu}{\partial}_{\sigma}\rho{\partial}^{\sigma}\psi-2\nabla_{\mu}\nabla_{\nu}\rho$		(2.15)
			$\displaystyle-{\textstyle{\frac{\scriptstyle 1}{\scriptstyle 2}}}{\rm tr}\big{(}A_{\mu}A_{\nu}-{\textstyle{\frac{\scriptstyle 1}{\scriptstyle 2}}}g_{\mu\nu}{\rm tr}(A_{\sigma}A^{\sigma})\big{)}\,,$

(after using (2.13) to simplify (2.15).

Using general coordinate transformations, any two-dimensional metric can be written, locally, as a conformal factor times the Minkowski metric. Thus we may now take $ds_{2}^{2}$ to be the Minkowski metric, with $e^{\psi}$ as the required conformal factor. It is convenient, furthermore, to introduce light-cone coordinates $x^{\pm}$, so that we have $ds_{2}^{2}=2dx^{+}dx^{-}$. Note that the $++$ and $--$ components of the Einstein equation (2.15) can be used to solve for $\psi$, since it gives

$$({\partial}_{+}\rho)\,{\partial}_{+}\psi={\partial}_{+}^{2}\rho+{\textstyle{\frac{\scriptstyle 1}{\scriptstyle 4}}}{\rm tr}(A_{+}^{2})\,,\qquad({\partial}_{-}\rho)\,{\partial}_{-}\psi={\partial}_{-}^{2}\rho+{\textstyle{\frac{\scriptstyle 1}{\scriptstyle 4}}}{\rm tr}(A_{-}^{2})\,.$$

(2.16)

Equation (2.14) can now be seen to be a consequence of (2.12), (2.13), (2.16) and (2.7), so the two-dimensional equations reduce to solving

	$\displaystyle d(\rho{*A})$	$\displaystyle=$	$\displaystyle 0\,,$		(2.17)
	$\displaystyle\qquad dA+A\wedge A$	$\displaystyle=$	$\displaystyle 0\,,$		(2.18)
	$\displaystyle\square\rho$	$\displaystyle=$	$\displaystyle 0\,,$		(2.19)

with $\psi$ then being found using (2.16).

The use of the Lax equation in this context was discussed in [7], and in a somewhat different, but related way, in [18]. We shall begin by following Schwarz’s discussion of the Lax-pair formulation in [21], except that we prefer to use differential forms where possible, rather than light-cone coordinates. The equations (2.17) and (2.18) can both be derived from the integrability condition for a solution $X$ of the Lax equation

$$\tau(d+A)X={*dX}\,.$$

(2.20)

By taking the appropriate linear combination of this and its dual, we obtain¹¹1Note that $*^{2}=+1$ when acting on 1-forms in signature $(1,1)$ spacetimes. A summary of some further useful properties of forms in two dimensions can be found in [6].

$$dXX^{-1}=\frac{\tau}{1-\tau^{2}}\,{*A}+\frac{\tau^{2}}{1-\tau^{2}}\,A\,.$$

(2.21)

Unlike the flat-space case where the spectral parameter $\tau$ is a constant, here it must be allowed to have a specific dependence on the two-dimensional spacetime coordinates. It is convenient in what follows to parameterise $\tau$ in terms of $\theta$ according to

$$\tau=\tanh{\textstyle{\frac{\scriptstyle 1}{\scriptstyle 2}}}\theta\,,$$

(2.22)

which implies we shall also have

$$s\equiv\sinh\theta=\frac{2\tau}{1-\tau^{2}}\,,\qquad c\equiv\cosh\theta=\frac{1+\tau^{2}}{1-\tau^{2}}\,.$$

(2.23)

In fact, one finds that the $x^{\mu}$ dependence of $\tau$ should occur through the function $\rho(x^{\mu})$, as follows:

$$d\tau=\frac{\tau}{\rho}\,(cd\rho+s{*d}\rho)\,.$$

(2.24)

Equation (2.22) then implies

$$d\theta=\frac{s}{\rho}\,(cd\rho+s{*d}\rho)\,.$$

(2.25)

With these preliminaries, it is now an elementary calculation to see that from (2.21) that the Cartan-Maurer equation $d(dXX^{-1})=(dXX^{-1})\wedge(dXX^{-1})$ implies

$$\tau\,(dA+A\wedge A)+\frac{1}{\rho}\,d(\rho{*A})=0\,,$$

(2.26)

from which the Bianchi identity (2.18) and the equation of motion (2.17) indeed follow.

A simple calculation also shows that the integrability condition that follows by taking the exterior derivative of (2.25) implies

$$d{*d}\rho=0\,.$$

(2.27)

This is indeed the correct equation of motion (2.19) for $\rho$. If we now introduce light-cone coordinates for the two-dimensional spacetime, and work in the gauge where the metric is flat, $ds^{2}=2dx^{+}dx^{-}$, then on any function $f$ we shall have

$$df={\partial}_{+}f\,dx^{+}+{\partial}_{-}f\,dx^{-}\,,\qquad{*d}f={\partial}_{+}f\,dx^{+}-{\partial}_{-}f\,dx^{-}\,.$$

(2.28)

Thus (2.27) becomes ${\partial}_{+}{\partial}_{-}\rho=0$, with the general solution

$$\rho=\rho_{+}(x^{+})+\rho_{-}(x^{-})\,.$$

(2.29)

The equation (2.25) that governs the $x^{\mu}$ dependence of $\theta$ becomes, using the light-cone coordinates,

$${\partial}_{+}\theta=\frac{1}{2\rho}\,(e^{2\theta}-1)\,{\partial}_{+}\rho\,,\qquad{\partial}_{-}\theta=\frac{1}{2\rho}\,(1-e^{-2\theta})\,{\partial}_{-}\rho\,.$$

(2.30)

These equations may be integrated to give

$$1-e^{-2\theta}=\rho\,f_{-}(x^{-})\,,\qquad e^{2\theta}-1=\rho\,f_{+}(x^{+})\,,$$

(2.31)

where $f_{\pm}$ are arbitrary functions of their respective arguments. Eliminating $\theta$ between the two expressions implies

$$\rho=\frac{1}{f_{-}(x^{-})}-\frac{1}{f_{+}(x^{+})}\,.$$

(2.32)

Comparing with (2.29), we can write

$$\frac{1}{f_{-}(x^{-})}=\rho_{-}(x^{-})+\frac{1}{2\lambda}\,,\qquad\frac{1}{f_{+}(x^{+})}=-\rho_{+}(x^{+})+\frac{1}{2\lambda}\,,$$

(2.33)

where $\lambda$ is a constant. From (2.29) and (2.31), the solution for $\theta$ is given by

$$e^{2\theta}=\frac{1+2\lambda\,\rho_{-}(x^{-})}{1-2\lambda\,\rho_{+}(x^{+})}\,.$$

(2.34)

It then follows from (2.22) that

	$\displaystyle\tau$	$\displaystyle=$	$\displaystyle\frac{1}{\lambda\rho}\,\Big{[}1-\lambda(\rho_{+}-\rho_{-})-\sqrt{(1+2\lambda\rho_{-})(1-2\lambda\rho_{+})}\Big{]}\,,$
		$\displaystyle=$	$\displaystyle{\textstyle{\frac{\scriptstyle 1}{\scriptstyle 2}}}\lambda\rho+{\textstyle{\frac{\scriptstyle 1}{\scriptstyle 2}}}\lambda^{2}\rho(\rho_{+}-\rho_{-})+{\textstyle{\frac{\scriptstyle 1}{\scriptstyle 8}}}\lambda^{3}\rho[\rho^{2}+4(\rho_{+}-\rho_{-})^{2}]+\cdots\,.$

Note that we have chosen the negative sign in front of the square root, so that $\tau$ is analytic in $\lambda$ in the neighbourhood of $\lambda=0$. Furthermore, $\tau=0$ when $\lambda=0$.

In what follows, the constant parameter $\lambda$ will play the rôle of the spectral parameter for the Lax equation. The solution $X$ of the Lax equation (2.20) depends on the spectral function $\tau$, and since $\tau$ is given by (2.2) it follows that we may characterise $X$ by the value of the arbitrary constant spectral parameter $\lambda$. Thus when we consider a solution $X$ of the Lax equation, we shall denote it by $X(\lambda)$. (Of course, $X$ also depends on the two-dimensional spacetime coordinates $x^{\mu}$, but we shall suppress the explicit indication of this dependence.) We shall choose the solution so that $X(0)=1$; this is clearly consistent with the Lax equation (2.21), since $\tau$ vanishes at $\lambda=0$.

The symmetries of the two-dimensional system are described by field transformations that preserve the equations of motion (2.17) and (2.19). The full set of symmetries will involve transformations of the coset representative ${\cal V}$ that are expressed using the solution $X$ of the Lax equation (2.20). It is therefore necessary also to find how $X$ itself transforms; this is determined by requiring in addition that the Lax equation (2.20) be left invariant.

We begin, following [20, 21], by constructing transformations of ${\cal V}$. These are of the form

$$\delta{\cal V}=w{\cal V}\eta+\delta h{\cal V}\,,\qquad\eta\equiv X(\lambda)\epsilon X(\lambda)^{-1}\,,$$

(3.1)

where $\epsilon$ an infinitesimal constant parameter taking values in the Lie algebra ${\cal G}$, and $\delta h$ is a $\lambda$-dependent and field-dependent compensating transformation in ${\cal H}$ that restores the original gauge. The quantity $w$ is a function that is a singlet under the Lie-algebra, and will be determined shortly. The meaning of (3.1) is as follows. The coset representative ${\cal V}$ itself is, of course, independent of the spectral parameter $\lambda$. The transformation $\delta$ is $\lambda$-dependent, and in fact we expand it as a power series in $\lambda$:

$$\delta(\epsilon,\lambda)=\sum_{n\geq 0}\lambda^{n}\,\delta_{{\scriptscriptstyle(}n)}(\epsilon)\,.$$

(3.2)

By equating coefficients of each power of $\lambda$ in the Taylor expansions of the two sides of equation (3.1), one therefore obtains a hierarchy of transformations $\delta_{{\scriptscriptstyle(}n)}(\epsilon){\cal V}$. One can take independent $\epsilon$ parameters for each $n$. The transformations at the $n=0$ order are just the original infinitesimal global ${\cal G}$ symmetries.

In the flat-space case discussed in [20, 6] $w$ is spacetime-independent, and one can take $w=1$, but in the gravity-coupled situation we are considering in this paper, one must choose a very specific $x^{\mu}$-dependence for $w$ in order to ensure that (3.1) describes a symmetry of the equation of motion $d(\rho{*A})=0$. The calculation is performed by first noting from (2.4) that (3.1) implies

$$\delta M=wM\eta+w\eta^{\dagger}M\,,\qquad\delta A=D(w\eta)+D(wM^{-1}\eta^{\dagger}M)\,,$$

(3.3)

where we define $Df\equiv df+[A,f]$ on any ${\cal G}$-valued function $f$. Note that $\rho$ will be inert under the Kac-Moody transformations.

Use of the Lax equation (2.20) shows that

$$D\eta=\frac{1}{\tau}\,{*d\eta}\,,\qquad D(M^{-1}\eta^{\dagger}M)=\tau\,{*d}(M^{-1}\eta^{\dagger}M)\,,$$

(3.4)

as in the flat-space case [6]. From this, it follows that

	$\displaystyle d(\rho{*\delta A})$	$\displaystyle=$	$\displaystyle\Big{[}d\Big{(}\frac{\rho w}{\tau}\Big{)}-\rho{*d}w\Big{]}\wedge d\eta+\big{[}d(\rho w\tau)-\rho{*d}w\big{]}\wedge d(M^{-1}\eta^{\dagger}M)$		(3.5)
			$\displaystyle+d(\rho{*d}w)\,\big{(}\eta+M^{-1}\eta^{\dagger}M\big{)}\,.$

and so $d(\rho{*\delta A})$ will vanish (and thus the equation of motion (2.17) will be preserved by the transformations) if [21]

$$\rho{*dw}=d\Big{(}\frac{\rho w}{\tau}\Big{)}\,,\qquad\hbox{and}\qquad\rho{*dw}=d(\rho w\tau)\,.$$

(3.6)

Using (2.25) one sees that this can be achieved, by taking $w$ to be a constant multiple of $s/\rho$. We shall take $w=s/(\lambda\rho)$, and thus the transformations (3.1) we shall consider are given by

$$\delta{\cal V}=\frac{s}{\lambda\rho}\,{\cal V}\eta+\delta h{\cal V}\,,\qquad\eta\equiv X(\lambda)\epsilon X(\lambda)^{-1}\,.$$

(3.7)

(The reason for choosing to divide out by the constant parameter $\lambda$ is that then, as can be seen from (2.23) and (2.2), the prefactor $w=s/(\lambda\rho)$ approaches 1 as $\lambda$ goes to zero.)

We now need to consider the transformations of $X$. These are determined by the requirement that the Lax equation must be preserved, with $A$ transforming as in (3.3). We shall need to know how $X(\lambda)$ with spectral parameter $\lambda_{2}$ transforms under variations $\delta(\epsilon,\lambda)$ with an independent choice of spectral parameter $\lambda_{1}$. We denote $X(\lambda_{2})$ by $X_{2}$, and $\delta(\epsilon_{1},\lambda_{1})$ by $\delta_{1}$. From (2.21) we therefore have

$$\delta_{1}(dX_{2}X_{2}^{-1})=\frac{\tau_{2}}{1-\tau_{2}^{2}}\,{*\delta_{1}A}+\frac{\tau_{2}^{2}}{1-\tau_{2}^{2}}\,\delta_{1}A\,.$$

(3.8)

($\tau_{2}$ means the solution for $\tau$ given in (2.2) with the spectral parameter $\lambda$ taken to be $\lambda_{2}$.)

Let us write the transformation of $X$ as $\delta_{1}X_{2}=UX_{2}$. It follows that $\delta_{1}(dX_{2}X_{2}^{-1})=dU+[U,dX_{2}X_{2}^{-1}]$, and so from (3.8) we have

$$dU+[U,dX_{2}X_{2}^{-1}]=\frac{\tau_{2}}{1-\tau_{2}^{2}}\,{*\delta_{1}A}+\frac{\tau_{2}^{2}}{1-\tau_{2}^{2}}\,\delta_{1}A\,.$$

(3.9)

We can write the solution for $U$ as the sum

$$U=U^{\rm{hom}}+U^{\rm{inhom}}\,,$$

(3.10)

where $U^{\rm{hom}}$ is the solution of the homogeneous equation with the right-hand side of (3.9) set to zero, while $U^{\rm{inhom}}$ solves the inhomogeneous equation with the variations of $A$ on the right-hand side providing the source.

It is easily seen that the homogeneous equation is solved by

$$U^{\rm{hom}}=X_{2}\epsilon_{1}X_{2}^{-1}\,.$$

(3.11)

For the inhomogeneous equation, we make the ansatz

$$U^{\rm{inhom}}=u\,\eta_{1}+v\,M^{-1}\eta_{1}^{\dagger}M\,,$$

(3.12)

where $u$ and $v$ are ${\cal G}$-singlet functions to be determined. Note that $\eta_{1}\equiv X_{1}\epsilon_{1}X_{1}^{-1}$, with $X_{1}\equiv X(\lambda_{1})$, and that therefore $d\eta_{1}=-[\eta_{1},dX_{1}X_{1}^{-1}]$. Substituting the variations of $A$, and the ansatz for $U=U^{\rm{inhom}}$, into (3.9) one finds, after some straightforward although slightly elaborate algebra involving the use of the Lax equation (2.21), (2.24) and (2.25), that there is a solution with [21]

$$u=\frac{s_{1}}{\lambda_{1}\rho}\,\frac{\tau_{2}}{(\tau_{1}-\tau_{2})}\,,\qquad v=\frac{s_{1}}{\lambda_{1}\rho}\,\frac{\tau_{1}\tau_{2}}{(1-\tau_{1}\tau_{2})}\,.$$

(3.13)

Here $s_{1}$ denotes $s=\sinh\theta=2\tau/(1-\tau^{2})$ with $\tau$ given by (2.2) for $\lambda=\lambda_{1}$. A remarkable property, which is absolutely crucial in what follows, is that $d(u-v)=0$ and hence

$$u-v=\hbox{constant}\,.$$

(3.14)

This can be verified using (2.24) and (2.25).

To summarise the results so far, we have shown that there exist symmetry transformations of $X$ of the form

	$\displaystyle\delta_{1}^{\rm{hom}}X_{2}$	$\displaystyle=$	$\displaystyle X_{2}\epsilon_{1}\,,$		(3.15)
	$\displaystyle\delta_{1}^{\rm{inhom}}X_{2}$	$\displaystyle=$	$\displaystyle\frac{s_{1}}{\lambda_{1}\rho}\,\frac{\tau_{2}}{(\tau_{1}-\tau_{2})}\,\eta_{1}X_{2}+\frac{s_{1}}{\lambda_{1}\rho}\,\frac{\tau_{1}\tau_{2}}{(1-\tau_{1}\tau_{2})}\,M^{-1}\eta_{1}^{\dagger}MX_{2}\,.$		(3.16)

The inhomogeneous transformations are accompanied by the transformations of ${\cal V}$, $M$ and $A$ given by (3.7) and (3.3), whilst the homogeneous transformations are completely independent, acting only on $X$ and leaving ${\cal V}$, $M$ and $A$ invariant.

The complete set of global symmetry transformations of $X$ will be read off by expanding the various $\delta_{1}$ variations, and $X_{2}$, as power series in $\lambda_{1}$ and $\lambda_{2}$ respectively, both around $\lambda_{i}=0$. Since $\tau_{i}$ vanishes at $\lambda_{i}=0$ (see (2.2)) this appears to present a difficulty for the inhomogeneous transformations (3.16), because of the pole associated with the denominator $(\tau_{1}-\tau_{2})$ in the first term. In the analogous treatment of the flat-space case (see [20, 6]) this was not a problem, because there $\tau_{1}$ and $\tau_{2}$ were themselves constant spectral parameters and so the pole could subtracted by including an appropriate constant multiple of the homogeneous transformation. Here, we cannot simply subtract the analogous multiple of $\delta^{\rm{hom}}$ with a prefactor of the form $(s_{1}/\lambda_{1}\rho)\,\tau_{2}(\tau_{1}-\tau_{2})^{-1}$, because this is not a constant, and so $(s_{1}/\lambda_{1}\rho)\,\tau_{2}(\tau_{1}-\tau_{2})^{-1}\delta_{1}^{\rm{hom}}X_{2}$ is not a symmetry transformation. At this point the remarkable property (3.14) comes to the rescue. In fact, by substituting (2.22) and (2.34) into (3.13) we find that

$$\frac{s_{1}}{\lambda_{1}\rho}\,\frac{\tau_{2}}{(\tau_{1}-\tau_{2})}=\frac{s_{1}}{\lambda_{1}\rho}\,\frac{\tau_{1}\tau_{2}}{(1-\tau_{1}\tau_{2})}+\frac{\lambda_{2}}{\lambda_{1}-\lambda_{2}}\,.$$

(3.17)

Thus, we may rewrite (3.16) as

$$\delta_{1}^{\rm{inhom}}X_{2}=\frac{s_{1}}{\lambda_{1}\rho}\,\frac{\tau_{1}\tau_{2}}{(1-\tau_{1}\tau_{2})}\,\big{(}\eta_{1}+M^{-1}\eta_{1}^{\dagger}M\big{)}X_{2}+\frac{\lambda_{2}}{\lambda_{1}-\lambda_{2}}\,\eta_{1}X_{2}\,.$$

(3.18)

This shows that, despite the original appearance in (3.16), the pole at $\lambda_{1}=\lambda_{2}$ in the inhomogeneous transformation rule actually has a pure constant coefficient, and so by subtracting the homogeneous symmetry transformation $\lambda_{2}/(\lambda_{1}-\lambda_{2})\,\delta_{1}^{\rm{hom}}$ we can easily remove the pole.²²2In Schwarz’s discussion in [21], the important point that one must remove the $\tau_{1}=\tau_{2}$ singularity in $\delta_{1}^{\rm{inhom}}X_{2}$, and furthermore that this can actually be done, for all spacetime points $x^{\mu}$ simultaneously, appears to have been unnoticed. In fact Schwarz instead made a subtraction such that $\delta_{1}^{\rm{inhom}}X_{2}$ vanished at a preferred point $x_{0}^{\mu}$ in the two-dimensional spacetime. However, this subtraction does not in fact cancel the singularity at $\tau_{1}=\tau_{2}$ for other points in the spacetime. The absence of the cancellation did not show up in Schwarz’s subsequent calculation of the commutator $[\delta_{1},\delta_{2}]$ because he evaluated it only on ${\cal V}$ (which is inert under the homogeneous variation in question) and not on $X_{3}$ (which is not inert).

With these points understood, we can now present the full set of global symmetry transformations of the two-dimensional system in the final form that we shall use in what follows. We shall denote them by $\delta_{1}$ and $\tilde{\delta}_{1}$, and their actions on the original sigma model fields in ${\cal V}$ (and hence on $M$), and on the fields in $X(\lambda)$, are as follows:

	$\displaystyle\delta_{1}{\cal V}$	$\displaystyle=$	$\displaystyle\frac{s_{1}}{\lambda_{1}\rho}\,{\cal V}\eta_{1}+\delta h{\cal V}\,,$		(3.19)
	$\displaystyle\delta_{1}M$	$\displaystyle=$	$\displaystyle\frac{s_{1}}{\lambda_{1}\rho}\,\big{(}M\eta_{1}+\eta_{1}^{\dagger}M\big{)}\,,$		(3.20)
	$\displaystyle\delta_{1}X_{2}$	$\displaystyle=$	$\displaystyle\frac{\lambda_{2}}{\lambda_{1}-\lambda_{2}}\,\big{(}\eta_{1}X_{2}-X_{2}\epsilon_{1}\big{)}+\frac{s_{1}}{\lambda_{1}\rho}\,\frac{\tau_{1}\tau_{2}}{1-\tau_{1}\tau_{2}}\,\big{(}\eta_{1}+M^{-1}\eta_{1}^{\dagger}M\big{)}X_{2}\,,$		(3.21)

where $\eta_{1}\equiv X_{1}\epsilon_{1}X_{1}^{-1}$, and

	$\displaystyle\tilde{\delta}_{1}{\cal V}$	$\displaystyle=$	$\displaystyle 0\,,$		(3.22)
	$\displaystyle\tilde{\delta}_{1}M$	$\displaystyle=$	$\displaystyle 0\,,$		(3.23)
	$\displaystyle\tilde{\delta}_{1}X_{2}$	$\displaystyle=$	$\displaystyle\frac{\lambda_{1}\lambda_{2}}{1-\lambda_{1}\lambda_{2}}\,X_{2}\epsilon_{1}\,.$		(3.24)

The $\delta$ transformations are the inhomogeneous transformations we discussed above, with the subtraction of the necessary homogeneous term in $\delta_{1}X_{2}$ to ensure analyticity when $\lambda_{1}$ approaches $\lambda_{2}$. The $\tilde{\delta}$ transformations are independent and purely homogeneous, thus leaving ${\cal V}$ and $M$ invariant. The inclusion of the specific $\lambda_{i}$-dependent prefactor in (3.24) is purely for convenience; it ensures that the final algebra obtained by calculating the commutators of the transformations arises in a simple and conventional basis.

It is evident from the transformations above that the expansions of the variations $\delta$ and $\tilde{\delta}$ will be of the forms

$$\delta(\epsilon,\lambda)=\sum_{n\geq 0}\lambda^{n}\,\delta_{{\scriptscriptstyle(}n)}(\epsilon)\,,\qquad\tilde{\delta}(\epsilon,\lambda)=\sum_{n\geq 1}\lambda^{n}\,\tilde{\delta}_{{\scriptscriptstyle(}n)}(\epsilon)\,.$$

(3.25)

There is an independent ${\cal G}$-valued infinitesimal parameter for each $n$ in $\delta_{{\scriptscriptstyle(}n)}(\epsilon)$, and for each $n$ in $\tilde{\delta}_{{\scriptscriptstyle(}n)}(\epsilon)$.

Having obtained the explicit expressions (3.19)–(3.24) for the $\delta$ and $\tilde{\delta}$ transformations of the fields, it is now a mechanical exercise to calculate the commutators of these transformations. Specifically, we calculate the commutators $[\delta_{1},\delta_{2}]$, $[\delta_{1},\tilde{\delta}_{2}]$ and $[\tilde{\delta}_{1},\tilde{\delta}_{2}]$ acting on $M$ and on $X_{3}$. After some algebra, we find

	$\displaystyle{[}\delta_{1},\delta_{2}{]}$	$\displaystyle=$	$\displaystyle\frac{\lambda_{1}}{\lambda_{1}-\lambda_{2}}\,\delta(\epsilon_{12},\lambda_{1})-\frac{\lambda_{2}}{\lambda_{1}-\lambda_{2}}\,\delta(\epsilon_{12},\lambda_{2})\,,$		(3.26)
	$\displaystyle{[}\delta_{1},\tilde{\delta}_{2}{]}$	$\displaystyle=$	$\displaystyle\frac{\lambda_{1}\lambda_{2}}{1-\lambda_{1}\lambda_{2}}\,\delta(\epsilon_{12},\lambda_{1})+\frac{1}{1-\lambda_{1}\lambda_{2}}\,\tilde{\delta}(\epsilon_{12},\lambda_{2})\,,$		(3.27)
	$\displaystyle{[}\tilde{\delta}_{1},\tilde{\delta}_{2}{]}$	$\displaystyle=$	$\displaystyle\frac{\lambda_{2}}{\lambda_{1}-\lambda_{2}}\,\tilde{\delta}(\epsilon_{12},\lambda_{1})-\frac{\lambda_{1}}{\lambda_{1}-\lambda_{2}}\,\tilde{\delta}(\epsilon_{12},\lambda_{2})\,,$		(3.28)

where $\epsilon_{12}\equiv[\epsilon_{1},\epsilon_{2}]$. Note that there are no poles at $\lambda_{1}=\lambda_{2}$, because in (3.26) and (3.28) the numerator on the right-hand side has a zero that cancels the denominator there.

Using (3.25), expanding the expressions (3.26)–(3.28) in powers of $\lambda_{1}$ and $\lambda_{2}$, and then equating the coefficients of each power, we can read off the algebra of the modes, finding

	$\displaystyle{[}\delta_{{\scriptscriptstyle(}m)}(\epsilon_{1}),\delta_{{\scriptscriptstyle(}n)}(\epsilon_{2}){]}$	$\displaystyle=$	$\displaystyle\delta_{{\scriptscriptstyle(m+n)}}(\epsilon_{12})\,,\qquad m\geq 0\,,\ n\geq 0\,,$		(3.29)
	$\displaystyle{[}\delta_{{\scriptscriptstyle(}m)}(\epsilon_{1}),\tilde{\delta}_{{\scriptscriptstyle(}n)}(\epsilon_{2}){]}$	$\displaystyle=$	$\displaystyle\delta_{{\scriptscriptstyle(m-n)}}(\epsilon_{12})+\tilde{\delta}_{{\scriptscriptstyle(n-m)}}(\epsilon_{12})\,,\qquad m\geq 0\,,\ n\geq 1\,,$		(3.30)
	$\displaystyle{[}\tilde{\delta}_{{\scriptscriptstyle(}m)}(\epsilon_{1}),\tilde{\delta}_{{\scriptscriptstyle(}n)}(\epsilon_{2}){]}$	$\displaystyle=$	$\displaystyle\tilde{\delta}_{{\scriptscriptstyle(m+n)}}(\epsilon_{12})\,,\qquad m\geq 1\,,\ n\geq 1\,,$		(3.31)

where in (3.30) it is to be understood that $\delta_{{\scriptscriptstyle(}n)}=0$ for $n\leq-1$ and $\tilde{\delta}_{{\scriptscriptstyle(}n)}=0$ for $n\leq 0$.

As in the flat-space case discussed in [6], these three sets of commutation relations can be combined into one by introducing a new set $\Delta_{{\scriptscriptstyle(}n)}$ of variations, defined for all $n$ with $-\infty\leq n\leq\infty$, according to

	$\displaystyle\Delta_{{\scriptscriptstyle(}n)}$	$\displaystyle=$	$\displaystyle\delta_{{\scriptscriptstyle(}n)}\,,\qquad n\geq 0\,,$
	$\displaystyle\Delta_{{\scriptscriptstyle(-n)}}$	$\displaystyle=$	$\displaystyle\tilde{\delta}_{{\scriptscriptstyle(}n)}\,,\qquad n\geq 1\,.$		(3.32)

It is then easily seen that (3.29), (3.30) and (3.31) become

$$[\Delta_{{\scriptscriptstyle(}m)}(\epsilon_{1}),\Delta_{{\scriptscriptstyle(}n)}(\epsilon_{2})]=\Delta_{{\scriptscriptstyle(m+n)}}(\epsilon_{12})\,,\qquad m,n\in{\mathbbm{Z}}\,,$$

(3.33)

with $\epsilon_{12}=[\epsilon_{1},\epsilon_{2}]$. This defines the affine Kac-Moody algebra $\hat{\cal G}$. If we associate generators $J_{n}^{i}$ with the transformations $\Delta_{{\scriptscriptstyle(}n)}(\epsilon^{i})$, where $\epsilon=\epsilon^{i}\,T^{i}$ and $T^{i}$ are the generators of the Lie algebra ${\cal G}$ satisfying $[T^{i},T^{j}]=f^{ij}{}_{k}\,T^{k}$, then (3.33) implies

$$[J_{m}^{i},J_{n}^{j}]=f^{ij}{}_{k}\,J^{k}_{m+n}\,.$$

(3.34)

In addition to the Kac-Moody symmetries that we discussed in section 3, which are the affine extension of the original ${\cal G}$ global symmetry of the sigma model, there is also a further infinite-dimensional symmetry that is a singlet under ${\cal G}$ [22, 23, 18]. This is related to the Virasoro algebra. It was discussed in detail for sigma models in flat two-dimensional spacetime in [20], but the generalisation to the gravity-coupled case was not found in [21]. Here, we use the methods developed in [20, 21], and extend them to obtain explicit Virasoro-like transformations in the gravity-coupled sigma models.

The action of the Virasoro-like transformations on the coset representative ${\cal V}$ will be taken to be³³3We should really include an infinitesimal parameter as a prefactor in the definition of $\xi$ in equation (4.1). However, since it is a singlet it plays no significant rôle, and so it may be omitted without any risk of ambiguity.

$$\delta^{V}{\cal V}=h{\cal V}\xi\,,\qquad\xi\equiv\dot{X}X^{-1}\,,$$

(4.1)

where $h$ is a ${\cal G}$-singlet function to be determined, and $\dot{X}$ means $dX(\lambda)/d\lambda$. From this it follows that

$$\delta^{V}A=D(h(\xi+M^{-1}\xi^{\dagger}M))\,.$$

(4.2)

We can show from (2.34) that

$$\dot{\theta}=\frac{s^{2}}{\lambda^{2}\rho}\,,\qquad\dot{\tau}=\frac{s\tau}{\lambda^{2}\rho}\,.$$

(4.3)

By taking the $\lambda$ derivative of the Lax equation (2.21), we can then show after a little algebra that

$$D\xi=\frac{1}{\tau}\,{*d}\xi-\frac{s^{2}}{2\lambda^{2}\rho\tau}\,(\tau{*A}+A)\,,\qquad D(M^{-1}\xi^{\dagger}M)=\tau{*d}(M^{-1}\xi^{\dagger}M)+\frac{s^{2}}{2\lambda^{2}\rho}\,({*A}+\tau A)\,,$$

(4.4)

and hence that

$$D(\xi+M^{-1}\xi^{\dagger}M)=\frac{1}{\tau}\,{*d}\xi+\tau{*d}(M^{-1}\xi^{\dagger}M)-\frac{s}{\lambda^{2}\rho}\,A\,.$$

(4.5)

We now examine the variation of the equation of motion for $A$, namely

$$\delta^{V}\big{(}d(\rho{*A})\big{)}=d\big{(}(\delta^{V}\rho){*A}+\rho{*\delta^{V}A}\big{)}=0\,.$$

(4.6)

Note that unlike the Kac-Moody transformations, which leave $\rho$ invariant, here we shall find that the Virasoro-like transformations must act also on $\rho$. Substituting (4.2) into (4.6), and using (4.5), we obtain

	$\displaystyle\delta^{V}\big{(}d(\rho{*A})\big{)}$	$\displaystyle=$	$\displaystyle\Big{[}d\Big{(}\frac{\rho h}{\tau}\Big{)}-\rho{*dh}\Big{]}\wedge d\xi+[d(\rho h\tau)-\rho{*dh}]\wedge d(M^{-1}\xi^{\dagger}M)$		(4.7)
			$\displaystyle+d(\rho{*dh})[\xi+M^{-1}\xi^{\dagger}M]-d\Big{(}\frac{hs}{\lambda^{2}}\,{*A}\Big{)}+d(\delta^{V}\rho\,{*A})\,.$

This leads us to require $h$ to satisfy

$$d\Big{(}\frac{\rho h}{\tau}\Big{)}=d(\rho h\tau)=\rho{*dh}\,,$$

(4.8)

which has as solution

$$h=-\frac{s}{\rho}\,,$$

(4.9)

(the constant factor is arbitrary, and we take it to be $-1$ for later convenience). Equation (4.7) now reduces to

$$\delta^{V}\big{(}d(\rho{*A})\big{)}=d\Big{(}\frac{s^{2}}{\lambda^{2}\rho}\,{*A}\Big{)}+d(\delta^{V}\rho\,{*A})\,,$$

(4.10)

which, in view of the fact that $d(\rho{*A})=0$, vanishes if we take

$$\delta^{V}\rho=-\frac{s^{2}}{\lambda^{2}\rho}+\alpha\rho\,,$$

(4.11)

where $\alpha$ is an arbitrary constant parameter. However, since by its definition $\delta^{V}$ acting on ${\cal V}$ has no $\lambda^{0}$ term in its Taylor expansion, we may choose $\alpha$ so as to remove the $\lambda^{0}$ term in (4.11). (We shall return to discussing the extra symmetry associated with $\alpha$ later.) This means setting

$$\alpha=1\,.$$

(4.12)

Since $\rho$ satisfies the free wave equation (2.13), we must also check that its variation (4.11) is consistent with this. This is easily done, by substituting (2.34) into (4.11), giving

$$\delta^{V}\rho=-\frac{2\lambda\rho_{+}^{2}}{1-2\lambda\rho_{+}}+\frac{2\lambda\rho_{-}^{2}}{1+2\lambda\rho_{-}}\,.$$

(4.13)

Since this is manifestly the sum of a function of $\rho_{+}(x^{+})$ and a function of $\rho_{-}(x^{-})$, it clearly satisfies ${\partial}_{+}{\partial}_{-}(\delta^{V}\rho)=0$. In fact we can read off from (4.13) that

$$\delta^{V}\rho_{+}=-\frac{2\lambda\rho_{+}^{2}}{1-2\lambda\rho_{+}}+\beta\,,\qquad\delta^{V}\rho_{-}=\frac{2\lambda\rho_{-}^{2}}{1+2\lambda\rho_{-}}-\beta\,,$$

(4.14)

where $\beta$ is another arbitrary constant parameter. Again, like the previous discussion which implied we could take $\alpha=1$, here too we could require that there be no $\lambda^{0}$ term in the expansion, and this constrains $\beta$ also, to

$$\beta=0\,.$$

(4.15)

We shall discuss the extra symmetry associated with a non-zero $\beta$ below.