数理解析研究所講究録1086 巻 1999 年 1-16

1

-Witten Model toExtension of
Dimensions and -Toroidal Lie Algebra*$\mathrm{W}\mathrm{e}\mathrm{s}\mathrm{S}-\mathrm{z}\mathrm{u}\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{o}$$2\mathrm{n}$$\mathrm{n}$Takeo Inami\dagger
Yukawa Institute for Theoretical Physics
Kyoto University, Kyoto 606, Japan

Abstract

A 4-dimensional (4D) extension of the$2\mathrm{D}$-Witten model has a$\mathrm{w}\mathrm{e}\mathrm{S}\mathrm{S}- \mathrm{z}_{\mathrm{u}\min 0}$

few remarkable properties. In particular, we have shown that this model has an

infinite-dimensional symmetry which generates 2-toroidal Lie algebra. Generalization of the construction of the model to higher dimensions $D=2n$ is also given.

at “Topological Field Theory and Related Topics”, held at RIMS, Kyoto,
December 16-19, 1996.
\dag er Present address : Physics Department, Chuo University, Kasuga, Tokyo 112.

$*\mathrm{T}\mathrm{a}\mathrm{l}\mathrm{k}$2
1. Introduction
Many Physical systems in nature have symmetries. If symmetries are continu-

ous, they generate Lie algebras (LA). Let me give two examples, one from atomic
physics and another from particle physics : Hydrogen atom has$\mathrm{O}(4)$symmet-ric energy spectrum. The (massless) quark model has $SU(2)_{L}\mathrm{x}SU(2)_{R}$ chiralsymmetry in addition to $SU(3)_{c}$ gauge symmetry.

A physical system becomes integrable, if the symmetry is large enough, namely,
dimension of LA$=$

number of degrees of freedom

A classic example is the Wess-Zumino-Witten (WZW)$\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{e}1^{[1]}$(1). It is a group-

valued non-linear sigma model (NLSM) on a spacetime of 2 dimensions with anaddition of an anomaly term (a certain 2-cocycle term in mathematical terms).
The dynamical variable is a mapping$g:X_{2}arrow\emptyset$

where$X_{2}$,(2)

is a 2-dimensional (2D) spacetime with coordinates$x^{\mu}$and$\mathfrak{G}$is a group.

As we will see in a moment, the WZW model has currents, which are composite
operators composed of the field $g(x)$ ,$J(x)=g^{-}\partial_{z}g1,\overline{J}(x)=\partial_{\overline{z}}gg-1$

where we have used complex coordinates $z=x^{0}+\dot{i}X^{1}$ and,(3)$\overline{z}$(or $x^{\pm}=x^{0}\pm x^{1}$ inlight-cone coordinates). The currents obey conservation laws :$\partial_{\overline{z}}J=0,$$\partial_{z}\overline{J}=0$.

(4)

The mode expansion of $J^{a}(z)=tr(taJ(Z))$ ,$J_{n}^{a}=n=- \sum_{\infty}^{\infty}Z-n-1Ja(_{Z)},$

yields an infinite number of conservation laws$(\mathrm{C}\mathrm{L})$$(5)$. Their commutation relations

3

define an affine Kac-Moody algebra,$[J_{m}^{a}, J_{n}^{b}]=\dot{i}fab\mathrm{c}_{J_{m}+\frac{1}{2}k}ca+nm\delta_{m}+n,0\delta b$.

(6)

The representation theory of affine Kac-Moody algebras was developed in
the eighties, especially in connection with$(\mathrm{Q}\mathrm{F}\mathrm{T})^{[2}]$$2\mathrm{D}$integrable quantum field theories. Beyond affine Kac-Moody algebras, new classes of infinite-dimensional

Lie algebras are known : hyperbolic Kac-Moody algebras and -toroidal Lie alge$\mathrm{n}$bras. It was conjectured that massive modes of superstring, which are infinitely

many, generate a hyperbolic Kac-Moody algebras called$\mathrm{s}\mathrm{h}\mathrm{o}\mathrm{w}\mathrm{n}[4]$that the symmetry of a$4\mathrm{D}$$\mathrm{E}(10)^{[}3]$. We have recently

analogue of the WZW model generates 2-

toroidal Lie algebra, a generalization of a loop algebra.
The WZW model has many remarkable physical and mathematical properties.

It is a finite QFT, i.e., the$\beta$-function vanishes. It has an infinite-dimensional

symmetry, as explained a moment ago, which is responsible for the model being
solvable. The model can be expressed as path integral of a fermionic$\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{e}1^{[5]}$

. One

.
can -deform the model to get a massive
If one could construct integrable QFT in dimensions higher than $D=2$ , itwould have enormous implications from th mathematical point of view and ap$\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{e}1^{[6]}$$\mathrm{q}$$\mathrm{b}.0$

plication to particle physics. In the past there have been various attempts in thisdirection. When Polyakov put forward CFT in late sixties, he meant 3+1 dimensional$\mathrm{s}\mathrm{y}\mathrm{S}\mathrm{t}\mathrm{e}\mathrm{m}\mathrm{S}[7]$. Cardy proposed to generalize modular invariance to $D>2^{[8]}$ . It

was conjectured that at level of classical equations of motion all integrable models
are related to self-dual Yang-Mills (SDYM) equation in $D=4^{[9]}$ .
WZW model are motivated by thisanalogue of the
Recent studies of away of thought, and they have revealed a few remarkable properties of thisK\"ahler WZW (KWZW) model. It has an infinitemodel, which we refer to$2\mathrm{D}$$4\mathrm{D}$$4\mathrm{D}$$4\mathrm{D}$dimensional (anti)holomorphic symmetry,and it is solvable in its algebraicin spite. The model has been shown to be one-loop on-shell$\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{o}\mathrm{r}[10,11]$$\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{e}^{[]}12$of apparent non-renomalizability by power counting.
In this talk we will concentrate on the infinite-dimentional symmetry of the
KWZW model. We will show that this symmetry generates a Lie algebra called4

2-toroidal Lie algebra, the central extension of two-loop algebra. A few mathemati-

cians have recently begun to study this class of algebra as a possible extension ofaffine Kac-Moody
model (and of the2.

$\mathrm{a}\mathrm{l}\mathrm{g}\mathrm{e}\mathrm{b}\mathrm{r}\mathrm{a}\mathrm{s}^{[}13$$4\mathrm{D}$$\mathrm{W}\mathrm{e}\mathrm{S}\mathrm{S}^{-}\mathrm{z}\mathrm{u}\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{o}$]. We will also mention an extension of the

KWZW model) to general$2\mathrm{n}$$2\mathrm{D}$WZW

dimensions.

-Witten model

Here we summarize the WZW model as a preparation for extension to $D=4$ .
The WZW model is a -valued NLSM with an addition of an anomaly term. We$\mathrm{g}$extends the spacetime from

The action$\mathrm{S}$$X_{2}$

to$X_{3}$

so thatis the boundary of$X_{2}$

is a functional of the -valued field$\mathrm{g}(\mathrm{x})$$\mathrm{g}$$X_{3},$ $X_{2}=\partial X_{3}$,$S[g]=- \frac{k}{8\pi}[\frac{1}{2}\int_{X_{2}}tr(g\partial-1g\wedge g\overline{\partial}-1g)+\frac{1}{3}\int_{X_{3}}tr(g^{-1}dg)]$

where$\partial=\partial_{-}d_{X^{-}}$(or$\partial_{\overline{z}}dZ$

).

$k$

.

,

(7)

is a coupling constant in the usual sense in particle

physics. It has to be an integer for the theory to be well-defined, and is to be

identified with the centre$k$appearing in-We compute the variation

field,$\delta g=g\epsilon$. The

to an integral over$\mathrm{v}\mathrm{a}\mathrm{r}\mathrm{i}\mathrm{a}\mathrm{t}\mathrm{i}\dot{\mathrm{O}}\mathrm{n}$$X_{2}$$\delta S$

defining a Kac-Moody algebra.

$\mathrm{e}\mathrm{q}.(6)$of the action (7) for an infinitesimal change of theof the anomaly term in (7) turns out to be reduced,$\delta\int_{X_{3}}tr(g^{-13}dg)=3\int_{X_{2}}d^{2}x\epsilon^{\mu}\nu tr(\epsilon g^{-}\partial g\mu g\partial_{\nu}g)1-1$

where$\epsilon^{\mu\nu}$is the so-called -tenson,$\epsilon$$\epsilon^{01}=-\epsilon^{10}--1$,(8). Thanks to this expression, theequation of motion takes a very simple form,$\partial_{+}(g^{-1}\partial_{-}g)=0$.

(9)This equation implies a conserved current, i.e.,$J_{-=}g^{-1}\partial_{-}g$$\partial_{+}J_{-}=0$

The last equation is the Cauchy-Riemann.

$(\mathrm{C}\mathrm{R})$

,

(10)
(11)
relation, whose solution is given by5].
. Note that the CR relation is equivalent to self-duality inEqs.(10) and (11) define a current in the right sector, a sector solely dependingandSimilarly one can define a current in the left sector,
on$2\mathrm{D}\mathrm{N}\mathrm{L}\mathrm{s}\mathrm{M}^{[14}$$J_{-}=f(x^{-})$$\overline{J}_{+}=\partial_{+gg^{-1}}$$x^{-}$$\partial_{-}\overline{J}_{+}=0$.

Eq.(ll) implies an infinitely many conserved quantities by making mode ex, where
to be the “time” coordinate, we havepansion. Choosing$\partial_{+}J_{n}=0$$x^{+}$(12)$J_{n}= \int_{x^{+}}dX^{-}(x-)-n-1J-\cdot$

3.

$4\mathrm{D}$K\"ahler WZW model

extension of the WZW model, we begin by writinga general NLSM in $D=4$ . The basic tool is a mapping
Aiming at constructing a$4\mathrm{D}$$\phi^{a}$:$x_{4}arrow \mathcal{M}$

(13)

.

. The target spaceand metric-manifold with coordinates
is an n-D manifold with coordinates(or embedding space in mathematics)$\phi^{a}(a=1, \ldots, n)$ and metric. We give an additional structure to bothThe

$X_{4}$is a$g_{\mu\nu}(x)$$x^{\mu}$$4\mathrm{D}$$\mathcal{M}$$X_{4}$$g_{ab}(\emptyset)$and A4 by assuming that there exist a 2-form on$\omega=\omega_{\mu\nu}dx^{\mu_{\wedge}}d_{X}\nu$

and another on$\mathcal{M}$$X_{4}$

,

,

(14)

.

(15)

,
$\Lambda$.

$B=B_{ab}d\phi^{a}$A$d\phi^{b}$

Under the assumption that the action is bilinear in$S[ \phi^{a}]=a\int_{X_{4}}$($-d\phi a_{\wedge dgb}*\emptyset ba+\kappa\omega$$\partial_{\mu}\phi^{a}$A$d\phi^{a}$Note that the theory contains two coupling constants,
dimension $-2$ and , respectively).
$0$, it consists of two terms,
A$d\phi^{b}B_{ab}$

$a$

and

$\kappa$).

(16)(they have mass

6

The action consisting of the first term of (16) alone defines a usual NLSM.
This theory is non-renormalizable, i.e., contains divergences which arize quantum

mechanically and cannot be handled properly. What is the role of the second
term, then? After a lengthy calculation of one-loop quantum effects, Ketov has

shown that the NLSM defined by (16) becomes one-loop on-shell finite, provided
the following three conditions are:$\mathrm{m}\mathrm{e}\mathrm{t}^{[12]}$a) The target space A4 is a group$\oplus$(parallelizable more precisely). Then

$dB=H$ ,(17)

is a torsion on 6, and $H_{ijk}=f_{ijk}$ provides the structure constant of 6.
b) The coupling constant$\kappa$

is to be tuned to the value$\kappa=2_{\dot{i}}$c) The spacetime$X_{4}$.

(18)

is a hyper-K\"ahler manifold (Ricci-flat). We can thenintroduce complex coordinates,$z^{1}=x^{0}+\dot{i}x1,$$z^{2}=x^{2}+\dot{i}x^{3}$

We will often use the notation $z^{1}=u,$ $z^{2}=v$ . The;$z^{\overline{1}},$$\omega$$z^{\overline{2}}$

.

(19)

is nothing but the K\"ahler

2-form,,(20)$\omega=\frac{\dot{i}}{2}h_{\alpha\overline{\beta}}dz^{\alpha}\wedge dz\overline{\beta}$$d\omega=0$

.

(21)

After taking account of the three conditions given above, the action (16) canbe reduced to the one which was constructed by Nair and Schiffilo] in extending
the Chern-Simons action to $D=5$ and by$\mathrm{D}\mathrm{o}\mathrm{n}\mathrm{a}\mathrm{l}\mathrm{d}_{\mathrm{S}}\mathrm{o}\mathrm{n}^{[}15$] in anviewpoint (DNS action). To see this, we proceed as follows.

algebraic-geometric7

We now have a map$g$

We extend the$X_{4}$

to$X_{5}$

so thatmeans that

$V_{a}^{i}$$X_{4}=\partial X_{5}$, to give an anomaly interpretation to$t^{i}$$V_{a}^{i}$$\mathfrak{G}$$\emptyset$$V_{a}^{i}$$V_{a}d \phi^{a}=-\frac{1}{2}$We have replaced the scalar fields
Using

(22)

be vielbein on ,be generators of and.
can be shown to obey the Maurer-Cartan equation. Thisis a pure-gauge,

the second term of (16). Let$V_{a}^{i}t^{i}=V_{a}\in \mathrm{g}$: $X_{4}arrow 6$ .

$\mathrm{e}\mathrm{q}\mathrm{s}.(17),$$\phi^{a}(x)$

igdg.

(23)

by $g(x)$ .

(18) (20) and (23).’ the two terms of (16) can be expressed in$,$terms of $g^{-1}dg$ and$\omega$, and we arrive at the DNS action

.

(24)

$S=- \dot{i}\int_{X_{4}}\omega\wedge\tau_{r}(g^{-1-1}\partial g\wedge g\overline{\partial}g)+\frac{\dot{i}}{3}\int_{X_{3}}\omega\wedge tr(gd-1)^{3}g$

Here,

of

$\partial=\partial_{\alpha}dz^{\alpha}$. We have absorbed the coupling constant.
We can prove the following identity, which is a$\omega$:$a$into the redefinition$\omegaarrow 2a\omega$

Wiegmann$\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{u}\mathrm{l}\mathrm{a}^{[}16$]$4\mathrm{D}$analogue$\dot{\mathrm{o}}\mathrm{f}$the Polyakov-(2-cocycle condition in mathematical terms).

$S[gh]=s[g]+s[h]-2_{\dot{i}} \int_{X_{4}}\omega\wedge tr$($g\partial_{\mathit{9}}-1$

A$\overline{\partial}hh^{-1}$

).

(25)

We see easily from this formula that the action is invariant under holomorphic
right$(\mathrm{H}\mathrm{R})$and antiholomorphic left (AHL) infinite symmetries,$garrow h_{L}(_{Z^{\overline{\alpha}}})ghR(z)\beta$

.

(26)

The equation of motion can be derived in the same fashion as the$2\mathrm{D}$case. Itis given by$\overline{\partial}$( A$g^{-1}\partial g$) $=0$ ,

(27)

( A

$\overline{\partial}gg^{-1}$) $=0$ .

(28)$\omega$or equivalently,$\partial$$\omega$

These equations allow us to identify the conserved currents$J(\overline{J})$which generate

8

the right(left)-action symmetry in (26).
$J=-\dot{i}\omega\wedge g^{-}\partial 1g$$\overline{\partial}J=0$,,$\overline{J}=\dot{i}\omega\wedge\overline{\partial}gg^{-1}$$\partial\overline{J}=0$

(29)

,

(30)

.

You may consider the dual (one-form) of , which we denote by$\mathrm{J}$$J_{\alpha}dz^{\alpha}$. Then, theconservation law (30) reads$\overline{\partial}J=\partial\overline{u}Ju+\partial\overline{v}Jv=0$

It is curious to note that the same value of$\kappa$(31).

assures one-loop finiteness on one

hand and the PW formula (25) and consequently the current conservation (30) on
the other.
The equation of motion (30) (or (.31)) can be shown to be equivalent to the$\acute{\mathrm{S}}$DYM equation. Quite some time ago, Yang cast the SDYM equation (in the flatspacetime) into the$\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}^{[17]}$$F_{\alpha\beta}=F_{\overline{\alpha}\overline{\beta}}=0$$\eta^{\alpha\overline{\beta}}F_{\alpha\overline{\beta}}=0$

The conservation low (31) is$\mathrm{e}\mathrm{q}.(33)$

(32)

,

(33)

.

generalized to a K\"ahler spacetime.

The DNS action may be cast into a more familiar form by using Gauss decomposition. For simplicity we consider the case of$\mathrm{g}=sl(2)$. Let $H,$

$E^{\pm}$

be Chevalley

generators of , obeying $[E^{+}, E^{-}]=H$ . The Gauss decomposition reads$\mathrm{g}$$g(x)=e^{x(x)}E^{+}ee\varphi(x)H\psi(x)E-$$=$

We have introduced three scalar fields$\varphi,$$\chi,$$\psi$.

(34). Using the PW formula (25), the9action is now written as an integral over$S[ \varphi, \chi, \psi]=-\dot{i}\int_{X_{4}}\omega$

So far$X_{4}$

is an arbitiary

consider the case of flat

$X_{4}$$4\mathrm{D}$A$X_{4}$.

$(\partial\varphi\wedge\overline{\partial}\varphi+\partial x\Lambda\overline{\partial}\psi e^{-2})\varphi$

.

(35)

K\"ahler manifold. In the following analysis we will.

$h_{1\overline{1}}=h_{2\overline{2}}= \frac{1}{2}$,$\omega_{1\overline{1}}=\omega_{2\overline{2}}=\frac{\dot{i}}{2}a$

.

(36)

The action is then given by$S=-a \int_{X4}d^{4_{Z}}\sum_{\gamma=1,2}(\partial_{\gamma}\varphi\partial_{\overline{\gamma}}\varphi+\partial_{\gamma}\chi\partial\psi\overline{\gamma}e^{-2\varphi})$

The currents$J_{\alpha}$and$\overline{J}_{\overline{\beta}}$

.

(37)

are expressed in terms of the scalar fields. By writing, we have$J_{\alpha}=J_{\alpha}^{0}H+J_{\alpha}^{-}E^{+}+J_{\alpha}^{+}E^{-_{\epsilon \mathfrak{G}}}$$J_{\alpha}^{0}=a(\partial\alpha\varphi+\psi\partial\alpha\chi e-2\varphi)$

,

,

(38)

$J_{\alpha}^{+}=-a(\psi\partial\alpha\varphi-2\partial\alpha\varphi+\varphi^{2}\partial\alpha\chi e-2\varphi)$$J_{\alpha}^{-}=a\partial_{\alpha}\chi e-2\varphi$.

4. Infinite dimensional Lie algebra
Since we are given a field theory, we should be able to compute various quantities and relations. We consider the question : What is the infinite-dimensional
symmetry that the currents$J_{\alpha}^{a}( \overline{J}\frac{a}{\beta})$generate?

To address to this question we use the canonical formalism and compute the
Poisson brackets (commutation relations in quantum theory) of the currents. To
this end we have to choose one of the four coordinates as “time” and the remaining$t$three are space coordinates. There are two alternatives of doing this : a) $t=$$x^{0},\vec{x}=(x^{1}, x, X^{3})2$ being space coordinates. b)
being space$t=\overline{u},\vec{x}=(u, v,\overline{v})$coordinates (light-cone scheme).

10Here we present the result in the light-cone scheme, taking$\overline{u}$as time. The

coordinates are assumed to take real values by using Wick rotation. We notethat the action (24) is first order in time derivatives, that is, it is already in
the Hamiltonian form. Hence we can define$\mathrm{P}.\mathrm{B}$. without introducing conjugate

momenta of the field $g(x)$ . The symplectic two-form of this model is given by(39)$\Omega_{ab}(_{\vec{X}},\vec{x})’=a\delta ab\partial u\delta 3(_{\vec{X}\vec{x}’)}-.$The Hamiltonian structure is obtained as the inverse of$\delta^{3}(\vec{x}-\vec{X}’)$. Here,$\partial_{u}^{-1}\delta(u)=\frac{1}{2}\epsilon(u)$$\Omega,$$\mathcal{H}^{ab}(\vec{X},\vec{x}’)=a^{-1}\delta^{ab}\partial_{u}^{-1}$. It is then straightforward to compute P.B., e.g.

$\{tr(Xg\partial_{Ag}-1)(\vec{x}), tr(Yg^{-1}\partial Bg)(\overline{X}^{\sqrt})\}$

where$\mathrm{A},$are$\mathrm{B}$$u,$ $v$

or$\overline{v}$

and $X,$$Y$

areRecall that the current is given by$\overline{u}$as time,we choose:$J_{u}^{a}$$garrow h_{L}(z^{\overline{\beta}})g$The$\mathrm{P}.\mathrm{B}$

(40)

.

$J_{A}= \frac{1}{2}ag^{-1}\partial_{Ag}$. In the present choice ofare the generators of the HR action symmetry :as time instead, then$u$$t^{a}$,$\overline{J}\frac{a}{u}$$garrow gh_{R}(z^{\alpha})$are the generators of AHL action symmetry.

. of current$J_{A}^{a}$can be computed from (40). We refer you to refs. [4,19] for the explicit result and derivation. Here we only write the result for$\mathrm{B})=(\mathrm{u}, \mathrm{u})$. If$(\mathrm{A}$,, which we will be concerned with hereafter.

$\{J_{u}^{a}(\vec{x}), J^{b}(u\overline{x}^{\sqrt})\}=\dot{i}f^{ab}CJ_{u}^{c}(\vec{x})\delta^{3}(\vec{X}-\overline{X}’)+a\delta^{ab}\partial_{u}\delta^{3}(\vec{x}-\vec{X})’$

that is, the currents$J_{u}^{a}$

,

(41)

satisfy a current algebra in $D=4$ with a -number anomaly$\mathrm{c}$term.

We make mode expansion of the currents$J_{u}^{a}(\vec{x})$to derive a more familiar-

looking Lie algebra from the current algebra (41). To this end, it is convenient
to compactify the three space coordinates$u,$ $v,\overline{v}$, by letting$u,$ $v,\overline{v}$take values in

11

The generators of HR action symmetry are given by$I=[0,2\pi]$ .

$Q^{a}= \int_{-}^{2}\mathrm{o}u\pi_{ddv\epsilon^{a}(u,v)J^{\overline{a}}(\overline{u},u,v)}$

where$J^{\overline{a}}$are zero-modes with respect to ,$\overline{v}$$J^{\overline{a}}( \overline{u}, u, v)=\int_{0}^{2\pi_{d}}\overline{v}J_{u}^{a}(\overline{u},\vec{X})$Note that$\overline{J}^{a}(u, v)$$\mathrm{e}\mathrm{q}.(41)$

(42)

,

$\partial_{\overline{u}}J^{\overline{a}}=0$, and hence that

$J^{\overline{a}}$

.

(43)

are holomorphic functions of. We use the notation $zarrow=(u, v)$ . The$\mathrm{P}.\mathrm{B}$. for$J^{\overline{a}}(^{arrow}z)$

$u$

and

$v$

,

can be derived from, and is given by$\{J^{\overline{a}}(\vec{z}),\hat{J}(^{arrow}b)z’\}=\dot{i}fab_{C}\overline{J}C(^{arrow}z)\delta 2(z^{arrow}-z)arrow\prime 2a\delta ab\partial_{u}\delta^{2\prime}(^{arrow}+z-z)arrow$

The parameter$\epsilon^{a}(^{arrow}z)$

.

(44)

are defined on the 2-torus, and hence it can be expandedas,$\sum\infty$(45)$\epsilon_{\vec{m}}^{a}e^{i\vec{m}}.z^{arrow}$$\epsilon^{a}(\vec{z})=$$m_{0},m_{1}=-\infty$

where$J^{\overline{a}}$$\vec{m}=(m_{0}, m_{1})\epsilon Z_{2}$and $\vec{m}\cdot zarrow=m_{0}u+m_{1}v$ . Correspondingly, the currents

are mode-expanded as$J_{\vec{m}}^{\overline{a}}= \int dudve^{i\vec{m}\cdot z_{J^{\overline{a}}}}(Zarrowarrow)$

,

(46)

so that we have$Q^{a}= \sum_{\vec{m}}\epsilon_{\vec{m}\vec{m}}aJ^{a}$

We obtain from

.

(47)

$\mathrm{e}\mathrm{q}.(44)$,$\{J_{\vec{m}}^{\overline{a}},\hat{J}_{\ell+}^{bbc}arrow\}=\dot{i}f^{a}\overline{J}C\lambda 0m0\delta^{a}\vec{m}\ell^{arrow+}\ell b\delta_{\vec{m},-}arrow$

where$\lambda_{0}=$(area of$T^{2}$(48)) $\cdot a/8=\pi^{2}a/2$ . This relation defines a 2-loop algebra withcentral extension, which is 2-toroidal Lie algebra.

12

In

$\mathrm{e}\mathrm{q}.(48)$consist ofthere has appeared one centre,$\lambda 0m_{0}+\lambda_{1}m_{1}$$\lambda 0m_{0}$. A general central term should(and perhaps many more terms).

In mathematical language, an affine Kac-Moody argebra is defined by using

Laurent polynomial expansion. In an analogous way, a 2-toroidal Lie algebra isexpressed by double Laurent polynomial expansion as$\mathfrak{G}_{2-tor}=\mathfrak{G}\otimes \mathrm{C}[s, t, s-1,-t]1\oplus$

centre$\oplus \mathrm{C}_{0}d_{0}+\mathrm{C}_{1}d_{1}$

.

(49)

We should mention that the representation theoretic content of 2-toroidal Lie
algebras has been poorly understood. Let us compare the affine Kac-Moody algebra$\mathrm{g}$ofand the 2-toroidal Lie algebra$\mathrm{g}$$9_{2-\iota_{\mathit{0}}}r$

in the case of$\mathrm{g}=sl(2)$. The root system

is two-dimensional, being infinite in one (affine) direction. That ofthree-dimensional, being infinite in two directions. The Cartan matrix of

$K=$

.

$\mathrm{g}_{2-tor}$$\mathrm{g}$

is

is
(50)

The use of “Cartan matrix” in toroidal Lie algebra is yet to be understood completely. For the$\mathrm{g}_{2-to}r$root system we have tentatively chosen, the “Cartan matrix”

is

$K–$
We note that in (54) one of the off-diagonal elements.

$\mathrm{i}\mathrm{s}+2$(51), a feature not shared byfinite-dimensional and affine Lie algebras. Extension of highest-weight representations to$\mathrm{g}_{2-tor}$

is a diffucult problem because of the two-dimensionality of infinite

directions of its root system.

5. Generalization of the KWZW model to$\mathrm{D}=2\mathrm{n}$It is an intriguing question whether the KWZW model allows a generalization
to $D=2n$ possessing many of its remarkable properties at $D=4$ . Here we present

one possible model of this$\mathrm{t}\mathrm{y}\mathrm{p}\mathrm{e}^{[18}$

].

13

Let

$X_{2n}$NLSM on-dimensional K\"ahler manifold with 2-form
with a target space furnished with a 2-form B.

be a$X_{2n}$$2\mathrm{n}$$\omega$. We consider a$S[ \phi^{a}]=a\int_{X_{2n}}(-d\phi^{a_{\wedge}*}d\phi^{b}g_{a}b+\kappa\omega\wedge n-1d\emptyset^{a}\wedge d\phi bb_{a}b)$

.

(52)

By repeating an argument analogous to that we have made in the case of $D=4$ ,.
we arrive at a a group -valued NLSM with an addition of an anomaly$\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{m}^{[19]}$$\mathfrak{G}$$S=- \dot{i}\int_{X_{2}}\omega^{n-}\mathcal{R}1\wedge tr$Here$\partial=\partial_{\alpha}dz^{\alpha}$($g\partial-1g$. We have tuned

A$\kappa$$g^{-1}\overline{\partial}g$)$+ \frac{\dot{i}}{3}\int_{X_{2n+}}1\omega n-1$

A $tr(g^{-1}dg)^{3}$ .

(53)

to the value$\kappa=2^{n-1}\dot{i}/(n-1)!$

(54)

,

so that the PW of the form (25) holds true.
The equation motion can be derived from (53) in the same fashion as the WZW
case.
$\overline{\partial}J=0$(equivalently$\partial\overline{J}=0$

(55)

),

where we have already introduced conserved currents$J=\dot{i}\omega^{n-1_{\wedge}}g-1\partial g$

For the dual of , which we denote by$J$$(\overline{J}=-\dot{i}\omega^{n-}\wedge\overline{\partial}1gg^{-})1$$J_{\alpha}d_{Z^{\alpha},\mathrm{e}}\mathrm{q}.(55)$$\partial_{\overline{\alpha}}J_{\alpha}=0$

.

(56)

reads
(57)

.

It is very curious to note that the same equation as (57) has previously been
written in connection with a moduli problem of gauge fields in algebraic geometry.

On a K\"ahler manifold$X_{2n}$the curvature 2-form$F$of a gauge field is decomposed

14

into $(2, 0)$ , $(1,1)$ and $(0,2)$ components.

$\mathrm{D}_{0}\mathrm{n}\mathrm{a}\mathrm{l}\mathrm{d}\mathrm{S}\mathrm{o}\mathrm{n}^{[}15$], Uhlenbeck and$\mathrm{Y}\mathrm{a}\mathrm{u}^{[19}$]wrote the conditions that a holomorphic vector bundle is stable.

$F_{\alpha\beta}=F_{\overline{\alpha}}=\overline{\beta}0$$h^{\alpha\overline{\beta}}F_{\alpha\overline{\beta}}=0$

,

(58)

.

(59)

The connection of the DUY equation to the equation of motion of the KWZW
model is as follows. We set$A_{\alpha}=h_{R}^{-1}\partial_{\alpha}h_{R},$$A_{\overline{\beta}}=\partial_{\overline{\beta}}h_{L}h_{L}^{-1}$eqs.(58) and (59) can be shown to be equivalent to$\mathrm{e}\mathrm{q}.(57)$.

and $g=h_{R}h_{L}$ . Then

15

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