The geometric structure of W N gravity

N UCLEAR

P HY SI

Nuclear Physics B413 (1994) 296—318 North-Holland

CS B

________________

The geometric structure of

~“N

gravity

C.M. Hull Physics Department, Queen Mary and Westfield College, Mile End Road, London El 4NS, UK Received 30 July 1993 Accepted for publication 13 September 1993

The full nonlinear structure of the action and transformation rules for ~2~~gravitycoupled to matter are obtained from a nonlinear truncation of those for w 0, gravity. The geometry of the construction is discussed, and it is shown that the defining equations become linear after a twistor-like transform.

1. Introduction Classical v-gravity theories [1—81are higher-spin gauge theories in two dimensions that result from gauging 7T algebras [91,which are higher-spin extensions of the Virasoro algebra. One motivation for studying two-dimensional matter coupled to 7f gravity is that such systems can be interpreted as generalisations of string theory in which the two-dimensional space—time is regarded as a world-sheet, in much the same way that matter coupled to ordinary gravity in two dimensions leads to conventional string theory. In particular, the ~ algebras play a central role in such v-string theories, just as the Virasoro algebra plays a central role in string theory. The actions for 7~gravity coupled to matter have a complicated nonpolynomial dependence on the gauge fields. In the case of gravity, this nonlinear structure is best understood in terms of riemannian geometry and this suggests that some higher-spin geometry might lead to a better understanding of ~ gravity. A number of approaches to the geometry of v-gravity theories have been considered [6,11—201.In refs. [18,191,the complete nonlinear structure of the coupling of a scalar field on a world-sheet M to w~gravity was given in terms of a function F on the cotangent bundle of M that satisfied a certain nonlinear differential equation, which is sometimes referred to as a Monge—Ampère equation [211or as one of Plebanski’s equations [221. Such equations also arise in the study of four-dimensional self-dual gravity [221; other connections between ~ algebras and gravitational instantons, which may be related, were described in refs. [23,241. In particular, it was shown in refs. [18,191 that the function F could be interpreted as giving a family of Kähler potentials for Ricci-flat metrics on ~ with self-dual curvature. The purpose of this paper is to extend the results of refs. 0550-3213/94/$07.00 © 1994 — Elsevier Science B.V. All rights reserved SSDI 0550-3213(93)E0480-N

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[18,19] to the case of ~ gravity; some of the results to be derived here were announced in ref. [201. It will be shown here that the coupling of a scalar field to ~t~N gravity can be given as a nonlinear truncation of the action for the coupling to w~gravity. The lagrangian is a function F which, in addition to satisfying the Monge—Ampère equation, satisfies an (N + 1)th order nonlinear partial differential equation, and it is a nontrivial fact that this constraint is consistent with the Monge—Ampère equation. This differential constraint can be interpreted geometrically as a condition on the family of self-dual metrics on l~.For the fourth order differential equation satisfied by the Kähler potential can be written as ~

~

+ Ta~T~p + T~Tapff+T~&.~Tap~v)~ (1.1)

where G~-~ is the Kähler metric and is a certain third rank tensor that is given in terms of the Kähler potential K by ~ in certain special coordinate system. It is interesting to note that similar, but distinct, geometrical constraints arise in the study of “special geometry”, i.e. the geometry of the moduli space of Calabi—Yau manifolds, and in the geometry of N 2 supersymmetric gauge multiplets in four and five dimensions [271. For 7ttN with N> 3, the differential constraint can be written as a restriction on the (N 3)th covariant derivative of the curvature tensor. =

=

—

1.1. LINEARISED

~‘

GRAVITY

Before proceeding to the nonlinear theories, it will be useful to review unearised 7’N and w~gravity. Consider the action for a free scalar field in two dimensions

s

0

=

~fd2x ~W4.

(1.2)

This has an infinite number of conserved currents, which include n=2,3,...,N,

[31 (1.3)

and these satisfy the conservation law ~J4’~ 0. The current W2 T is a component of the energy—momentum tensor and generates a Virasoro algebra. The 7~Ncurrents algebra (1.3) generate a current algebra which is a certain classical limit of the of ref. [25] for finite N, and in the limit N the classical current algebra =

=

—÷ ~,

*

Flat two-dimensional space M

2 = ~, dxi’ dx’~= 2dz d2, where z =(1/V~Xx1+ is2) 0 has metric ds I = (1/~Xx’ — is2) are complex coordinates if M 2), 2 =(1/~Xx’ — x2) are null real 0coordinates. is euclidean, 8=3~and while, ifI=~. M0 is lorentzian, z = (1/~/~Xx’ +x

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becomes the w~algebra [23]. Similarly, the currents l’V~ (1/nX94Y’ generate a second copy of the ~N or w~algebra. Adding the Noether coupling of the currents J4’~,, W~to corresponding gauge fields h~,h~gives the linearised action =

N

1

S

=

fd2x(a~

+

[~o~y

+~~(a~I

(1.4)

+ O(h2)),

which is invariant, to lowest order in the gauge fields, under the transformations 1+A~(~, 2)(o~’}, 2)(a~y

n2~

oh~=—2aA~-l-O(h), ~5h~=—28A~-i-O(h).

(1.5)

This gives the linearised action and transfonnations of ~‘N or (in the N limit) w~,gravity. The full gauge-invariant action and gauge transformations are nonpolynomial in the gauge fields. —~ ~

2. Nonlinear w~gravity The nonlinear structure of the coupling of a scalar field to w~gravity [18,19] will now be reviewed. The two-dimensional manifold M, which will sometimes be referred to as the world-sheet, can have any topology and has local coordinates x~. The action is a nonpolynomial function of and can be written as =

for some

E, which has the

J

E(x, ô~),

(2.1)

following expansion in y,~

=

P(x, y)

=

~

_~t~2/~(x)y,Ly~ 2...

y~,

(2.2)

n=2

where ~~iL2 ~(x) are symmetric tensor (density) gauge fields. The gauge fields .~(2)’ g(3), g(4),... are required to satisfy an infinite set of constraints, the first few of which are det(~~)=,

(2.3) (2.4)

~~~( ~vpu~ 4)

3pva.r 1 38~a~v13~(3) g(3)

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~‘N gravity

where ~ is the inverse of ~ ~ = 6~)and = ±1 is the signature of the world-sheet metric. For the n = 2 gauge field, the constraint (2.3) can be solved in terms of an unconstrained metric tensor g,,.~as = ~ where g = det(g~.), so that the term 4d,4 becomes the standard minimal coupling to gravity. If = + 1, this metric has euclidean signature while if E = 1 the signature is lorentzian. Alternatively, the single constraint det(~~) = e on the three components of can be solved in terms of two unconstrained functions h2(x), 12(x) g~V

—

which correspond to the two spin-two gauge fields of the previous section. Similarly, the constraints on the spin-n gauge field ~~2• ~“(x) can be solved either in terms of tensor gauge fields satisfying algebraic trace constraints, or in terms of two unconstrained functions, which can be identified with the gauge fields h~(x),i~~(x) [19]. The full set of constraints are generated by the following constraint on 2F(x y) det 8

(2.6)

=. yl).

YV

Expanding (2.6) in y generates the full set of constraints. This is the condition that F satisfies the real Monge—Ampère equation [21]. The action (2.1) is invariant under the local w 0, transformations ~4

=A(x,

34),

(2.7)

~ m,n=2

—(n

—

1)~$L1~2 ô~A(~)P~)

(m—1)(n—1) +

a(A~1~2...

g(~MP)_g(~1~2

(2.8)

y1

(2.9)

where A(x~,y)

=

~A~2_1(x)y~y~...

for some infinitesimal symmetric tensor parameters A~2~~~_1(x) which are required to satisfy the set of algebraic constraints generated by expanding

2

F,~~a 33A(x, y) =0 -

(2.10)

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in y, where ~ written as

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Geometric

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y) is the inverse of the matrix 1~~(x, y)

=

2/ay~dy~)E(x, y)

(a

~

and can be

y).

(2.11)

Using (2.6), the constraint (2.10) can be rewritten (for infinitesimal A) as

a a2a

det

-

(F+A)(x, y)

E.

(2.12)

y!.~ YP This constraint is necessary for the transformations to be a symmetry of the action [18,19]. As will be seen in the next section, the constraints (2.6), (2.10) can be solved to give a theory which, in the linearised limit, reproduces the linearised theory of the previous section. The action is also invariant under the local symmetries with parameters for q

(2.13)

~

~2~q

with all other fields inert. These are the analogues of the “Stuckelberg” symmetries of ref. [3] and reflect the reducibility of the one-boson realisation of w~. Nevertheless, most of the structure of the one-boson realisation developed in this paper carries over immediately to multi-boson realisations [28]which are nontrivial and do not have Stuckelberg symmetries. For further discussion, see refs. [18— 20,7,28].

3. The solution of the constraints The constraint (2.6) can be given the following geometrical interpretation [18,19]. Let ~, ~ (j.~.= 1, 2) be complex coordinates on ll~’.Then, for each xlL, a solution P(x, y) of (2.6) can be used to define a function ~ ~) on R” by ~

~)=F(x~,

~

(3.1)

For each x, K 1 can viewed as theofKähler potential for a Kähler metric 2Kx/a~~a~v on be R’~.As a result (2.6), each K~ satisfies the Monge—

G~= a equation det(G~)= e and so the corresponding metric is Kähler and Ampere Ricci-flat, which implies that the curvature tensor is either self-dual or anti-selfdual. In the euclidean case (~ = + 1), the metric has signature (4, 0) and is

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hyper-Kähler with an SU(2) holonomy, while in the lorentzian case (e = —1) the metric has signature (2, 2) and holonomy SU(1, 1). As the Kähler potential is independent of the imaginary part of the metric has two commuting (triholomorphic) Killing vectors, given by i(a/a~~a/a~).Thus the lagrangian F(x, Y) corresponds to a two-parameter family of Kähler potentials K 1~for (anti-)self-dual geometries on ~ with two Killing vectors. The parameter constraint (2.10) implies that F + A is also a Kähler potential for a hyper-Kahler metric with two Killing vectors. Two solutions of the constraint (2.6) were discussed in refs. [18,19] and both are related to twistor transforms. The first is the Legendre transform solution of ref. [26]. Writing y1 =~, y2 = ~, F(x~,~, ~) can be written as the Legendre transform with respect to ~of some A’, so that ~,

—

E(x,

~) =7r~—Z(x,

~,

IT, ~),

(3.2)

where the equation

ax-’ (3.3) gives ~r implicitly as a function of x, ~, Taking the Legendre transform has the remarkable property of replacing the complicated nonlinear equation (2.6) with the Laplace equation [26] ~.

2z

a

a2z

(3.4)

and the general solution of this is ~r+

f~~) +f(x,

~r-

f~),

(3.5)

where f, f are arbitrary independent real functions if = —1 and are complex conjugate functions if e = + 1. Then the general solution of (2.6) is the Legendre transform (3.2), (3.3), (3.5) and the action can be given in the first order form S=fd2xF(x, Y)

=

fd2x[IT~_f(x~,

IT+a,~)+f(x~, IT—a~)],

(3.6)

where T = x1 and o = ~/-~x2. The field equation for the auxiliary field IT is (3.3) and this can be used in principle to eliminate IT from the action, but it will not be possible to solve eq. (3.3) explicitly in general. The constraints (2.10) can be —

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~N

solved similarly. Expanding the functions f, f gives the hamiltonian form of the w,~ action [8]

S

=

—[h~(IT+a~~ +~fl(IT_a~)nJ).

Jd2x(ITa~ — ~

(3.7)

A related solution [18,19] that involves transforming with respect to both components of y~and maintains Lorentz covariance was suggested by the results of ref. [2] and their generalisation [3,4]. It will be useful to introduce a background “metric” h’-”(x) on the cotangent space, satisfying the constraint

det(h~”’)=.

(3.8)

This constraint can be solved in terms of an unconstrained background “metric” hp” by = [E

det(h~)] “2h~”.

(3.9)

(Note that J~”only determines h’~’up to a Weyl rescaling.) In refs. [18,19], this “metric” was chosen to be the flat metric J~”(x) =

h~’(x)=

v”,

(3.10)

but here it will be useful to allow a more general choice. Different choices will give equivalent results, but a judicious choice in which hg” transforms as a tensor density will be seen later to lead to manifestly covariant results. F(x’1, Y~)is written as a transform of a function H(x~,jr”) as follows: P(x~, Y~)~

(3.11)

where the equation

aH

y~,=—~

(3.12)

implicitly determines IT~ = IT~(x’~, ye). The transform again linearises (2.6) and F satisfies (2.6) if and only if its transform H satisfies 2

a2H a7T’~’a7r”= 1

______

(3.13)

and h~ satisfies (3.8). This is true for any “background metric” hg” satisfying (3.8).

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It will be useful to introduce a zweibein e~(a = 1, 2) such that h~ = = 1, IT2 by IT’~~ where 2e~e~2with = (1/ ~Xe~ ±e~),and define IT e = det(e~),together with the null coordinates IT±=(1/v~XIT1±~I—IT2) which are independent real coordinates for lorentzian signature (e = 1), and are complex conjugate coordinates for euclidean signature (e = + 1). The general solution of (3.13) can now be written as —

H—~e[IT~IT+f(x,IT~) +J(x,

(where J = f * if e = 1, but f, f are independent real functions if solution can be used to write the action S

=

~

—

—2ef(x,

IT~)

(3.14)

IT)],

~

=

—

1). This

/1,J~,,IT~ITV — ~-h’~~Y!.LYV

—

2ef(x,

IT)],

(3.15)

where is the inverse of h’”. The field equation for IT~ is (3.12), and using this to substitute for IT gives the action (2.1) subject to the constraint (2.6) (details are given in the appendix). Alternatively, expanding the functions f, f as 001 —h~(x)(IT~)~,

f= ~

f= ~

n=2~

—h~(x)(~)~

(3.16)

n= 2n

gives precisely the form of the action given in ref. [3], following the approach of ref. [2]. The parameter constraint (2.10) is solved similarly, and the solutions can be used to write the symmetries of (3.15) in a form similar to that given in ref. [3]. For example, the variation of 4. given by 64 = A(x, y) with Y~= becomes ~çb=A(x~, IT”) =A(x, 2IT1

—

Y(IT)),

(3.17)

h’”Y~.The constraint (2.10) on

whereY)Y(IT) found by aF/aY~ = linear Laplace equation constraint on A(x, then isbecomes thesolving following simple A(x, IT):

-

a2A alT aIT

~ =0.

(3.18)

The calculation leading to this result is given in the appendix. The part of H quadratic in IT is ~th~v1TMITv+h 2(IT+)2+h2(IT_)2],

(3.19)

and the terms involving h2, h2 consist of a background part (&‘~“)and a perturbation involving h2, h2. Different choices of IV”” correspond to expanding the full

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metric about different background metrics. The action (3.15) is invariant under spin-two transformations for any choice of background hILv; different choices lead to different transformation rules. For example, with the choice kL~= the action (3.15) becomes precisely that of ref. [3] and the transformations are those given in ref. [3]. If instead h is chosen to be a tensor density transforming as ~“,

~“

6h~”= kPa~).’-~~

+

—

h’~”a~k~

(3.20)

under spin-two transformations with parameter kM = A(2) and lT1~ is also chosen to be a tensor density, then hM” transforms as a tensor, ITa is a coordinate scalar, the first three terms in (3.15) are manifestly coordinate invariant and the remaining terms will be invariant if the gauge fields h~,h~are chosen to transform as scalars under coordinate transformations and as spin-n tensors under two-dimensional local Lorentz transformations. Then the part of H quadratic in IT is given by (3.19) and the terms involving h 2, h2 can be absorbed into a shift of hM”. After this shift, the action is given by (3.15), (3.16), in terms of the new shifted hMv, which is again a tensor density, but now with

h2=h2=0. As a result, by

~

(3.21)

the spin-two gauge field in the expansion (2.2) of F, is now given

(3.22) giving a formulation similar to that of ref. [2]. This has the advantage that the invariance under diffeomorphisms is manifest, although the shift of variables leads to a formulation in which the spin-two gauge fields are no longer on an equal footing with the higher-spin ones. For each xM, the variables h~(x),h~(x) parameterise the space of Kähler potentials K1 (given by (3.1)) which are solutions of the Monge—Ampère equation, so that the h~,/z~can be taken to be the moduli of self-dual metrics on II’~with two commuting Killing vectors. For the family of geometries labelled by the world-sheet coordinates XM, the moduli become functions h~(x),h~(x)of x~and these functions are interpreted as the gauge fields of w00 gravity.

4. Nonlinear

~‘N

gravity

tN gravity the discussion the in introduction, the linearised action ~ (i.e.From the action to linear in order the gauge fields) is an Nth orderfor polynomial in aM4~ given by (1.4). However, the full nonlinear action is nonpolynomial in the

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gauge fields and in aM4, but the coefficient of (a4Y’ for n > N is a polynomial function of the finite number of fundamental gauge fields that occur in the linearised action. The simplest way in which this might come about would be if the action were given by (2.1), (2.2) and F satisfies a constraint of the form

a’~P aYMaYM 2.. .ayMNl

-

=

0 + 0(F2)

(4.1)

where the right hand side is nonlinear in F and its derivatives, and depends only on derivatives of F of order N 7~N or less. It will be shown in this section that this is gravity is given by (2.1) where F satisfies (2.6) indeed the and case;the the right actionhand for side of (4.1) will be given explicitly. Just as the and (4.1), nonlinear constraint (2.6) had an interesting geometric interpretation, it might be expected that the nonlinear form of (4.1) should also be of geometric interest. It is essential that (4.1) should be consistent with the Monge—Ampère constraint (2.6). In the last section, the action for w,, gravity was given in terms of a function IT, ~) satisfying (3.4) or a function H(xM, ITM) satisfying (3.13). It follows from the results of refs. [2—4,8]that these same actions can be used for “N gravity provided that the functions A’ or H are restricted to be Nth order polynomials in IT or 1PM. The canonical first order form of the ~~-gravity action is then given by (3.6) where A’ (3.5) satisfies (3.4) and

a~‘A’ aIT~’~=0,

(4.2)

so that expanding the functions f, f gives the action (3.7), but with the summation now running from n = 2 to n = N [8] so that there are only a finite number of gauge fields h~,h1., where n = 2, 3,. N. Similarly, the covariant first order form of the action is given by (3.15), (3.16) where H satisfies (3.13) and . .,

a’~‘H 02 .

alT MIaITI.

. .

aIT~”÷~ =0

(4.3)

so that H (3.14) is given by (3.16), with the summation running from n = 2 to n = N [2—4].Again, this leaves a finite set of gauge fields, h~,h~where n = 2, 3,. . N for 7~~Ngravity, in agreement with the linearised analysis. It is remarkable that the constraints defining 71’N gravity (3.4), (4.2) or (3.13), (4.3) are simple linear equations when written in terms of the IT variables. This can be understood in terms of the relation [26] between the transform from E(x, Y) to A’(xM, IT, 40 or H(xM, ITM) and the Penrose transform, which translates the condition that a geometry be self-dual into a linear twistor-space condition. .,

—

—

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The Laplace equations (3.4), (3.13) become the Monge—Ampère equation (2.6) when written in terms of F and it is this equation which characterises w,,, gravity. The 71”N condition, which is a complicated nonlinear constraint on E, becomes the simple linear constraint (4.2) or (4.3) that the transform A’ or H is an Nth order polynomial in IT. 4.1. THE CONSTRAINTS ON F

Eqs. (3.11), (3.12) give F implicitly in terms of the function H and these can now be used to relate derivatives of F to those of H. It will be useful to introduce the notation

a”H H

=

aITMIaITM2. . .8IT~

FM1M2

M,,

=

a”F ayMaYM.. •3~M

(4.4)

and to define the inverse HMV of the “metric” HMV(x, IT), so that HMVHV~= Differentiating (3.11) twice with respect to y and using (3.12) and aITM

(4.5)

ay~

gives FTM”

= _M”

which can be used to give the “metric”

+

1~1M”

2hM”,

(4.6)

in terms of F and the metric hM”: (4.7)

Further differentiation yields FM”P = FMVPff = FMVPUT =

_2HM~HPPHPYHapy, —

2HMäHVl!~HPYH~~Hai~yo + ~HapFa(M~FP~~P,

—

2HMaHvPHP7H0&HTEHa~y~,+ 4Ha~Fa(MvFP~~)P

—

~HapHy,~Fa(MvFPo~IYIFT)l~,

(4 8) (4 9)

(4.10)

and it is straightforward to extend this to any number of derivatives (see the appendix for details).

G.M Hull

Consider first ~T3 gravity. For N

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3, eq. (4.3) becomes =

0,

(4.11)

and using this (4.9) becomes FTM”PO~= ~

(4.12)

or, using (4.7), FMVP~=

3(h~’~ + FaP)’Fa(MvFP~)P.

(4.13)

This is the required extra constraint for 7f3 gravity. Thus the action for 7/’~ gravity is given by (2.1), (2.2), where F is a function satisfying the two constraints (2.6) and (4.13). Similarly, for ~ gravity, ~ = 0 and (4.10) becomes T

=

4Ha~Fa(MvFPUT)P— ~HapHyaFa(TMvFPUIYIF~)Pa,

(4.14)

FM”PI

c

so that the 7J’~ action is (2.1) where satisfies (2.6) and (4.14), and HTM” is given in terms of F by (4.7). Similar results hold for all N. In each case, taking the transform of the linear constraint (4.3) yields an equation of the form (4.1), where the right hand side is constructed from the nth order derivatives FM!.. Mo for 2 N, the coefficient of the nth order interaction can be written in terms of the coefficients ~(m) of the mth order interactions for 2 ~ m ~ N. For 7/~,the n-point vertex can be written in terms of three-point vertices for n > 3, so that .

~~Pu2(haP

(4.15)

~

s(~~ +g~)1(J~8~

(4.16)

TM~VP~T_

etc., while for vertices, e.g.

all vertices can be written in terms of three- and four-point

~,

=

g(

5( jafi +

5)

af3~—1

g(2) ) -

a(TM” prr)$

g(3) g(4)

s(~+~)‘(~

~

(4.17)

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v

h~:I~ ~I

Fig. 1. The four-point interaction in ~r

3 gravity can be written in terms of three-point interactions. The diagram represents this factorisation, with the symmetrization over the indices corresponding to the “sum over channels”.

Fig. 2. This diagram represents the factorisation of the five.point interaction into three-point interactions in ~ gravity, with the summation over channels suppressed.

These “factorisations” can be illustrated in Feynman-style diagrams. (4.15) is depicted in fig. 1, where the “propagators” represent contraction of indices using the metric j~jTM”~Similarly, (4.16) and (4.17) are depicted in figs. 2 and 3 respectively, where the ‘summation over channels’ is not shown explicitly.

Fig. 3. This diagram represents the factorisation of the five-point interaction into three-point and four-point interactions in ~ gravity, with the summation over channels suppressed.

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4.2. THE CONSTRAINTS ON A

From the linearised analysis, it is expected that the ~~-gravity invariant under transformations under which 64=A(x,a4),

action should be

(4.18)

where A(x, Y) is of the form (2.9) and satisfies constraints whose linearised forms are

a2n ayTMayV

=0+...

(4.19)

and aNn aYTM1aYTM2.

=0+....

(4.20)

.

The full nonlinear form of the constraint (4.19) is given by (2.10), while the nonlinear form of (4.20) will give the parameters A 12 Mo-i for n > N in terms of the parameters A~2 TM.o-! for m ~ N and the gauge fields, so that the number of independent symmetries is the same as in the linearised theory. The full non-linear form of these constraints will now be found by transforming the corresponding constraints in the covariant first order form of the theory. The covariant first order form of the w 00 action, given by (3.15), (3.16) where H satisfies (3.13), is invariant under transformations given explicitly in ref. [3] and which include (3.17) where A(x, 11-) satisfies the constraint (3.18). It follows from the results of refs. [2—4]that the truncation to ~~“N gravity is obtained by imposing the constraint (4.3), so that H is an Nth order polynomial in IT, together with the constraint .

aITM1aITTM2.

on A(x,

=0 . .

(4.21)

aIT’~’

so that A(x, IT) is an (N— 1)th order polynomial in IT. In addition, the constraints on the gauge fields given by (4.3) or (4.13), (4.14) etc. are not invariant under the A transformations, but they become invariant if the A transformations are supplemented by compensating ‘Stuckelberg’ transformations, as in ref. [3]. Now, using the chain rule and (4.5), it is straightforward to express derivatives of A(x, IT) with respect to IT in terms of derivatives of A(x, y) with respect to y. IT),

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For example,

an

an =

a2n aITMaIT”

a3A aITMaIT”al,.P

a2n

=

an HH+—H ayaaYp aY~ ~“

a3n =

an

~

+

+3

a2n H H ay~aY~a(M” p)!3

a4n

a4n

an

aITTMaIT”aITPaIT~ = ayaaypaYyaYa HaMHpvHypH~+ ~~haTMvPcr

+

a3n 6aaa

+

a

Consider first the case of ~ on n(x, IT) given by

Ya Y!3

Ha(TMVHI ~

(3Ha(TMVHpU)!3 + 4Ha(~vpH,j.)p).

(4.22)

gravity. In this case, HTMVPO. = 0 and the constraint

a3n aIrTMaIT”aIT~ =0

(4.23)

can be rewritten in terms of A(x, y) and E(x, y) using (4.22) as

a3n aYaaY!3ayy

=

3 —

a2n

(4.24)

~

2 aYTMaYP P

TM”

where ~jTM” = 2(hTMv + FTM”)~ as before. This constraint gives all of the parameters A(~)for n = 4, 5, 6,... in terms of the gauge fields, the diffeomorphism parameter A~)and the spin-three parameter A~which satisfies the tracelessness constraint (~~‘Y ‘A~= 0. For example, for spin-four the constraint implies that =

2(~P~

(4.25)

G.M Hull

/ Geometric

gravity

~‘N

311

The 7J7~constraint (4.13) gives a sequence of constraints on the gauge fields, the first two of which are given by (4.15) and (4.16). These constraints are not invariant under the gauge field transformations, which are now given by (2.8), but with all the parameters for A~ for s > 3 given in terms of the s = 2 and s = 3 parameters. They do become invariant, however, if the spin-two and spin-three transformations are supplemented by a compensating Stuckelberg transformation of the form given by (2.13) with p = 4, q = 2. The transformations of g~VPare unmodified, but the spin-three transformation of ~, which previously was zero, now becomes modified to =

(4.26)

~A(M”PU)Y~Y,J.,

where ATM”P°— ~TM”3 —

~°~—

3 (3) a g(3)

(3)

~

aTMvpo~~ a!3~(3) g(2) ~

+

~

+

2G~A

(3)

3g(3)

a (3)

—

3

3 “~ (3) a g(3)

yTM),~a~, ap~(2) (3)

3

a~

±ILvP~9 ~o~a L

~~‘~-!-

(3) 1aV...2G

3a YMPO1 y

+2G 2 3

lOaMva 3 g(3) a

pua

~a~(2) ~Gapg

9 P0P_~G aTMvcrpa ~ ~g(3) 3 a!3~(3) g(2) y (3)

(Mv~ya a~~(3) (3)

~“(A~a~gj3j~’ + 2A~a~~~)

g~a~g~ + ~

+

2A~r)aa~~.

(4.27)

Here GM~=HTM~~Yo= 2(&TM”

(4.28)

+g~)’.

This unpleasant form of the transformations simplifies dramatically if we choose a general hTM”, and absorb h2, h2 into field redefinitions, so that hTMv = and = (,~TM”Y Then the frame components A’~’~of (4.27) are given by ~.

A++±±’,~++r +±±‘)±±+17 ~+± — ~./L(3) 1’ +g(3) g(3) V +1t(3)

A~~=2A~V~ ~ In the chiral limit Similarly, for ~ constraint

—

,

2~V_A~, 0.

(4.29)

= 0, the transformation rules of ref. [1] are 4n/aITTMaIT”aITPaIT~ = 0recovered. leads to the gravity, the constraint a

= 3HTMvFTM(*!3A~~ —

+

~

2HTMV FTM(a!3Yn~)v 3HMP Ha,, F~~~p(aF!3YI In&)TM —

(4.30)

312

G.M Hull

/

Geometric ~‘N gravity

where

a”n nTMIM2

.100

(4.31)

.

YM!

The only independent parameters are the constraints

Y~2... Y100

A~),A~and ~

and these are subject to

1TM”fl 5(2)TM~ (3)

-

3g(3) (3)g(2)10~g(2)~,.,

ATMvP=TM”P~ (4)

where g(2)10~ (j~) ~. As in the ~ case, the transformations of the gauge fields are again modified by compensating Stuckelberg-type transformations. 4.3. THE GEOMETRY OF THE CONSTRAINTS

To attempt a geometric formulation of these results, note that while the second derivative of P defines a metric, the fourth derivative is related to a curvature, and the nth derivative is related to the (n 4)th covariant derivative of the curvature. The ~ constraint (4.13) can then be written as a constraint on the curvature, while the ~F’N constraint (4.1) becomes a constraint on the (N 3)th covariant derivative of the curvature. One approach,4 (for motivated byEthat of sect. 3, is to each xM M) given in terms of introduce a second Kähler metric K1 on P the potential K 1 introduced in (3.1), by —

—

K1 = K1 + haP~~

(4.33)

The corresponding metric is given by GTM~=

Then if F satisfies the (4.34) satisfies =

(4.34)

hTMI~+ G10”.

constraint (4.13), the curvature tensor for the metric

~

~

+ ~

+ Tp~TMTaP~ + Tl~TMT~oPv),

(4.35)

where TTM”P

a~i~ =

-

,

Ti”’” =

a~j~ ~

.

(4.36)

G.M Hull

/

313

Geometric ~‘N gravity

This is similar to, but distinct from, the constraint of special geometry [27]. Note that (4.35) is not a covariant equation as the definitions (4.36) are only valid in the special coordinate system that occurs naturally in gravity. However, tensor fields TTMV!3, ~ can be defined by requiring them to be given by (4.36) in the special coordinate system and to transform covariantly, in which case eq. (4.35) becomes covariant, as in the case of special geometry [27]. For ~N’ this generalises to give a constraint on the (N 3)th covariant derivative of the curvature, which is given in terms of tensors that can each be written in terms of some higher order derivatives of the Kähler potential in the special coordinate system. For each XTM, the solutions to the constraints for ~“N gravity are parameterised by the 2(N 1) variables, h~,h~for 2
—

—

—

constraint. For the x-dependent family of solutions, the moduli become the fields h~(x),~~(x) on the world-sheet. Further properties and generalisations of these actions will be given elsewhere.

7’N

Appendix A The background “metric” ~~M”’ which satisfies det(hTM”) = terms of an unconstrained “metric” hTM” as =

where

h=e

det(hTM”).

~/)~h~”

,

can be written in

(A.1)

Then F can be written as

P

=

v~(2~MY 10 ~hTM”Y10Y,, 2A’), —

—

(A.2)

where A’=h’~2H.

2IT10,

(A.3)

~.TM =h’/

It is useful to introduce a zweibein e,~(a such that

hTM” where

~qcth

=

1, 2) (with inverse e~,and e = det(e~))

= flabe1Le~=

2e~ei2,

(A.4)

is the flat metric given by diag(E, 1), and

=

±I~e~).

(A.5)

We define flat null coordinates IT’~= e,~frM= ee~ITTM,so that IT ± are independent real coordinates if the signature is lorentzian ( = 1), and are complex conjugate —

314

/

G.M Hull

Geometric ~N gravity

coordinates (IT~= (IT_)*) if the signature is euclidean (e = flat metric in the IT ± coordinate system is ?lab~ (0

~

+

1). In either case, the

i\ ot

(A.6)

The lagrangian P(x, Y) is given by a transform of a function H(x, general H(x, IT), the second derivative

2H

HP.”

a aIr~ IT” =e~A’10”

can be written in terms of

IT).

For

a2A’

A’10”

(A.7)

x-cb’

h

______

H 10~=eA’abe~e~,A’ab= aITaaIrb

=

(a

a

~)‘

(A.8)

for some a(x, IT), h(x, IT), h(x, IT). The constraint (3.13) becomes 1,

!~abx-~~...A’_

(A.9)

and the general solution of this is H=eA’=e[ITr+f(IT~)

+f(ir)J,

(A.10)

TM has been suppressed. Differentiating twice with respect where dependence on x to IT gives A’ab which, in the IT ± frame, takes the form A’ab=(h

(A.11)

~),

where a2f

a~j

±2 a(IT)

and a

=

(A.12)

a(IT)

1. It will be convenient to define

4=1—h/i. In the

IT ± frame,

(A.13)

the determinant of (A.11) is det±(A’,~b) =

—4

=

—1

+hh,

(A.14)

G.M Hull

while in the

IT1, IT2

/

Geometric

315

7~Ngravity

frame the determinant is det(A’ab)

= —E

det±(Hab)

(A.15)

ELI.

(The sign changes result from the fact that the jacobian for the change of coordinates from IT1, IT2 to IT ± is f.) The inverse of (A.11) is —

1—h~(

A’=ele~e~HTM”=

—h)’

~

HTM”~(H

1.

(A.16)

10~)

From (4.6), (A.6), (A.11), the second derivative of F is

FTM”

=

ee~eb”F,

(A.17)

where Fab=

_~ab+

2h 1+hh

1 1—hh

2A’ab

—

1+hh —2h

(A.18)

and the inverse of this matrix is (Fab)’

=

(

1 2h 1—hhU+hh -

1 +hh

2h

(A.19)

The determinant F”~’isUsing —1 in the ITthis ± coordinate so is satisfied. in the 1, IT2 coordinateofsystem. (A.17), implies thatsystem (2.6) and is indeed IT Now, using (A.16), (A.6), A’aCdA’~

1

=

—

(1—h/i)

2

—

2~ 1 + hh

1+hh

(A.20)

—2h

so that (A.18) gives =

4A’a~~CdA’~Th

(A.21)

and (F~’)~=4_lA’ac7lCdA’db.

The variation of çb is given by a function n(x, the constraint (3.18). This can be rewritten as

ITTM)

(A.22)

by (3.17), where

n satisfies

a2n ~ab

a

aIT aIT

b =

0.

(A.23)

316

/

G.M Hull

Geometric VN gravity

Now the second relation in (4.22) can be written as

a2n

a2n

aITaaIrt~

=

aYCaYd

an

A’ A’bd

+

a3~ .

ay~aITh2aITt~aITc

ac

(A.24)

(Ya = e~Y

10),and taking the trace gives 2n a2n flabb flabZA’bd a

(A.25)

using (A.9). Then the constraint (A.23) becomes

a2n

,

(F~’) aYaaYb

=

0,

(A.26)

using (A.22), and this is equivalent to the parameter constraint (2.10). We now turn to the identities satisfied by the derivatives of F. From

P(xM, y~)=

2ITTMY 10

~/iTMVY10Y~ 2H(x,

—

—

IT)

(A.27)

and

aH y10

~j—~

(A.28)

TM—hM”y~,

(A.29)

=

it follows that

aF

-

FM~ YTM =2IT ~—

alT10

a2F FTM”

=

3y

10ay~ FTM”~

3P

a aY

—

__________

aY10...aY10

(A.30)

a2ITM =

2

(A.31) ay~ay~

a~E TM~

hTM”

ay~

10aY~aY~

•

-

2—

a’~’ITTM1 =

2

aY10...aY10

.

(A.32)

Then differentiating (A.28) gives aY aIT”

=

2H ~H a aITTMa7J-” TM”

(A.33)

G.M Hull

/

Geometric ~‘N gravity

317

and hence aITTM —1 aY~ = (HTM” )

HTM”.

(A.34)

Differentiating this gives aITTM

a

_ 1aIT~

aY~aY~ aIT~ TM” 3H

=

_HMaH~PHPY a . aITc0aITPaIT~~

(A.35)

Substituting this in (A.31) gives (4.8). Differentiating (A.35) gives aITTM

a =

—

aY~aY~aY~ aITT =

—

HMaH”PHPV

a3~

aIrT

aIT”aI7-~aIT~ aY~

HTMaHv!3HPYH~~Hapy~ + 3HK(a!3 ~

(A.36)

and this leads to (4.9). It is straightforward to generalise these relations to higher derivatives, and also to represent the results graphically in figures similar to figs. 1—3 in which HTM 1TMZ •M~ is represented as an n-point vertex and HTM” is represented as a propagator.

References [1] CM. Hull, Phys. Lett. B240 (1990) 110 [2] K. Schoutens, A. Sevrin and P. van Nieuwenhuizen, Phys. Lett. B243 (1990) 245 [3] E. Bergshoeff, C.N. Pope, L.J. Romans, E. Sezgin, X. Shen and K.S. Stelle, Phys. Lett. B243 (1990)

350 [4] G.M. Hull, NucI. Phys. B353 (1991) 707 [5] G.M. Hull, Phys. Lett. B259 (1991) 68; Nucl. Phys. B364 (1991) 621; A. Mikovié, Phys. Lett. B260 (1991) 75 16] K. Schoutens, A. Sevrin and P. van Nieuwenhuizen, NucI. Phys. B349 (1991) 791; Phys. Lett. B251 (1990) 355; E. Bergshoeff, C.N. Pope and KS. Stelle, Phys. Lett. B249 (1990) 208 [7] C.M. Hull, in strings and symmetries 1991, eds. N. Berkovits et al. (World Scientific, Singapore, 1992); Lectures on W-gravity, W-geometry and W-strings, Trieste Summer School Lectures 1992 (World Scientific, Singapore) to be published [81 A. Mikovi~,Phys. Lett. B278 (1991) 51 [9] P. Bouwknegt and K. Schoutens, CERN preprint CERN-TH.6583/92, Phys. Rep., to appear [10] C.N. Pope, L.J. Romans and KS. Stelle, Phys. Lett. B268 (1991) 167; B269 (1991) 287 [11] E. Witten, in Proc. Texas A&M Superstring Workshop 1990, eds. R. Arnowitt et al. (World Scientific, Singapore, 1991) [12] A. Bilal, Phys. Lett. B249 (1990) 56; A. Bilal, V.V. Fock and I.!. Kogan, NucI. Phys. B359 (1991) 635

318

G.M Hull

/

Geometric

71tN gravity

[13] G. Sotkov and M. Stanishkov, Nucl. Phys. B356 (1991) 439; G. Sotkov, M. Stanishkov and C.J. Zhu, NucI. Phys. B356 (1991) 245 [14] M. Berschadsky and H. Ooguri, Commun. Math. Phys. 126 (1989) 49 [15] P. Di Francesco, C. Itzykson and J.B. Zuber, Commun. Math. Phys. 140 (1991) 543 [161J.M. Figueroa-O’Farrill, S. Stanciu and E. Ramos, Leuven preprint KUL-TF-92-34 [17] J.-L. Gervais and Y. Matsuo, Phys. Lett. B274 (1992) 309; Ecole Normale preprint LPTENS-91-351 (1991); Y. Matsuo, Phys. Lett. B277 (1992) 95 [18] CM. Hull, Phys. Lett. B269 (1991) 257 [19] C.M. Hull, Commun. Math. Phys. 156 (1993) 245 [20] CM. Hull, Geometry and v-gravity, QMW preprint QMW-92-21 (1992) [hep-th/9301074], Proc. 16th John Hopkins Workshop on Current problems in particle theory, Gothenborg, 1992, to be published [21] T. Aubin, Non-linear analysis on manifolds. Monge—Ampère equations (Springer, New York,

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579 [26] U. Lindstrom and M. Roëek, Nucl. Phys. B222 (1983) 285; N. Hitchin, A. Karlhede, U. Lindstrom and M. Roèek, Commun. Math. Phys. 108 (1987) 535 [27] S.J. Gates, Nucl. Phys. B238 (1984) 349; M. Gunaydin, G. Sierra and P.K. Townsend, Nucl. Phys. B242 (1984) 244; B. de Wit, P.G. Lauwers and A. van Proeyen, NucI. Phys. B255 (1985) 569; A. Strominger, Commun. Math. Phys. 133 (1990) 163; L. Castellani, R. D’Auria and S. Ferrara, Phys. Lett. B241 (1990) 57; Class. Quantum Gray. 7 (1990) 1767 [28] C.M. Hull, in preparation