22 September 1994

PHYSICS LElTERS B

Physics Letters B 336 (1994) 183-189

A canonical approach to duality transformations

Enrique Alvarez ‘, Luis hvarez-Gaum6, Yolanda Lozano ’ Theory Dtvtslon, CERN, 1211 Geneva 23, Swrtzerland

Recemd 1 July 1994 Edltor R Gatto

Abstract

We show that Buscher’s Abehan duality transformation rules can be recovered in a very simple way by performmg a canomcal transformation first suggested by Glveon, Rabmovicl and Venezlano We explore the properties of this transformation, and also discuss some aspects of non-Abehan duality

1. Introduction

(Target space) duality, the generahzatlon of the well-known R-duality in toroldal compactlfications m strmg theory [ 11, remains a promlsmg avenue to fur- ther understand the underlying symmetnes of strmg

theory. The most powerful description of this symmetry was

first introduced by Buscher [ 21, and then further elab-

orated upon by many others [ 4-81. It starts wltb the sigma-model formulation of the correspondmg conformal field theory (“string vacuum”), and it works when the target space metric has at least one continuous uometry. Even m this case, however, the procedure looks unnecessarily complicated the isom- etry 1s gauged, the (non-propagating) gauge.fields are constramed to be trivial, and the Lagrange multipliers themselves are promoted to the rank of new coordmates once the Gaussian integration over the gauge fields 1s performed One suspects that all those complicated intermediate steps could be avoided, and that it should

’ On leave of absence from Departamento de F&a Te6nca. Umv- emdad Autdnoma de Madnd, 28049 Madnd, Spam

0370-2693/94/$07 00 0 1994 Elsevler Science B V All nghts reserved

SSDIO370-2693(94)00982-l

be possible to pass directly from the ongmal to the dual theory.

Some suggestions have indeed been made m the literature pomtmg (at least m the slmphfied situation where all backgrounds are constant or dependent only on t) towards an understanding of duality as particular instances of canonical transformations [ 91.

Our aim m this paper IS to follow this lead. We shall find that It works well when the lsometry is Abelian, laying the theory on a simpler settmg than before, namely as a ( pnvleged) subgroup of the whole group of (non-anomalous, that 1s implementable m quantum field theory [ lO-121) canonical transformations on the phase space of the theory.

We shall be able to perform the transformation start- mg from arbitrary coordmates (and wrltmg the dual manifold m arbitrary coordinates as well) and, besides, we will also follow explicitly the transformations of the currents of both the initial and the dual model. In

particular in the case of WZW models [ 131 it becomes rather simple to prove that the full duality group is given by Aut(G),X Aut(G),, where L, R refer to the left- and right-currents on the model with group G, and Aut(G) are the automorphisms of G, both inner and outer. Due to the choral conservation of the currents m

184 E Alvarez et al /Physm Letters B 336 (1994) 183-189

this case, the canomcal transformatton leads to a local expression for the dual currents In the case where the currents are not chnally conserved, then those currents associated to symmetries not commutmg with the one used to perform duahty become genertcally non-local

m the dual theory and this 1s why they are not manifest m the dual Lagrangtan [ 2 11.

It is, however, somewhat disappomtmg to have to report that we have not been able to understand non- Abehan duality m terms of canonical transformations. Once again m the case of WZW-models, due to the

presence of chorally conserved currents one can com- pute the dual theory m terms of a set of non-local variables introduced m [ 81. This procedure unfortu- nately only works for theories with chnal currents, and even for that case we lack a Hamtltoman formulatton of the transformatton

2. Abelian duality and canonical transformations

We start wtth the bosomc sigma model wntten m

arbitrary coordmates on a manifold M with Lagrangian

L=~(g,,+b,,)(~)a+~“a-~“, (2.1)

where X* = (rfg)/2, p, V= 1, , n=dtm A4. The correspondmg Hamiltoman 1s

+gcLyFP#“) 3 (2.2)

where c$‘@= dcpc”lda We assume, moreover, that there 1s a Ktllmg vector field kw, _TkgPy=O and I&= - du for some one-form v, where (zfi) I.Ly=

kPHp,, and H = db locally This guarantees the exts- tence of a parttcular system of coordinates (“adapted coordmates” ), which we shall denote by xa = ( 0, x’), such that k = a/ a 8. We shall denote the Jacobtan matrix by e> = &Pla+r In practtce, this means that we have to complete the Ktllmg vector k = kPala+r wtth (n - 1) other commuting vectors m order to form a holonomtc n-hem. [k, e,] = [e,, e,] = 0

This then defines a point transformation m the ortg- ma1 Lagrangtan (2.1) which acts on the Hamiltonian as a canonical transformatton with generatmg function @=x”( +)p,, which yields.

(2.3)

Once m adapted coordmates we can wnte the sigma model Lagrangtan as

L=;G(~*-P*)+(8)+@)J_ +(&/Y)J+ +v,

(2 4)

where

G=g,=k*, v=;(g!,+b,)a+xla_x-l,

J.- =~(g,,+b,,,)a-xl, J, =f(go,-b&3+x

(2.5)

In finding the dual with a canonical transformatton we can use the Routh function with respect to 0, 1 e. we only apply the Legendre transformatton to ( e, 6). The canonical momentum 1s given by

p@=Gi+(J+ +J_) (2 6)

and the Hamtltoman

H=p,&L=$G-‘p2,-G-‘(J, +J_)p,+$Ge’*

+$G-‘(J, +J_)*+e’(J+ -J-)-V (2 7)

The Hamilton equattons are

i= $f =G-‘(pe-J, -J_), e

co= - f$ = (Ge’+J+ -J_)‘, (2.8)

where the last equation is equivalent to current conser- vation

The current components are:

Y+ =~Ga+e+J+ =+pe+$Ge’+f(J+ -J_),

AC = +Ga _ e+J_ = ;ppB- ice’- f(J+ -J_)

(2.9)

Then a-s, +a+s_ =Oede= -SHISe The

generator of the canomcal transformation we choose 1s:

F=f

I

di?Ade= 4 9

(erg- e@)du,

D,JD=Sl Sl

(2.10)

that is,

SF 6F pe=SB=-Jr, pa=-z=_e~ (2.11)

E Alvarez et al /Physm Letters B 336 (1994) 183-189 185

This generating functional does not receive any quan- tum corrections (as explained m [ lo] ) since it is linear m 6 and 8. If 13 were not an adapted coordmate to a contmuous isometry, the canonical transformation would generically lead to a non-local form of the dual Hamiltoman Since the Lagrangian and Hamdtoman m our case only depend on the time- and space-derivatives of 19, there are no problems with non-locahty. Below we will show how to transform wave functionals from the original to the dual theory

The dual Hamiltonian is:

H=$G-~P+G-~(J+ +J_)&+;G~$

-(J+ -J_)ps+fG-‘(J, +J_)*-V. (2.12)

Since:

6= g =Gps-(J, -J_) , 0

(2.13)

we can perform the inverse Legendre transform*

~=~G-‘(8*_~r*)+G-1J+(B-8;))

-G-‘J_@+@)+V-2G-‘J+J_. (2.14)

When translated m terms of gab, b,, this is nothing but a Buscher transformations *

1 I

b 01 ho’ -9 go,= - -9

ho ho

s =g

I/ u

_ go,go,-both,, g 01

= _ g,,

&3x3 &ho ’

g =b _ go,bo,-ad’,

13 V &lo

(2 15)

For the dual theory to be conformal invariant the dilaton musttransformas a’=@-logg, [2,3].

Some useful mformation can be extracted easily m the approach of the canonical transformation:

-From the generatmg functional (2 10) we can learn about the multivaluedness and periods of the dual var- iables [ 61. Smce 8 is periodic and m the path integral the canonical transformation is implemented by [ lo] :

&[i&T)] =N(k) j ,e(,)e~~[~,~(~)l~~[e(~)] ,

(2 16)

* The mmus signs m go, and &, can be absorbed m a redefinmon 3 Note that the canonical transformed current does not comclde m e-+-e general wth the dual current

where N(k) is a normalization factor, 4k(e+a) =4kk(e) impliesfor S &(T+~T) -8(o) = 47r/a, which means that fi must live m the dual lattice of 8. Note that (2.16) suffices to construct the dual Hamtltoman. It is a simple exercise to check that acting with (2.12) on the left-hand side of (2.16) and pushing the dual Hamdtoman through the integral we obtam the original Hamiltonian acting on +J e( (T) ] :

(2.17)

This makes the duality transformation very simple con- ceptually, and it also imphes how it can be applied to

arbitrary genus Riemann surfaces, because the state $,J 0( @ ] could be the state obtained by integrating the original theory on an arbitrary Riemann surface with boundary. It is also clear that the arguments gen- eralize straightforwardly when we have several com- muting isometries.

- We can easily see that under the canonical trans- formation the Hamilton equations are interchanged

de= - ; =(GW+J+ -J-)’

+8=GpB-J+ +J_ ,

i= E =G-‘(pe-J, -J_) e

+ds=(G-l(@+~+ +J_))’ (2.18)

The canonical transformed current conservation law is m this case equivalent to the first Hamilton equation*

(2.19)

(where we denote the canonical transformed current by 7 y ) whereas the dual current 3 conservation law is equivalent to the second Hamilton equation.

- In the chual case J- = 0 (i.e. got = -b,) and G is a constant, therefore we can normahze 8 to set G= 1 and:

L=~(i*-er*)+(&-ey+ +v

The Hamiltoman is

(2.20)

186 E t&zrez et al /Physrcs Letters B 336 (1994) 183-189

H=;p&T+pB+f(J+ +8’)*-V. (2.21)

The action is mvartant under S8= a(~+), a U( l)L.

Kac-Moody symmetry. The U( 1) Kac-Moody alge- bra has the automorphism Y+ + - Y+ This is precisely the effect of the canomcal transformatton. The equation of motion or current conservation is

a_(a+e+J+)=o (2 22)

Y + = a + 0 + J, =pe + 8’ transforms under the canon- ical transformation m SC: = - 3’ -pi = - 7 +

- We can also follow the transformation to the dual

model of other contmuous symmetries. The simplest case 1s as usual the WZW-model [ 131 which 1s the basic model with chnal currents. Consider for stmphc-

tty the level-k SU( 2)-WZW model with action

S[g] = -$ d*oTr(g-‘a+gg-‘a-g) I

+ ST Tr(g-‘&)3f I (2.23)

parametrrzmg g m terms of Euler angles,

g=e ram/* &3a/2 erYd2 9

the left- and right-choral currents are

(2 24)

~-=-+-g.

Exphcttly for 7 +

(2 25)

s: = &3+a+cosga+y), (2.26)

and simtlarly for the right currents If we perform dual- ity wtth respect to (Y+ (~+const. we know that s: + -y-+, 7 3 -tY 3. For these currents it is easy to find the action of the canomcal transformatton because only the dertvattves of (Y appear For 7 ‘;” there is an exphcn dependence on cz and it seems that

the transform of these currents 1s very non-local. How- ever, due to the choral nature of these currents, we can show now that there are similar chtrally conserved cur-

rents m the dual model To do this we first combme the currents in terms of root generators,

s(,+,=s: +19-:

=e -‘“(la+p-sinpa+r)=e-‘“j(,+),

s(,-) =y’ -19-2 + +

= -e’“(la+p+sinpa+~)=e’“~~-), (2.27)

from choral current conservation a _ 7 !+* ) = 0 we obtam

a-J:*‘= fza_c!j\*), (2.28)

m these equations only 6, (Y’ appear, and after the canonical transformatton we can reconstruct the dual non-Abelian currents (m the previous equations the canomcal transformatton amounts to the replacement (Y + 6) which take the same form as the ortgmal ones except that with respect to the transformed 7% the roles of posmve and negattve roots get exchanged One

also verifies that the 7 ?. are unaffected This tmphes therefore that the effect of duahty with respect to shifts of (Y 1s an automorphtsm of the current algebra amount- mg to performmg a Weyl transformatron on the left currents only while the right ones remain unmodified. This result although known [ 51 IS much caster to derive m the Harmltonian formahsm than m the Lagrangran formahsm where one must introduce external sources which carry some ambigumes. The constructton for SU(2) can be stratghtforwardly extended to other groups This imphes that for WZW-models the full duality group is Aut( G), X Aut( G)R, where Aut( G) 1s the group of automorphisms of the group G, mclud- mg Weyl transformations and outer automorphrsms For instance, tf we take SU(N), the transformatton .I++-J’,, 1.e charge conlugation, follows from a canonical transformatton of the type discussed It suf- fices to take as generating functions for the canomcal transformation the sum of the generating functions for each generator m the Cartan subalgebra. External auto- morphtsms have been used recently to extend the notion of duality [ 18-201 It is important to remark that the choral conservatton of the currents is crucial to guarantee the locality of the dual non-Abehan currents If the conserved current with respect to which we dual-

E Alvarez et al / Physm ktters B 336 (1994) 183-189 187

tze is not chually conserved locahty is not obtained. The simplest example to verify this is the principal chual model for SU(2). Although it IS not a CR, tt serves for illustrative purposes. The equations of motion for this model imply the conservatton laws

a-9:a+a,9-:=o,

where

(2 29)

Sf= $+gg-i (2.30)

If we perform duahty with respect to the mvarrance under (Y translations we know how 7: transform, since they are the currents associated to the isometry. With the canonical transformation is possible to see as well which are the other dual conserved currents. Smce the dual model is only U( 1) -mvartant [ 61 one expects the rest of the currents to become non-local [ 211 if they exist at all. In terms of the root generators introduced in (2.27) the conservation laws

a_s$*)+a+s!_*)=o (2.31)

are expressed as:

a _ j’+” +a +j!.*) Ti(a _ a]$*) +a+aj!_*))=o. (2.32)

Performing the canonical transformation we obtain that the dual conserved currents are given by:

9 2” =exp i (1

do (&+cos Py’) )

W~a.p-w3a*r),

9 k-’ = -exp -i ( J

do(&+cos P-y’) 1

x(la.fl+sw3a.-h, (2.33)

which cannot be expressed in a local form The dual manifold A( 6, n’) is automatically

expressed m coordmates adapted to the dual Killing vector k = a/a 6 We can now perform another pomt transformation, with the same Jacobian as before

-(I ep=e> (2.34)

to express the dual manifold m coordinates which are as close as possible to the original ones.

The transformations we have performed are: first a point transformation +“+ ( f3, x’), to go to adapted coordmates in the original manifold. Then a canomcal transformation { 0, xl} --) { 8, x’}, which is the true dual- ity transformation. And finally another point transfor- mation { 8, xl} + I$ /L, with the same Jacobian as the first point transformation, to express the dual manifold m general coordmates.

It turns out that the composition of these three trans- formations can be expressed in geometrical terms using only the Killing vector k “, wp = e% and the corre- sponding dual quantities 4.

It is then quite easy to check that the total canomcal transformation to be made in ( 2.1) is Just

(2 35)

whose generating function is 5

(2 36) D D

One then easily performs the transformations m such a way that the dual metric and torston can be expressed in geometrical terms as 6

gjLLv=gflLv - j$ (k&k.--(u,-o,)(u.--w,)) 9

(2.37) . 1

g -w=gpy+ (l+ku)2 [(k2+(u-u)*)kL”k”

-2(1+ko)k”‘(u-w)“‘],

and

(2.38)

(2.39)

which are the generalization of the Buscher formulae (2.15) in arbitrary coordinates, where

4 Please note that we must raise and lower mdtces with the dual metnc, I e &=&&‘,, t?‘=~%?;, and tlus implies 6, = oc, but (5~=k@(k*+u*)+e’*o (where e’ze,“), i@=k”, but iTM=(mll-(ell u))/k’, we have moreover i5z=kz+u2+g%,oz and A’= l/k’ 5 The one-form o= w,d4’ IS dual to the &lhng vector k o(k) = 1, o(e,) = 0, but it nofcoursed~fferent fromk= k,/k*d@’ (the former IS an exact form, whereas the latter does not even m general sattsfy F&emus condmon 5 A dk = 0)

6 In checkmg that the total result IS equivalent to Buscher’s trans- fomons, one needs to use the fact that 02= (1 lk*) + gvk,k,/k4

188 E Alvarez et al /Physm Letters B 336 (1994) 183-189

3. Non-Abelian duality

In view of the stmphcity of the canomcal approach to Abehan duality, one could be tempted to thmk that the correspondmg generalization to the non-Abehan case would not be very difficult. Unfortunately this is

not the case, the reason being that there are no adapted coordmates to a set of non-commutmg tsometrtes, and therefore one 1s led to a non-local form of the Hamtl-

toman In [ 81 we could carry out the non-Abehan dual- ity transformatton due to the existence of left- and right-chnal currents and as a consequence of the Poly- akov-Wtegmann [ 141 property satisfied by WZW- actions Although m the intermediate steps it was necessary to mtroduce non-local variables (a conven- ient representatton of the auxiliary variables setting the

non-Abehan gauge field strength to zero), the final result led to a local action m the new variables as a result of the special properties of WZW-models men- tioned. The computattons could be carried out exactly until the end to evaluate the form of the effective action m terms of the auxthary variables needed m the con- struction of non-Abehan duals Although a glance at the action could lead us to think that the non-Abehan dual of a WZW-model with group G with respect to a non-anomalous subgroup H 1s locally a coset theory G/H times a WZW-theory for the subgroup H, a careful analysts of the BRST constraints that appear m the

construction together with a computation of the duality transformatton for torotdal world-sheets show that the model 1s self-dual We have so far been unable to express these functional integral mampulations m a Hamtltoman setting as m prevtous section We should mention, however, that the use of non-local variables introduced m [ 81 m cases where the currents are con- served but not chtraly conserved leads m general to non-local actions, hence they are not very useful m those cases A setting where they are stall useful 1s m the case of ( I,1 ) -supersymmetnc WZW-models [ 15 171 We will not spell out the details here because the results are exactly the same as m the bosomc case [ 81

except that fields are replaced by ( 1, 1) -superfields, and that the shift of level k -+ k+ cH ( cH IS the dual Coxeter number of the subgroup H) does not take place due to cancellattons between bosomc and fermtomc determinants Apart from these cases a sattsfactoty treatment of non-Abehan duality m non-choral cases m the functional integral or Hamtltoman approaches still remams elusive

4. Conclusions

We have presented m this paper a canonical approach to (target space) duality m string theory. We believe that this is a “mmimal” approach m the sense that no extraneous structure needs to be introduced, and all

standard results m the Abehan case (and more) are easily recovered using tt. In parttcular, the behavior of currents not commutmg with those used to implement duality 1s clartfied All the generators of the full duality group 0 (d, d, Z) can be described m terms of canonical transformattons. This gives the tmpresston that the duality group should be understood m terms of global symplecttc dtffeomotphtsms The configuration space of the two-dimenstonal field theories considered 1s the loop space of a manifold with isometrtes, the phase space is the cotangent space to this loop space It would be useful to formulate the notion of duality in the con- text of some analogue of the group of dtsconnected dtffeomorphisms, but for the time being we have not found such a constructton. Finally non-Abehan duality seems to fall beyond the scope of the Hamtltoman point of view advocated m this paper. Whether this IS lust a question of technical mgenmty or whether tt is due to some conceptual difficulty m the very notion of non- Abehan duality remains for the time being undecided

Acknowledgements

We would like to thank E Ktritsts and M.A Vazquez-Mozo for useful dtscussions E A. and Y L were supported m part by the CICYT grant AEN 93/ 673 (Spain) and by a fellowshtp of Comumdad Auton- oma de Madnd (YL)

E Alvarez et al /Physm Letters B 336 (1994) 183-189 189

Note added

After completton of thts work we learned that m Ref. [ 2 1 ] a canonical transformation was built relatmg the SU( 2) prmcrpal chual model and its non-Abehan dual

with respect to the left actron of the whole group. We thank T. Curtrtght and C. Zachos for bringing this mfor-

mation to our attention and for other drscussrons. We also learned that this non-Abehan dual model was stud- ted m [22].

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