Volume 136B, number 1,2 PHYSICS LETTERS 23 February 1984

WEAK-COUPLING ANALYSIS OF THE SUPERSYMMETRIC LIOUVILLE THEORY ~

Thomas CURTRIGHT and Ghassan GHANDOUR l Physics Department, University of Florida, Gainesville, FL 32611, USA and Fermi National Accelerator Laboratory, Batavia, IL 60510, USA

Received 12 September 1983

A weak-coupling expansion is developed for the supersymmetric Liouville quantum field theory defined on a circle. The zero-mode system (supersymmetric quantum mechanics with an exponential potential) is first solved exactly, and then nonzero-mode effects are incorporated as perturbations. The theory is translationally invariant, conformaUy covariant, and has a zero-energy ground state with ( ~ ) ~ 0.

The supersymmetric LiouviUe theory is a two-di- mensional modelofreal scalar (4) and Majorana spinor (~) fields with exponential interactions. In superfield form, the classical action is given by

A = f d 2 0 d2x [½ dRqbdL qb -- (2Mi/g2)egCb], (1)

with

~ = ~b + Ot~ +~OOF, (2)

and

dR, L= O/OOR, L - iOR,L(O r-+ 0o) . (3)

Here 0 -(OR)is the usual Majorana Grassman variable with 0 2 = ~ = (OR, OL} = O. Our other conventions are:

70 = 0 i 0 '

Note that the auxiliary field, F, may be eliminated

¢' Research supported in part by the Department of Energy, Contract No. DE-AS-05-81ER40008, and by the Research Council of the University of Kuwait, Project No. SP012.

1 On leave from the Department of Physics, University of Kuwait.

from (1) since it only appears quadratically. The result is F = -(2M/g)eg¢.

The model defined by (1) was proposed in ref. [1] as a supersymmetric extension of the pure Liouville theory (obtained by setting ff = 0) which would maintain the latter's complete classical integrability. More recently the model has been extensively analyz- ed [ 2 - 7 ] as a result of the Polyakov approach [8] to the covariant quantization of the spinning string. A goal o f this analysis (as yet unachieved) is to compute quantum correlation functions involving arbitrary ex- ponentials o f superfields, for all values of the coupling constant, g.

In this paper we shall consider the weak-coupling limit, g ~ 0, o f the supersymmetric Liouville theory defined on a circle. We shall extend to the supersym- metric case some results which were previously ob- tained [9] for the pure Liouville theory.

We show that a weak-coupling expansion can be deffmed for the supersymmetric theory by first solv- ing the zero-mode problem exactly and then perturb- ing in nonzero modes. Within this framework, we compute the first nontrivial corrections to matrix elements o f exp(c~gcb) between states whose energies are O(g2). We also check the conformal covariance o f the model using perturbation theory, and confirm some results obtained from a formal operator analysis.

Consider the model in component field form. In that form, a close analogue of the hamiltonian analysis

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Volume 136B, number 1,2 PHYSICS LETTERS 23 February 1984

of the pure Liouville theory, as given in ref. [9], is obtained through the use of the supercurrent. Classi- cally, the supercurrent is

S u = [i~¢ -(2M/g)eg4)]Tut) +iC[Tu,Tv]~v~, (4)

where the last term is a conformal improvement. The classical supercurrent is traceless on-shell, i.e. 7" S = 0, if the improvement coefficient is C = 1/g. For the quantized theory, we anticipate that C will require corrections, as is true o f the conformal improvement coefficient for the pure Liouville theory [10]. Indeed, we shall argue below that

C = g - l ( 1 +g2/47r) , (5)

is necessary for the conformal covariance of the model. We now quantize the supersymmetric Liouville

theory on a periodic spatial interval (i.e. a circle), 0 ~< cr ~27r, using operator methods. At time r = 0 ,we de- fine canonical ~ and ff fields using the mode expan- sions

~ n - l (ane - in° +bn ein°) (6a) ¢ ( O = q + . * 0

r r (o )=-P- - - -+-~ l ~ (ane- in°+bn ein°) (6b) 27r N / ~ n :#0

¢(o) = ~, g,L(O)] , ~(a) = i(ffL(O ), --~pR(O)) , (7a)

__2_1 1 ~ ein°un, (7b) R ( o ) = r + / , / :~0

1.__~ l+ 1_~ ~ e_inod n . (7c) ~/L(O) = X//~ ~ n¢0

All these operators are by construction periodic in or: ¢(o) = q~(o + 270, etc. One may also contemplate ff's which are antiperiodic in o (cf. ref. [5] and references therein). However, we shall not discuss that possibility here.

The bosonic mode operators, (q, p, an, bn) , satisfy the usual commutat ion relations (cf. ref. [ 10]) such that [q~(o), rr(o')] = i6(o - a'). The fermionic non- zero-mode operators satisfy the ant icommutat ion re- lations

{u n , U k} = 8 n+k, 0 = (d n , dk } . (8)

The zero-mode fermionic operators satisfy

r 2 = I 2= 1, {r,/} = 0 . (9)

Thus we have {t~R(O), ~R(O')} = 6(0 -- o') = {~L(O), ffL(O')). All other anticommutators vanish. In addi- tion the fermionic mode operators commute with the bosonic operators.

Next we define the supercurrent operator as in eq. (4), but with ego replaced by :egg':, where colons de- note normal-ordering with respect to the a n and b n operators, as in ref. [10]. It turns out that this simple normal-ordering prescription leads to a calculational scheme (at least for weak-coupling) which is ultravio- let finite.

It is also convenient to take spatial Fourier trans- forms o f the left- and right-handed projections of the supercurrent operator. So we define

2rr + 1 ( d o e±iNo 0 G~¢=2 - . . / I ( 7r (O) + ~'~-U ¢(°) ) ~L,R (O)

0

-T- 2C~--o ~L,R(O):r-2~Mg :egO(°):tldR,L(O)] . (10)

The usual supercharges are obtained for N = 0, for which case we write

Z -+ ---Z~ + l ~ , ( l l a )

= z ± + R ± , ( 1 1 5 )

u - n , (1 ld)

2rr Z ~ = - T - - ~ J do(:eg¢(O:--egq)~R,L(O ) . ( l l e )

O Hence Z~ represents all interactions involving non- zero-modes.

A weak-coupling expansion is now defined by per- turbatively constructing exact eigenstates o f Z + or Z - , treating Z~ as the perturbation. This is analogous to constructing exact eigenstates o f H + P for the pure Liouville theory using the Lippman-Schwinger formalism (cf. ref. [9]). Consider the case of Z - . We have

Z-I Ik )=gk l l k ) , (12)

Volume 136B, number 1,2 PHYSICS LETTERS 23 February 1984

where

1 Ilk) - Ik) + Z~-Ik)

gk - Z - +ie o o

( )n = ~ 1 Z~- Ik). (13)

n=0 g k - Z ~ + i e

Here we have chosen lk)to be a nonzero-mode vacuum, but a nontrivial zero-mode eigenstate. Thus

R- l k )=O, R+lk)=O, (14a)

z - [k) = gk [k), - ~ < k < ~ , (14b)

Zff Ik> = gklk>. (14c)

Note the spectrum of z - is continuous, nondegener- ate, and includes k = 0. Note also that it was not ne- cessary to choose Ik) to be a nonzero-mode vacuum. This was done only to simplify the analysis to follow.

The zero-mode solutions in (14b)may be explicitly obtained in the position representation, where p = - i d / d q and where

r = ( 0 i - i ) l=(01 10) i l r= ( ; 1 ~ )

We find

iX cosh 7rk~ 1/2 ( Kx]2_ik(X ) + K1/2.fik(x ) ~ (15) (q lk )=

27r2 ] \ iK1/2_ik(x)-iK1/2+ik(X)]'" "

where K u is a modified Bessel function and x - (4nM/ g2)egq. The normalization of the states is as appropri- ate for a continuum with

(k'lk") = g - 1 6 ( k ' - k " ) . (16)

Note that the [k = 0) state is particularly simple in this representation.

(eLk=0) = Vz r ~ l (10) exp[-(arrMjg2)egq] . (17)

The state [k = 0) is in fact the ground state of the unperturbed system governed by Z0-, in the usual sense that it has lowest energy (zero) and it is transla- tionally invariant (with zero momentum). This can be understood from

~(z~) 2 =h+ 2 ~, (b_.b. +nu_.Un), (18) n=l

with

h = (1/470 ( z - ) 2 = p2/4n + (47rM2/g2)e 2gq + Milregq. (19)

The nonzero-mode operators (18) annihilate Ik), by construction, while h [k) = (1/41r)g2k 2 Ik). The opera- tor h in (19) is easily seen to be the hamiltonian for a supersymmetric quantum mechanical model with an exponential interaction, as previously discussed in ref. [ 11 ]. The supersymmetric model has a zero-energy solution, unlike the pure Liouville theory, because ilr[k = 0) = - 1 Ik = 0), and the last potential term in h is therefore attractive. Since ~ff = (1 [27r)ilr + nonzero modes, another statement of the above is that the ground state of the supersymmetric Liouville theory on a circle consists o f a negative fermion-pair conden- sate.

The exact ground state Ilk = O) defined by the series in eq. (13) also has the above properties of translational invariance (zero momentum) and zero energy. This follows from the usual definition of the momentum operator,

P = ~__J ( b _ n b n - a _ n a n ) + n(U_nU n - d _ n d n ) , (20) n=l

+ + '

which commutes with Z 6 and Z i , and from the fact that the hamiltonian for the full theory, H, is defined for supersymmetric models by

H -T-P = (1/47r)(Z±) 2 . (21)

Note that H + P is trivially finite when acting on the states defined by (13).

Next we consider the effects o f the Z 1 terms in (13) by examining specific operator matrix elements, as was done for the pure LiouviIle theory in ref. [10]. We have calculated (k"lle ag~ I lk ' ) to lowest nontrivial order in the limit g -+ 0, with k" and k ' fixed. The basic methods are straightforward extensions of those in ref. [ 10] and only the results of the calculation will be given. It is necessary to normal-order the com- ponents of e c~g~ to remove ultraviolet divergences. We find

(k "]] : eagC) : [lk ')

= e-ix(k")(k,, [¢uxgq [k')eix(k')f(o~; k", k'), (22)

(k"llffR :eag~: Ilk ') = e-ix(k")(k" [(r/x/~) ec~gq ]k')

X eix(k')f(c~; k", k ' ) , (23)

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Volume 136B, number 1,2 PHYSCIS LETTERS 23 February 1984

(k"l[ ~L :eag4~: Ilk') = e- ix(k")(k"l( l /vC~)e agq Ik')

X eix(k')f(a; k", - k ' ) , (24)

where the phase x(k) is

o o

×(k) = (7g6~3/768rr 3) (k + 4k 3) + O(gS), ~'3 =,~1 -~3'

(25) and where f i s a polynomial in a, k", and k ' when cal- culated to any finite order. For example,

f ( a ; k" , k ' ) = 1 - (g6~'3/256rr3) [4(k "2 - k '2) 2

+ (a - 1)(4a + 3 ) ( k " - k ' ) 2 + a ( 4 a - 7)(k"+k') 2

+ a ( a - 1)(7 + 16a + 12a2)/3] + O(g8) . (26)

It is necessary to consider terms up to and including n = 3 in the series o f e q . (13) in order to obtain these results.

The net effect o f nonzero-mode perturbations on the matrix elements in (22 ) - (24 ) is to multiply the zero-mode (supersymmetric QM) matrix elements by simple polynomials. Exactly the same effect appeared in the pure Liouville case [10]. Also, the arbitrariness o f the ie prescription in eqs. (13) is completely ex- pressed by the phases ?((k") and x(k') in eqs. ( 2 2 ) - (24), since the QM matrix elements are real.

Note that the structure of the polynomial in (26), in particular the presence of c~ and (a - 1) factors, implies that matrix elements of the spinor field obey the appropriate equations of motion. That is,

i ~ = 2M:ege: ~ . (27)

The change of sign o f k ' in the polynomials in eqs. (23) and (24) is crucial to verify eq. (27) for matrix elements o f that equation. Also, the results eqs. (22) and (23) are compatible with the supersymmetry transformation obtained from using canonical com- mutation relations, [ Z - , :e ag~ :] = - i a g v ~ R :e._ ag~': •

Next we comment on matrix elements of the 00 component of the superfield e ag~ . T o be compatible with supersymmetry (cf. ( Z - , ~L :eagO: )) we must obtain

(k"l[ :ec~gq~: (1 a g f ~ - F) l lk ' ) = e - ix (k" )

X (k"leagq [(1/4rr)~gilr + (2M/g)egq] Ik ~)

X eix(k')f(a; k", - k ' ) , (28)

and we indeed find this result. However, some care is

needed in removing ultraviolet divergences to obtain (28) by direct calculation. The problem is that (k"ll :eag~: fffl~k') and (k"ll :eage~:FIIk t) are not sepa- rately UV finite. A proper definition of :eag~': F must be given to cancel the UV divergence in :eag~: f f f and subsequently give the finite result in (28). We have found that the naive choice, :e~g~:F = -(2M/g):eag4~: :eg~:. with the mode sums for ¢ and ¢~ cut-off symmetrically, is not an acceptable defini- tion. A prescription which does yield (28) is to sym- metrically cut -of f the mode sums for ~b, ~k, and F. Hence F differs from :egg: at order g2 and beyond* l This acceptable cut-off treats all components o f the superfield q5 on a more equal footing, and for this reason we believe it respects the supersymmetry of the model.

We also note that eqs. (22) and (28) imply that these matrix elements are consistent with the bosonic equations of motion

O2q~ = M:eg¢: ( 2 F - g f f f ) , (29)

where the RIdS is defined as the limit as the afore- mentioned cut-off is removed.

Finally, we consider the conformal covariance o f the supersymmetric Liouville theory. A formal, cano- nical operator calculation leads to the superconform- al algebra [12]

{G~, Gn) = L~+ n + (~ + 2rrC2)k28k+n,O , (30a)

[G;c,Ln] =(k 1 +_ -~ n)C~+ n , (30b)

= (k - + + C2)1,38k ,0, (30c1

where L n is defined as the spatial Fourier transform of the conformally improved [ 10] energy-momen- tum tensor, L n = ½fdae±inO(To0 +- T01). To arrive at eq. (30), one may naively take F = - (2M/g) :eg~': and ignore cut-off subtleties. One then finds that the alge- bra does not close unless the condition in eq. (5) is imposed. A careful investigation o f cut-off schemes which preserve the algebra in (30) is in progress [13].

Here we report on a check o f the algebra eq. (30), and the relation in eq. (5), using the weak-coupling analysis defined above. An immediate consequence of eq. (30b) is that

4=1 Replacing :egq~: in the supercharges Z~ (lle) by this cut-off F does not produce any changes to the order of perturba- tion theory we have calculated.

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Volume 136B, number 1,2 PHYSICS LETTERS 23 February 1984

G~vIIk) = 0, f o r N > 0, (31)

or else the positivity o f the hamiltonian [cf. eq. (21)] for the supersymmetric Liouville theory would fail. To test this, we compute directly for N > 0

(k"l:eag~ : GN[Ik') = - ~ [ (1 + g2/41r - gC)

an 2 ( 1 ~ .NN N 2 ) ÷ - ( g c - 1)

n>N n 2 n=l n "1

O(gS)l (k"le(c'+l)gql [k') +

+ agM [(gC- 1) + O(g2)] (k"le ~gq [h, legq] Ik'>. 2N 2 (32)

To the order calculated [again, up to and including n = 3 in eq. (13)] we find that the matrix element vanishes as expected, given eq. (5). Thus the weak- coupling expansion reveals the conformal covariance o f the model, aswas also the case for the pure LiouviUe theory [ 10].

In conclusion, we have developed a weak,coupling expansion for the supersymmetric LiouviUe theory which preserves translational invariance and conformal covariance, and which exhibits the spectrum of the theory. It remains to develop an exact quantum solu- tion to the model, and to find strong coupling ap- proximations.

It is a pleasure to thank E. Braaten and C.B. Thorn for useful discussions about this work.

References

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[2] P. DiVecchia, B. Durhuus, P. Olesen and J.L. Petersen, Nucl. Phys. B207 (1982) 77;B217 (1983) 395.

[3] Y.Y. Goldschmidt, Phys. Lett. 112B (1982) 359. [4] A.N. Leznov and V.V. Khrushchev, Serpukhov preprint

IFVE-82-162. [5] J.F. Arvis, Nucl. Phys. B212 (1983) 151; B218 (1983)

309. [6] R. Marnelius, Nucl. Phys. B221 (1983) 409. [7] E. D'Hoker, Phys. Rev. D28 (1983) 1346. [8] A.M. Polyakov, Phys. Lett. 103B (1981) 207,211. [9] E. Braaten, T. Curtright, G. Ghandour and C. Thorn,

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