Conformal extension and bosonisation of the super-Liouville theory

Physics Letters B 267 ( 1991 ) 188-192 North-Holland

PHYSICS LETTERS B

Conformal extension and bosonisation of the super-Liouville theory H.C. Liao Theoretical Physics Group, The Blackett Laboratory, Imperial College, London SW7 2BZ, UK

and P. Mansfield Theoretical Physics Department, Oxford OX1 3NP, UK

Received 5 June 1991

We generalise the N= 1 super-Liouville theory so as to describe the super-conformal minimal series, and construct operator solutions for the Neveu-Schwarz super-fields. By employing a novel form of bosonisation we show that the c= 1 theory can be rewritten as the B2-Todatheory and construct the corresponding Ramond fields.

1. Introduction

2. Super-Liouville theory and its conformal extension

The N = 1 super-Liouville theory [ 1] is a simple super-conformally invariant lagrangian field theory for which operator solutions were obtained in ref. [ 2 ]. It appears naturally in Polyakov's approach to the non-critical superstring. There is evidence [2] to suggest that when its coupling is taken to be imaginary, it can provide a lagrangian realisation of the minimal super-conformal series, generalising the case of the purely bosonic theory [3]. This provides an alternative to the super-Coulomb gas approach and the manifestly unitary but non-lagrangian G K O coset construction [4]. In ref. [2] we observed that the full super-conformal grid o f Neveu-Schwarz primary fields for a modular invariant theory does not emerge naturally, and it is the purpose o f this letter to remedy this by exploiting a quantum symmetry of the stress tensor to generalise the model. We outline the construction of operator solutions for this "conformally extended theory". We will also exploit a novel form ofbosonisation to construct Ramond fields, and discuss the application of this technique to the recently proposed B (0, n )-Toda theories of Watts [ 5 ].

The super-Liouville theory consists of a scalar 0 interacting with a Majorana fermion ~u, and has a single coupling constant ft. As described in ref. [ 2 ] we take hfl 2 negative to make contact with the super-conformal minimal series. To preserve the reality of fl we choose h = -4zr. We lose manifest unitarity but we hope to recover it by a truncation similar to the BRST approach of Felder [ 6 ]. The quantum lagrangian

Vp--- - flq~g :eP~: m p~

( 1)

differs from the classical one [ 1 ] because a term in e 2po is suppressed by quantum effects, and anyway would break conformal invariance, m is an infrared regulator which is essential because we are working in non-compact two-dimensional Minkowski space [ 7 ]. The stress-energy tensor and super-current are T+±=~ :0_+00+_0:-½(1/fl-#)020 +¼i :~_+ 0 + ~ + : ,

J+ =½~+_ O ± O - ½ ( l / , 8 - f l ) O ± g/+_.

(2)

By tuning fl= x / ( m + 2 ) / m , m >/3 the central charge coincides with the super-conformal unitary series 188

0370-2693/91/$ 03.50 © 1991 Elsevier Science Publishers B.V. All rights reserved.

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~= 1-8/m(m+2). Operator solutions to the Heisenberg equations can be constructed through the Lax pair formulation [2] providing the quantum mechanical counterpart to the famous solution of the classical Toda theory by Leznov and Saveliev [8]. These solutions are primary fields, falling into supermultiplets with leading bosonic piece :e-2Ja°: where j is a spin labelling highest weight representations of Osp (1, 2). These fill out the first column of the allowed Neveu-Schwarz representation of the corresponding super-conformal grid in the minimal models [ 9 ]. Note that the stress-energy tensor and the supercurrent (2) are invariant under fl--,- 1~ft. This implies that the super-Liouville theory with coupling constant - 1/fl and potential

V-I/B= (lift) ~)V :e-°/B: m l/a2

12 September 1991

super-Liouville theory in its original sense. We will call it the "conformally extended" theory following ref. [ 10 ]. Operator solutions to this new theory can be constructed from those of the old theories by working in the interaction picture. First, divide the hamiltonian into free and interacting parts

H=Ho+ H,, Ho = ½0uOOUO+mZfb2 + i~ ~ ' , H l = VB+ V_I/p .

(Again m is an infra-red regulator. ) As usual the time evolution of any operator is generated by the free hamiltonian while that of the states is generated by the interacting hamiltonian: (~1 (~.) =e --iH0r ~r'21( 0 ) eiU0~,

(3) ]o~l('c)>=e

has the same stress tensor, super-current and central charge as before. However the operator solutions of this new theory occupy the first row of the super-conformal grid instead of the first column. These two super-Liouville theories can be "combined" as follows to produce a model in which all the NeveuSchwarz primaries of a modular invariant theory appear naturally. From ref. [2] it follows that the theory with lagrangian 5°= ½[ (0q~)2+i¢#~ - V]

(8)

where z is a time parameter, and the superscript I is used to denote operators in the interaction picture. Now, as in ref. [ 10], V~ and Vt_~/a commute and T

W = T e x p [ ! dr ( V ~ + Vl._,/a)]

=[Texp(idzVlp)][(Texp(idzV~-~/P)] ~ WaW_I/fl-~- W

(9)

I/flW a .

Consider the following Heisenberg operator with 09, half integers or integers:

Tu,, = :0uq~ 0,O+ ~i(¢y, 0 ~ u - OvCYuV)

:e -2 [a,o- ( l/a)~/o(o : m 4[aco- ( l/a)~l 2

0V

--gu"50 + g~'"-8nn O00tk : --CB(O,,O,,--gu,,02)O,

e - i H z : e -2[aoJ- ( 1/a)os]o(o) : m4la¢o- ( 1/a)~D]2 eiHr

(5)

where : : denotes normal ordering on the light cone. This is traceless provided h OV T~u=2:V:+4n.000O.

-iHl(*) l O L l ( 0 ) > ,

(4)

has a conserved stress tensor

h

(7)

OV +CB:~:=0,

(6)

ensuring conformal invariance. This condition is satisfied with the same value of CB by V-- Va or V= V_ ~/a, o r b y linearity by V= Va+ V_ wa. So we are led to consider the theory with V= Va+ V_ ~/a. It is straightforward to check that it is super-conformally invariant for the same reason that it is conformally invariant. This new theory is not the

= W -1 :e-2aw#(,): :e-(2/a)~01(r):

X mm 4[~-(l/a)°12 e 4w°aa(°) .

(10)

Since the highest weights 09, ~Dare half integers or integers, it can easily be shown that [ Villa

, "e-2ae°¢l(r) : ] = 0

,

[ :e~2/a)a'01~*):, V~] = 0 so ( 10) can be factorised as

e4,Oa,aA~o) (W~-I :e-2ao~o~): Wa) × ( W= ]/a :e <2/a),~,o,.): W_ l/a) .

( 11 )

189

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This is a product of two operator solutions belonging to the two distinct super-Liouville theories with coupling constants fl and - 1~ft. These may be expressed in turn in terms of free fields on the light cone using the results of ref. [ 2 ] finally providing the operator solutions to the conformally extended super-Liouville theory. If we write the conformal weight of a degenerate super-conformal primary field as

hp,,~= l~ ( c - 1 ) + ½(pa+ +qo~_ )2

12 September 199l

Z = ~ ~ ( 0 , @, ~')

We may do the @, ~ integral first and expand the interaction term in a power series ~ ~0 ~exp(-

1 (,B~'~ ep~) X ~ ~.

+ ~ [ 1 - - ( - - 1 ) 2tp-")] , c~_+= ~ 1

[(1--~)W2+(9--~) 1/2] ,

p, qe½7_,

~ d2xi~)

(12)

with ½~
n,

(14)

which is just a sum of Green functions of a number of energy operators @~ of a free Majorana fermion theory in the presence of the external field ~. The free Majorana fermion is the c = ½member of the conformal minimal series that can be identified with the critical point of the Ising model. According to ref. [ 3 ] this may be represented by the Liouville theory, which is the same as the Aj Toda theory, with lagrangian ~Liou ----½(a~) 2 - e~p'¢ •

(15)

We have chosen the length of the only simple root a to be x/~ and the coupling constand fl' can be either fl'=x/~/2 or - 2 / x / ~ The energy operator, @~, is then identified as :eU¢: with M= - 3x//~ or x/~/3 so we can replace (14) by f c2-~exp(- /d2X~#Liou)

3. B o s o n i s a t i o n

It is well known in the case of superstring theory that Ramond sector operators may be obtained after bosonising the fermions. To obtain such primaries in the conformally extended super-Liouville theory we will use a novel form of bosonisation in which the Majorana fermions are replaced by the c= ½ Liouville theory. For (unfortunately just) one member of the minimal series the resulting purely bosonic theory is a Toda theory for which operator solutions can be constructed. This theory can again be conformaIly extended. Returning to the super-Liouville theory of ( l ), the partition function reads

190

1

× ~ ~. (fleUee#~) n .

(16)

The power series can then be summed again and we obtain the bosonised lagrangian

=½0,~.0"~-

Z ea"*,

(17)

7

where ~=- (~, ~+ fl- ' In fl). The vectors o~#are given by Aa~ = (x/~fl', 0),

2a2 = (/2, fl).

(18)

We have preserved conformal invariance in this reformulation as may be checked by employing the method ofref. [ 12 ] to construct a traceless stress ten-

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sor with the same central charge as before. It is an interacting theory, and in general it is difficult to solve, although for special values offl it reduces to the bosonic Toda theory for some Lie algebra and then the operator solutions of ref. [12] can be constructed. The necessary condition is that the two vectors ( 18 ) corresponds to a set of simple roots of some semi-simple Lie algebra g. The only possibility is to take g = B2 = C2 with m = 4. Given the various choices that can be made for fl' and/t, the only solutions are

fl'=v/3/2,

# = - ~

fl'=-Z/x/~,

/t=~

fl2=3/2; fl2=2/3.

Both solutions give the same value of the central charge for the total theory, and in fact they correspond to the fl~ - 1/fl invariance of the stress tensor for the B2 theory that allows it to be conformally extended. Note that interchanging the two solutions interchanges the lengths of the two simple roots of B2, (18). The operator solution [ 12 ] takes the form : e-X°~°: where co is the highest weight of an arbitrary irreducible representation of the algebra B2. It can be written in terms of the fundamental weights with non-negative integral coefficients (9= m(gl +/'1(92. The resulting conformal weights for the spectrum of solutions can be computed as

h((9) = ~(mZ + 2mn+ 2n 2) + In.

(19)

As (m, n ) = (0, 0), (1,0), (0, 1), we obtain the conformal weights 0, ~, 1 which are the first column of the allowed representations in the m = 4 super-conformal grid. A new feature appearing here after the bosonisation is the construction of the Ramond field with weight 3. If we make use of the fl' ~ - 1/13' symmetry in the original Liouville theory ( 15 ), the bosonised theory can be conformally extended as well. In the case of m = 4, operator solutions to the extended theory can be similarly constructed as in the unbosonised theory ( 11 ). The spectrum of solutions is as shown in the diagram below: *NSI *R3/8 *NSo

R7/16 NSI/j6 RI/16

*NZi/6 *Rl/2a *NS1/6

Rl/16 NSl/16 R7/16

*NSo *R3/s *NSI

(marked with • are solutions to bosonised N = 1

12 September 1991

super-Liouville theory in the m = 4 super-conformal grid). The conformal primaries all take the form :e-~'+'./~'/~. q~:, where (9, & are weights of B 2 with simple roots al, a2 and a~/x/2, X//2a2 respectively. (9, & are restricted by the condition that they are the highest weights of unitary representation of K a c Moody algebras of levels one and two respectively. Truncation of the operator spectrum can be shown using the approach of ref. [ 10 ] based on a "picturechanging" operator associated to Weyl reflections in the plane orthogonal to the root oq + %. Although the spectrum of solutions does not fill out the super-conformal grid, they form a modular invariant combination of a partition function in the ( A - D ) series as classified by Cappelli [ 11 ].

4. Bosonisation of the B(0, n)-Toda theory The super-Liouville theory is a special case of the B (0, n) theory which is described by the lagrangian

[5] 5¢=~1 ((0~,)2 + i~ ~ , + fl~7~'e P ~ " ° - ~ eP~J~)

(20) where aj, j = 1..... n - 1 are the long simple roots of Bn, o~, the short simple root. The stress-energy tensor of this theory has been given by ref. [ 5 ] from which we can deduce the central charge n + ½- 3 [ lZ+ 32+ 5z+...+ ( 2 n - 1 )2] ( 1/fl_fl)2. It can be bosonised as before,

1(

)2__ e - / ~ o

+ fl e ~ 0 , - ~ 7 ~ 0....

~ eP~J0) ,

....

(21)

and we can define a new vector field 0 - ( ~ ..... ~,+ ~). The lagrangian (21) then takes the form o f a generalised Toda theory with a "root system" a,=fl(1,-1,0

.... , 0 ) ,

a2=fl(O, 1 , - 1 .... , 0 ) , i

(22) 191

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.....

a n + , =/3(0, ..., 0, x / ~ ) ,

(22 c o n t ' d )

which is linearly independent. Then a traceless stressenergy tensor which generates the conformal symmetry o f ( 2 1 ) can be computed. In particular, i f / 3 = (22) defines the simple root system o f B,+~ up to a constant. Hence the bosonisation o f B(0, n) super-Toda theory gives the B,+ Lbosonic T o d a theory for this particular value o f ft. F o r n > 1, it produces a series o f central charge - 2 . 5 , - 1 4 , - 3 7 . 5 , ...,

(23)

which does not correspond to any m i n i m a l model. Moreover, the central charge is in a region where we do not expect a unitary theory will arise. In respect o f this point the n > 1 theory is less interesting.

5. Discussion It is well known that the T o d a theory is conformally invariant and the central charge coincides with that o f the m i n i m a l models if we choose an imagi, nary coupling constant, or equivalently, a negative h. The conformally extended version o f the Toda theory provides a spectrum o f operator solutions which is just the conformal grid o f the corresponding minimal models. It is natural to consider the super-symmetric generalisation. O u r previous work shows that although the super-Liouville theory produces a central charge which mimics the super-conformal discrete series, the o p e r a t o r solution gives the N e v e u Schwarz representation of the first row or first column o f the super-conformal grid only. In this letter we have conformally extended the theory, the spectrum o f solutions then fills out all the NS representation o f a m o d u l a r invariant partition function. F o r r e = o d d this belong to the ( A - A ) series and for r e = e v e n it belongs to the ( A - D ) series. The Ram o n d representation can hopefully a p p e a r by bosonising the fermions o f the theory. In the case m = 4 the resulting lagrangian can be interpreted as a T o d a theory for the Lie algebra B2 and the spectrum o f solutions does reproduce all the allowed R as well as NS representation in the ( A - D ) series. It will be o f interest to find solutions for other values o f m, i.e., that

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o f a generalised T o d a theory and examine the resulting spectrum. These solutions also provide an alternative way to c o m p u t e correlation functions o f the m i n i m a l models as in the bosonic T o d a theory.

Acknowledgement H.C.L. would like to thank the Croucher F o u n d a tion for a fellowship. P.M. would like to thank the Royal Society for a Fellowship. We would also like to acknowledge stimulating conversation with Professor D a v i d Olive.

Note added We saw that there were two B2 theories with the same central charge differing by an exchange o f the simple roots, and we r e m a r k e d that this allowed a conformal extension o f the theory. This generalises to the B, theories in which case the simple roots are exchanged with those o f C,. Thus the conformal extension requires the a m a l g a m a t i o n o f the Bn and Cn theories.

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