Supersymmetric Ising field theory

Nuclear Physics B154 (1979) 140-156 © North-Holland Publishing Company

SUPERSYMl~{ETRIC ISING FIELD THEORY B. SCHROER *, T.T. TRUONG and P. WEISZ Institut fiir Theoretische Physik, Freie Universitiit Berlin, Arnimallee 3, D-1000 Berlin 33, BRD

Received 29 December 1978

We construct the simplest solvable non-trivial (S ~ 1) supersymmetric model: the supersymmetric Ising field theory. The supersymmetric generalization of the duality order/disorder algebra is discussed. Finally we attempt to "double" the model and comment on the connection between this doubled model and the unsolved problem of the supersymmetric sine-Gordon kink structure.

1. Introduction

Soliton-quantum field theories (SQFT) are quantum field theories in two dimensions possessing an infinite number of conservation laws which imply absence of particle creation and factorization of the S-matrices [1 ]. Many such models are known, the knowledge of their particle spectrum enables the exact determination of the Smatrix. The "bootstrap" program [2] for the reconstruction of the field theory (i.e., correlation functions) has only been carried out for the minimal S-matrix of a selfinteracting Bose system [3] which is S (2) = - 1 . The natural (minimal) interpolating local field turns out to be the order parameter field (for T > Tc) of the continuous two-dimensional Ising model. This field was previously obtained as the scaling limit of the lattice field theory [ 4 - 6 ] , using as underlying fields the lattice fermions of Schultz, Mattis and Lieb [7] which in the scaling limit define a relativistic Majorana field [6]. The model leads to a Z 2 duality structure [8,9]. The duality algebra for the relativistic fields has precisely two inequivalent representations: the T > Tc representation found by Sato et al. [4] and the T < Tc representation in terms of a Z 2 spurion [10]. The correlation functions of the relativistic (dis)order variables continued to the Euclidean region are precisely those of Mc Coy, Tracy and Wu [11,12]. In addition there are mixed order/disorder correlations [6,17] with interesting prope~ ties, they are outisde the Wightman framework of quantum field theory because they lead to a breakdown of the Bargman-Hall Wightman unicity property of analyt-

* Supported by DFG under contract Az 160/5. 140

B. Sahroer et al. / Supersymmetrie Ising field theory

141

ically continued correlation flmctions. The scale-invariant correlation functions can be worked out [13] explicitly by the technique of doubling [6]. Since this simplest non-trivial (S 4= 1) field theory has already such a rich topological structure which manifests itself in the validity of a duality algebra, the desire to generalize this model by supersymmetrization seems natural and hopefully amounts to more than just an academic exercise. The S-matrix of the supersymmetric generalization is that of an interacting boson-fermion system of two types of particles with S (2) = - I . In sect. 2 we propose the Heisenberg fields of this model. These fields are obtained by making an educated guess rather than by the systematic "bootstrap" approach. All supersymmetric properties are checked. In sect. 3 we try to imitate the process of doubling which worked for the simple Ising model [6]. The doubling we propose here is perhaps not the most natural or useful one. The most natural doubling is expected to generalize the ordinary sine-Gordon kink to the supersymmetric sineGordon kink. However the latter has not been understood up to now, the reason for this difficulty is the impossibility of interchanging the process of supersymmetrization with the equivalence transformation of Coleman [14-16].

2. Construction of a supersymmetric extension of the Ising model in the scaling limit As already well-known, the underlying field theory of the Ising model is the field theory of a free massive Majorana field [6] given by

~(x)=-~ f dOp{v(p)ei°x+i~/4e~ +u(p)e-'°x-i'~/4cp},

(1)

where Op is the rapidity of the momentum p = (m cosh Op, m sinh Op), and {c~, Cq } : 6(Op - Oq). The field ~k(x) satisfies the Dirac equation

(i'y"~. -m) ~(x) =O, with the choice of 7 matrices ,yo= IO ~ l '

,./1 = I--~ : 1 '

7 s = I--:

011"

Sato et al., have demonstrated that the order and disorder variables a(x) and It(x) are local quantum field operators [18] and they are constructed as follows: ae(x) = * qSo(X)exp Me(x) *,

(2a)

Ite (x) = **exp Me(x ) •,

(2b)

where ,~ **is the normal ordering of creation/annihilation operators in momentum

B. Schroer et al. /Supersymmetrielsingfield theory

142 space and (e = -+)

M e (x)

i =- -~fdOpdOq

{2 cotanh (½(Op - Oq) + eiO)e+pcq e i(p-q)x

+ tanh(½(0p - Oq)[Cp eq e i(~+q)x + Cpeq e-i(p+q)x I }, 1

+

~)O(X) = ~ f d O p { c p e

ipx + Cpe ipx}.

(3) (4)

The ie0 prescription determines the choice of the boundary condition for x I ~ oo. The operators % (x), and/le (x) satisfy the Z2 duality algebra [ 10]. The auxiliary field 0o 0c) is clearly non-local since {0o (x), Oo (Y)} = 2x], (x - y ) . However, it does have a conserved current

j . (x) = i*

(s)

3. Oo (x) . ,

its "charge" being simply the fermion number N F :

NF(x)

= ½rio(O, x') dx'.

It can be verified that the field Me(x), bilinear in the fermions, is precisely the pseudopotential of the current ].(x), i.e.,

a"MAx) = e"%(x).

(6)

One could say, although the underlying field theory is that of a massive Majorana field, the Ising model in the scaling limit can be completely understood in terms of a non-local auxiliary field 0o(X). This non-locality should not mislead one to expect the expression (2.a) to be non-local, for it can in fact be obtained from a local shortdistance expansion [ 14,15 ] : lim t)a(x)lxe(y) =

lim

x--'y

x~y

( (

l fe ip'(x-y)+Op,[2 2rr J

dOp,} oe(x )

(x-y) 2 < 0

+ regular operators. c~ labels a component of the Majorana field. Although the 2-point function of the pseudopotential Me(x) has a worse shortdistance singularity (log2 m ~ L - ~ ) than the free field, the operator oe(x) has dimension ~ (which can be read off the 2-point function (oe(x)oe(y)) [18]) and is an interpolating field for bosons with S-matrix S = ( - 1 ) N~(Ni3-1)[2

B. Sehroer et al. / Supersymmetric Ising field theory

14 3

in the N' B particles sector [ 19], of asymptotic particles. In view of our supersymmetric extension we consider now a free boson field of mass m q~(x) = ~ - Jl d 0 p["{ C - p~ + e ipx+iTr'4 / + C'~pe-ipx-irr/4},

(7)

with

U~p, ? ' ; ] -- 8(0p - 0 q ) . Let us introduce now in analogy with (4) an auxiliary non-local two-component field: 1

~"

_

i

x~+

"

~o(X) = - ~ n JdOp {v(p)e p C p + u(p)e-'PX7p } .

(8)

Again we observe that the boson number operator NB is appropriately described by the space integral over the time component of the conserved current

*~(x) 0

1"~ # = *

')'ta

~ 0 (x)** .

The pseudopotential of this current is

Me(x = i f ( -2 ~+~pCq C -~ j dOpdOq,sinh [½(0p - Oq) + eiO--~ +

1 cosh l(Op - Oq)

~

ei(p-q)x

-

I

~'C;~; e i(p+q)x -- CqCpe t(p+q)x]j .

(9)

Again eiOis related to the choice of the boundary condition for x I -+ o o . Note that (Me(x)Me(y)) shows the same short-distance singularity as a free field. Then we can define the corresponding order/disorder variables by

"oe(x) = **~o(X) exp Me(x) **,

(10a)

"~e(x) = ] exp ME(x) **.

(10b)

In fact these operators (e = +) have been constructed by Sato et al. [18] as the solution to the commutation relations:

~+(X)(~(y) = qS(y)~+(X),

yl > x l ,

--~(y)'~+(X),

y~ < X ~,

"a+(x)(~(y) = O(y)'o+(x),

yl > X 1 '

--~(y)'O+(X),

y l
~+ (X)'O+(y) = "~+(y)'~+(X),

y l > X 1'

--a'+ (y)~+ (X),

yl < X 1 '

(lla)

(lib)

(1 lc)

B. Sehroer et al. / Supersymmetric Isingfield theory

144

and {~+(x), "~+(y)} = O,

for ( x - y)2 < O,

(lld)

which are the symplectic versions of the orthogonal case, namely

t~(y)lJ+(X),

yl > x l ,

--~(y)la+(X),

yl < X l ,

(11e)

O+(X)~(y) = --~(y)o+(x),

yl >X 1'

(11f)

@(y)o+(x),

y l < x 1'

(1 lf)

o+(y)ll+(X),

yl >X 1'

//+ (X) ~ ( y ) =

tl+(X)O+(y) =

--o+(y)12+(X).

yl
(llg)

Here also the boson theory is completely described by its non-local auxiliary field. The choice (e = - ) would add a factor ( - 1 ) to the right-hand side of all the equations (11). Let us state now our main result. The fields: r/(x) = * q~o(X) exp [M+(x) - M_ (x)] * ,

(12a)

p ( x ) = ~ $ o ( x ) e x p [ M + ( x ) - M_ (x)] *,

(12b)

belong to a supersymmetric multiplet (13a)

[Q', r/] = ip ,

(13b) with ~" = m r / + t . ( ~ o 7 s ~ o ) r / * ,

(14)

and [Q2, ~'] = 1

-+ia_+Ol

,

(15)

2

The supersymmetric charge is defined by

Q = i fdOp (o(p)Cp~p - utp)'~LCp ), Q, = 7SQ,

~), = _~).),s.

(16) (17)

The proof of the commutator/anticommutator, (13a) and (13b), is given in appen dix A in order to keep the main ideas of the paper clear of technical details. As shown by Sato et al. [ 18], both o and'o have asymptotic limits and the S-

B. Sehroer et al. / Supersymmetric Ising field theory

145

matrix of the multiplet (r/, O) is (--1) N(N-1)I2 where N = N ) + N v . At first sight the possibility of obtaining a non-trivial (S :~ I) S-matrix from a local but not manifestly local function of a free Majorana field seems surprising. The intuitive reasoning for in of O(X) is as follows: for large positive t the relevant Fourier different limits aout component in o(x) x

c~(x)

" ~o(X) exp in f

(the definition of the

* ~bt~b2 * dx 1 "

is given in ref. [21 ]) is

tt a+(p)expiTr f vt +

o(t, v t ) ~

(18)

'

,

,2 d(v't)

a (p)a(p )Po 2tin 2

_oo

1

+

~ a

P

(p) exp in f

a+(p')a(p ') dp'

(19)

2Po"

_e¢

For t ~ - ' , o the stationary phase method leads to

1 + a+(p')a(P') _P~°-~d(vt ). o(t, vt) ~ it~i~a (p) exp - icr f t 2m" ,2

(20)

of

Therefore o in :/: o °ut.

3. Attempt of "doubling" the supersymmetric Ising field theory It is known that the ordinary Ising model in two dimensions can be equivalently or even advantageously described by its "doubled" version [6]. The price to pay for this formulation in terms of sine-Gordon potentials affiliated with a free massive Dirac field is the square relationship between the original order/disorder correlation functions and the new correlation functions• In this way we lose the dual algebra (eqs. (11 d - f ) ) but gain a remarkable simplification [6] for the construction of the scale-invariant limit (m ~ 0) of the Ising model. The necessity of drawing square roots out of the new Wightman functions does not pose any problem, the correct Wightman boundary prescription fixes the sign of the branch of the square root. In this section we shall attempt to imitate such a construction for the supersymmetric model. We first expound the theory of massive free charged spinor and scalar fields in two-dimensional space-time. It is well-known that the electric current of these two fields are conserved and their respective pseudopotentials explicitly constructed. Non-polynomial functions of these physical potentials properly triple dots ordered [21], are in turn well-defined operators. The conversion from the local "" ordering to the non-local creation/annihilation of momenta denoted by * * leads us

B. Sehroer et al. / SupersymmetricIsingfield theory

146

in a natural way to the auxiliary fields of the type (4) and (8) and their pseudopoten tials. The simultaneous conservation of the currents of the original fields and the cur rents of the auxiliary fields reflects simply the separate conservation of both types of charge.

3.1. Fermion theory Let (a+p,aq) and (b+p,bq) be the creation /annihilation of the 2 types of charge entering in the expression of the Dirac field ~I,(x) and its charge conjugate qzC(x), for example 1

*(x) = - ~

f dOq(u(q)e-iqX aq + o(q)e'qXbq }.

(21)

Then a sine-Gordon type of pseudopotential ~o-+(x)is given by ~--/'di (P.V. -1 [ei(p_q)Xa;a q + e_i(p_q)Xb~bp] ~o+-(x ) -4X/n oOpdOq [ sinh l(Op - Oq)

+

1

cosh l(op - Oq)

[ei(p+q)x apbq + + + e_"'(P+q)Xaqbp] } -+ !x/n'q2 ~e, (22)

with the boundary condition lim ~-+(x) = +-½,v/~Qe,

(23)

X 1.--+~

Q2 being the electric charge. Lehmann and Stehr [20] have observed that " ex p[ --2t"XX/~~0" +(x)]": = *, ex p ~ } _( x ) * , ,

(24)

where ./~+X_(x)is the following bilinear operator: _e_k(Op_Oq ) ~+X(x ) _ sin rrX fdOpdOq P.V. 2rr s i n h l(Op - Oq) e-MOp-Oq) X [a; aq e i(p-q)x ~ ink + b.~bp e- i(p--q)x +-irrk] + cosh ~(Op - Oq) "t X [a;b; e i(p+q)x + aqbpe-i(p+q)x ] / - i sin 7rX[-7-e~i~rXNa + e -+irrkNb]. I (25)

Na, Nb are the number operators of both types of charge. The observables of the doubled Ising model in the scaling limit are related to the X = ~e (e = +1) value of the exponential (24) namely [6]: ta+-(x) = i cosx/-nSo+-(x)i,

(25a)

o ± (x) = i sinx/-~ ¢-+(x)i.

(25b)

B. Sehroer et al. / Supersymmetric Ising field theory

147

Then we realize that.t2~/2 (x) can be simply expressed in terms of a pair of non-local charged auxiliary fields (anticommuting Bose fields) q5~ (x) (and its charged conjugate

4~-+(x)) d~(X) = ~ l

f dOp{ape_ipx;izr/4 + O pe'+ ipx+-iTr/4~ ,

(26)

and the pseudopotential, tp~(x), of their conserved current:

+

~p~(X ) :

+ ; d•

Oq)[a~ aq e i(p-q)x - bqbpe -i(p-q)x] Op dO q ~cotanh ~(OpI

+ i tanh ½(Op

Oq) [apb + q+e i(p+q)x + aqb p e - i(p +q)x ] - ½x/~N,

(27)

N =Na + Nb. This expression of J2e+_/2(x) is now ~?e+_/2 (x ) = + ~

2

+

~o~(x ) - 2ie * ~+- (x ) COo(X) *.

(28)

Note that here we follow the definition of the pseudocurrent as (1/x/r) 3u~0-+ in ref. [21 ]). In contrast to eqs. (2a, b) we have 2 + U-+(x) = * exp - + - - ~ 0 ~ ( x ) : ,

(29a)

y ,,

o±(x) = * 2q~o c-+(x)q~o(x)exp -+ 2 + , + ~-~-~06(x),,

(29b)

where there is a factor 2 in the exponential corresponding to the doubling of the number of degree of freedom and the -+ sign corresponding to the choice of the boundary conditions indicates that ~-+(x) commute with qffy) when y is left/right o f x in the relative space-like region. The presence of a charge structure permits the construction of two new charged scalar fields of dimension ¼

T-+(x) = ,* q,6(x) + exp -+~ ¢ ~ ( x ) ,*,

(30)

and its charged conjugate T c-+(x) by a local short-distance expansion lim x 1---~y 1

I ,ein[4 ,\ *(x)la+-(y)~ K1/2(m Ix 1 - Yll) { J T-+(x) + less singular terms. .

e--iTr] 4

d

(31) Both components of the spinor field yield the same leading operator up to constants The existence of this scalar non-canonical charged Bose field implies the most concise consistency check on a formal non-local expression for spinors in terms of

B. Schroer et al. / Supersymmetric Ising field theory

148

the axial potential ¢(x) which for free massive fermion reads: ~ ( x ) ~ exp ix/n(3,S¢(x) + f

~(x')dx').

The short-distance expansion of this qz(x) with exp +ix/~¢(x) should yield as the leading operator the exponential of the line integral. This is expected to be a solitoncharge carrying boson-field of dimension ¼. In this form the formula of Mandelstam [14] seems to be not appropriate, however eq. (31) reveals that the short-distance expansion with the self-adjoint/2±(x) = ' cosx/rn¢±(x) 'has the desired property of giving a Bose field with dim T*- = 4" !

3.2. Boson theory The massive scalar charged field cb(x) (and its charge conjugate qSC(x))

a~(x) _- x / ~1J d O; p ( a p~e -ipx + b'~+peipX),

(32)

has a conserved current which allow the construction of an axial potential analogous to (22): ~+¢_(x) = 4 ~~ i

f

dOq (P.V. cotanh ½(Op - Oq) [a;"aqe i(p-q)x + b~bpe -'(p-q)x]

+ tanh ~(Op 1 + ~ + i(p+q)x + "aq'bpe-i(p+q)x I -+ lx/nQ'e • (33) -- Oq)[apbqe Qe is the electric charge of the Bose field. Triple dots ordered exponential functions o f ~ ± ( x ) exist and an analog of the Lehmann-Stehr theorem [20] can be worked out namely •

*

~)k

*

e x p - 2iXx/n~ ± (x)i = , exp .6? ± ( x ) , ,

(34)

with

~ ~(x) =

dOpdOq (e-X(°p-°q cotanh ½(Op - Oq)[ei(p-q)x=inX ~p~q

+ e - i ( p - q)x*-i~X~q~p] + e-~.(0p- Oq) tanh ½(Op - Oq) X t ~[~i(p+q)x'~+'~+~pvq+e-i(p+q)X'aqb'~p] } + i sin rrX [-7-e~/TrX/va + e+-irrXNb].

(35)

Again at X = ~e, e = -+1, the exponential operator at x commutes with d/,(y) when y is left/right o f x in the space-like region of (x - y). In complete analogy with the fermion case one can introduce non-local charged auxiliary fields in terms of which ~/2 (x) can be expressed.

B. Schroer et al. / Supersymmetric Ising field theory

149

Consider the 2-component Bose field 'I%(x) (and its charged conjugate ~ ( x ) ) :

q%(x) =

~ f dOq{u(q)e

oiqx±iTr/4~qj. "h+ q"

iqx~i~/4~q + v(q)~

(36)

Its corresponding axial potential is constructed as

~+ i ~a(x) = @ +

fdOpdOq{P.V. i

- c o s h ½(Op -

-1

sinh l(Op - Oq)

~+p~dqei(p_q)X_e_i(p_q)~bqbp]

^ , ['~;'b+qei(p+q)x + e i(p+q)x"aqb"pl - ½x/~N,

%)

(37)

N = Na + Nb (total Bose number). Consequently an alternative expression for ~e+_/2(x) is 2 ~+ ~-e±/2(x) = + - ~ ( x ) -

-+ _ _ l - e 7s q%*. 2ie **q~ 2rn

(38)

Since only ~'~(x) is relevant to the exponential commutation relation with qb(x) we define in analogy to eq. (29a). ~'-+(x) = *, exp - + ~2¢ g~+( x ) * .

(39)

There is no natural ~'± in this case, but there exists a charged spinor field fZ±(x), (and its charge conjugate f2±C(x)) obtained by a short-distance expansion lim ~(~)~'±(0) ~ (order 1 non-singular operator) ~o + (av'~+ (a* )_ + bx/~_(a±)+)

+ ....

(40)

~e are the light-cone coordinates of ~. In fact Sato et al. have initially constructed the field g2±(x) and shown that it is a local operator [18,19] having an asymptotic limit. Finally we observe that both ~± (x) and ~ (x) are solutions of the LehmannStehr [21] equation E3~p(x) = 2m! ~(x) (cos 2X/~(x) - 1)i.

(41)

This follows from the study of both eq. (34) and eq. (35) in the limit )t ~ 1 and the comparison with [] ~p(x). To carry out the supersymmetric extension we must first introduce the supersymmetric charges by defining their action on the free underlying charged fields. We propose the following choice:

Q = i fdO,

['g,b; v(p) -

apu(p)],

QC = i f dOp [~pap u(p) - b"~bp v(p)],

(42a) (42b)

150

B. Schroer et al. / Supersymmetric Ising field theory

which act in the following way on the basis fields:

E°E,E;II,i04,

EoE:clI:,E:I

(43a)

The supersymmetric charges among themselves obey (Q,~, ~3~) = (QC, ~)~} = O,

(44)

but (Qa, C)~}= {Ts [rn(Qe + ~)e) -- 3'U(Pu + Pu)}a~,

(45a)

(Qe, 0t3} = (3'5 [ - m ( Q e + Qe) - 3'u(P'u + P u ) } ~ ,

(45b)

where now Pu and Pu are the translation generators of the charged Fermi and Bose fields. Note the appearance of the total electric charge in the supersymmetric algebra the presence of such charges has been observed by Olive and Witten [22] in the supersymmetric sine-Gordon model. In fact this property remains if we take the other independent choice of supersymmetric charge: R = i fdOp['bpa+pv(p) - "apbpu(p)], R c=i

loop [~pb+po(p) -

~ ap u(p)].

(46a) (46b)

When we replace Q by R then the right-hand sides of eqs. (43) change into their charge conjugate. We are now in a position to introduce the possible candidates for interpolating fields. As scalar fields we take (e = +) 2 • . ••(x) = * *~)(x) exp e ~ - ~ [~Po + ~ ] . ,

(47)

and its charge conjugate r/C(x), and for spinor field 2 • . p • ( x ) =*. qzg(x)exp e ~--~ [~0o + ~'g] . , together with pC(x), its charge conjugate. We recall that e = -+ corresponds to a choice of boundary conditions in the local axial potentials of the original underlying free field. Assume now e = +.

(48)

B. Sehroer et al. / Supersymmetric Ising field theory

151

Then the action of the supersymmetric charge Q is described by [75Q c, r/] = - i T S p - X ,

[QC, r/c] = 0,

(49a)

[Q, r/c]

[Q, 77]

(49b)

= - i T S p c - x c,

: 0,

{(~)cTs)~, p~) = - i [ ( 7 0 - im)TS]~7) c + ~-c ,

{0a, P~) = 0,

(49c)

{~?c, Pt~) = 0,

(49d)

with

~-.~ -- 2 * [(~o)~(q'o)~ - ~ ( * ~ . q ' ) ]

r/:,

(5o)

X = 2 *•~ o c~ o P * .*

(51)

The computation of those anti/commutators are parallel to those in the neutral theory. We do not include here the details since they are tedious and do not present difficulties. We observe the appearance of additional fields X and Xc as well as ~ and ~-c, representing the interaction. Finally for e = - we have a similar algebra, and again it can be shown that the algebra closes. The set of interpolating field operators (r/, r/c) and (p, pC) also obey composition rules of the Baker-Hausdorff type. The derivation of these rules is beyond the limited scope of this paper but because of the structure of exponential functions of bilinear form, such rules are most natural for example in the study of the "doubled" Ising model [6], when it is shown that they play a crucial role in demonstrating the factorized structure of the correlation functions. Here we do obtain a factorization of correlation functions also, in fact it was our main motivation for pursuing the idea of doubling. To get an idea for such factorization let us write down as examples the simplest correlation functions, the non-vanishing two-point functions:

( r/c (x) r/(y)~ = (o(x) o(y)~ (u(x) u(y)~ ('Y(x) "ff(y)>~,

(52a)

(pC (x)On(y)) = ( 7~ ( x ) ' ~ ( y ) ) ( ~ ( x )

(52b)

~(y))(U(x)/~(y))2.

It is not quite the square relationship found in the Ising model, hence zero-mass statements cannot be made without care. On the other hand, the short-distance behavior of each of these functions has been already evaluated by Sato et al. [18]. For the convenience of the reader we merely quote here their results: ( o ( t ) o(0)) ~ (U(t) u(O)) ~ (-X/C~) -1/4,

/ 1 \-1/2 (~(t) ~ ( 0 ) ) ~ (-X/L-~)- 1 / ' [in ~ } , \ y-t-/

t 2 < O,

(53a) (53b)

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B. Schroer et al. / Supersymmetric Isingfield theory

/

1 \--1/2 ,

(53c)

.

(53d)

(~+(t)~+(O))-(+~-t2)-5/411n~_t2)

[ 1 (~'+ (t)~'_ (0)) ~ (-X/C~) -s/4 [ln ~ ]

]-3/2

The short-distance behavior of (52a, b)is then known. The procedure of "doubling" for the ordinary Ising field theory leads to a particular case (~2 = 4n) of the sine-Gordon model [61. Hence one would expect that the doubled supersymmetric Ising model can be related to a special case of the supersym metric sine-Gordon model as discussed by Shankar and Witten [23]. However, the main picture about the kinks as consisting of a supersymmetric complex Dirac and Bose particles although being in agreement with the supersymmetric breather spectrum, seems to give the wrong multiplicity and the wrong supersymmetric breatherS-matrix * as a result of"fusing" the complex fermion-boson S-matrix of the kinks. So perhaps the correct kink S-matrix is more complicated (possible involving objects of ~ Lorentz spin and the corresponding exotic statistics) and the process of the sine Gordon massive Thirring equivalence [16] does not commute with the process of supersymmetrization. Because of this unsolved problem we are presently unable to make a qualified comment on how our doubling is related to the kink theory of the supersymmetric sine-Gordon equation. We thank R. Schrader for pointing out that the lattice version of the Ising model does not admit a natural supersymmetrization. We have supersymmetrized the Ising field theory.

Appendix P r o o f o ; [Q, ~1] = i75p.

Let us first write for arbitrary r, r' = +1 and e, e' = + ,

r / = , ~bo exp L(x)* ,

withL=rMe+rMe,.

F

Then by elementary algebra we obtain the following contractions:

[Q*' * q~o exp L ( x )

*] = * [Qq~o - q~oQ] exp L ( x ) * + 2 * QOo exp L ( x ) * + **(QL + L_Q)cko(X) exp L ( x ) * .

(A.1)

Now observe that (a)

Q£o + ~boQ = i7 s Co(X),

(A.2)

(b)

r = "r(M _ eNF)+ r'(M _ e N' ~B ) ,

(A.3)

* This is a consequence of preliminary investigations of V. Kurak, private communication.

B. Schroer et al. / Supersymmetric Ising field theory where M and M are the expressions Me,

Me'

in which the

ie prescription

153

is replaced

by principal values. But

QNF~ + NFQ = Q,

QNB~ + NBQ, , = Q.

(A.4)

Thus the 2nd term in r.h.s, of (A.1) combines with (A.3) and (A.4) to give (2 -

re - r'e') **Q~o(X)

exp

L(x) **.

This term vanishes for re = r'e' = 1 (c) we calculate now the remaining contractions:

~O = 1 f dOpdOq [tanh ½(Oq - Op)cqe -i(q+ p)x + cotanh --I

½(Oq - Op) Cq ei(q-p)x] v(p)'C'p,

"

QM : ~ J dOpdOq [tanh ~Op - 0 q)Cq e'(p +q)x + cotanh

Mk~ Q~ ~ +

f

{(Op - Oq)cqe i(p-q)x] u(p)"cp,

[ 1 ~ --i(p+q)x "~;ei(q~p)x dOpdOq cosh l(Oq - Op) Cqe +

sinh

l(Oq1 --

Op)

"~q ei(q-P)

"1Cpu(p),

1 ~+qei(p+q)x OT4=- -~1 f dOodOqI c o s h ~(Oq 1 -- Op) -t sinh where

v(p)

1

x]

~Oq - Op) ~qei(P q) CpO(p),

and u(p) are the spinors / e-Op/2\

v(P)=v~(_eOp/: ),

/e-Op/2

u(P)=x/m(eop/: )"

The singular factors cotanh 0 and 1/sinh 0 are defined as principal values. Thus we have

{ l fdOodOq?p4d,p+q)xl/

= - ~-~

+ r tanh

~(Op -

Oq)u(p

)1 -

-~g l

cosh

i ½(Oq - Op)v(p)

fdopdOq?+pCqei(p q)x

B. Schroer et al. / Supersymmetric Ising field theory

154

X [r cotanh l(Op _ Oq)u(p) + 7-,

if dOpdOq'~pC+qe-i(p-q)x

+ -2n ,

-

X

1

r sinh l ( o p - Oq) v(q

~

1

sinh ½(Op -

Oq)

cotanh

_

~(Oql

u(q

)1

Op)V(p)

1 fdOpdOq.~pCqe_i(p+q) x

+

tanh l(Oq - Op)o(p)

7- cosh ½(On - Oq) u(q .j! "

But for r = - r ' the integrals simplify thanks to the identifies 1

tanh ½(Op - Oq)u(p) - cosh ~(Oq 1 - Op) v(q)

1 = cotanh {(Op - O q ) u ( p ) - sinh ½(Op - Oq) u(q) = - v ( p ) ,

cotanh l(Oq - Op)v(p) +

1

sinh l(Op - Oq)

v(q)

1

= tanh ½(Oq - Op) v(p) + c~osh ½(Oq - Op) u(q) = u(p), namely T(QM + M Q - QM - M ~ ) = 2r~o(X)Oo(X). This completes the proof of the commutators (13a), since *• O o2,t X ) , *• =

O.

Proof o f eq. (13b) Again algebraically, in a similar way we have (C)~, **~oa(X) exp L(x) * } = * [ ~ ) ~ o a + ~boa__Q~]exp L(x) *

+ 2* O - ~ o a e x p L ( x ) * +

:[O~L(x)+L(x)Q~]__ 4,o~expL(x):.

(A.5)

Computing now the contractions separately (a) Q~_._~o~+ ~boa_Q~= ~

-

i

f dOp {CpVa(p)g~(p)e ipx - CpU~(p) g~(p)e -ipx }

i N/~;dOp{cp~

+

[(3" P - m ) 3 ' s ] ~ etpx

+ ca [(--')' • p - m)TS]~t3 e -ipx} = -i[(i'gO + m)yS]a~Oo(X).

B. Schroer et al. / Supersymmetric 1sing field theory

15 5

(b) As before the 2nd term of the r.h.s, of (A.5) cancels with terms of type (A.3), (A.4) contained in the 3rd term for 2 - re - r ' e ' = 0. (c) Choose now r = - r ' , then using the previous calculations

r {Q_dM + M ~

- Q~ ,M - M_@3}= 2r~o~(X )Cbo(X ).

Hence collecting all the results we obtain: {Oe, * ~o~ e x p L ( x ) : }

= ** [(3'" 3 - im)3`s]c~e~oexpL(x) * + 2r **( f o ) ~ ( f f o ) ~ o

exp L(x) **.

Making use of 3`' a3`SL(x) = 3`"3`Sr%~ (J~ - 7"") = r % ( T " - j u ) , 26~e'6a'~

= (3`")a~(3`u)a'y + (3'5)aS(3`s)~'a ' + 6~6~,~,,

: ( ~ o ~ o ) : = o = ; ~2 : , we have finally {~9~, P~ ) = [(3" ~ - im) 3`s] a~ r/+ .~5 **r ~o 3`s ff o q~o exp L (x) **, i.e.,

{Oe, p~) = (3`. a3`s)~en - i 3 ` ~ - , with S- = * ~o( m + i r ( ~ o 7 s ~;o)) exp L ( x ) *. The fact that the algebra closes follows from usual considerations. Taking a = t3 in the above relation one has

{ 0 1 , p l } = z4". 2

2

Now the c o m m u t a t o r [Q, ~-] can be calculated using the algebra

to obtain

[Q2, ~] = +i3+-pl . 1

2

156

B. Schroer et al. / Supersymmetric Ising field theory

References [11 A.B. Zamolodchikov and A1.B. Zamolodchikov, Nucl. Phys. B133 (1978) 525; Phys. Lett. 72B (1978) 481; M. Karowski, H.J. Thun, T.T. Truong and P. Weisz, Phys. Lett. 67B (1977) 321. [2] B. Schroer, T.T. Truong and P. Weisz, Phys. Lett. 63B (1976) 422. [3] B. Berg, M. Karowski and P. Weisz, FUB preprint 16/78, Phys. Rev., to appear. ]4] M. Sato, T. Miwa and M. Jimbo, RIMS Kyoto preprint 207 (1976). [5] B. Schroer and T.T. Truong, Phys. Lett. 72B (1978) 371 ; 73B (1978) 149. [6] B. Schroer and T.T. Truong, Nucl. Phys. B144 (1978) 80. [7] T.D. Schultz, D.C. Mattis and E.H. Lieb, Rev. Mod. Phys. 36 (1964) 859. [8] L. Kadanoff and M. Ceva, Phys. Rev. B3 (1970) 3918. [9] G. 't Hooft, Nucl. Phys. B138 (1978) 1. [10] B. Schroer and T.T. Truong, Nucl. Phys. B154 (1979) 125. [11] B. McCoy, C.A. Tracy and T.T. Wu, Phys. Rev. Lett. 38 (1977) 793; D.B. Abraham, Phys. Lett. 61A (1977) 271. R.Z. Bariev, Phys. Lett. 64A (1977) 169. [12] B. McCoy and T.T. Wu, Phys. Rev. D18 (1978) 1243, 1253, 1259. [13] A. Luther and I. Peschel, Phys. Rev. B12 (1975) 3908. [14] S. Mandelstam, Phys. Rev. D l l (1975) 3026. [15] B. Schroer and T.T. Truong, Phys. Rev. D15 (1977) 1684. [16] S. Coleman, Phys. Rev. D l l (1975) 2088. [17] M. Sato, T. Miwa and M. Jimbo, Proc. Japan Acad. 53A (1977) 153. [18] M. Sato, T. Miwa and M. Jimbo, RIMS preprint 263 (1978). [191 M. Jimbo, RIMS preprint 269 (1978). [20] M. Lehmann and J. Stehr, DESY preprint Hamburg (1976). [21] A.S. Wightman, in Carg~se Lectures in Theoretical Physics 1964 (Gordon and Breach, New York, 1966). [22] E. Witten and D. Olive, Supersymmetric algebras that include topological charges, University of Harvard preprint A013/78. [23] R. Shankar and E. Witten, Harvard preprint 77/A076.