A Kallosh theorem for BF-type topological field theory

Physics Letters B 273 ( I99 1 ) 67-73

North-Holland PHYSICS LETTERS B

A Kallosh theorem for M-type topological field theory

Richard Gibbs and Soussan Mokhtari I’h)xm Lkparir~?ent, Louislana Tech C:niwr.srr)~, Ruston. L.4 71272. VS.4

Received 30 September I99 1

A Kallosh theorem is established for the case of BF-type theories in three dimensions, including a coupling to Chern-Simons

theory. The phase contribution to the one-loop off-shell effective action is computed for a two-parameter family of local covariant

gauges. It is shown that the phase is independent of these parameters. and thus equals the “no Vilkovisky-DeWttt” gauge result.

The field space metric dependence of a corresponding calculation for generalized Bb’theory is briefly discussed.

1. Introduction

Recently, progress has been made in understanding the structure of the one-loop off-shell effective action in three-dimensional BF-type topological field theory [ I-31. The central object of interest in such a study is the I/- function of the determinant of a first order operator. The most natural framework for examining these theories is provided by the Vilkovisky-Dewitt effective action [ 4,5].

According to the tenets of this program, one must ensure that the geometry on the space of fields is accounted for in an appropriate manner. It was observed in ref. [ 31 that the presence of the Jw term in the functional measure had a significant effect on the result obtained for the ?I-function, despite its field independence. Having incorporated this factor, it was found that the f/-function contribution to the effective action was proportional to the pure Chern-Simons term, in agreement with a diagrammatical analysis of ref. [ I]. The calculation in ref. [ 31 was performed in the so-called “no VD” gauge; in such a gauge one can show that potential VD correction terms vanish. As such, it represents the unique (i.e. gauge fixing independent) result within this program.

In the present paper, we would like to demonstrate explicitly the gauge-fixing independence of the above results. Our aim is to establish a Kallosh-type theorem [ 61 for the r/-function of this theory. This involves the introduction of a two-parameter family of covariant gauges; in other words, the “no VD” gauge is implemented in the path integral with the use of two arbitrary parameters. Since we are dealing with off-shell contributions to the effective action. the Kallosh theorem states that the r/-function should be independent of these gauge fixing parameters.

The outline of this article is as follows: The theory under study. namely, BF theory [ 7-91 coupled to Chern- Simons theory [lo] is first presented, together with the relevant details of its quantization. We then recall the construction of an appropriate field space metric and summarize the calculational method being employed. The Kallosh theorem is then explicitly verified, and we conclude with some brief remarks on similar calculations in a generalized BF theory [ 1 1 1.

’ Supported by “Commission des Communautes Europeennes” (DG XII-CCR).

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Volume 273, number 1,2 PHYSICS LETTERS B 12 December 1991

2. The model

The classical action of the theory we wish to study is given by

f Sc= d3xtr e~/S:'B~F/~:.+ ~ d3xtre"/#(A,~ O/~A..+ ~A,~A/~A:.) , (1)

which corresponds to a coupling of BF theory to the Chern-Simons term. The results we establish are also valid in the limit in which the integer-quantized Chern-Simons coupling k is set to zero. The field content consists of a gauge connection A~ with curvature F~/3=-O~A~-O~A~+ [A~, A/j], while B=B~,TOdx" is a one-form in the adjoint representation of the gauge group. We shall write all group traces in the fundamental representation, and take the theory to be defined on R 3, where the momentum space calculational procedure being used is valid. Our conventions are that the structure constants are real and completely antisymmetric, with [ T °, T b ] = f ~'~ Tq For the fundamental representation of SU (n), the matrices T ~ are skew-hermitian and we take tr T " T t ' = - ½c~ "~', while for the quadratic Casimir we havef~"dfl"d=c,.fi "l'.

The local symmetries of ( l ) correspond to the usual Yang-Mills gauge invariance and, in addition, a further local 0-symmetry due to the presence of the B field. Specifically, we have

8A,,=D~o). 8B, =D,~0+ [B~, o)] . (2)

The first step in a one-loop background field calculation is to split the fields into a background plus a quantum part as follows:

A - ~ A + A q, B - , B + B q . (3)

We stress that the backgrounds of interest here are necessarily off-shell, in compliance with the rules of the Kallosh theorem [ 6 ]. With this decomposition, the symmetries now read

8A~=D~oJ, 8Aq=[Aq~,~o], 8B~-=D,~O+[B~,co], 8Bq=[Bq, o)]+[Aq~,o], (4)

where the covariant derivative is with respect to the background field. In order to quantize the theory, we need a suitable set of gauge fixing conditions, where suitable in this instance is defined to mean that they are covariant with respect to (4). The following set can be deemed appropriate:

Go = D.Aq=0, G,~-D.Bq+[B, ,Aq"]=O, (5)

and indeed it can be checked that they transform as

8Go=[Go, o-~], 8G,~=[G~,co]+[Go, O]. (6)

Our aim now is to implement these covariant gauge fixing conditions in the quantum action with the use of two arbitrary gauge fixing parameters, which we denote by a and a ' . The part of the quantum action which is quadratic in the quantum fields is given by

S~q2)= d3xtr e~/~: ' 2BqD/~Aq+B,[A~},A~]+-4~Aq~DI~A q

+ 20(D 'A q - ½o~0) + 2zt(D.Bq + [B~, A q~ ] - ½oF z r ) ]+ghos t s , (7)

where one recalls that the ghost and multiplier fields do not possess classical backgrounds. The "no VD" gauge corresponds to choosing both of these parameters to be zero. We are concerned here with the first order matrix H J connecting the gauge fields and multipliers; specifically this is given by

68


d~;,S.'D<,,, <x:,l, .<.h <. D~'~' f""bB<~ [ B } I f - 6 i f B..+ (k/4~)D~/'] S¢q'-~=-~ 2 d~x(Bq~ a ~ O ~)" - " ' ' (8)

0 - DJ' o~5 "l' 0 ' "

- D}*' -J" "<'t'B'is 0 o~'5 "t,

Eq. (8) defines a rank two-symmetric object H 0, which lies between the fields ~ ' and ~ , where we adopt the collective notation ¢b'_= ( B q " ( x ) , A q J ( x ) , ¢ " ( x ) , ~z"(x) ).

The theory is defined via the partition function

Z = ~ x ~ t G,, 1-[ d ~ ' e x p ( i ~ H , , q~j) , (9) 1

the result of which is

det'/2 [G,i ] det- i /2 [H,,] =de t - , / 2 [H~] . (10)

in ref. [3], the presence of the xfdet G o is crucial for ensuring the field reparametrization invariance As shown of the path integral, or in other words, the background gauge invariance, even when this factor is field independent.

At this juncture, therefore, we must address the issue of which field space metric is appropriate. For the case of the gauge fields, we first rewrite the transformations (2) in condensed notation [4,5] as

80'=K'o<d ,< . ( 11 )

Here, the field ¢* labels the classical fields A and B, the symmetry generators are denoted by K~, and the infini- tesimal gauge parameters are e<~= (aJ, 0).

An acceptable field metric is defined by the requirement that it admits the gauge generators as Killing vectors [4,5], that is

0=G,a O, KX<~ +Gja O, Ka,~+ (0kG,,)K~, (12)

for all c~. In the case at hand, one can establish that a constant, field independent, metric which satisfies (12) does exist. However, we must also obtain a suitable metric on the multiplier space. Eq. (6) defines the transformations of the gauge fixing conditions; together with the requirement that ~DG~+ 7~G~ be invariant with respect to (4), we find that the multiplier fields transform as

8¢= [g$, co] + [~, 0], 8~= [~z, ~o] . (13)

Given these transformations, we can determine the multiplier metric by requiring that 8(G<~2<~2 n) = 0, where ,t<<_- - (¢, ~).

Under these conditions, the most general solution is given by

0 ,a.a~, n 0 0

G,j= J.6~/~ a6<~/j 0 0 15 , t ,~(x_y ) (14) / 0 0 0 2'

\0 0 2' o"

where 2 and 2' are strictly non-zero. Rescaling A q and B q by 1/x/X, and ~ and ¢ by 1/x/27, we find that the inverse metric takes the form

G,,= 6<~p 0 0 U ' d ( x - y ) (15) 0 0 - r ' 0 0 1

69


with r = o-/2 and r' = or'/2' . Having obtained all the necessary ingredients, we can now proceed with the calculation. In order to regularize

(10) we define the Cand q functions of the operator t t=-HS=G*aHaj via its eigenvalues 2,,:

G , ( s ) = Z 12,,I -~, ~lu(S)= Z (sign 2,,)12,, I ~ (16)

This leads to the result that

Z,~.g =exp[ ½~'u(0) + ~im/u(0) ] . (17)

We are interested in establishing a Kallosh theorem lbr the phase contribution to the path integral: this phase is given by the q-function which has the following integral representation:

i 1 d t / c ' J ~ / ' ~ T r [ H e x p ( - I t 2 t ) ] . (18) ~l.(S) - F(½ (s+ 1 ) ) {)

In general, q(s) is diff icult to calculate for arbitrary values of s; fortunately, our interest is in its value at s=0 which has a more manageable representation, To see this, one decomposes H as H o + l t ~ , where Ho is independent of the background fields, and H~ contains the interaction terms. One can then employ a trick due to Gilkey [ 12 ], see for example ref. [ 13 ], to show that

i qu(0) = l im --s d t t ~ , _ , ~ / 2 ( T r o + T r ~ + T r , + . . . ) , (19) , .,, F ( ~' ( s + 1 ) )

0

where

T r o = T r [ H ~ e x p ( - t t o l ) ] ,

I

= j " q , H , I + ' H ~ ) ] Tr. - t d u T r [ e x p ( - H o u t ) t t , e x p [ - H o ( 1 - u ) l ] ( ~ . . . O

I I

T r ~ = I t - , ~ u d u ~ j d z ~ T r [ e x p ( - ~ u ~ t ) H ~ e x p [ - H ~ ( ~ - u ) t ] ~ H ~ ' ~ e x p [ - H ~ u ( ~ - t ~ ) t ~ ` H ' ~ ] . (20) 0 0

The calculation thus amounts to an evaluation of the those terms which contribute a pole in s. so as to cancel the explicit s factor in ( 19); these can be performed most readily in momentum space with the definition

I d"p f d"p d " v d " v < p l x } < x l W r ' e l Y } < v l P > (21) T r [ ( ] = J ( 2 = ) , ~ < p l T r ' ( l p } = ~ , . , .

Here, the prime indicates a trace over any other indices carried by the operator ( . Our conventions are such that (p Ix } = exp ( - ipx) and ,4 ( p ) = j-d"x exp ( - ipx)A (x) .

3. A Kallosh theorem

From (8) and ( 15 ), we see that we must now compute the q-function for the operator

( - ~ , D:., ~"/~ ~D " / ' - f " ~ / ' B ' ~ r~,,I, _ . , u~ , J

,~ ,t, . I 0 - e'wlq)ot';., 0 b~,!( (G t / a ' ) ~ v ( : < Y ) = l n,,/, r 'n"t ' "~""~ - r ' a6 "t' ee'6"" a ( x - y ) 0

0 -- DSe~ aa"" 0

(22)

70


where ?= r - k / 4 7 r . In momen tum space, we have the following representat ions for the free (Ho) and interacting

(Hn) parts o f / l :

(Pl Ho I q}3~,/J= i4, t , (P-q) -ps i r 'p,, ir'o~ - O ' J '

0 --Pts -- iol

and

(23)

n, ,V~ r.<~,, 0 -~.<'/s.4~i 0 ,4{~ (24) ( P I H , i . . . . . =.- ' , -.47, (r',4'is-sh,) 0 0

o -.~</, o o

where .4 above implici t ly denotes ,4 ( p - q ) . The next step is to de termine the structure of H0, and one finds

<s,i H ,lq>J = [p'-S<<'s+ X<<"U,) ] a<,,,a U , - q) , (25)

where

0 [ --2['((5~/sl)e-pa.p/~) - (r+r')l)<~l)/s] - io~(r+r ' )p<,. io~'p<~

0 0 io~p~ 0 ( 26 ) A'<*/~(P)= ioer'p/~ - i o e ' p / j - i o o =P/s oe2r'2+oeo~' - ( r + r ' ) P z - r ' om~ '

\ - io~p/~ ioo"p/~ - oe 2r' oeol'

The impor tant point to notice here is that tto is not diagonal: some care is then needed in order to deal with the factors o f e x p ( - H o t ) which appear in the traces in (20) .

For the purposes of verifying the Kallosh theorem, it suffices to consider the H~ {Ho, H~ } part of Tr. in (20). This can be seen by using a simple power counting argument, where one observes that the opera tor ,3 produces terms of order t - ~/-'. Equivalently, each power of momen tum p is of order l - ~/2. In performing the relevant trace, one encounters the exponent ial factor e x p ( - t t o t ) in various combinat ions . An unders tanding of the structure of this term is therefore impor tan t for de termining which terms have the potential of contr ibut ing a pole in the trace. Given (25) , one can expand the exponential in powers of X. However, using the power counting argument one can see that it is only necessary to retain terms in this expansion which are at most O ( t i/2); higher order terms do not contr ibute a pole in s. Due to the presence of O ( ? 3) terms in (26) , it is necessary to expand the exponent ial to third order. Fortunately, the matr ix structure of Xcon l r ives to terminate the series at this point. The relevant piece is then given by

(p} exp( - t t o l ) b q )'J/ =exp( -p=t ) [ l"l~ + X'~/~(p ) ]~,/,~5(p-q) , (27)

where

t-~\ <~/J "<~/J(p)=(-,¥t+X~-~I - X - ~ . . ) (p ) . (28)

The non-zero entries in ~{; are as follows:

( 1 , 2 ) : 2p~t(c~,~/,pe--p<~p/,)+(r+r')lp~&,,

( 1 . 3 ) : ioet(r+r')p,~(l--~tp2),

( 1 , 4 ) : --ioz'tp<~+½ieel=p'-p<,.(r+r')=(l--~lp'-), (29)

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Volume 273, number 1,2

(2, 3): - i eap~,

(2 ,4 ) : -½ic~t2p2p.(r+r ' ) ,

(3 ,1 ) : - i c~ tp~[r ' -~ t ( r+r ' )p 2],

(3, 2): ic~,tp/s+io~tp/j[r,2_tp2r,(r+r, )+~tZp4(r+r, )2] ,

( 3 ,4 ) : ( r + r ' ) t p : ,

(4, I) : ic~tp/s,

(4 ,2 ) : - icxtpl j[r ' -½tp2(r+r')] .

PHYSICS LETTERS B 12 December 1991

(29 cont 'd)

It remains only to compute the trace. Upon expanding the exponential factors, one finds that there are four pieces which contribute non-zero pole terms. These can be seen upon examinat ion of the relevant trace, namely

I

Tr~ = - ½t J du d3p d3k exp[ - p 2 u t - ( p - k ) 2 ( l - u ) t ] (Pl ( I + X ) H , ( I + X ) [ p - k ) ~ f f ( p - k l {Ho, H~ }]p),,/s.. o

(30)

The contributions to Tr~ are listed in turn as

Ht --, - A " ( k ) . A " ( - k ) (2r+4r ' ) + 4A" (k ) .B" ( - k ) ,

)(H, - + A " ( k ) . A " ( - k ) [7ur+9ur' - 5u2(r+r ' ) ] - 4 u A ~ ( k ) ' B " ( - k ) ,

H j R - ~ A " ( k ) . A " ( - k ) [ 2 r + 4 r ' + 3 u r + u r ' - 5 u 2 ( r + r ' ) ] + 4 ( u - 1 ) A " ( k ) . B " ( - k ) ,

X H , . g ~ A " ( k ) ' A " ( - k ) [ - lOu(r+r ' )+ lOu2(r+r' ) ] , (31)

where each piece in (31 ) is multiplied by 1

2(47t)3/2 d du d3k e x p [ - k 2 u ( 1 - u ) t ] , (32) o

and ) (on the left, and right, in (31 ) correspond to (28) with the replacements t-+ ut, and t ~ ( 1 - u) t, respectively. We can now sum the different contributions, and one finds that all the cg terms which are present do indeed

sum to zero. Perhaps it is worth remarking the absence of any c~' and B.B terms in the above calculation. In addition, the result is independent of the scale parameters r, r ' , and ~. We have thus verified the Kallosh theorem for the q-function phase of this theory. In other words, the tl-function contribution to the off-shell effective action is independent of the gauge fixing parameters c~ and c~'. The result thus coincides with the "no VD" gauge result obtained in refs. [ 1,3], namely

qH (s = 0 ) = 2f}~ f d3X tr e~e:'(A. 0,A:, + ~A.A/~Ar) . ( 33 )

4, Generalized BF theory

One can also consider a generalized system with classical action [ 11 ]

,f lj S= -~ d3xe"/~;'tr(½B.F/~:,+ lB.B/sB;,)+ -~ d3x~"/Srtr(A~ ~/sA~,+ 2A.A/jA:,+B.D/~B,,)-S~ +$2. (34)

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Volume 273, number 1,2 PHYSICS LETTERS B 12 December i991

In this case, the local s y m m e t r i e s are g iven by

8A,~=D(~o)+ [B~, 0], 8B~=D,~O+ [&~, {o] ,

wi th the fo l lowing set o f c o v a r i a n t gauge f ixing condi t ions :

Go=_D.Aq+[B, , ,BO{~], G , ~ - D . B q + [ B , ~ , A q ' ~ ] .

These t r ans fo rm as

(35 )

(36 )

8G~,, = [G,~, o0] + [G~, 0], 8 G ~ = [G~, {o] + [Go, 0] . (37)

Fol lowing the same strategy as was e m p l o y e d in the p rev ious example , we find the t r ans fo rma t ions o f the mul-

t ipl ier fields:

8 0 = [0, c o l + [Tr, 0], 87r= [Tr, ~o]+ [0, 0] . ( 38 )

• The resul t ing field space me t r i c turns out to be o f the general form

(;,i = a 0 (39 ) 0 or' '

0 2'

where the only res t r ic t ion is that cr 2 - 2 2 ¢ 0, and s imi lar ly for the p r i m e d variables . Various s tudies o f the effec-

t ive ac t ion tbr this mode l have been p e r f o r m e d in refs. [ 1,14]. In par t icular , the system S~ was s tud ied in ref.

[ 14] with a d iagonal field space metr ic , a = a ' = 1 and 2---2' = 0 . It is clear that such a me t r i c is an acceptable

Kil l ing met r ic for this theory. However , the above analysis indicates that there is more f r eedom here in choos ing

a field metr ic , and d i f fe ren t choices will lead to di f fer ing results for the q- funct ion o f the theory. For example ,

one may s tudy the SI system with ~ = a ' = 0 , and 2 = 2 ' = 1. One can then check that the phase con t r ibu t ion is

p ropo r t i ona l to the $2 act ion. Such a s i tua t ion s imply means that the q u a n t u m theory under s tudy depends on

the choice o f f ield metr ic , and d i f ferent choices can lead to inequ iva len t q u a n t u m theories; see ref. [ 11 ] in this

regard.

Acknowledgement

S.M. apprec ia tes the hospi ta l i ty o f the T h e o r y D iv i s ion at C E R N .

[ 1 ] 1. Oda and S. Yahikozawa, Effective actions of 2 + 1 dimensional gravity and BF theory, ICTP preprint IC/90/44 (April 1990). [2] D. Birmingham, H.T. Cho, R. Kantowski and M. Rakowski. Phys. Left. B 264 ( 1991 ) 324. [3] D. Birmingham, H.T. Cho. R. Kantowski and M. Rakowski, Operator phases in BfLtype topological field theories II, CERN preprint

CERN 6160/91 (July 1991 ), Phys. Lett. B. lo appear. [4] G. Vilkovisky, Nucl. Phys. B 234 (1984) 125. [5] B. DeWitt, in: Architecture of fundamental interactions at short distances, Proc. Les Houches Summer School 1985, eds. P. Ramond

and R. Stora (North-Holland, Amsterdam, 1987). [6] R. Kallosh, Nucl. Phys. B 78 (1974) 293. [7] G.T. Horowitz, Commun. Math. Phys. 125 (1989) 417. [8] M. Blau and G. Thompson, Ann. Phys. 205 ( 1991 ) 130. [ 9 ] A. Karlhede and M. Roeek, Phys. Lctt. B 224 ( 1989 ) 58;

R.C. Myers and V. Periwal, Phys. Left. B 225 (1989) 352. [10] E. Witten, Commun. Math. Phys. 121 (1989) 351. [1 I ] E. Wiuen, Nucl. Phys. B 311 (1988) 46. [ 12 ] P.B. Gilkcy, Invariance theory, the heat equation and the Atiyah-Singer index theorem ( Publish or Perish, Wilmington, DE, 1984). [ 13] D. Birmingham, H.T. Cho, R. Kantowski and M. Rakowski, Phys. Rev. D 42 (1990) 3476. [ I4] D. Birmingham, R. Gibbs and S. Mokhtari. Phys. Left. B 263 ( 1991 ) 176.

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A Kallosh theorem for BF-type topological field theory