Operator phases in BF-type topological field theories

Volume 264, number 3,4 PHYSICS LETTERS B 1 August 1991

Operator phases in BF-type topological field theories

D a n n y B i r m i n g h a m

Dublin IAS, I0 Burlington Road, Dublin 4, Ireland

H.T. Cho, R. K a n t o w s k i a n d M. Rakowsk i Department of Physics and Astronomy, University of Oklahoma, Norman, OK 73019, USA

Received 22 August 1990; revised manuscript received 21 May 1991

We discuss the presence of operator phases in topological field theories whose classical action is given by Tr[B A F]. These phases are given by the q-function of certain first order operators which appear in a one-loop analysis. Using general arguments, we show that there is no phase for these theories when the spacetime dimension is even, and by explicit computation in three dimensions, we find a phase which violates one of the local symmetries of the classical action. Although the calculation is per- formed in the background field formalism with a gauge which is covariant with respect to all local symmetries, we nevertheless find an anomalous contribution to the phase which is of the form Tr [ B ̂ B ̂ B ].

1. Introduction

The interest in the moduli spaces of flat connections has led to the construction of arbitrary dimensional topological field theories describing these structures. These models are called BF and super-BF models [ 1-8 ], and have been analysed from several points of view.

In the study of Chern -S imons field theory [9], which describes flat connections in three dimensions, one finds that the de terminant of a certain first order operator H, appearing in the one-loop analysis of the theory, possesses a non-zero phase. This phase is given by the q-function of the operator H, evaluated at s = 0 , and is proportional to the Chern -S imons action.

The natural question which arises is whether such phases appear in other topological field theories. Our aim

here is to study this question for the BF-type theories in arbitrary dimensions. In the one-loop analysis of these systems one also finds first order operators whose determinants can potentially have non-zero phases.

The plan of this paper is as follows. In the next section we define the q-function for a general first order

hermit ian operator and outline the calculational technique. Following this we present the results for the two- and three-dimensional BF models, and discuss the structure of the phase for arbitrary dimensions. We will see that this phase vanishes in even dimensions, while a detailed analysis in three dimensions reveals that one-loop effects violate a local symmetry of the classical theory. We conclude with some remarks on the possible implications of this calculation and on the analysis of super-BF models.

2. Background

In a path integral formulat ion of quan tum field theory, one is often confronted with formal expressions like

Z= ~ [Dfb] exp(i ~ f~H~)) . (1)

324 0370-2693/91/$ 03.50 © 1991 - Elsevier Science Publishers B.V. (North-Holland)


Here ~ is a generic field and the functional integral is assumed over all physically distinct field configurations. The operator of interest H is typically a hermitian operator. To derive a regulated expression for ( 1 ) we assume the fields are defined on some compact manifold so that the spectrum of H is discrete. We could for example be considering an integral over the fermionic degrees of freedom in QCD so that H = Jp, the Dirac operator coupled to a Yang-Mills field.

One standard approach to defining Zreg [ 10 ] is to decompose the fields ~ into eigenfunctions of H:

¢ ( x ) = ~ a , ~ n ( x ) , HCn=2n~, . (2) n

The measure [D0] is then taken to be 1-In d a J v / ~ , and with the eigenfunctions appropriately normalised (~,, ~,,) = ~ . . . . Z becomes

co

Z = H ~ d-d--~ exp(ia~2~)= ~ ]--~2x/-Zl exp(¼i~tsgn2~) . (3) _ o o ~ ~

Such an expression needs to be regulated, and the standard procedure is to define

~(s)= ~ 12~1 -s , r /(s)= Z (sgn2n) 12,1 -s , (4) n n

so that a natural regulated definition of Z becomes

Zreg= exP[½(' (0) +¼in q ( 0 ) ] . (5)

This analysis has assumed that the original fields ¢ are bosonic, but one can likewise deal with the fermionic case.

It is the q-function which measures the spectral asymmetry of an operator, that is, the mismatch of positive and negative eigenvalues. Witten has shown that the q-function plays a key role in Chern-Simons quantum field theory [ 9 ], and other applications can be found in ref. [ 11 ]. In ref. [ 12 ], we presented a simple momentum space procedure for evaluating q(0), and we subsequently gave a more detailed analysis on the gauge dependence of 1/(0) in Chern-Simons field theory [ 13 ]. The method used here is in the spirit of McKeon and Sherry [ 14 ], and in fact extends their results to deal properly with first order operators.

For the purposes of calculat; '" is most convenient to begin with the integral representation of eta [ 15 ],

1 ; ~H(s)= F(l(s+l)) d t t ( . . . . . T r [ H e x p ( - H a t ) ] • (6)

0

Although tiE(S) is difficult to calculate in general, it is frequently possible to evaluate qH(0 ), which as we have seen, is the phase of interest.

Following Gilkey [ 15 ], one can introduce a one-parameter family of operators H(2) such that H( 1 ) = H and H(0) has a kernel which can easily be evaluated. By differentiating with respect to 2, one finds

co

d2 - F ( ½ ( s + l ) ) dt t ( s -~) /2Tr e x p [ - t H 2 ( 2 ) ] ' (7) 0

showing that l im~o is given by integrating (f~ d2) the residue of the pole at s=0 in the above t integral. By writing

M = H Z ( 2 ) = M o +M~ , (8)

one then makes a Schwinger expansion of exp [ - Mt ]:

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e x p ( - M t ) = ~ K,(t), tl

(9)

Ko(t) = e x p ( - M o t ) ,

1

K l (t) = - t f du exp[ - M o ( 1 -u)t] MI exp( -Mout), 0

t

Kn+l(t) = - f dt' exp[-Mo(t- t ' )] M,K,(t ' ) . 0

(10)

On E" (or a torus T n) we can evaluate the trace in the s = 0 limit by a simple calculation in momentum space. We take

Idn T r [ C ] = j ( ~ ) ~ ( p l T r ' g ' [p) = d ' x d " y (ply) (xlTr'g~ly) (YIP), ( l l )

where Tr ' d' refers to the trace over any finite dimensional space. [Note that our conventions are such that (plx) = exp ( - i p x ) and A (p) = f d"x exp ( - i p x ) A (x). ] We will illustrate the use of this method by explicit computations in the next section.

3. Applications: BF models

3.1. Even dimensions

In this section we will explicitly compute qn(0) for the two-dimensional BF system. We begin in two dimensions with the quantum action [2 ]

S= f d2xTr(BeOF;j+2c)O;A;+bO.Dc). (12)

Here F;j=O;Aj-OjA;+ [A;, As] is the curvature of the connection A;; B is a scalar field (zero-form); ~ is the Lagrange multiplier enforcing the gauge constraint, and (b, c) are the usual ghosts. The choice of the Landau gauge is not well suited to a background field calculation. To perform a one-loop analysis, we begin by making a background field expansion for each classical field

A-~A+Aq, B~B+Bq. (13)

Here, A and B are now considered to be the background fields, while the subscript q denotes a quantum field which occurs in the functional integral. We can impose the background field gauge DiA ~ = 0 by using the quantum action

Sc[A+Aq, B+Bq] + ~ dZx Tr[20 D.Aq + b D . I ) c ] , (14)

where D ab = 0;Jab+f aCbA~ is the covariant derivative with respect to the background connection A, and l) rep- resents the derivative covariant with respect to the total field A q-Aq. The one-loop quantum action is then given by the part of this expression which is quadratic in the quantum fields,

d2x Tr( -e°Aq[B, A q ] q-2Bq¢, ij D;Aq + 2~ D~Aiq+ b DZc) . ( 15 ) 3 ( 2 ) =

We wish to study the operator H which appears sandwiched between the B q, A q, and q~ fields in S ~ 2)

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I "

S = ½ 1 d 2 x ( B q A q O) a -e i rD~ b ~.ijfacbBc . (16) tl

0 - D jab

We are using the conventions where the structure constants are real and completely antisymmetric with [ T a, T h] =f~bCTC. For the fundamental representation of SU(n) , the matrices T ~ are skew-hermitian and we take T r ( T a T b) = -½6 ab, while the quadratic Casimir is defined byf~CafbCa=Cv6 ~b. Note also that t°dk= - 6 ik and tirEPk = (6~P6 r~- 6~k6rP ).

K~ defined in (10) can now be evaluated by using ( 16 ) and defining Ho as H with background fields A = 0 and B=0, H , - H - H o , M o - H ~ , giving

T r ( e x p [ - H ~ ( 1 - u ) t ] {Ho, H, } e x p ( - H ~ u t ) H,)

exp[ - p Z ( 1 - u ) t - (p+ q)Zut ] 4c~ ¢J[ Ba( - q )iq,A~( q) ] . (17) dZp dZq

= - (21r)2 (2/~) 2

It is now easy to see that there is no pole term in the t-integral, and hence K~ does not contribute to t/n(0). Similarly, the terms which are of higher order in the background field are even less singular and also do not contribute; t / , (0) = 0.

We are now in a position to discuss the general case. I f we consider the n-dimensional BF action, then since B and A are respectively ( n - 2 ) - and one-forms, their dimensions are B~,~...j,,_2(x) ~ p , - 2 and A~(x) ~p. Hence Bu,...u,_2 (p) ~ p - 2 and A~ (p) ~ p - " + ~. We also note that p ~ t- , /2. A glance at the structure of the n-dimensional equivalent of ( 17 ) and (28) reveals that the trace has a dimension of t -~/2. Combining this with the remaining powers of t in the definition of q, one sees that the integral yields a factor of F( ½ ( s - n + 3) ). This is singular at s=O only in odd n dimensions. Thus we can conclude that I/,1(0) = 0 in even dimensions, while in odd dimensions the possibility exists for a non-zero phase.

3.2. Odd dimensions

We begin with the classical action given by

So= ~ T r [ B ^ F ] = f d3xTr[¢aP'BaFpe] (18)

in three dimensions, where F is the curvature two-form and B is a one-form in the adjoint representation of the gauge group. This action has two local symmetries which take the form

8A, = D , og, 8B, =D,~0+ [B,, o91 . (19)

The usual og-symmetry of the action is manifest, while the 0-symmetry is a consequence of the Bianchi identity. In the background field formalism, where we expand each classical field about a background,

A ~ A + A q - A , B - - , B + B q - B , (20)

these symmetries take the form

8A=Dog, 8Aq=[Aq, og], 8 B = D 0 + [ B , og], 8Bq=[Bq, og]+[Aq, 0 ] . (21)

Notice that the quantum fields transform linearly as vectors, and that the covariant derivative is with respect to the background field A. We seek a set of gauge conditions which are covariant (transform as vectors) with respect to these transformations, and one such set of conditions is

0 = D . A q = G I , O=D.Bq+[B, Aq]=-G2 . (22)

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It is straightforward to check the transformations of these conditions under the above variations, and one finds

6Gi = [Gl, to] , ~SG2=[G2,og]+[G~,O]. (23)

The BRST quantization of this system is standard, and one generates the quantum action Sq given by

S~[A,/~] +2 f Tr[~ D.Aq + n(D.Bq + [B, Aq] ) +b D.Oc+b' {D. ( l )c+ [/~, cl ) + [B., l)c] }] , (24)

with the BRST symmetry,

8Aq=eOc, 8Bq=e(f)c '+[B,c]) , 8c=-½e{c,c}, ~c '=-e{c ,c ' } ,

8 b = - ~ , ~ b ' = - ~ n , 5 ~ = ~ n = ~ A = ~ B = 0 . (25)

As usual, I3 denotes the derivative covariant with respect to the total field A. At the one-loop, where we consider the above action to second order in the quantum fields, we note that the off-diagonal ghost terms can be elimi- nated by the simple shift

1 c'--,c'- ~5 ([D.B,c]+Z[B. ,Dc]) . (26)

A phase can then only arise from the H determinant, where H is given by the expression sandwiched between the vectors in S ~2),

0 -e~aYD~ b

S(2)=½ f d3x (Bg A q ~ 7~) a ~ ~ ~ acbn ~

0 --D~ b - D'~b f acb R --d u T

Proceeding now as in (17) we find

Tr(exp [ - H 2 ( 1 -u) t ] {Ho, H~ } e x p ( - H ~ u t ) Hi )

0 D '~ab facbB~

o o o} 0 0

(27)

= _ f d3p d3q (2~Z)3 ( '~3 exp[ _p2( 1 --U)t-- (p+q)2ut] 8Cv ~"PY[B~,(-q)iqaA~(q) ] , (28)

for the lowest order trace kernel that contributes. The complete phase involves two additional trace kernels in the Schwinger expansion ( T r [ H 3 ] and Tr[H~{Ho, H~} 2] ) which contribute cubic order terms in the background field. Only two types of structures arise in the evaluation of t//•(0); the terms proportional to BAA com- bine with the above BOA piece to give a covariant expression proportional to Tr[B ^ F], while the remaining terms are all proportional to Tr [B ̂ B A B]. Although tedious, it is straightforward to establish the final expression,

qH(0) = ~Cv ~ ~p~ (Tr[B.Fpr] +_~ Tr[B.BpB~]) (29)

where both traces are in the fundamental representation. The first term proportional to the classical action could be expected to arise in this calculation and can be absorbed into a renormalization of the coupling parameter; however, the second term is not invariant under the 0-symmetry. One-loop effects violate this unusual local symmetry. The solutions of the one-loop corrected equations for the background fields are consequently not identical to those of the classical theory. Although we have not considered the case of a curved three-manifold here, the B 3 term is metric independent and would be expected in the more general case. It is unclear whether the metric independence of this phase is sufficient to render a consistent topological quantum field theory.

A natural issue that arises at this point concerns the gauge dependence of these results; i.e., does the phase depend on the particular choice of gauge condition? The answer is that it does, and the easiest way to see this is

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to perform the same calculation in other gauges. One convenient choice is the condition

0=D.Aq =D.Bq , (30)

though it is not covariant in the sense of our first choice. The only difference in this calculation is that our previously defined operator H no longer has the two off-diagonal entries proportional to B. In fact, the computation is even easier, and we find the result that r/(0) is given by

Cv e '~r (Tr [B, Fpy] + ~ Tr[B,~BpBy] ). (31)

27t 2

Similarly, one can impose the gauge condition 0 = D . B q = D . A q + [B, Bq], though here we find a phase that is identical to the one obtained in our covariant gauge.

In summary, we have shown that the one-loop corrected phase in the partition function is proportional to

~ E ~p~ Tr [B,~Fp~ +½2ZB~, [Bp, By] ] , (32)

where ,l is a gauge dependent real number. There is one intriguing point to be made here. If we look at the conditions for the extrema of this functional, we find the two conditions

0 = F , p + 2 2 [ B ~ , Bp] , 0=~"P~DpBy. (33)

I f we now look at the curvature of the connection As + 2B,, we see that it is flat,

F,~p(A+2B) =F ,p (A) +22[B~, Bp] +2[D,~(A)Bp-Dp(A)Bc,] = 0 . (34)

Although suggestive in some ways, the meaning - if any - is still a mystery.

4. Conclusions

We have shown that the determinants of certain first order operators in BF-type topological field theories possess non-zero phases. Whereas this phase vanishes in even dimensional theories, the structure of the phase in odd dimensions is troublesome. We considered the three-dimensional case in detail, and discovered a contribution to the effective action which violated one of the classical local symmetries, even though a "covariant" gauge condition was employed. Although the anomalous piece is metric independent, the one-loop corrected equations for the background fields are modified, and the implications of this are not yet clear. It would be useful to check these results with a Pauli-Villars-type regulator along the lines presented in ref. [ 16 ] ~ for the case of the pure Chern-Simons theory in three dimensions.

Conventional wisdom about quantum field theory suggests that one treat a theory with an anomalous gauge symmetry with suspicion, but it is also true that topological field theories survive in spite of features that would not be acceptable in a physical theory. For example, TQFTs are not generally unitary.

A likely source of this anomalous behavior can be traced to an unusual feature in the geometry of the space of fields. In a geometric approach to the effective action [ 18,19 ], one first constructs a metric on the total space of fields which admits the gauge transformations as Killing vectors. Typically, one can choose this metric to be the constant unit matrix, as in the case of Yang-Mills theory. If we let ~ denote the collective field content of our theory, and denote a gauge transformation by ~q~i= i Q,E , then the Killing equation can be written as

0 = Gik ojO k + Gjk O, a~ + Ok Gij Q~. (35 )

~t While this manuscript was being reviewed we have discovered such a calculation. See ref. [ 17 ].

329


In the case at hand , a un i t mat r ix does no t satisfy this cond i t ion . The only admiss ib le field i n d e p e n d e n t met r ic takes the form

GA~,(x) Abs(y) : (:7" 1, Go~,(:,) B~(y) = O, GA~(x) ,~(y) = 2 ' 1 , (36)

where 1 denotes the express ion 5abg,~p d (x - -y ) , a n d a a n d 2 are cons tants , 2 # 0. The unusua l feature is that G is no t pos i t ive def ini te , so we see that the Hi lber t space i nne r p roduc t which was impl ic i t in the de f in i t i on of ou r opera to r H is no t i n v a r i a n t u n d e r gauge t r ans fo rma t ions . It wou ld be in teres t ing to try to cons t ruc t a sui table measure based on the i n v a r i a n t met r ic G.

In this paper , we have been conce rned solely with the phase of a cer ta in first o rder operator , bu t one mus t also

cons ider the m a g n i t u d e o f its d e t e r m i n a n t , a n d this is g iven by the der iva t ive o f the l - func t i on at s = 0 . Al though no a n o m a l o u s behav io r o f the type we have descr ibed appears at lowest order ( two powers o f the backg round

f ie ld) in this func t ion , there does no t seem to be any th ing that p roh ib i t s gauge n o n - i n v a r i a n t t e rms at higher order.

There is ano the r class of topological theories which are, in a sense, supe r symmet r i c ex tens ions o f the ones

cons idered here. In three d imens ions , this theory is related to the Casson i nva r i an t [ 20 ], and can also be cons idered as the d i m e n s i o n a l r educ t ion of topological Y a n g - M i l l s theory [ 21,22 ]. These supe r -BF theories are, however, very d i f ferent [ 3 ], s ince the unde r ly ing gauge s y m m e t r y is m u c h larger. In fact, in add i t i on to the boson ic d e t e r m i n a n t t rea ted here, one mus t also cons ider an ana logous f e rmion ic d e t e r m i n a n t . We expect that a careful analysis o f the phase in this case will lead to a qui te d i f ferent result.

References

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and R. Stora (North-Holland, Amsterdam, 1987). [20] E. Witten, Nucl. Phys. B 323 (1989) 113. [21 ] D. Birmingham, M. Rakowski and G. Thompson, Nucl. Phys. B 315 (1989) 577. [ 22 ] L. Baulieu and B. Grossman, Phys. Lett. B 214 ( 1988 ) 223.

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Operator phases in BF-type topological field theories