Index theorems on open infinite manifolds

Nuclear Physics B269 (1986) 131-169 North-Holland, Amsterdam

INDEX THEOREMS

ON OPEN INFINITE MANIFOLDS A.J. NIEMI1f

Institute for Advanced Study, Pnnceton, NJ 08540, USA G.W. SEMENOFF2 Department of Physics, Unwers~tyof British Columbta, Vancouver, BC, V6T 2A6, Canada Received 21 May 1985

A new index theorem for Dtrac operators defined on open infimte manifolds is derived. The derivation is based on the use of trace identities and assumes the validity of the standard chiral anomaly equation. The theorem generalizes the Callias index theorem for arbitrary background fields and xs analogous to the Atiyah-Patodi-Singer index theorem for Dirac operators on open compact manifolds. The essential differences between compact-and infimte-manifold index theorems is clarified by demonstrating that on an open compact mamfold the standard form of the chiral anomaly is altered A family index theorem is then derived and used to devise a technique for the computation of the ,1-invariant of arbitrary hernutian Dirac operators. This provides a complete solution to the mathematical problem of fermion number fractionizatmn. Finally, elements of cohomology are developed by showing how the adiabatic approximatmn, when supplemented by a spectral flow analysis, is related to the index theorem. 1. Introduction V a r i o u s i n d e x t h e o r e m s [1] p l a y a central role in m o d e r n theoretical physics. I n p a r t i c u l a r t h e i n d e x of a D i r a c o p e r a t o r i n d e x ® d=ef d i m k e r ® - d i m c o k e r ®

= d i m ker ® - d i m k e r ® f

(1.1)

h a s b e e n u s e d extensively to o b t a i n r e l a t i o n s h i p s b e t w e e n the a n a l y t i c a l a n d t o p o l o g i c a l a s p e c t s o f q u a n t u m field theories. G e n e r i c a l l y , a n i n d e x t h e o r e m relates the n u m b e r o f n o r m a l i z a b l e zero m o d e s to a n u m b e r that characterizes the b a c k g r o u n d field t o p o l o g y . P r o v i d e d we u n d e r s t a n d the classical structure o f the t h e o r y this r e l a t i o n c a n t h e n yield v a l u a b l e i n f o r m a t i o n o n the q u a n t u m structure. 1 Research supported in part by Department of Energy grant no. DE-AC02-76ER02220. t On leave from Helsinki University of Technology, Espoo, Finland. 2 Supported in part by the Natural Sciences and Engineering Research Council of Canada. 0550-3213/86/$03.50©FAsevier Science Publishers B.V (North-Holland Physics Publishing Division)

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Applications of index theorems are many. In instanton physics the Atiyah-Singer index theorem [2, 3] can be used to enumerate the number of independent parameters needed to characterize the most general multi-instanton configuration [4]. The resolution of the U(1) problem and the derivation of chirality selection rules is also related to this index theorem [5], and the connection between anomalies and index theorems has provided valuable constraints in model building [6]. Most notably, recent extensive mathematical investigations on the structure of anomalies and index theorems [7, 8] have been crucial in pinpointing the 0(32) [9] and E 8 x E 8 [10] superstring theories as strong candidates for ultimate unification [11]. A widely studied application of index theorems, with consequences both in particle physics and condensed matter physics, arises from fermion number fractionization [12]. Here, the nontrivial interplay of index theorems and topology has already resulted in experimentally observable phenomena in the electric properties of polymers such as polyacetylene [13]. In the special case of field theory models with a conjugation symmetry (e.g. charge conjugation) the fermion number of a topological soliton is proportional to the number of zero modes of the pertinent Dirac hamiltonian. This allows an index theory analysis of the problem: Indeed, the open infinite space index theorem by Callias [14] was originally motivated by the Jackiw-Rebbi [15] analysis of fermion number fractioni_zation in conjugation symmetric models. In "the absence of a conjugation symmetry more general topological techniques must be introduced [12]. In the course of this analysis it has been established that besides the zero modes, the spectrum of a Dirac operator also reflects other, more universal topological conservation laws [16]. In the most general case the fermion number of a topological soliton is proportional to the Tl-invariant (spectral asymmetry) of the pertinent Dirac hamiltonian [12], Q = ½f dx (vac[ [ '/,f, ~/,][vac) = - ½~z,

(1.2)

where we have defined

~.= h~mo~.(s)d~ ~im ~sign(X.)[X~[ -~

(1.3)

and X, are the energy eigenvalues of the hamiltonian H. (If the operator H has a continuum spectrum the summation in (1.3) must be replaced by a spectral integral.) The properties of the functional ~(s) have been studied extensively on both compact and noncompact manifolds [17-20]. In particular, it has been established that quite generally ~(s) is a meromorphic function of s with a finite s-* 0 limit. Furthermore, for a large class of Dirac operators ~(s) is a topological invariant for arbitrary values of s [16, 20]. Since ,/(s) is essentially the Mellin transformation of the odd

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133

part of the spectral density of a Dirac operator, this identifies the odd part of the spectral density as a topological invariant, generalizing the concept of an index and reproducing standard index theory results in particular limits. Besides fermion number fractionization, the ~-invariant has also found various other applications in physics. For example, both local and global anomalies can be related to the ~-invariant of an operator defined on a higher dimensional manifold [21]. The ,/-invariant also arises in the analysis of fermion zero modes in the background of gravitational instantons [1, 22]. The recent progress in the chiral bag model and other phenomenological models of nuclei can be largely attributed to advances in our understanding of their topological aspects [23, 24]. Finally, new condensed matter physics applications [25] have emerged from the phenomenology of quantized Hall conductance [26], electric properties of graphite [27], and the angular momentum paradox in 3He - A [25, 28]. It is needless to say that progress in our understanding of the topological aspects of Dirac operators and index theorems will most likely have an enormous impact in the future of theoretical physics. From the physics point of view index theorems such as the Atiyah-Singer theorem and the Atiyah-Patodi-Singer theorem are somewhat restrictive. These theorems are only applicable on compact manifolds and do not provide any direct information on the topological aspects of scattering states (i.e. continuum spectrum). Open infinitemanifold index theorems such as the Callias index theorem do not have this limitation. However, such theorems have only been derived for a very special class of Dirac operators. In this paper we shall present a new index theorem for Dirac operators on open infinite manifolds. Our theorem applies to fairly arbitrary operators and provides a nontrivial extension of both the Callias index theorem and the Atiyah-Patodi-Singer index theorem. We extend Callias original discussion to a much wider class of operators. In addition to the pertinent characteristic classes, we find that the index is also related to a certain ~/-invariant and in a sense our index theorem then resembles the Atiyah-Patodi-Singer index theorem. However, since we deal with infinite manifolds the operators that we consider are in general inaccessible to the Atiyah-Patodi-Singer theorem. The original motivation for our work is drawn from our recent investigations on fermion number fractionization [12]. The present paper is a culmination of that project in the sense that now we can provide a practical technique for the computation of the fermion number in the most general case. Our approach to the index theorem is analytical and field theoretical in nature. We hope that this will make our results more accessible to the physics community. This paper only requires knowledge of certain elementary aspects of Green functions and chiral anomalies, and the pertinent facts can be found from standard field theory textbooks [29]. Using the trace identities introduced in [20] we convert an integral over the local index density to an integral over the pertinent chiral anomaly and to a surface integral which we then identify as an 71-invariant of another Dirac operator. Our technique is an adaptation and generalization of the technique

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A.J. N:emt, G. W. Semenoff / Index theorems

originally introduced by Callias [14] and Weinberg [30] to analyze a particular class of Dirac operators, and as special cases we recover their index theorems. Independently, a similar index theorem has been derived by Stern [31]. He uses somewhat different techniques and his work is motivated by certain problems that arise in the theory of locally symmetric spaces and arithmetic groups. We refer to [31] for a more rigorous treatment and to an analysis of certain mathematical aspects not discussed here. In sect. 2 we review the Atiyah-Patodi-Singer index theorem for a Dirac operator defined on a compact spin manifold with a boundary. We explain the spectral boundary condition that plays a central role in this theorem. We then introduce the asymptotic heat kernel expansions that are used to identify an expression for the index. No details are presented: This section is intended as a background for readers who are not familiar with the Atiyah-Patodi-Singer index theorem. We also introduce various quantities and definitions that we use in the later sections. Otherwise the rest of the paper is largely independent of this section. However, since our index theorem is formally equivalent to the Atiyah-Patodi-Singer index theorem we find it instructive to discuss both theorems in detail, in order to clearly expose their differences. In sect. 2 we also explain how the zero modes of the "boundary operator" affect the index theorem. This discussion generalizes trivially to the infinite manifold case and will not be repeated there. In sect. 3 we derive our index theorem. We first explain why the Atiyah-PatodiSinger index theorem cannot be applied on open infinite manifolds. We then discuss the role of boundary conditions. Finally, we derive the open infinite-manifold index theorem by extending the formalism developed in [20]. Recently there has been much interest in the boundary effects on chiral anomalies [23], [32]. Most notably, in the chiral bag model the baryon number is a direct consequence of a boundary effect [23, 24]. In sect. 4 we use the Atiyah-Patodi-Singer index theorem to analyze how boundaries modify chiral anomalies on compact manifolds. Since the derivation of our infinite-manifold index theorem is based on the chiral anomaly equation, this section then serves as a comparison between the two theorems. On open infinite manifolds an index theorem has much more flexibility than on open compact manifolds. The "boundary" of the manifold is at infinity and to a certain extent the index is insensitive to variations at large enough proper distances. In sect. 5 we explore this fact by deriving a family index theorem that relates the ~-invariant of a Dirac operator on an open infinite manifold to the index of another Dirac operator defined on a manifold with one more dimension. This relation provides a practical scheme for the computation of the ~-invariant for arbitrary Dirac hamiltonians as exemplified in sect. 6. In sect. 7 we discuss some elementary cohomology aspects of our index theorem. However, since we are mainly interested in relating the index theorem to the

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adiabatic approximation of Goldstone and Wilczek [33] we do not introduce formal differential geometric language. We show how the various terms in our index theorem admit a direct physical interpretation and demonstrate that corrections to the adiabatic approximation can be accounted for by the index of a family operator.

2. The Atiyah-Patodi-Singer index theorem Consider a Dirac operator ~ defined on a compact d-dimensional euclidean spin manifold % with boundary ~. The Atiyah-Patodi-Singer index theorem [17] relates the index of ~ to the spectral asymmetry (1.3) of an operator H obtained by restricting g to ~ as follows: We assume that H is a self-adjoint and elliptic operator on ~. We also assume that near ~ the manifold 9C can be parametrized as the product ~ × R + where R + is the half-line x >/0 with positive x in the direction of the inward normal of ~. On 9C we choose a metric which for x ~- 0 (i.e. near the boundary) has the product form ds 2 = d x 2 +

g,j(x = 0; y) dy'dyJ,

(2.1)

where y ' are the coordinates on ~. We extend the background fields of the operator H to the manifold 9~ and introduce an operator L and its formal adjoint L t which near x -- 0 are of the form L = 0x + A x + H , L t = - 0x - A x + H .

(2.2a) (2.2b)

Here A~ is the normal component of the background gauge field near ~ and we choose the gauge A x = 0. Near ~ the operator ~ is then related to H by

o__[oo]

<,,,

The eigenvalue equation for g is

0,.[o

(2.4)

and D is a selfoadjoint operator if and only if

(ulLlo)= (vlLtlu).

(2.5)

Since H is sdf-adjoint on ~, (2.5)implies

f vt(y, x)u(y,

x)L,= o-- O.

(2.6)

The self-adjointness o f / ~ depends on the boundary conditions imposed on u and v

A.J. Nieml, G. W. Semenoff / Index theorems

136

on ~. In general local boundary conditions such as a Dirichlet condition may become incompatible if the background fields have a nontrivial topology. According to a theorem by Atiyah and Bott [34] local boundary conditions can lead to "topological obstructions". In order to avoid these obstructions one introduces a nonlocal boundary condition compatible with the topology. This is the spectral boundary condition obtained as follows: We introduce the eigenfunctions of H on ~,

Hex = hex.

(2.7)

For simplicity we assume here that H does not have zero modes on ~. We postpone the discussion of zero modes of H to the end of this section. In the function space spanned by {lex)} we define the projection operator P that projects onto the positive eigenvalues, Pe x = 0 for h < 0. (2.8) We then require that

Po(x=O)=O** f e~(y)v(y,x=O)

20

(h>0),

(2.9a)

(1-P)u(x=O)=Oc, f e~(y)u(y,x=O)=O

(h<0).

(2.9b)

Near the boundary we introduce the expansions

v(y, x) = Efx(x)ex(y),

(2.10a)

u(y, x) = Egx(x)ex(y)

(2.10b)

h

h

and if we substitute (2.10) into the eigenvalue equation (2.4) we obtain (d~x +

~)fx(x)=Egx(x) '

(2.11a)

Efx(x' ) .

(2.11b)

- ~xx + h gx(x) =

Combining these with the obvious relations fx(0)= 0 for h > 0 and gx(0)= 0 for h < 0, we then get the spectral boundary condition fx(0) = 0

for h > 0,

- - 2

fx

-h

gx(0) = 0

for h < O,

-- = h gx

for h < 0,

(2.12a)

for }~> O.

(2.12b)

With this boundary condition L* is the adjoint of L, i.e. (2.5) is valid.

A J Nlemt, G. W. Semenoff / Index theorems

137

The boundary condition (2.12) has the following motivation: If we attach to ~ the semi-infinite tube R - corresponding to x < 0, construct the manifold ~ u (~ × R - ) and extend the operator ~ to ~ by a constant extension of the background fields from the boundary ~, both (2.11) and (2.12) remain valid in the cylinder. The equations for the zero modes on the manifold ~ are Ltu = 0,

Lo = O,

(2.13)

where L and L t are now extended to ~ . Since the background fields are continuous on ~ , the solutions of these equations are continuous extensions of the original zero mode solutions on ~ with continuous first derivatives across ~. For a zero mode of L, (2.12) implies that it corresponds to h < 0 with a positive logarithmic derivative at x = 0. Consequently v(y, x) is an exponentially damped function at x = - oo. Similarly, (2.12) implies that a zero mode of L t is also exponentially damped at x = - oo. These extensions are manifestly square integrable. Thus we may interpret the spectral boundary condition (2.12) as a constraint that in solving for the zero modes of L and L t on %, only solutions with a natural square integrable extension across the boundary ~ are acceptable. Indeed, one can show that the index of J~ computed on % with the spectral boundary condition (2.12) agrees with the index of the extension of D to ~ restricted to the space of square integrable functions. With the domains of L and L t specified by the boundary conditions (2.12) one can show that both L L t and LtL are self-adjoint positive operators on %. If we iterate (2.4) we obtain LLtu = E2u,

(2.14a)

L t L v = E 2v .

(2.14b)

For the index of J~ we find index ~ = def=dim ker L t - dim ker L = d i m k e r L L t - dimker L t L .

(2.15)

Since % is compact the spectra of LtL and L L t are discrete and their nonzero eigenfunctions are paired. Indeed, if u is a properly normalized solution of (2.14a), then L t L ( L t u ) = E2( L t u )

(2.16)

and as a consequence v=

1

IE[

is a properly normalized solution of (2.14b).

Ltu

(2.17)

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AJ Nzemz,G.W. Semenoff/ Indextheorems

The Atiyah-Patodi-Singer relation between the spectral asymmetry (1.3) of H and the index (2.15) is obtained as follows [17]: We first introduce the heat kernels

( ~ + LLtx)KLL*(x, Y; fl) =O,

(2.18a)

('

(2.18b)

)

" ~ + LtLx KL, r(x, y;fl)=O,

subject to the

spectral boundary condition (2.12)

and to the initial condition

rrL,( X, y; O) --- KL, L( X, y; O) = 8( x - y ) .

(2.19)

Clearly, the index (2.15) equals the fl - * oo (low temperature) limit of the heat kernel a ( O ) -- tr(xle - a L v - e-aL*LlY)

f

=- d x t r [ K L L * ( X , X ; fl) -- KL*L(X,X;

B)].

(2.20)

Since the eigenfunctions of LL* and LtL are paired, za(fl) must be independent of 13. In particular, its fl ~ oo limit must equal its fl --* 0 limit. In the fl ~ 0 limit the heat kernels (2.18) admit (asymptotic) high-temperature expansions of the form oo

tr[Krr,(x, x; ,8)] - ~LL*(fl) + y'~ (fl)(n-d)/2b~L*(X),

(2.21a)

n=O oO

tdrvr(X, x ;

~)1 -

~v,(P) + E

(/~)("-~)/:b~,L(x),

(2.21b)

n~O

where d is the dimension of the manifold %. The scalar functions b~r,(x) and b~,t(x) are constructed from the coefficients of the operators LL t, LtL and their derivatives. These coefficients have the same functional form as the pertinent coefficients evaluated on a compact manifold. In particular, they are independent of the boundary conditions. The boundary dependence is summarized by the additional terms ~VLr,(fl) and q~L,r(/3) that satisfy [17]

~LLt (fl)--~r*L(fl) = ½ Z sign(h)erfc([h[V~),

(2.22)

where 2 r~

erfc(x) = ~

.

L dt e

--t 2

(2.23)

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A.J. Ntemi, G. W. Semenoff / Index theorems

is the error function. The h's are the eigenvalues of the boundary operator H and we recognize (2.22) as a regularization of the spectral asymmetry of H, 7/H = lirn E sign(X )erfc(lX [1/fl) • x

(2.24)

If we define

and take the fl + 0 limit of (2.20), by the fl independence of A(fl) we then get the Atiyah-Patodi-Singer index theorem

indexD=_A(oo)=A(O)=

[~dtt --

~dL,t] + l$,lU,

(2.26)

from which we obtain for */n ~tt = 2 index ~ + 2[ ~d, z -- ~dL, ].

(2.27)

The index is by definition an integer. Consequently the fractional part of ~ln arises from the second term. In particular, for a Dirac operator in a gravitational and gauge field background we have ~ d L , - ~dL,L= f [ A(%)eh(V)] vol,

(2.28)

where the "vol" subscript indicates that only the term proportional to the volume form of the manifold % contributes. i

i2

ch(V)=tre'F/2"=r+~-~trV+2(-~)2trV2+

...

(2.29)

is the Chem character of the gauge field (r is the dimension of the representation of the gauge group) and

=H

- xo sinh(½x.)

(2.30)

is the Dirac genus of the manifold %, with x a the 2-form skew eigenvalues of the curvature 2-form, Xl --X 1

1 "~-'~ R ab =

X2 --X 2

° . °

° . °

° . .

° ° °

. ° .

° ° °

...

° ° °

I

° . °

J

° ° °

Xn --X n

m

(2.31)

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A.J. Nleml, G. HI. Semenoff / Index theorems

Finally, we consider the possibility that the boundary operator H in (2.2) has h zero modes. One option is to add h to (2.26). In this way we obtain [17]

indexD--A(oo)=A(O)=[~dLt--~d,L]+½(h+~ln).

(2.32)

This means that the projection operator P in (2.8) projects both onto the positive and zero eigenvalues of H. However, in certain cases we might want to adopt another procedure: Suppose that the operator H itself has the off-diagonal form (2.3). Define the operator Hz by

H~ = H + zF c -=

(0

+z

M*

o)(z

0

=

-1

M*

-z

,

(2.33)

where z ~ 0 is a real parameter. Since H 2 = n 2 + z 2 >/z 2 ,

(2.34)

H z does not have zero modes. If we apply (2.27) we get d

T/nz = 2 [index ~ + ( ~a, L - ~LL* )] "

(2.35)

Notice that the r.h.s, of (2.35) is manifestly independent of z (excluding a step-function behaviour). Hence its 1.h.s. must also be independent of z. Indeed, if Hz~x = X~x

(2.36)

is a normalized eigenfunction with X ~ + z, then 1 ~x= 2 ~

1

[ H ' r¢l'/'x

(2.37)

is also a normalized eigenfunction with H ~ x = - ~'~x.

(2.38)

On a compact manifold this yields a one-to-one mapping between the positive and negative eigenfunctions of H r Consequently only the zero modes of H can contribute to ~ and we obtain [20] nn, = sign(z )index H

(2.39)

and from (2.35) we find

indexH=2sign(z)[indexDz+(~a,t--~dL,)].

(2.40)

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141

Consequently ~LL* a -- ~L*L d is necessarily a half-integer. However, since the manifold has a boundary this difference does not have to be invariant under local variations of the background fields and weobtain "anomaly cancellation" i.e. d d ~L*L -- ~LV =0

(2.41)

and the index of the two operators satisfy index ~ = ½sign(z )index H .

(2.42)

Comparing (2.32) and (2.42) we conclude that these two prescriptions agree if and only if h = sign( z )index H.

(2.43)

This relation is true e.g. in QED on S 2 and in SU(2) Yang-Mills theory on S 4. However, in general it does not have to be true. The two index theorems (2.32) and (2.42) are then different: in the present case the projection operator P in (2.8) projects onto the positive eigenvalues of H and onto the zero modes of either M or Mr, depending on the sign of z. The above discussion generalizes directly to open infinite manifold index theorems. Consequently in the following it will be sufficient to consider only boundary operators with no zero modes.

3. An open infinite-space index theorem The Atiyah-Patodi-Singer index theorem (2.26) is valid for a Dirac operator that satisfies the spectral boundary condition (2.12) on a ~ m p a c t manifold with a product metric of the form (2.1). On a more general manifold, e.g. if the metric is of the form

dsZ=dr2+r2dI2 2,

r ~ [ 0 , A],

(3.1)

with finite A and dI2 = a compact metric independent of r, (2.26) is not valid. A generalization to metrics of this form has been derived [1]. In addition to the characteristic classes and the ~-invariant, the r.h.s, of (2.26) then also involves the second fundamental form of the boundary. The asymptotic high-temperature expansions (2.21) of the heat kernels (2.18) are essential in the proof of the Atiyah-Patodi-Singer index theorem. Using the 13 independence of (2.20) the 13---, 0 limit of A(fl) can be related to its 13 --* 00 limit which yields the index. This is valid on a compact manifold where the spectrum of is discrete. However, in many physics applications one deals with background fields that have a nontrivial behaviour at infinity. As a consequence a ( f l ) might be a nontrioial function of ft. In such cases the index theorem (2.26) might not be

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A.J. Nlemz, G.W. Semenoff / Index theorems

applicable since a compactification can lead to an inequivalent index problem. For example the index of any elliptic differential operator on an odd dimensional compact closed manifold vanishes [2] even though there exist open infinite space operators with a nonvanishing index [14,15]. Furthermore, in certain physics applications the spectral boundary condition might be undesirable. In this section we shall derive an index theorem for a Dirac operator D defined on an open infinite manifold 9(~ with euclidean metric ds 2

~__.

g/~vdx~ dx ~.

(3.2)

For these Dirac operators the heat kernel A ( f l ) is in general a nontrivial function of fl [14,16, 35]. Consequently the index can not be obtained from its fl ~ 0 limit. In order to find an expression for the index it becomes necessary to evaluate the fl ---> limit of A (fl) directly. We shall assume that as the proper distance r tends to infinity the metric (3.2) approaches e.g.*, ds

2 r---~ oo

'~ dr2 + r 2 g , j d y ' d y j,

(3.3)

where g,j is a metric on a compact manifold ~ (the "boundary" of ~ at infinity). The Dirac operator D has the form

(0

~=

L*

'

where L is some elliptic differential operator of the form (for notation see appendix A) L = i'ta W ~ + Q ( x ) = i~lae~ V ~ + Q ( x )

(3.5)

and L* is the hermitian conjugate of L. The covariant derivative V ~ is defined using the spin connection on % and Q ( x ) includes all other background fields (including background gauge fields). We assume that ~ is a Fredholm operator. The "ta matrices are constant matrices that satisfy "~a~Yb-I- yb~Ya~-- 28~b ,

(3.6a)

"~a~/b ~ -I- ~/bVa ~ = 28~b.

(3.6b)

= ira v'~ + K ( x ) - iFae ~ W ~' + K ( x ) ,

(3.7)

We write (3.4) as

* This assumption ts here made only for discussion. From what follows it is obvious that the final result will be valid for a much larger class of metrics. The only essential restriction arises from the assumption (3.44).

A.J. Ntemt,G.I4. Semenoff/ Indextheorems

143

where we have defined

r°=v 0 ~¢a) 0' K ( x ) = ( Qt(x) 0

(3.8)

Q(x)) 0 "

(3.9)

From (3.6) we conclude that the F-matrices satisfy the euclidean Dirac algebra

{ra, rb}=28ob,

{rC, ra}=O,

( r c ) 2 = 1,

(3.10)

where F c is the constant matrix introduced in (2.33). The operator D anticommutes with F c, {~,r c } =0.

(3.11)

Consequently the spectrum of ~ is symmetric, i.e. if

~ k e = E~e

(3.12)

0(TClke) = -E(FCd/e).

(3.13)

then

For E = 0 the eigenvalue equation for ~,

(UE)=Ed/E

(3.14)

L*u = 0

(3.15)

becomes

1_,v= O,

and the zero modes of L and L* correspond to eigenmodes of F c,

Notice that the isomorphism (2.16), (2.17) between the nonzero eigenfuncfions of

LL t and LtL remains valid on an open infinite manifold. However, since (2.17) does not have to preserve the densities of the continuum spectra, in general the heat kernel A(fl) will be a nontrivial function of ft. On an open infinite space the spectral boundary condition (2.12) becomes meaningless. Instead we impose the following boundary condition [36]: since J~ is hermitian, it can be interpreted as a Dirac hamilt0nian on a minkowskian manifold

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A.J. Ntemi, G. W. Semenoff / Index theorems

Rx X ~ with a flat time direction. The usual causal (Feynman) boundary condition then defines a unique causal propagator on this manifold, and for time independent background fields the propagator determines the function space where O acts. Intuitively this means that we impose standard "scattering theory" boundary conditions. We wish to find an expression for the index of D. We define z

+ z rC=

L*

L )

(3.17)

"

Clearly, ~2 = j~2 + z 2 >/z 2

(3.18)

and as in (2.34) we conclude that all eigenvalues of (3.17) ~)fl" = ~z [ ~] =h~l,

(3.19)

are nonvanishing (~2 >/z2). We introduce the following regularization of the *iinvariant of D,: (3.20) oo

Here p,(h) is the spectral density of D~ [16] and we expect that the fl ---,0 limit of (3.20) exist, def

*1~-- lira ~ ( f l ) . B~o+

(3.21)

In the z --* 0 limit the operator ~z approaches ~ and we conclude that in this limit ,/z yields the index of ~, lim ~ -- - sign(z )index

z--*O

= sign(z) ( dim ker L - dim ker L* }.

(3.22)

In order to compute (3.22) we write (3.20) as 1

n

(fl)

=-f ,r

oo

o0

do e-'a / dXpAX)V -oo

~k

+,o 2

=-,rf°° dtoe_,a~ofgcdxleltr(xl~ l x )

(3.23) (3.24)

A.J. Nteml, G. W. Semenoff / Index theorems

145

Here (as always) the trace "tr" denotes a trace over all internal indices and the integration over the manifold % yields a trace over the function space. Since (3.24) is a regulated quantity, the various orders of the traces commute (see appendix B). Consider the integrand in (3.24): tr(xl ~2 + ,02 lY) = z tr
1

= i-tr(xlFC~, ff /// +

ia

lY),

(3.25)

where we have defined o--- ~

+ z2

(3.26)

and we have also introduced a point-splitting x #: y. We shall now generalize the trace identity derived in [20]: At this point we find it convenient to define the various traces using a ~'-function regularization. In this approach the chiral anomaly arises from the nonvanishing of the trace of pC in the function space where (3.17) operates. This trace is defined by analytic continuation (x is here fixed)

tr(xlFClx) def ~ m t r ( x l F c =

[lel(/~

1

+ io

)]' Ix >'

(3.27)

where the r.h.s, is a meromorphic function of s which is analytically continued from the complex-s half-plane with large positive Re(s). We write the r.h.s, of (3.27) in the equivalent form 1

tr
(3.28) 1

= ½tr(xlrClel ( i r . v. + K + ia) [lel( D + ia)]s+z lx )

(3.29)

1

+ ½tr(xlFC [lel ( ~ + io)]

s+l

lel (iF~ V~ + K + io)lx).

(3.30)

Using the identity w~ le[/"' --- telr" w,,

(3.31)

and the explicit form (A.18) of the spin connection (remember that the gauge field

146

A.J. Niemz, G. IV. Semenoff / Index theorems

A~, is included in K) we write (3.29) in the form 1

½tr( xlyC( i w~£ ~ + K + io )lel [lel (4~ + io ) ] s+ l Ix) = ½i tr(x I a,£c r*lel

1

[lel(~ + ia)] ,+11 x) 1

+ {tr(xlYC(i~,,£ " + r+io)lel [lel(D + io)]'+z Ix).

(3.32)

Similarly, (3.30) can be written as

½tr(xl Fc

[lel( I~ + ia)] "+~

i a,,£* lel Ix)

1

+ ½tr(xlrC [lel (~0 + io)] s + l

+/~+ ia)lel Ix)

(3.33)

and using cyclicity of trace this becomes 1

= - ai tr(xl£CF~'lel [lel(~ + io)] ,+z O~,lx) 1

- ½tr(xl£C(i%P ' + K - ia)lel [lel(J~ + ia)] s+t Ix) •

(3.34)

Combining (3.28)-(3.30), (3.32) and (3.34) and using


(3.35)

we then find in the s ~ 0 limit the trace identity "

C

def.

tr(xlY I x ) =

limtr(xlr c

• -.o

1 [lel(~+ia)] "lx)

= ~o (½i%tr(xlr~rc[j~ + 1io]s+z Ix)

1

+ia tr(xlr c [a + ia],+x Ix) 1

def. {i 171~t r ( x l r ~ r C ~ l x )

) 1

+ io tr(xlrC~b + io Ix).

(3.36)

A.J. Memt, G. W. Semenoff / Index theorems

147

Notice that (3.36) is the standard chiral anomaly equation and for a vanishing l.h.s, it reduces to a continuity equation for a fermionic flavor current. Combining (3.36) with (3.24), (3.25) we find

~ ( f l ) = sign(z)e-~lzlf 1

~

+

dx leltr(xlrClx) z

1

d~e-'0~' .--2T'T~-/, _2 0~ d y ~ t r ( x l i r ~ ' r c ~ - - - i x > ,

=

to

+z

.%

O+io

(3.37)

which reproduces previously derived results [14, 20, 22, 30] in special cases. Consider now the first term on the r.h.s, of (3.37). This term is the integrated anomaly and under "reasonable" assumptions* on the background fields it is proportional to the integral (2.28) over the characteristic classes. Indeed, if we introduce the fl ~ 0 asymptotic expansions [3, 38] oo dx [e[tr(xle -aLL* Ix ) - ~-~0fl<"-d)/2~LLt'

(3.38a)

oo

f%dx leltr
(3.38b)

n~O

with ~ L t and ~ t L defined in (2.25), for fl = 0 and large values of z we find [16]: 1 d n~-~-Izl sign(z)~ tzl-"r[½+~(n-d)]{ ~7.v- ~*L}"

(3.39)

hid

In odd dimensions the z ~ oo limit of (3.39) vanishes [3] while for even values of d and for proper background fields we find lilrl ~z = { ~ LdL *

g--~ oo

- - ~ Ldt L

}"

(3.40)

Under mild assumptions on the operator ~ the second term on the r.h.s, of (3.37) does not contribute to (3.40) and we get

fg dx leltr(xlFCl x) = za(O) ={~L,--~atL}=f

[.4(%)ch(F)]vol.,

(3.41)

* "Reasonable" here means that we assume the standard form of the chiral anomaly. This is guaranteed if the continuum eigenfunctaons asymptotically approach plane waves so that e.g. Fujikawa's technique [37] becomes applicable

148

A.J. Nzeml, G. W. Semenoff / Index theorems

The fact that the surface integral in (3.37) is nonvanishing shows that in general the heat kernel A(fl) is a nontrivial function of ft. Consequently its 13~ 0 limit cannot be identified with the index of J~. In order to obtain an expression for the index we must compute the z ---}0 limit of this surface integral: We define

n"(X)=e:(x).a(x)

(3.42)

to be a unit normal to the surface ~ with its induced metric. We introduce the normal and tangential components of the Dirac matrices,

r=..r.,

= r.-."c...

(3.43)

Similarly we use n ~' to define the normal and tangential parts of all other quantities. We assume that asymptotically the normal direction is geodesically flat and that all background fields are asymptotically independent of the normal coordinate, r--~

n~%

,0,

r--*~

n t' O~,n" n ~

(3.44a)

, O,

(3.44b)

Or (all background fields) r-* "~ oo Or

(3.44c)

where % is the pertinent connection and r the proper distance on the manifold %. The metric tensor (3.3) is certainly consistent with (3.44). However, these assumptions are more general and consequently our final index theorem will be valid for a larger class of backgrounds. Using (3.42), (3.43) we write the surface integral in (3.37) as 1

f ~ oo

~odto 1

e_,a,0 ~ Z [ d/~(Z)tr(xliYY 1 ^ c o~2 + z 2 Jo Ib + i-------~ Ix) o¢

= ~-~ f~

Z

d ~ e -'~'° 602 + oo

× f dt~(Z)tr(xliFc

Z2

•~ + i ~ , F T v r1+ ~ , K + i o l X ) ,

(3.45)

where d # ( Z ) is the measure on the asymptotic surface ~. Without loss of generality we can assume that on ~ the tangential operator d ' r r x7T + ~ K

(3.46)

A.J. Nteml, G IV. Semenoff / Index theorems

149

does not have zero modes. We then use 1

8(~)---

z

tim z2

~r z-,0+

+

(3.47)

'° 2

tO take the z --) 0 + limit of (3.45). The result is 1

d/~ ( ~ ) tr( xliF c 18

• + i~F r VT +

(3.48)

~ K Ix)"

Defining i ~ = ig"l'*TgrT + ~Qt,

(3.49a)

- 1®* = i ~ , y T Wr + ~*Q,

so that the operator (3.46) becomes

(3.49b)

0) _i@,

(3.50)

'

we can write (3.48) as i d/~(Z) tr(xl ~

i _i®,+i~[x).

(3.51)

Since the background fields are independent of the normal variable ~, we can introduce the following Fourier representation of (3.51),

1 ~ m. + f]° ~ = 4~r

d~e-"~f ~ d#(~)tr(x[

~ 1+

1 ®,_i-------~]x).

(3.52)

In order to interpret this we introduce the eigenmodes of ® and ®*, ®~ = X~,

®*~0 = X*~0.

(3.53)

For simplicity we assume that the spectral densities for ® and ®* are equal. (3.52) then becomes -i

~o oo

(

1

1

~ - iX

o~+ iX*

= ½ f d # ( X ) sign[Re(X)] de~ 1,/[Re(®)]

)

(3.54)

i.e. a spectral asymmetry constructed from the real part of the eigenvalues of @.

150

A.J. Ntemi, G. W. Semenoff / Index theorems

Using this result we now take the z--* 0 + limit of (3.37) and combining (3.22), (3.41) and (3.54) we find the following open-space index theorem index ~ - a(oo) = a(0) + ½.[Re(°~)]

= {

} + ½.[Re(¢)],

(3.55)

which should be compared with the Atiyah-Patodi-Singer index theorem (2.26). Notice that by (3.6) Sy~ +'/tT$ = O,

(3.56a)

~tyT + yT~ t = 0.

(356b)

If we assume that ® = ®t and that ® anticommutes with ~ and use (3.24), (3.52) reduces to 1 :~ . ® def lim - - [ d & e - " ~ d/~(Z)tr(xl Ix) --- ½.,¢ • --,o+ 2¢r J-oo ~ O~2q" (~2

(3.57)

i.e. the spectral asymmetry of a hermitian operator. The index theorem (3.55) then becomes index D = { ~ ' L - - ~OL* } + ½"¢

(3.58)

in analogy with (2.26). 4. A comparison: chiral anomaly and manifolds with boundaries Despite the obvious differences, the index theorems (2.26) and (3.58) have an equivalent structure when expressed in terms of the characteristic classes. In the case of the infinite-manifold index theorem the integral over the characteristic classes arises from the chiral anomaly equation (3.37). In the derivation of (2.26) we did not encounter the chiral anomaly equation. Since these two index theorems are different one might then inquire whether the spectral boundary condition (2.12) has modified the chiral anomaly equation. We consider the massive Dirac operator i~7 - im - iFae~ gr~ - im,

(4.1)

with the covariant derivative (A.18) on a compact d-dimensional euclidean spin manifold % with boundary ~. We choose a representation of the y-algebra where

o

(42,

A.I. Nzem,, G.W. Semenoff / Index theorems

151

We introduce the eigenfunctions

i~r~be = E~be

(4.3)

and we assume that close to the boundary ~ the metric on ~ is a product metric so that L = ~. ~r + H,

(4.4)

where h is the unit normal to the surface and H is the operator H = i'tr Vr"

(4.5)

The subscript "T" denotes the tangential part. H is hermitian if the matrices ~'r are hermitian. On the boundary ~ we introduce the complete set of states

He x = ~e x

(4.6)

and for the eigenfunctions (4.3) we impose the spectral boundary condition (2.12). On ~ we then have the expansions ½(1 +

rc) E =

E g~ex(x) c ~ > o ,

(4.7a)

h>0

½(1 - FC)~kE = E f~ex(x) c ~ < o ,

(4.7b)

h<0

where ~ZIC>o and ~ < 0 denote the eigenspaces of H with eigenvalues ~ > 0 and h < 0 respectively. For simplicity we assume that H does not have zero modes. The propagator 1 (i¢ r - im)S(x, y; m) = ~_, 8(x - y ) , lel

(4.8)

subject to the boundary condition (2.12), admits the spectral resolution

1 S(x, y; m) =- (xl ~ I Y )

= ~-, E

~bE(x)~te(y ) E -im

(4.9)

We introduce the Pauli-Villars regulated propagator

SR(x,y;m)=Alim ~ (S(x,y;m)-S(x,y;A)},

(4.10)

152

A.J. Nterm, G. W. Semenoff / Index theorems

which satisfies the chiral anomaly equation

dof½iv"tr
v~'J; =

1)

i(7-im

{1

= im

tr
i ~ 7 - iA

Ix)

1 }

i~7 - i m

Ix)

i~7 - iA

iA

- tr,

(4.11)

We are mainly interested in the integrated version of (4.11). For this we first consider the integral of the l.h.s. Using the spectral resolution (4.9) we obtain

=-~if~v"tr
1

i~7- ia

=x-iY"'£d°~+*~(x)r~r%E(X) E 2

E ~

1 __

]Ix>

,m

E

1 _

iA

]

(4.12) "

Since rcqJe ----f f - r and

n~r~%>oC%
n"r~%o

(4.13)

this integral vanishes, i.e. ~ do~j~=0.

(4.14)

Consider now the r.h.s, of (4.11): Using the spectral resolution (4.9) the first term yields im

~Sy.~~trFC~be ( E - i1r a

E-

1iA ) "

(4.15,

Since / ' c ~ E = W-E, only the zero modes of i # can contribute to (4.15). In the A ~ oo limit we then obtain

,m a--, 0,

E - im

,m ) E - iA

= index 6 -

(4.16)

A.J. Nzemt, G. W. Semenoff / Index theorems

153

Using the explicit form (4.9) the second term on the r.h.s, of (4.11) can be written as

iA

A2

A2

F.,f * rql,EE_ia=ftr(xl L . t L + A 2 - L L t + A 2 ] X ) . E

(4.17)

Applying the identities derived in [39] we find A2

liraoo A-.

f tr(xl LtL + A2

_- m0(Tr[0

A2

L L t + A2Ix)

Trte

oo

)

d

+ la o fo dBe-aa-- [tr(xle-aL"-e-a *lx)].

(4.18)

Under mild assumptions on the heat kernel

Za( fl ) = Tr[e-/JL' L -- e-I~LL* ] ,

(4.19)

the integral in (4.18) vanishes and we get

lira ~ f ~ t E F C ~ e ,A = za(0)= [ ~ d L , - ~d,L] + ½ ~ . a-~ E - iA

(4.20)

(Actually on a compact manifold (4.20) follows trivially from the fl independence of A(fl).) In (4.20) we have used (2.20)-(2.24) in the last step. Combining (4.12), (4.16) with (4.20) we then conclude that the Atiyah-Patodi-Singer index theorem (2.26) is a manifestation of the chiral anomaly on a compact manifold with a boundary. However, unlike in the case of an open infinite space, now the chiral anomaly is not an integral over the pertinent characteristic classes but in addition it also includes the ,/-invariant of the boundary operator.

5. Spectral asymmetry and family index theorems In many physics applications we are interested in the computation of the ,/invariant of a hermitian Dirac operator H in the presence of topologically nontrivial background fields. For example, the fermion number of a topological soliton is proportional to the 7/-invariant of the pertinent Dirac hamiltonian [12]. In this

154

A.J. Niemz, G. W. Semenoff / Index theorems

section we shall show how the index theorem (3.55) can be used to compute such ~-invariants. We consider an arbitrary Dirac hamiltonian H defined on a d + 1 dimensional open infinite manifold °)IL with minkowskian signature and metric of 'the form ds 2 = dt 2 + g,j d x ~ d x j

(5.1)

i.e. we assume that g0o = 1 and go, = 0 so that topologically ~ L - R x ~. The hamiltonian H is of the form 1 H = -7T°'yaE~ V, + Q ( x ) , 1

(5.2)

where V, is the covariant derivative on ~ (with its induced metric) and Q ( x ) includes all background fields (including the background gauge fields). We assume that the background fields are static and without loss of generality we also assume that the eigenvalues of H H ~ , -- E,+,

(5.3)

are nonvanishing. In order to compute ~H we introduce a one-parameter family of Dirac hamiltonians H, with ~"~ ( - o o , co) by extending the background fields in (5.2) (including g,j) t o ~--dependent fields. This corresponds to extending the manifold o~ to a cylinder ~ - R x o~ with euclidean signature. We assume that lira H, -- H ,

(5.4a)

"r---~ -4- o 0

lim

H, - H o

(5.4b)

"i" "--} - - 0 0

and we assume that H 0 is some conveniently chosen hamiltonian for which the ~-invariant can be computed independently (e.g. a free field hamiltonian). On the cylinder X we introduce the elliptic operators L = i't ° O, + iT°H~, L f = i7 ° O~ - i H : t °

and the following (euclidean) Dirac operator

We define the matrices

(5.5a) (5.5b)

A.J. Nlemz, G. Be. Semenoff / Index theorems

155

that satisfy the d + 1 dimensional euclidean Dirac algebra ( F ~ , F ~) =2~ ~

(a, fl--1 ..... d + l )

(5.8)

and write (5.6) as f) = i F ° a, +

iFaE'oV, + K(x),

(5.9)

where K--

( 0 iy°Q( x ) ) . -iQ(x)y ° 0

(5.10)

In analogy with (2.33), (3.17) we introduce the constant matrix F c and define

(5.11)

& = ~ + ~r c.

We substitute (5.11) into (3.37) and divide the explicit surface integral into three separate integrals:

*lz = sign(z)fdx [e[tr(x[FC[x) 1

o~

z

+ --2¢r~--.o+limf_oo da~ e-'~'~ ~ 2 + z 2 {Ix(z)+I2(z)+I3(z)}

- A(O) + --',(z) + --'2(z) + ,-'3(z),

(5.12)

(5.13)

where we have defined A (0) = s i g n ( z ) f 0 x [e Itr(xlFclx)

(5.14)

and

1

/I(Z)

=f*=oo dx

I2(z)

=- f

1

~

Ia(Z) =~R

le]tr(xliF°FC~) + i--------dlx),

--

00

dx le[tr(xliF°FC ~ + Ix), io 1

do~ t r ( x l i F v F C ~ l x ) . ,=oo ~ + io

(5.15a) (5.15b)

(5.15c)

The parameter o is defined in (3.26) and the integrals (5.15) are evaluated over the caps at • = + c~ and the spatial boundary of the cylinder 9¢- R x 2, respectively; the minus sign in (5.15b) is a consequence of the orientation.

A.J, N:erm, G. 14/.Semenoff / Index theorems

156

Consider first the integral 11: We write it in the equivalent form

leltr(xlF°r c fp21+ o2 (I~ -

II(z) = f = oodx

io)lx >.

(5.16)

Using the explicit representation (5.7) of the "/-matrices this becomes 1

Ii(z)= ~

1

dxleltr(xli~,°L,L + o 2 L * - i ~ ° L L ,

+o2LIx ) .

(5.17)

Ix>.

(5.18)

~00

Substituting this into (5.12) and using (3.47) we find 1

11(0) = ½ f dx l e l t r ( x l - - + • =oo H - 8,

1

H+-~

Since H and the trace are both manifestly independent of % in analogy with (3.52) we can introduce a Fourier representation for a~ and write (5.18) in (5.12) as 1 oo H "-'~(0)=-- ~m d b e - ' ~ e ° / d x l e l t r ( x ] H2 IX> 2~r~ o+ -oo Jo~ +g~2

f

= - -1 ~m f_:° d~ e-'~"'~ E~ En + ~2 2~r ~ 0+ ~ n = {,~0+ E sign(En)e-~te't-- !~/2t/.

(5.19)

n

Similarly, we find that the contribution from the bottom z = - oo of the cylinder is Z2(0 ) = - ½71,0.

(5.20)

Finally, as in sect. 3 we fred that the contribution (5.15c) that arises from the spatial boundary of % is of the form ~3(0) = ½71[Re(®)].

(5.21)

Here ® is in general neither a hermitian nor an antihermitian operator and ~l[Re(®)] is constructed from the real parts of its eigenvalues, in analogy with (3.54). Combining (5.19), (5.20) and (5.21) with (3.22) and (5.12) we find for the index of the family operator - index I~ -- A (0) + 1~1n - ½71no+ ½~1[Re( ®)].

(5.22)

A J Nteml, G. 14/. Semenoff /

Index theorems

157

From (5.22) we find for 71H ~7, = - 2 index ~ - 2Zl (0) + ~/Uo(0) - 71[Re( ® ) ] .

(5.23)

This relates the spectral asymmetry of the Dirac hamiltonian H to the index of the operator D, the integral A(0) over the characteristic classes and to the spectral asymmetries of the operators H 0 and ®. Provided the quantities on the r.h.s, can be computed (5.23) can then be used to obtain ~7~,.

6. An example In this section we show how the family index theorem (5.23) can be used to compute ~7~- We consider the following 1 + 1 dimensional lagrangian

L = ~( t ~ - dpl - iys~2)~b.

(6.1)

We assume that the background fields 4h(x) and ~2(x) are arbitrary static functions of x with the limits ~bl(___oo) =a_+,

~2(___oo) = b ± .

(6.2)

If we identify

70 = 03,

.rl = _ iol,

~/s = ~O,)tl = 0 2

(6.3)

where the o's are the Pauli matrices, the pertinent hamiltonian has the form ( H =

~1

-Ox'[-t~2 )

Ox q- ~b2

__ d~1

(6.4) .

We wish to use (5.23) to compute the ~7-invariant of (6.4): As in the previous section we introduce the parameter ~"~ ( - o¢, oo) and extend the fields #h(x), ~2(x) to smooth functions of x and ~-. We choose these extensions in such a way that both ~l(x, - oo) and ~2(x, - oo) are independent of x, t~l(X , - oo) ~ a + ,

~2(x, - oo) = b _ .

(6.5)

The comparison hamiltonian (5.4b) then becomes

( H°=

a+ tgx+b_

-Ox+b_ ) -a+ "

(6.6)

On the boundary x = - oo we choose qh to be an arbitrary function of "r and q~2 to be a constant, q h ( - oo, ~ ~ ± oo) ~ a :~,

~ 2 ( - oo, ~') - b _ .

(6.7)

158

A.L Ntenu, (7.14/.Semenoff / Index theorems

Similarly, on the boundary x = + oo we choose ~l(Oo, "r) -= a+ ,

,2(oo,'r---> + ~ ) ~ b : ~ .

(6.8)

The operator L defined in (5.58) is then of the form L

= i o 1 tg~ + io 3 Ox -

o2~a + i~2

(6.9)

and the boundary operator ® in (5.21) is an hermitian operator with ®=io2 O,-al~l(x

) -o3b_ ,

@= --io2Or+ola++o3dp2,

(6.10a)

x= -o0,

(6.10b)

x= +oo.

Consequently (5.15c), (5.21) becomes

f~

-'~(0) -- ~1 f ~_ d~ _ d~" ( tr(r, oo I 032® + g~2I% °0 ) 03 - t r ( ' r , - oo [ ~ l ~ "

-oo)/

(6.11)

by (3.24); this is half the difference in the 71-invariants of the operators (6.10a) and (6.10b). Clearly, both operators (6.10) can be represented in the form (3.17). The ~/invariants for operators of this form have been computed in [20]. Instead of reproducing the details we give the result: 1[ b+ arctanb_ + a r c t a n a_ -[Re(03)] = 2.~3(0 ) = ~r ~arctan a+ a+ b_ -

=

--

¢rt

arctan

a+

- arctan

a_ )

.

a+} arctanff-_

(6.12)

Similarly, the operator (6.6) is also of the form (3.17) but with constant background fields. Hence its ~l-invariant vanishes. Combining these results we conclude from (5.23) that the ,/-invariant of the operator H in (6.4) is given by

~ n =

- -- arctan

- arctan a_ ) - 2 index ~ .

(6.13)

In the general case the index in (6.13) is quite difficult to compute. However, since

A J. Nlem:, G.W. Semenoff / Index theorems

159

the index is an integer the second term yields the fractional part of 71it exactly. The result (6.13) reproduces the earlier approximative results [33] in the pertinent limits and agrees with the result obtained in [19]. Additional applications of the family index theorem (5.23) are discussed in [12] and [40].

7. Elementary cohomology: fermion fractionization and the adiabatic theorem In this section we shall develop elementary cohomology for the index theorem (3.55), (5.23). However, instead of using the language of differential forms we find it more instructive to use concepts that have a direct physical interpretation. In particular, we wish to estabfish a connection to the field theory analysis of fermion number fractionization [12] and exhibit a relation to the adiabatic approximation by Goldstone and Wilczek [33]. We also demonstrate how corrections to the adiabatic approximation can be accounted for with the family index theorem (5.23). Without loss of generality we consider a Dirac hamiltonian H defined in the background of a topological soliton (such as a kink, vortex, magnetic monopole .... ). The fermion number N of this soliton is proportional to the ~-invariant of H [12], N = --t7/2I-I"

(7.1)

The adiabatic scheme [12, 33] for the computation of n is implemented by first introducing a background field configuration for which the fermion number is exactly known. The desired soliton configuration is imagined as being built up from this comparison configuration by slow (adiabatic) changes of the fields in space and time. The local fermion number density is then calculated in an expansion in powers of derivatives of the background soliton and, by invoking current conservation the fermion number of the soliton state can be extracted [33]. Often the comparison configuration is the normal vacuum or some conjugation symmetric background soliton, and the result obtained using the adiabatic technique can be expected to be reliable for final solitons with small spatial gradients. However, in practice the result is correct only for the fractional part of the fermion number. A modification of the background field will in general modify the spectrum of the Dirac hamiltonian and an important issue that arises is, whether the state reached by the adiabatic construction is indeed the ground state of the final soliton [12, 23, 41]. This can only be the case if a spectral flow contribution to the final fermion number is absent. Suppose that during the adiabatic process an energy level of the Dirac hamiltonian crosses zero, say from negative values to positive values. During the adiabatic evolution the fermion number operator

N = f dK { b~b~ - dt~d~ } - ½./.

(7.2)

160

A.J. Ntemz, G. W. Semenoff / Index theorems

remains normal ordered with respect to the original comparison state since in an adiabatic process the definition of this operator can only change continuously. As a consequence the resulting positive energy state will be interpreted as an antifermion and in the expansion (7.2) it is still associated with a creation operator d +. Hence the adiabatic result for the fermion number will differ by one from the fermion number of the "true" ground state and the adiabatic computation can only yield a reliable prediction for the fractional part of the fermion number. In physical applications where the integer part plays an important role (such as the fermion number of the skyrmion) the computation must then be supplemented by an analysis of the spectral flow [12, 23, 41]. In order to relate the adiabatic method to the family index theorem (5.23) we introduce an interpolating family of Dirac hamiltonians H, with the asymptotes lim H, = H , "i" - ~

(7.3a)

-'}- 0 0

lim H , = H 0. T---~ --

(7.3b)

O0

Here H is the hamiltonian corresponding to the background soliton whose fermion number we wish to compute and H 0 is a comparison hamiltonian corresponding to a background field whose fermion number is assumed to be known. Suppose that the evolution of H, as a function of z is such that the euclidean Dirac equation 0

-~zq,(x, ~-) = H,q,(x, ~')

(7.4)

is obeyed for all values of z. This implies the continuity equation a.<

) = a.L = o

(7.5)

for the fermion number current. However, it is possible that H+ interpolates through background fields for which this fermion current has an anomalous conservation law. In such cases we must modify (7.5) to

O~,j~,= za(x, ~'),

(7.6)

where A(x, ~-) is the appropriate anomaly term i.e. a source for fermions. We first consider the case where there is no spectral flow, i.e. we assume that for all values of r the eigenvalues E('r) of the Dirac hamiltonian H+ are nonzero. The

A.J. Ntemz, G.W. Semenoff / Index theorems

161

fermion number is then obtained as follows:

N -N_ =fdxjo(x, )-fdxjo(x,- ) =

=f_

0

-- - f= d,96dS'j,(x, >+

f dxa(x, ),

(7.7>

where we have used eq. (7.6) in the last step. Notice that the result depends only on the asymptotic properties of the current j~(x, ~). All local dependence on the background arises from the anomaly term. If we recall that the fermion number is proportional to the pertinent ~/-invariant and identify

f~_ dz~dS'j,(x,,)

= - ½,/[Re(®)],

(7.8a)

oo

-~ de f dxA(x, ~') = A,

(7.8b)

the eq. (7.7) becomes 0 = a +

1

1

+

1

[Re(®)],

(7.9)

which is formally equivalent to the index theorem (5.23) with index ~ = 0. We shall now show that a nonvanishing index corresponds to a nontrivial spectral flow during the adiabatic evolution of the hamiltonian H,. For definiteness we assume that at times ~'1 < 72< "'" <~'~ some number ~(~'1) . . . . . # ( ~ , ) of the eigenvalues E(~') cross zero. Now ~n, is a discontinuous function of "r with discontinuity + 2 for each eigenvalue E(¢) that crosses zero from negative values to positive values or vice versa. Whenever an eigenvalue E(~') of H, crosses from negative to positive (positive to negative) values the expansion (7.2) for the second quantized field operator must be modified accordingly. A state which was quantized as an antifermion (fermion) for negative values of E(~') must be reinterpreted as a fermion (antifermion) for positive values of E(¢). As a consequence the fermion number operator (7.2) has a discontinuity for each eigenvalue crossing. The definition of the normal ordered fermion current will then be different for each of the intervals (¢,, %+1) and the continuity equation (7.6) can only be a valid piecewise. Hence the relation (7.7) is not globally valid. However, since there is no eigenvalue crossing during the interval (%, ~',+1) we may still introduce relations analogous to

162

A.J. N~emi, G. W. Semenoff / Index theorems

(7.5), (7.6) for each of these intervals and the fermion number at 1- is still given by

(7.10) with a limiting description for the points ~'1..... %. If we write ~H, as a sum of its continuous part ~ t { H, } and discontinuous part ~d~c( H, }

~H, = ~cont{ H* } + ~disc{ H~"},

(7.11)

we obtain for the fermion number N~-N_~=

- :! / :

o~

d dl- .~_~=ont{H, } _ _ ½E,a~c{ H,, }

oo

(7.12)

l

If we now identify oo d ~f] d, ~,~t{

H, } = - ~ , [ R e ( ® ) l - a ( 0 ) ,

(7.13)

1 E~disc{ H,,} = - i n d e x J~

(7.14)

I

in addition to (7.1) we conclude that (7.12) is formally equivalent to the index theorem (5.23). Furthermore, comparing (7.7), (7.12) with (3.37) we conclude that we can choose za(x, ~') = tr(x, 1 =

,),

,irClx,

+

f

oo

_d

(7.15) z

z-q----+

1

2tr(x,,lir,rC,+iolx,,

>.

(7.16)

Hence (7.12) also provides a physical interpretation of the various terms in the index theorem.

8. D i s c u s s i o n

We have derived a new index theorem for open infinite-manifold Dirac operators. Our derivation is based on a manipulation of trace identities and employs the standard chiral anomaly equation. The index theorem is valid for a very large class of operators, the only essential restriction arising from (3.44). In particular, we emphasize that while we do assume the validity of the chiral anomaly equation, we do not make any assumptions of the detailed form of the pertinent anomaly. Indeed, if the standard form of the anomaly is for some reason modified e.g. as a

A~J. Nzemt, G. W. Semenoff / Index theorems

163

consequence of "large" background fields, our formal manipulations still remain valid. We would of course have to reconsider our identification of ~(0) with an integral over the characteristic classes. The trace identifies that we employ resemble those used by Callias and Weinberg. However, our index theorem is much more general: The index theorems derived in [14, 30] assume a very particular coupling of fermions to background Higgs fields. Here we consider arbitrary background fields. In [14, 30] the index is related to a particular asymptotic property of the I-Iiggs field on the sphere at infinity. Here we identify the asymptotic contribution with the ,/-invariant of a Dirac operator and find that such a contribution arises whenever this ,/-invariant is nontrivial. In particular, the pertinent background field does not have to be a I-Iiggs field and the asymptotic surface does not have to be a sphere. As an example, our index theorem is applicable e.g. to Dirac operators in the background of gravitational instantons where the asymptotic surface generically has a nonspherical geometry. (Further examples of operators with a nontrivial ,/ can be found in the literature; see e.g. [1,17-23, 33,41].) Formally, our index theorem is very similar to the Atiyah-Patodi-Singer index theorem, see eqs. (2.26) and (3.58). However, these two theorems are also very different: The spectral boundary condition introduced in [17] does not make much sense on open infinite manifolds. Instead, we use a causal (Feynman) boundary condition imposed on a related Dirac operator in one more dimension. Furthermore, our index theorem arises in a different manner: The Atiyah-Patodi-Singer theorem relates the index of a Dirac operator to the fl --> 0 limit of the heat kernel z~(fl). Such a relation only exists on a compact manifold and for a particular choice of boundary conditions. On an infinite manifold A(fl) is in general a highly nontrivial function of fl and a priori its fl ~ 0 limit does not have anything to do with the index. However, using the chiral anomaly equation we find that there is a connection but only if one in addition considers an ~/-invariant contribution which in the AtiyahPatodi-Singer case is already included in the anomaly A(0). In sect. 4 this fact was used to exemplify the formal differences between the two index theorems. We found that on open infinite manifolds the index theorem has some flexibility which is not available on compact manifolds. To a certain extent the index is insensmve to the asymptotic topology of the manifold. Since such asymptotic changes also leave the integral over the characteristic classes invariant, the corresponding ~-invariant must also be invariant. We explored this freedom to develop a technique for the computation of the ,/-invariant on an open infinite manifold by interpreting this manifold as an infinite "cap" of a "cylinder". This technique yields a practical scheme for the exact computation of ~-invariants and provides a complete solution to the mathematical problem of fermion number fractionization. In particular, we find that 71 is always a topological quantity in the sense that its dependence on local variations in the background can be related to an integral over a characteristic class and to the index of a Dirac operator defined inside the

164

A.J. N w m l , G. W. S e m e n o f f /

I n d e x theorems

"cylinder" i.e. on a manifold with one more dimension. Notice that in the case of open compact manifolds such deformations are not possible: even an infinitesimal variation on the boundary will in general affect the integral over the characteristic classes. Finally, we analyzed our index theorem in the adiabatic limit. We identified the pertinent currents that can be used to develop a formal cohomology for our index theorem. We also showed how the adiabatic approximation by Goldstone and Wilczek can be interpreted in terms of this index theorem and demonstrated that spectral flow corrections to the adiabatic approximation can be accounted for by an index of a family operator. A. Niemi thanks E. Seiler and M. Stern for useful discussions. Appendix A In this appendix we introduce the notation that we have used in the text. We consider a d-dimensional euclidean-signature riemannian spin manifold 9E with local coordinates x ~' and metric tensor g ~ ( x ) ,

ds 2 = g~,( x ) d x ~ d x ~.

(A.1)

The vielbein fields { e a ), a -- 1 . . . . . d form an orthonormal basis of one-forms on 9 g and are locally given by e a = %~d x ~',

(A.2)

where a is a frame index and # is a coordinate index. The metric tensor can be a decomposed into the vielbeins %,

rI ahe~a ( X ) e~b ( x ),

(A.3a)

~ab = g ~ ( x)e~, ( x )eb( x ) .

(A.3b)

gp.v(X) =

Here ~/ab is the flat euclidean metric, (A.4)

~ab = - - ~ a b

and Greek indices /L, v. . . . are raised and lowered with g~,~ while Latin indices a, b . . . . are raised and lowered with ,/~. The inverse vielbein is defined by E ~ -- -

rlabg

,~e~b .

(A.5)

It obeys E t.~-a. b -_ ,~b -a

,

E . et~ .

a

= 6~

and the Greek and Latin indices are interconverted using e~, E~.

(A.6)

A.J. Nzemt, G. IV. Semenoff / Index theorems

165

The spin connection co is an SO(d) valued one form on 63L satisfying %b = --t°b. de ~ + ~g/x e b = 0

(metricity),

(A.7)

(no torsion).

(A.8)

In local coordinates, (A.9)

~0~- ~o~ d x ~' = O~cec , a

__

a

v

it,

v

~ob,- e,eb, ~ = e:( O~Eb + Cxe b ) ,

(A.10)

where the semicolon refers to a covariant differentiation with respect to the LeviCivita connection Fix. In tensor notation, (A.7) and (A.8) translate into g~,., ~ = O . g ~ . - Fx.~,gx.- FX.g~,x = 0

F~-

F~, - 2Tf~ = 0

(metricity),

(no torsion),

(A.11) (A.12)

where

F~- ~g ( 8,g,~ + 8~g,,- 8,g,~). __

1

I.I,v

(A.13)

With o~ we associate the curvature 2-form . . . b+6o . l r, a b A e c. R b=dOo c. A6%--gl~bcde

(A.14)

Similarly, the gauge field (A.15)

A =- AffhBdx ~

is locally a Lie-algebra valued one-form on the manifold cAlLwith { hB } antihermitian generators of the corresponding (compact and connected) Lie group ~ satisfying

[X~, XB] =f~Bcxc.

(A.16)

The field strength F is a Lie-algebra'valued two-form F = dA + A 2.

(A.17)

The gauge fields ~0 and A define a covariant derivative for any field tk transforming according to some representation PM of SO(d) and p~ of 9,

(5

..)

(A.18)

166

A.J. Nlemi, G. W. Semenoff / Index theorems

In particular, for a spinor field pab _ t [.ya, `tb ] M--'4

(A.19)

{ `ta, `tb } = -- 27lab

(A.20)

where the "t-matrices satisfy

and defining `t"(x)

=

°

(A.21)

we obtain the 631Lcounterparts of the fiat-space Dirac matrices, { ¢ ' ( x ) , `t'(x)} = - 2 g ~ " ( x ) .

(A.22)

Appendix B In this appendix we discuss the definition of functional traces used in the text. Consider the eigenvalue equation H ~ n -- En~ n

(B.1)

for a hermitian elliptic differential operator H defined on the manifold 9IL. An example is provided by the Dirac equation,

H=i~r + r(x) =i~V~, + K(x),

(B.2)

where V~, is flae covariant derivative defined in (A.18) and K(x) is an arbitrary (hermitian) background field. The eigenfunctions 0n(x) form a complete set of states*

f d°x lel~*~(x)'km(x)=On,..

(B.3)

The operator (B.1) defines a mapping on the Hilbert space K spanned by the eigenfunetions { ~, }. In the cases that we are interested in, this space is a direct sum of N copies of another Hilbert space ~ , N

K = ~ 0C.

(B.4)

1=1

For example in the case of a Dirac operator the index set N runs over the Dirac indices. * T h e i n d i c e s n , m . . . . r u n o v e r b o t h the b o u n d s t a t e s a n d the c o n t i n u u m ~states; see [16].

A.J. Ntemt, G. W. Semenoff / Index theorems

167

In the Hilbert space ~ we introduce the complete set of states { Ix)}, 1

We also define a set of states

(xly)=~8(x-Y),

(B.5a)

f d x let tx)(xl = I .

(8.5b)

{ In;/)}'by ep'~(x ) = (n; ilx ) .

(B.6)

(B.5a) and (B.5b) then imply the completeness relations

(n; ilm; j ) = 8~J,~,

El.; i) n; il = I,

(B.7a) (B.7b)

n,!

so that { In; i)} is a complete set of states in K. The trace of an operator ~9 in the Hilbert space K is defined by TR{~)} = ~(92~,

(B.8)

#t

where (i } runs over the index set N in (B.4) and/~ runs over the complete set of states in the Hilbert space 0C. Usually the operator 0 is not trace class in % and consequently (B.8) does not have to be well defined. However, it might still be possible that at least one of the following traces exist: tr{0} = Y'~( ~ ) ~ ) ,

(B.9a)

(B.9b) (Notice that in (B.9a) we first evaluate the trace over the internal indices and then a trace over a complete set of states in %, while in (B.9b) we first evaluate a trace over a complete set of states in % and then the internal trace.) However, if the operator ~) is not of trace class, these two traces do not have to be equal i.e. it is possible that E E q=E E .

(B.10)

If this happens the underlying theory suffers from an anomaly. Indeed, suppose that

168

A.J. Nteml, G. W Semenoff / Index theorems

(B.2) is of the form (B.11) so that it anticommutes with the matrix

C:(o _o) 1

(B.12)

If we compute the trace of F ¢ in the function space spanned by the eigenfunctions (B.1) using the prescription (B.ga) we find t r ( r c } = 0,

(B.13)

since the internal trace of F c vanishes. However, if we compute the trace using the description (B.9b) we find (after a proper regularization) that T r ( F C } = A ( O ) = lim 8--,0+

tr{e-aLL*--e-aL*L}.

(B.14)

This is nonvanishing only if there is a chiral anomaly; see sect. 3. References [1] T. Eguchi, P. Gilkey and A. Hanson, Phys. Reports 66 (1980) 213 [2] M.F AUyah and I.M. Singer, Ann. Math. 87 (1968) 485;.87 (1968), 546; 93 (1971) 119; 93 (1971) 139; M F. Atiyah and G B Segal, Ann. Math 87 (1968) 531 [3] P. Gdkey, The-index theorem and the heat equation (Pubhsh or Perish, 1974) [4] L. Brown, R. Carlitz and C. Lee, Phys. Rev. D16 (1977) 417; R. Jackiw and C Rebbi, Phys. Rev. D16 (1977) 1052; M.F Atiyah, N. I-htchin and I. Singer, Proc Natl. Acad. Scl. USA 74 (1977) 2662; A. Schwarz, Phys. Lett B67 (1977) 172 [5] G 't Hooft, Phys. Rev. Lett. 37 (1976) 8; Phys. Rev. D14 (1976) 3432 [6] H. Georgl and S. Glashow, Phys Rev. D6 (1972) 429; D. Gross and R Jackiw, Phys Rev. D6 (1972) 477 [7] L. Alvarez-Gaum6 and E. Witten, Nucl Phys. B234 (1984) 269 [8] B. Zumino, m Relativity, groups and topology, Les Houches 1983, ed. B.S. DeWitt and R. Stora (North-Holland, 1984); B Zummo, Y.S. Wu and A. Zee, Nucl. Phys. B239 (1984) 477, W. Bardeen and B. Zumino, Nucl. Phys. B244 (1984) 421; L. Alvarez-Gattm6 and P. Ginsparg, Nucl. Phys B243 (1984) 449; Ann. of Phys., to be published [9] M.B. Green and J.H Schwarz, Phys. Lett. B149 (1984) 117 [10] D.J Gross, J Harvey, E. Martmec and R. Rob.m, Phys. Rev. Lett. 54 (1985) 502; P G.O. Freund, Phys. Lett. B151 (1985) 387 [11] P. Candelas, G T. Horowaz, A. Stronunger and E. Witten, Nucl. Phys. B258 (1985) 46 [12] A.J. Nierm and.G.W Semenoff, Phys. Reports, to be published [13] W P. Su, m Handbook of conducting polymers, ed T. Skotheim (Marcel-Dekker, 1984)

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