Topological invariance in supersymmetric theories with a continuous spectrum

Nuclear Physics B 2 4 2 ( 1 9 8 4 ) 81-92 © North-Holland Publishing Company

TOPOLOGICAL INVARIANCE IN SUPERSYMMETRIC THEORIES WITH A C O N T I N U O U S SPECTRUM Camillo IMBIMBO

Ecole Normale Supbrieure, Paris, France, and International School for Adt,anced Studies, Trieste, Italy Sunil MUKHI*

Max Planck-lnstitut fiir Physik und Astrophysik, Werner Heisenberg-lnstitut fitr Physik, D-8000 Munich 40, Federal Republic of Germany, and International Centre for Theoretical Physics, Trieste, Italy Received 9 January 1984

We calculate the superpartition function t r ( - 1 ) % t~ff for (Fredholm) supersymmetric hamiltonians with a continuous spectrum. This object depends explicitly on /3 although it is a topological invariant with respect to compact perturbations of the potential. We show that it can be evaluated by reducing the relevant functional integral to an integral over constant configurations. The relationship with the open-space trace theorem of Callias, Bott and Seeley is discussed.

1. The problem The Witten index [1] of a supersymmetric theory is defined as a = ng

°-,ff =°,

(1)

E=o is the number of bosonic (fermionic) eigenstates of the hamiltonian with where n B(F) eigenvalue zero. It can be expressed as the index (in the usual mathematical sense) of an operator related to the hermitian supersymmetry generator, and is hence a topological invariant. For a theory (quantized in a box) with a discrete spectrum, it has been shown [1, 2] that the index is equal to the "superpartition function": A = Z s -= t r ( - 1)Fe-fin

=

( F = fermionic number)

fperiodic boundary[dq0 d~ conditions

]exp( - Seuc(cP, ¢)).

* Address after February 1, 1984: Tata Institute of Fundamental Research, Bombay, India. 81

(2)

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C. Imbimbo, S. Mukhi / Topologicalinvariance

This is because, in the trace, the contributions from eigenstates with nonzero energy cancel exactly between bosons and fermions due to supersymmetry. It follows that the superpartition function is invariant under changes of/3 and the dimensions of the box, and also of the coupling constants and masses in the theory - so long as the asymptotic form of the potential is kept fixed. If the theory has a continuum part in the spectrum, the cancellation in the trace may no longer be valid. Hence zl may no longer equal Z S~ t r ( - 1 ) % ~H which could now depend on/3. However, it is easy to see that

A = lim Zs(/3 ) /~--,,oc,

(3)

as long as the hamiltonian is a Fredholm operator [3] (this means, in particular, that the zero eigenvalues should be finite in number and isolated points in the spectrum. In other words the continuum part, if any, is separated from zero by a finite gap. In the following, we always limit ourselves to this case). One may ask whether Zs(fl) is, in fact, a topological invariant, even for finite values of/3 - in the sense that it continues to be invariant under compact perturbations of the potential. By this one means perturbations which fall off to zero outside a sufficiently large sphere in field space*. We henceforth take the above as a definition of "topological invariance". Recently, a related problem was discussed by Hirayama [6] in the case of supersymmetric quantum mechanics with one degree of freedom: 2 H = p 2 + qo2(x) q- ~._x 1 ) 0(q_o

F

( F = fermion number).

(4)

(cp(x) is chosen such that limx_, + ~ ( x ) exists and is different from zero, which ensures that H is Fredholm and has a continuum part in the spectrum.) He observed that, for this theory, the hamiltonian can be expressed as

2H=L+L(1-

F ) + LL+F,

(5)

where

1 d L =--[ -d~x + i~( x ) .

(6)

t r ( - 1)Fe -I~H= tr(e -~L÷ L / 2 _ e-~LL+/2).

(7)

So that

* For a more precise mathematicaldefinition,see ref. [4].

C. Imbimbo, S. Mukhi /

83

Topological int,ariance

Taking fl --+ 00 in this equation, it follows that

(8)

A = index L = d i m k e r L - dimker L + .

(dimkerL = the number of zero eigenvalues of L.) The trace (7) can be obtained from the following quantity, calculated in a more general context, by Callias, and Bott and Seeley [4, 5] in 1978: tr

z

z

z + L+L

z +-LL +

)

=

1

(9) t ~ ( + 00)2 _1_ Z

tq0( -- 00) 2 q- Z

where z is an arbitrary complex (not real negative) parameter. Noting that the LHS of the above equation is proportional to the Laplace transform of eq. (7), we find that Z s ( f l ) - t r ( _ 1) Fe-~H _1[

~ ( + 0 0 ) P(lcp(+00)l ~2~)

¢P(- 00) P(Iq0(- oo)[ ~/1/3 )]

2 1 7+ 00)1

00)1

(10)

where p ( x ) = _ 2 f ~ dt

-I0 v~- e ,2

(11)

is the probability integral function. This expficitly shows that the superpartition function depends on/9, but that it is nevertheless a topological invariant - it depends only on ~( _+ 00). Further, we have: ;irn Zs(fl) = 1 [ ep(+ 00)

1 (+00)1

q°(- 00) I

I (-oo)1

'

(12)

which is the Witten index of this model. In this paper, we consider tr(-1)Fe -~u for supersymmetric quantum mechanics with an arbitrary number of degrees of freedom: 2 H =p2 + ( W i ( x ) ) 2 _ i x ~ x J W i . ( x ) '

(13)

where W ( x ) is the superpotential,, i denotes differentiation with respect to x* and X~, X~ are fermionic operators which satisfy i Xa, X~ }

=

28ag6 ij,

A,B=I,2 ) i,j=l

..... n "

(14)

C. Imbimbo, S. Mukhi / Topological invariance

84

W ( x ) is chosen to satisfy the asymptotic condition: a homogeneous function of order zero, bounded below by a positive constant,

I W, i(x)l 2lx[--->°~---'

(15)

which ensures that H is Fredholm. Physically, one expects a continuum portion in the spectrum and so an explicit dependence of Z s on ft. In a subsequent section we will explicitly construct the operator L which satisfies eq. (5) with the above hamiltonian. In the special case where n is odd and the superpotential satisfies the stronger condition I W~(x)[ 2 = 1 the object tr

(z z + L+L

for all x,

(16)

z) z + LL +

has been calculated by Callias and shown to be a topological invariant for each value of z. It is conjectured by him that this property continues to hold in the more general case eq. (15), and some explicit calculations support the conjecture [4, 7]. Accordingly, in the following, we will assume that Zs(fl) is a topological invariant for each value of/3. Using the functional integral representation of the superpartition function (see eq. (2)) we will show that it can be explicitly calculated as an ordinary integral over constant configurations. Our results are in agreement with those of Callias in the region of overlap, constituting an independent derivation of his formula. They also provide some generalizations.

2. Constant configurations for ~-dependent superpartition functions For supersymmetric theories with a discrete spectrum, the superpartition function does not depend on fl, as we have already pointed out. This property has been exploited [8, 9] to reduce the functional integral representation of this object to an ordinary integral over constant configurations. The demonstration uses the fact that one can take the limit fl ~ 0 (the "high-temperature" limit). An expansion in Fourier modes then implies [8] that the leading contribution is given by the zero frequency modes, and they give, in fact, the exact value of tr( - 1)% /~" since all the other (fl-dependent) terms must vanish. When there is a continuous spectrum, and therefore the possibility of a fl-dependence in the supertrace, this argument is obviously invalid. However, Z~(fl) is still invariant under compact perturbations of the potential, and this can be used to calculate the functional integral:

Zs--.cldxidxlexp( dtE)

C. lmbimbo, S. Mukhi /

85

Topological invariance

where (X)) 2 --

l ' i 2 _ L l i "i 1 E=~X " 4XAXA + 2(Wi

X;e A&'ivjW 4" AAAB",ij

(x)

(17)

is the euclidean lagrangian corresponding to the hamiltonian (13), and W~(x) satisfies the asymptotic condition (15). We choose the particular class of compact perturbations

W i(x ) ~ W,i(?`x )

where X is an arbitrary parameter > 0,

(18)

which corresponds to a "stretching" of W~(x), leaving the value at infinity fixed. Invariance under this perturbation implies that

Zs()=fec[dx'dx']exp(-fiat

+ ¼x d( ,

+~( 1 W i, (?`x))2-¼ieaBxiax~Wq(?`x)])

(19)

Performing a change of integration variables with unit jacobian: ?`X i --. x i '

~-X~4 ---' X],

(20)

this reduces to =

[d#dx;41exp-

at

~--~ 4

?`

+ ½(Wi(x))2--~ie~BXiAx~W,o(X)J).

(21)

At this point one may carry out a decomposition into Fourier modes as in ref. (8); it is seen immediately that the constant mode gives a contribution of O(1) in ?`, while the non-constant modes are at least O(?`). From the overall ?,-independence, one concludes that, in the present case also, the superpartition function is given by the functional integral reduced to constant configurations:

Z~(fl) = f

[Ii(dxi2dx~

dx~) (2~rfl),/2

exp(_fl(½(Wi(x))2_¼ieA.xiAxJBWij(X))). (22)

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C. Imbimbo,S. Mukhi / Topologicalinvariance

Integrating out the fermions

Zs(/3)=f(tTidxi)(2-~)n/2detlJWij(x)llexp(-~(Wi(x))2).

(23)

Making the change of variables y;= W,(x)

(24)

the final result for the superpartition function is

tr(-1)Fe-eU----NfB.([Ii dyi)(2~) ~/2e /3(v'):/z,

(25)

where B" is the image space of the mapping (24). N is the winding number of the map from S" 1 to S"-a given by the restriction of (24) to the sphere at infinity. An integral representation is given by: N=~

f dl~/1A..-

AdW,,

(26)

where l~i -= Wi/I W,I and ~2(") is the solid angle in Rn:

2rrn/2

~(") F(½n)"

(27)

To obtain the Witten index, one has to take the limit of eq. (25) as/3 ~ oo. Clearly in this limit the gaussian integral equals 1 (recall condition (15)) and hence A = N.

(28)

One can easily find a more explicit formula for the integral in eq. (25) when Ixl--,~ IWi(x)[ ~ v, a constant independent of the angles, which is a special case of (15). The image space B" becomes a sphere of radius v, and thus

Zs(/3)=gfoV(2~)n/2dDpn-l~?(n)e

/302/2

= N~(n) fo Bgr~2v _dt t" le t2.

(29)

For n = 1, this formula agrees with eq. (10) when [q)(+ oo)1 = Iq)(- oo)l, providing a rederivation of Hirayama's result (eq. (9)).

C Imbimbo, S. Mukhi / Topological invariance

87

3. Relationship with the Callias-Bott-Seeley theorem In this section we construct the first-order differential operator L on R n related by eq. (7) to the superpartition function just calculated, and discuss the relationship with the Callias-Bott-Seeley [4, 5] index theorem, which deals with a class of such operators. The hamiltonian (13) for our problem: 2 H = p i2 + ( W , i ( x ) ) 2 -- i x ] x J 2 w i j ( x ) ,

with

( X'A, xJe ) = 26A# 3is

(14)

satisfies the supersymmetry algebra

{QA,QB} = 26ABH,

(30)

where Qa are the (hermitian) supersymmetry generators: Q1 = ~[~( P iXl i + W~X~), =

(31)

PX2

In a basis in which (32) the supersymmetry generators take the form: 0 )

(similarly for (22)-

(33)

Therefore the superpartition function can be expressed as tr(_l)ee

CH=tr(e ~ L + L / 2 e ~rc-/2),

(34)

from which A = index L. We now construct the operator L in the two distinct cases n odd and n even.

(35)

C. Irnbimbo,S. Mukhi / Topologicalinvariance

88 3.1. n ODD

It can be checked that an explicit matrix representation for the fermionic operators obeying the 2n-dimensional Clifford algebra (14) is:

X~ = ~ i ~ ~p@Ol, X~ = ~p ~ .yi@ 02 ,

(36)

where 7 i, i = 1 ... n are the p = 2 ('-1)/= dimensional matrices representing the n-dimensional Clifford algebra (7i, 3,j } = 28 ij while 01 and o2 are 2 × 2 Pauli matrices. To find the representation matrix for ( - 1 ) F, one has to look for an operator K which satisfies

(K,x])

=0,

K2=~.

(37)

(Hence K is a generalized "'/5" for the 2n-dimensional Clifford algebra.) The unique solution is: K = -ix~

• " X InX 21" " X 2

n

=

~p ®'~p ® 03,

(38)

which is of the desired form (32). In this basis,

pl = ~ ('ytpi ® ~p ® O1-}'- W i[lp®,~i®o2) ,

(39)

from which it follows that:

L = ~'ipi ® lp + idp ® "~iwi.

(40)

This belongs to the class of operators considered by Callias, who studied the more general type:

L = 7ipi ® ~,n + ilp ® 6p(x),

(41)

which is the static Dirac operator with the space components of the gauge field eliminated (as they are assumed to go to zero at infinity and hence do not affect the index). ~, the scalar potential, is a hermitian m x m matrix. Callias obtains:

1

(i

](" 1)/2ftr(dU(x))n

index L = 2((n - 1 ) / 2 ) ! ~ 8~r } with U(x) ~

I@(x)l-~@(x),

I@(x)l = (@+(x). @(x)) ~/=,

(42)

C. Imbimbo, S. Mukhi / Topological invariance

89

and (dU) n represents dx q/x . . . A d x io OhU... O~U. Notice that the integral in the above reduces to a surface integral: fa t r ( d U ) ' =

lim fs ~ xtr ( U ( d U ) ' - I )' 8--,oo

(43)

(where S ~ - 1 is the ( n - 1)-sphere of radius R) from which the topological invariance of the object is evident. When • is unitary, Callias also provides a trace formula: ( 2z tr 2z + L + L

2z

2z + L L +

)

1 index L. (1 + 2 z ) n/2

(44)

Substituting in (42) ~ ( x ) = ,flW ~(x), one obtains exact agreement with the Witten index A given by eqs. (26, 28). The Laplace transform of the left-hand side of eq. (29) with respect to fl is: Zfomdfle-¢Ztr(_l)Fe

Z

CH=tr(_l)F z+H 2Z =tr

2z+L+L

2z

)

(45)

2z+LL + '

while the right-hand side gives oo

pn

(~.),/2

(~2 + 2z)n/2 N '

which agrees with (44) for v = 1. Thus our result constitutes a supersymmetric functional integral derivation of the open-space index and trace formulae for operators of the form (41), as long as ~ ( x ) = "/iWi(x). In fact, we obtain explicit trace formulae even when • is not unitary. 3.2. n EVEN

The operators of the class (41) considered by Callias all have vanishing index in an even number of dimensions. On the other hand, our result for the Witten index (28) is not in general zero in any number of dimensions. It follows that the operator L related to the hamiltonian in an even number of dimensions cannot be of this class. In fact, the representation of the 2n-dimensional Clifford algebra, for even n, in

X~ = / ~ i ® o1 '

• IFi+" ® °1 X~2 = ~ ~ q ® OZ,

i = 1 ..... n, i= 1..... n-

i = n,

1,

(47)

C. lmbimbo, S. Mukhi / Topologicalinvariance

90

where F a, a = 1 . . . 2 n - 1, are the q = 2" 1 dimensional matrices satisfying the 2n - 1 dimensional Clifford algebra. It follows that ( - - 1 ) F = --ix] "'" X n-llx2"''X2n

= Uq ® o3.

(48)

In this representation n-1

L = ~ Fipi + ~_~ Fn+iw, i(x) + iW.n~q. i=l

(49)

i=1

This is not a static Dirac operator, hence Callias formula does not apply to it; in any case, our results correctly give its index and the related trace*. L can be thought of as a "static" 2n-dimensional Dirac operator with n space and n "time" dimensions, in which the spatial components of the gauge fields are set to zero.

4. Comments and discussion We have calculated t r ( - 1 ) F e -tin for the general class of supersymmetric WessZumino-type quantum mechanics models with an arbitrary number of degrees of freedom, whose hamiltonian is a Fredholm operator. This quantity turns out to be non-trivially B-dependent, and hence gives information about the entire spectrum, besides reproducing the Witten index (a "zero-energy" quantity, n~* = ° - n~ =°) in the limit B -~ oo. Traces of this type have been used, for instance (in the case of one degree of freedom) to derive fermion fractionization in the presence of solitons [10] and to obtain sum rules relating various expectation values [6,10]. As pointed out earlier our calculation constitutes a derivation of the Callias-BottSeeley trace theorem (for a special class of potentials). The idea of using supersymmetry to derive mathematical results has been expounded by Witten [11,12] in the context of Morse theory, and also used by Alvarez-Gaum6 [9,13] to derive the Atiyah-Singer index theorem for operators defined on compact spaces. We note, en passant, that in the latter case the B-independence of the supertrace facilitates the calculation as it enables one to have recourse to the well-known high temperature (B ~ 0) limit of a partition function. For non-compact manifolds, one has to use the invariance under compact perturbations of the potential, as we have done in this paper. The generalization of this trick to derive trace theorems for arbitrary static * Bott and Seeley have discussed a very general class of operators on R ~ which includes (49). The expression quoted by them for the index (eq. (1.15) of their paper) would involve a rather complicated calculation to derive an explicit form which could be compared with ours.

C. lmbimbo, S. Mukhi / Topological im,ariance

91

Dirac operators on open spaces (with the asymptotic conditions specified by Callias) seems to present additional technical difficulties, and we are presently studying this problem. One may hope that in the future this kind of technique may be used not only to rederive known mathematical results, but also to provide new trace formulae for operators of interest to physicists. It is clearly of interest to enquire whether similar calculations of t r ( - 1)Fe '-~H are valid also in supersymmetric quantum field theories. In fact, if one assumes that the object is a topological invariant even in field theories, the very same formulae obtained by us hold also there, and constitute genuinely non-perturbative information about the spectrum. The generalization of this assumption to field theory is mathematically non-trivial because the relevant differential operators are defined on an infinite-dimensional space. It seems to us that the same problem exists, in principle, in the generalization of the Witten index from quantum mechanics to field theory - which is, however, usually believed not to present problems. The "physical meaning" of the fi-dependence of the superpartition function is discussed in a very recent preprint by Cecotti and Girardello [14]. Ttfis article predominantly concerns the supersymmetric Liouville theory, and hence nonFredholm hamiltonians, with which we have not dealt at all. However, it contains a discussion to the effect that a fl-dependence in the supertrace implies the presence of an anomaly in the supersymmetry Ward identity, at zero frequency, relating to OZ~(B)/Ofl. (This anomaly presumably disappears in the limit fl --, ~ for the class of models with Fredholm hamiltonians, as one recovers the Witten index with the usual properties). After reading the Cecotti-Girardello paper, we have realized that it should be possible to construct a "supersymmetry-proof" of the topological invariance (with respect to compact perturbations) of Z~(fl) for the theories studied by us. The authors of ref. (14) use the existence of a local Nicolai mapping for the real super-Liouville quantum mechanics to show that the fl-anomaly is only present in the zero-frequency mode. Since our models also admit a local Nicolai mapping, one could, in principle, derive in an analogous manner that Z~(fl) is a topological invariant and can be calculated via constant configurations. This offers the hope of extending the proof to at least one field theory - the N = 2, D = 2 Wess-Zumino model [15,16]. Other supersymmetric field theories probably do not have local Nicolai mappings, but perhaps even the existence of a non-local mapping [17] is sufficient for the required proof.

We would like to thank the Theory Group of the Max Planck-Institut fiar Physik und Astrophysik for their hospitality, and for providing the congenial circumstances in which this paper was written.

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C. Imbimbo, S. Mukhi / Topological invariance

CI acknowledges financial support from the International School for Advanced Studies, Trieste, and the Government of France, and the warm hospitality of the Lab. de Physique Throrique de l'Ecole Normale Suprrieure, Paris. SM is grateful to Professor Abdus Salam, IAEA and UNESCO for hospitality at the ICTP, Trieste, and the Physikalisches Institut, Universit~tt Bonn, for a fruitful visit. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]

E. Witten, Nucl. Phys. B202 (1982) 253 S. Cecotti and L. Girardello, Phys. Lett. ll0B (1982) 39 S. Goldberg, Unbounded linear operators (McGraw-Hill, New York, 1966) C. Callias, Commun. Math. Phys. 62 (1978) 213 R. Bott and R. Seeley, Commun. Math. Phys. 62 (1978) 235 M. Hirayama, Toyana Univ. preprint 44 (June 1983) E. Weinberg, Phys. Rev. D20 (1979) 936 L. Girardello, C. Imbimbo and S. Mukhi, Phys. Lett. 132B (1983) 69 L. Alvarez-Gaum6, Commun. Math. Phys. 90 (1983) 161 M. Hirayama and T. Torii, Prog. Theor. Phys. 68 (1982) 1354 E. Witten, J. Diff. Geom. 17 (1982) 661 E. Witten, Holomorphic Morse inequalities, Princeton preprint (1982) L. Alvarez-Gaumr, Harvard Univ. preprint HUTP-83/A035 (1983) S. Cecotti and L. Girardello, Nucl. Phys. B239 (1984) 573 G. Parisi and N. Sourlas, Nucl. Phys. B206 (1982) 321 S. Cecotti and L. Girardello: Ann. of Phys. 145 (1983) 81 H. Nicolai, Phys. Lett. 89B (1980) 341