called “topologial invariance” of the Witten index. More speci¬cally,

the Witten index is insensitive to the variations of the parameters

entering the potential so long as the asymptotic behaviour of W (x)

does not show any change in signs. For example, we can deform the

function W (x) = »x(x2 ’ c2 ) by changing the parameters » and c or

even adding a quadratic term without a¬ecting its zeros. However,

if we add a quartic term we run into an undesirable situation where

W (x) has an extra zero. This causes a jump in ∆. Note that such

deformations of W (x) are disallowed.

The de¬nition (4.8) for ∆ suggets T r(’1)Nf to be a natural

representation for it

∆ = T r(’1)Nf

= T r(1 ’ 2Nf ) (4.9)

where Nf is the fermion number operator. It is clear that (’1)Nf

assumes the value +1 or ’1 accordingly as there are even or odd

number of fermions.

However, the above de¬nition of the trace in terms of the fermion

number operator needs regularization. This is because the trace is

taken over the Hilbert space and issues relating to convergence may

arise. In the following we must ¬rst look into the anomalous be-

haviour of ∆ in a ¬nite temperature theory when (4.9) is used naively.

Afterward we take up the regularization of ∆.

© 2001 by Chapman & Hall/CRC

4.3 Finite Temperature SUSY

An obvious place to expect [7,8] breaking of SUSY is in ¬nite temper-

ature domains where the thermal distribution of bosons and fermions

1

are di¬erent. The connection of ∆ to β (≡ kT , k the Boltzmann con-

stant) gives a clue to SUSY breaking one expects ∆, starting from a

nonzero value at zero-temperature, to vanish at ¬nite temperature.

In the literature the subject of thermo¬eld dynamics [9] has

proved to be an appropriate formulation of the thermal quantum

theory. As has been widely recognized, the two-mode squeezed state

is some kind of a thermo¬eld state whose evolution can be described

by the Wigner function [10]. A novel aspect of two-mode squeez-

ing [11,12] is the creation of thermal-like noise in a pure state. The

problem lacks somewhat in uniqueness since there arise inevitable

ambiguities in the precise identi¬cation of the relevant operators per-

forming squeezing. Neverthelss, it is important to bear in mind that

squeezing is essentially controlled by a generator that is bilinear in

bosonic variables and that all the essential features of squeezing are

present in the state obtained by operating the generator on the vac-

uum.

It is important to realize that the two-mode squeezed state can

be associated with the following basic commutation relations satis¬ed

by the coordinates and momenta

[x1 , p1 ] = i, [x2 , p2 ] = ’i (4.10)

In terms of the oscillators bi (i = 1, 2), the pairs (x1 , p1 ) and (x2 , p2 )

are

1 1

x1 = √ b1 + b+ , p1 = √ b1 ’ b+ (4.11a)

1 1

2 i2

1 1

x2 = √ b2 + b+ , p1 = ’ √ b2 ’ b+ (4.11b)

2 2

2 i2

That the two quantum conditions in (4.10) need to di¬er in sign

arises from the necessity to preserve the so-called Bogoliubov trans-

formation

b1 (β) = b1 cosh θ(β) ’ b+ sinh θ(β) (4.12a)

2

b2 (β) = b2 cosh θ(β) ’ b+ sinh θ(β) (4.12b)

1

Note that (4.12a, b) have been obtained from the transformations

© 2001 by Chapman & Hall/CRC

b1 (β) = U (β)b1 U ’1 (β)

b2 (β) = U (β)b2 U ’1 (β)

U (β) = exp ’θ(β)(b1 b2 ’ b+ b+ (4.13)

21

The two-mode squeezed Bogoliubov transformation is also often

referred to as the thermal Bogoliubov transformation. Note that

the sets (b1 , b+ ) and (b2 , b+ ) continue to satisfy the normal bosonic

1 2

commutation relations

b1 , b+ =1

1

b2 , b+ =1

2

[b1 , b2 ] = 0

b+ , b2 =0 (4.14)

1

Denoting the vacuum of the system (b1 , b2 ) by |0 > and that of

(b1 (β), b2 (β)) by |0(β) > it follows that

++

0(β) >= e’ln cosh θ(β) e’b1 b2 tanh θ(β)

0> (4.15)

with bi |0 >= 0, bi (β)|0(β) >= 0, i = 1 and 2. It is clear from the

above that the b1 b2 pairs are condensed.

With this brief background on two-mode squeezing, let us de-

¬ne the thermal annihilation operators ai , i = 1 and 2 for fermions

namely

a1 (β) = a1 cos θ(β) ’ a+ sin θ(β)

2

a2 (β) = a2 cos θ(β) + a+ sin θ(β) (4.16)

1

Analogous to (4.15), one can write down the thermal vacuum for

fermionic oscillators which consists of a1 a2 pairs.

The boson-fermion manifestation in a supersymmetric theory

suggests that the underlying thermal vacuum is given by

0(β) >= exp ’θ(β)(a1 a2 ’ a+ a+ ) ’ θ(β)(b1 b2 ’ b+ b+ ) 0 >

21 21

(4.17)

’β/2 .

where |0 > is the vacuum at T = 0 and tan θ(β) = e

© 2001 by Chapman & Hall/CRC

If one now uses the de¬nition (4.9) for ∆ corresponding to NF =

a+ a1 in thermal vacuum, one ¬nds [7] from (4.16)

1

∆ = < 0(β)|(1 ’ 2a+ a1 )|0(β) >

1

1 ’ e’β

= (4.18)

1 + eβ

It transpires from (4.18) that as T ’ 0, the index ∆ ’ 1 while

as T ’ ∞, ∆ ’ 0. However for any intermediate value of T in the

range (0, ∞), ∆ emerges fractional and so the de¬nition (4.9) is not

a good representation for the index.

We now look into a heat kernel regularized index. We wish to

point out that even for such a regularized index the β dependence

persists when one considers the presence of a continuum distribution.

4.4 Regulated Witten Index

There have been several works on the necessity of a properly regu-

larized Witten index. Witten himself proposed [3]

∆β = T r (’1)NF e’βH (4.19)

while Cecotti and Girardello considered [13-21] a functional integral

for ∆β

[d¦] e’Sβ (¦)

∆β = (4.20)

where the measure [d¦] runs over all ¬eld con¬gurations satisfying

periodic boundary conditions and S the Euclidean action. It has

been found that one can evaluate ∆β both with and without the use

of constant con¬gurations [22]. Other forms of a regularized index

have also been adopted in the literature. See, for example, [13-21].

The regularized Witten index ∆β is, in general, β-dependent

when the theory contains a continuum distribution apart from dis-

crete states [6]. This is in contrast to our normal expectations that

since E = 0 states do not contribute to the trace, the right hand side

of (4.19) should be independent of β. Of course ∆β is independent

of β if the Hamiltonian shows discrete spectrum. In the following let

us study the β-dependence of ∆β .

© 2001 by Chapman & Hall/CRC

Expressing (4.19) as

∆β = T r e’βH+ ’ e’βH’ (4.21)

and de¬ning the kernels corresponding to H+ and H’ to be

K± (x, y, β) =< y|e’βH± |x > (4.22)

we can write ∆β as

∆β = dx [K+ (x, x, β) ’ K’ (x, x, β)] (4.23)

It is also implied from (4.22) that

± ±

e’βEk ψk (x)ψk (y)

K± (x, y, β) = (4.24)

k

in which the contributions from the discrete and continuum states

can be separated out explicitly as

± ± ± ±

e’βEk ψk (x)ψk (x) + dEe’βEk ψE (x)ψE (x)

K± (x, x, β) =

k

(4.25)

It is useful to distinguish the continuum state by ψ(k, x). From (4.7)

we can deduce E(k) = 1 k 2 + W0 > W0 corresponding to W (x) ’

2 2

2

±W0 for x ’ ±∞ noting that one can construct solutions of (4.7)

of the types e±ikx as x ’ ±∞, respectively.

To evaluate the integrand of (4.23) we must ¬rst take help from

(4.24) to express

d

e’βE(k) Ψ(x, y, k)

’ [K+ (x, y, β) ’ K’ (x, y, β)] = (4.26)

dβ k

where accounts for the bound states as well as continuum contri-

k

butions and the quantity Ψ(x, y, k) stands for

— —

Ψ(x, y, k) = ψ+ (y, k)ψ+ (x, k) ’ ψ’ (y, k)ψ’ (x, k) (4.27)

Using the supersymmetric equations (4.7) we now obtain

© 2001 by Chapman & Hall/CRC

E(k)Ψ(x, y, k)

d2

11

’ 2 + W 2 (y) ’ W (y) ψ+ (y, k)ψ+ (x, k)

—

=

22 dy

d2

1

’ 2 + W 2 (x) ’ W (x) ψ+ (y, k)ψ+ (x, k)

—

+

2 dx

d d

—

’ + W (y) ψ+ (y, k) + W (x) ψ+ (x, k)

dy dx

d2 — d2

11 —

= ’ψ+ (x, k) 2 ψ+ (y, k) ’ ψ+ (y, k) 2 ψ+ (x, k)

22 dy dx

1

+ ψ+ (x, k) W 2 (y) ’ W (y) ψ+ (y, k)

—

2

1—

+ ψ+ (y, k) W 2 (x) ’ W (x) ψ+ (x, k)

2

—

dψ+ (y, k) —

’ + W (y)ψ+ (y, k)

dy

dψ+ (x, k)

+ W (x)ψ+ (x, k) (4.28)

dx

Putting x = y it follows that

1 1 — —

E(k)Ψ(x, x, k) = ’ ψ+ (ψ+ ) + ψ+ ψ+

2 2

1 1—

+ ψ+ (W 2 ’ W )ψ+ + ψ+ (W 2 ’ W )ψ+