Here,XandZare respectively a real and a complex column matrix of heightN. These variables
and their associated measures may be parametrized as follows,

X=







x 1

x 2

·

xN





 Z=







z 1

z 2

·

zN







dNX=dx 1 dx 2 ···dxN

dNZdNZ ̄=d^2 z 1 d^2 z 2 ···d^2 zN

(14.59)

The measure onzis defined byd^2 z=dRe(z)dIm(z). The measures are invariant under orthogonal
rotations ofXand unitary transformations ofZ. TheN×NmatricesMandHare respectively
real symmetric and Hermitian. BothMandHare assumed to be positive, i.e. all their eigenvalues,
respectivelymiandhiare positive. Finally, the complex parameterτis required to have Re(τ)> 0
for absolute convergence of the integral.
We shall now prove the above Gaussian integral formulas. Using the fact that any real symmetric
matrixMcan be diagonalized by an orthogonal change of basis, and that any Hermitian matrix
Hmay be diagonalized by a unitary change of basis, we have

∫

dNXexp

{

−πτXtMX

}

=

∏N

i=1

(∫

dxie−πτmix

(^2) i
)
∫
dNZdNZ ̄exp
{
− 2 πτZ†HZ
}

∏N
i=1
(∫
d^2 zie−^2 πτhi|zi|
2
)
(14.60)
It remains to carry out a single real or complex Gaussian integral. We do this first forτreal and
then analytically continue inτ. The complex integral is readily carried out in polar coordinates,
zi=rieiφifor 0≤ri<∞and 0≤φi< 2 π. We find,
∫
d^2 zie−^2 πτhi|zi|
2
= 2
∫ 2 π
0
dφi
∫∞
0
dririe−^2 πτhir
i^2

1

τhi

(14.61)

Recasting this complex integral in real Cartesian coordinates,zi= (xi+iyi)/

√

2, we see that the
complex integral is the square of the real integral evaluated formi=hi, and thus,
∫
dxie−πτmix

(^2) i

1

√

τmi

(14.62)

Using the definition of the determinants forτMandτHin terms of the products of their eigenvalues,
we immediately recover the desired integrals. Note that, asRe(τ)→0, we recover an integral which
is not absolutely convergent, but which is conditionally convergent. Its conditional convergence
prescription may be taken to be limit Re(τ)→0.

14.10Evaluating the contribution of Gaussian fluctuations

The contribution of Gaussian fluctuations around a given dominant path q 0 (t) is given by the
Gaussian functional integral,
∫
Dyexp

{

i

2

∫tb

ta

dty(t)M(t)y(t)

}

(14.63)