JHEP09(2019)064
where the numerator N(Λ) and the denominator D(Λ) read
N(Λ) =
Z
[dM]
N
e
−α
1
p+1
Tr [(M+Λ)
p+1
]
≥2
D(Λ) =
Z
[dM]
N
e
−
α
2
P
p−1
k=0
Tr MΛ
k
MΛ
p−1−k
(2.2)
The square brackets with lower index ≥ 2 indicate that one should consider only terms
that are of order two or higher in the N ×N hermitian matrix integration variable M. The
coupling constant α can be identified with one over the string coupling, α = −1/g
s
. The
partition function of the closed string is a function of the matrix source Λ which codes the
values of all couplings to the matter primaries and gravitational descendants. When the
couplings go to zero, the matrix source Λ goes to infinity, and the τ function approaches
one. The matter content is specified by the order p + 1 of the potential term. We have
that p = 2 for pure gravity.
1
Our set-up is elementary. We take the source matrix as well as the integration variable
to be an (N + 1) × (N + 1) matrix and wish to interpret the last diagonal entry of the
source matrix as (the dual of) an open string (or boundary insertion) modulus. The
matrix integral only depends on the eigenvalues of the source matrix Λ
z
, which we take to
be diagonal:
Λ
z
= diag(λ
1
, . . . , λ
N
, z) . (2.3)
We used the special notation z for the last diagonal entry, which we plan to single out. We
have the corresponding closed string partition function:
τ(Λ
z
) =
N(Λ
z
)
D(Λ
z
)
. (2.4)
To perform the integration over the 2N off-diagonal degrees of freedom, we integrate over
all N(N + 1) off-diagonal degrees of freedom, and then reinstate those that we wish to
keep. To perform the integration, our main tool is the Harish-Chandra-Itzykson-Zuber
integration formula [38, 39]. We integrate in two steps. Firstly, we concentrate on the
numerator, and then we perform the Gaussian integration in the denominator.
The exponent in the numerator (2.2) of equation (2.4) can be simplified by shifting
away the source Λ
z
, to keep only a term which has power p + 1, and a linear and constant
term in the integration variable M. Indeed, we have
[(M + Λ
z
)
p+1
]
≥2
= (M + Λ
z
)
p+1
− Λ
p+1
z
− (p + 1)Λ
p
z
M , (2.5)
and therefore upon shifting M −→ M − Λ
z
, we find that the numerator N(Λ
z
) takes
the form:
N(Λ
z
) =
Z
[dM]
N+1
e
−αTr
M
p+1
p+1
−MΛ
p
z
+
p
p+1
Λ
p+1
z
. (2.6)
1
We remark that these matrix integrals should be thought off as consisting of a Gaussian integration
plus an exponential of higher powers that are to be expanded as formal power series in the integration
variable. For a careful treatment of these and other aspects see e.g. [9, 19].
– 4 –
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