![](https://csdnimg.cn/release/download_crawler_static/12311923/bg5.jpg)
A. Mironov et al. / Nuclear Physics B 889 (2014) 757–777 761
with w
Δ
being complicated combinations of the expansion coefficients of F
R
in the basis of
ϕ
R
(Δ). Functions of such a type we call Hurwitz exponential since these are exactly the expo-
nentials entering the Hurwitz partition functions (12).
2.2. From Hurwitz to KP partition functions and renormalization group
Let us comment more on the two bases, multiplicative C
k
and additive ϕ
R
(Δ). Note that
the generic Hurwitz exponential spanned by the additive basis (13) gives rise to the generating
function which is not a KP τ -function [36]. However, if the exponential is spanned by the linear
basis (2),
F
R
= exp
k
t
k
C
R
(k)
. (14)
Eq. (12)at K = 1, 2is the Toda lattice τ -function [36,37] with respect to times p
(1,2)
k
. Moreover,
it is also the Toda chain τ -function at K = 1 [36] with respect to times t
k
.
When dealing with these Hurwitz e
xponentials, one may keep in mind the following analogy
with the renormalization group (RG) and completeness of basis [10]. Let us consider a quantum
field theory partition function
Z(G; ϕ
0
; t) =
A;ϕ
0
Dφ exp
1
2
φGφ + A(t; φ)
(15)
which depends on: (a) the background fields ϕ
0
; (b) the coupling constants t and (c) the metric
G.
The coupling constants parameterize the shape of the action
A(t; φ) =
n∈B
t
(n)
O
n
(φ), (16)
where the sum goes over some complete set B of functions O
n
(φ), not obligatory finite or even
discrete. The space M ⊂ Fun(A) of actions parameterized by the coupling constants t
(n)
, is re-
ferred to as the moduli space of theories. The actions usually take values in numbers or, more
generally, in certain rings, perhaps, non-commutative. The space Fun(A) of all functions of φ
is always a ring, but this needs not be true about the moduli space M, which could be as small
a subset as one lik
es. However, the interesting notion of partition function arises only if the
completeness requirement is imposed on M. There are two different degrees of completeness,
relevant for discussions of partition functions. In the first case (strong completeness), the func-
tions O
n
(φ) form a linear basis in Fun(A), then M is essentially the same as Fun(A) itself. In
the second case (weak completeness), the functions O
n
generate Fun(A) as a ring, i.e. an arbi-
trary function of φ can be decomposed into a sum of multiplicative combinations of O
n
’s. In the
case of strong completeness, the notion of RG is absolutely straightforward, but there is no clear
idea how RG can be formulated in the case of weak completeness (which is more relevant for
most modern considerations).
In the strongly complete case, the non-linear (in coupling deri
vatives) equation, even if occurs,
can be always rewritten as a linear equation. In fact, one can easily make a weakly complete
model strongly complete, by adding all the newly emerging operators to the action A(t; φ), then,
if the product O
m
O
n
is added with the coefficient t
(m,n)
, one has an identity ∂
2
Z/∂
(m)
∂t
(n)
=
∂Z/∂t
(m,n)
.