# The four dimensional Yang–Mills partition function
in the vicinity of the vacuum

Gábor Etesi
Department of Geometry, Mathematical Institute, Faculty of Science,
Budapest University of Technology and Economics,
Egry J. u. 1, H ép., H-1111 Budapest, Hungary 111E-mail<EMAIL_ADDRESS>
###### Abstract

The partition function of four dimensional Euclidean, non-supersymmetric ${\rm
SU}(2)$ Yang–Mills theory is calculated in the perturbative and weak coupling
regime i.e. in a small open ball about the flat connection (what we call the
vicinity of the vacuum) and when the gauge coupling constant acquires a small
but finite value.

The computation is based on various known inequalities, valid only in four
dimensions, providing two-sided estimates for the exponentiated Yang–Mills
action in terms of the $L^{2}$-norm of the derivative of the gauge potential
only; these estimates then give rise to Gaussian-like infinite dimensional
integrals involving the Laplacian hence can be formally computed via zeta-
function and heat kernel techniques. It then turns out that these formal
integrals give a sharp value for the partition function in the aforementioned
perturbative and weak coupling regime of the theory.

In the resulting expression for the partition function the original classical
coupling constant is shifted to a smaller one which can be interpreted as the
manifestation, in this approach, of the existence a non-trivial trivial
$\beta$-function and asymptotic freedom in non-Abelian gauge theories.

AMS Classification: Primary: 81T13; Secondary: 81Q30, 57M50, 35K08

Keywords: Non-supersymmetric Yang–Mills partition function;

Zeta-function regularization; Heat kernel

##  1 Introduction and summary

Computing the partition function is a central problem of Yang–Mills theory.
For in Feynman’s path integral quantization framework it is equivalent to the
highly non-trivial task of taking summation over all vacuum Feynman graphs,
the computation of the partition function is the first and most difficult step
towards the construction of the underlying relativistic quantum field theory.
In the exposition of the problem mainly found in physicist’s textbooks (cf.
e.g. [8]) the difficulties are usually attributed to the presence of a huge
(namely gauge) symmetry of the theory alone; they have however certainly much
deeper roots connected e.g. with our unsatisfactory 18-$19^{\rm th}$ century
concept of convergence and the non-existence of a natural measure theory in
infinite dimensions [7], too. Nevertheless, because of its central importance,
a permanent effort has been made to calculate the partition function during
the past decades. These approaches are based on introducing additional
structures in order to “tame” the bare Yang–Mills partition function i.e. to
increase its computational accessibility. Very roughly speaking these
techniques hit the field in three powerful waves: In the 1970-1980’s various
supersymmetric and higher dimensional extensions of pure Yang–Mills theory
have been introduced making it possible to calculate their corresponding
partition functions via Atiyah–Bott-like localization techniques, cf. [11]
(especially [11, Chapter 10]). Then topological twisting, an additional
modification was introduced by Witten [19] which together with many other
ideas such as the Chern–Simons and conformal field theory correspondence and
various duality conjectures, etc. led in the 1990’s to revolutionary
discoveries connecting quantum field theories and low dimensional differential
topology [19, 20, 14] thereby clearly demonstrating the indeed deep, not only
physical but even mathematical, relevance of Yang–Mills partition functions.
However, eventually together with Nekrasov’s $\Omega$-deformation approach
[10] from the early 2000’s, these twisting and deformation techniques, as a
price for computability, gradually converted the Yang–Mills partition
function, an originally certainly highly analytical object, into a rather
purely combinatorial structure; in this way at least in part covering or
mixing the original physical content of Yang–Mills theory with auxiliary
mathematical structures.

In this paper, as a continuation of our earlier work on the Abelian case [4],
we make an attempt to return to the original setup and to compute the
partition function of the non-supersymmetric, four dimensional Euclidean, non-
Abelian pure gauge theory. The sacrifice we make for not using any
supersymmetric, etc. support is that unfortunately we shall neglect all non-
perturbative (like instanton, etc.) effects which are however certainly key
features of non-Abelian gauge thoeries; that is we shall consider the
perturbative regime only. It is worth briefly mentioning here that part of our
approach which in our opinion is the most interesting (and well-known) because
works only in four dimensions. The curvature of a connection $\nabla={\rm
d}+A$ looks like $F_{\nabla}={\rm d}A+A\wedge A$ i.e. consists of a derivative
${\rm d}A$ and a quadratic (interacting) term $A\wedge A$ of the gauge
potential. In four dimensions there is a delicate balance between these terms
as a consequence of the Sobolev embedding $L^{2}_{1}\subset L^{4}$ which is on
the borderline in four dimensions. Indeed, this embedding allows one to
compare the $L^{2}$-norm of the ${\rm d}A$ and $A\wedge A$ terms. Phyisically
speaking this means that precisely in four dimensions the energy content in
the Yang–Mills field strength is equally distributed between its derivative
and interacting terms.222One is tempted to say that although in dimensions
different from four classical Yang–Mills theory can be formulated, its
underlying quantum theory will be governed by ${\rm d}A$ or $A\wedge A$ alone;
hence it exhibits a different, perhaps less complex, behaviour. From the
mathematical aspect the existence of $L^{2}_{1}\subset L^{4}$ allows one to
estimate the $L^{2}$-norm of the curvature of a connection from both below and
above by various, at most quartic, expressions involving the $L^{2}$-norm of
the exterior derivative of the gauge potential alone. These estimates can be
re-written as Gaussian-like expressions for the Laplacian hence can be
formally Feynman integrated using $\zeta$-function and heat kernel techniques
providing a two-sided estimate for the partition function. After adjusting the
physical and technical parameters involved in this procedure, this “scissor”
about the partition function closes up giving rise to an expression for it.

Our main result can be summarized as follows. For more details we refer to
Section 3 below.

######  Theorem 1.1.

Consider a non-supersymmetric ${\rm SU}(2)$ gauge theory with complex coupling
constant $\tau\in{\mathbb{C}}^{+}$ over the Euclidean $4$-space
$({\mathbb{R}}^{4},\eta)$. Take a constant $0<\varepsilon<1$ and consider
those ${\rm SU}(2)$ connections $\nabla$ which are in the vicinity of the
vacuum $\nabla^{0}$ in the sense that
$\|F_{\nabla}\|_{L^{2}({\mathbb{R}}^{4})}<\varepsilon$. Let
$Z_{\varepsilon}({\mathbb{R}}^{4},\tau)$ denote the corresponding truncated
partition function of the theory obtained by formally Feynman integrating over
gauge equivalence classes of ${\rm SU}(2)$ connections in the vicinity of the
vacuum.

Provided the vicinity parameter $0<\varepsilon<1$ is small enough and the
complex coupling constant $\tau\in{\mathbb{C}}^{+}$ has large enough imaginary
part, using $\zeta$-function regularization and heat kernel techniques, the
truncated partition function can be computed and

$Z_{\varepsilon}({\mathbb{R}}^{4},\tau)=\left(\frac{{\rm
Im}\tau}{8\pi^{2}N_{*}^{2}}\right)^{-\frac{11}{20}}{\rm
e}^{\frac{3}{2}\zeta^{\prime}_{\Delta_{1}}(0)-3\zeta^{\prime}_{\Delta_{0}}(0)}$

where

$N_{*}=\frac{1-2c\varepsilon-\sqrt{(1-2c\varepsilon)^{2}-4c\varepsilon}}{2c\varepsilon}$

with a technical constant $0<c<+\infty$ and $\zeta_{\Delta_{k}}$ are the
$\zeta$-functions of Laplacians acting on $k$-forms over the round $4$-sphere
$(S^{4},g_{R})$, the conformal one-point compactification of the Euclidean
space $({\mathbb{R}}^{4},\eta)$, with a sufficiently small radius
$0<R<+\infty$ determined by $\tau$.

Moreover the truncated partition function
$Z_{\varepsilon}({\mathbb{R}}^{4},\tau)$ depends on $R$ only through its
determinant term ${\rm
e}^{\frac{3}{2}\zeta^{\prime}_{\Delta_{1}}(0)-3\zeta^{\prime}_{\Delta_{0}}(0)}$.
More precisely if $(S^{4},g_{R_{i}})$ with $i=1,2$ are two conformal one-point
compactifications of $({\mathbb{R}}^{4},\eta)$ then the corresponding
partition functions are related by

$Z^{2}_{\varepsilon}({\mathbb{R}}^{4},\tau)=\left(\frac{R_{2}}{R_{1}}\right)^{\frac{11}{10}}Z^{1}_{\varepsilon}({\mathbb{R}}^{4},\tau)$

demonstrating that the conformal invariance of classical gauge theory breaks
down.

######  Remark.

1\. As a technical comment the regularized determinant expression ${\rm
e}^{\frac{3}{2}\zeta^{\prime}_{\Delta_{1}}(0)-3\zeta^{\prime}_{\Delta_{0}}(0)}$
can be further expressed over the unit $4$-sphere in terms of the derivatives
of the standard Riemann and Hurwitz $\zeta$-functions (cf. e.g. [3, 9, 12]);
however the result is not promissing hence omitted. One might hope to obtain
nicer determinant expressions by introducing Dirac fermions into the theory,
too. Also cf. [1].

2\. The actual values of the exponent $-\frac{11}{20}$ or $N_{*}$ in
$Z_{\varepsilon}({\mathbb{R}}^{4},\tau)$ bear no direct physical meaning, for
they depend on the particular regularization scheme used to make sense of
infinite dimensional integrals here. Concerning $N_{*}$, it is essentially a
good choice for the constant $N$ (see (8) below) in Uhlenbeck’s fundamental
gauge fixing theorem [17] and the only relevant point is that $0<N_{*}<1$
always must hold in order our method to work. But at least $N_{*}$ is
universal in the sense that always $N_{*}\rightarrow 1$ as
$\varepsilon\rightarrow 0$.

3\. Nevertheless Theorem 1.1, among other things, admits an interesting
physical interpretation in the context of the existence of a non-trivial
$\beta$-function and asymptotic freedom which are key properties of non-
Abelian gauge theories. Referring for further details in Section 3 below
recall that the complex coupling constant of a gauge theory is
$\tau:=\frac{\theta}{2\pi}+\frac{4\pi}{e^{2}}\sqrt{-1}$ where $\theta$ is the
so-called $\theta$-parameter and $e$ is the coupling constant of the gauge
theory. The complex coupling constant enters the theory at its classical level
i.e. it appears already in its defining action. However it is well-known that
in a non-supersymmetric four dimensional gauge theory, meanwhile $\theta$ is
unaffected hence is a true quantum parameter, $e$ is subject to quantum
corrections (i.e. the theory has a non-trivial $\beta$-function). Therefore it
is intriguing to physically interpret the appearence of the purely technical-
mathematical constant $N_{*}$ in Theorem 1.1 as a quantum correction of the
classical gauge coupling. That is, recalling the analogous expression in the
Abelian case [4], we cannot resist the temptation to re-write the partition
function as

$Z_{\varepsilon}({\mathbb{R}}^{4},\tau)=\left(\frac{{\rm Im}\tau_{\rm
eff}}{8\pi^{2}}\right)^{-\frac{11}{20}}{\rm
e}^{\frac{3}{2}\zeta^{\prime}_{\Delta_{1}}(0)-3\zeta^{\prime}_{\Delta_{0}}(0)}$

i.e. absorb $N_{*}$ into $\tau$ in such a way that the classical $\tau$ is
modified to $\tau_{\rm eff}=\frac{\theta}{2\pi}+\frac{4\pi}{e_{\rm
eff}^{2}}\sqrt{-1}$ with $e_{\rm eff}:=N_{*}e$ being considered as an
effective, perturbatively quantum corrected coupling constant. However from
the expression on $N_{*}$ above we easily see that

$N_{*}\approx 1-c\varepsilon<1$

hence $e_{\rm eff}<e$ rendering the effective i.e. perturbatively quantum
corrected gauge coupling constant smaller than its classical value. This is
qualitatively consistent with the phenomenon of asymptotic freedom in pure
non-Abelian gauge theories, the net effect of the highly counter-intuitive
Yang–Mills charge anti-screening caused by virtual charged gauge bosons
floating around the real ones.333Note that $\tau$ remains uncorrected in the
full partition function in the Abelian case, cf. [4].

3\. We can also make a comment regarding $S$-duality [18]. In Theorem 1.1 it
is assumed that $\tau$ has large (but finite!) imaginary part that is, the
gauge coupling $e$ is small. This assumption is physically clear because in
this weak coupling regime the existence of convergent perturbation series is
reasonable. The weak and the strong coupling regimes of a gauge theory are
related by $S$-duality transformations. Supposing that $\tau_{\rm eff}$ is
already meaningful at the quantum level, more precisely after taking into
account at least small perturbative quantum corrections provided by the
vicinity of the vacuum and recalling the identity ${\rm
Im}\left(-\frac{1}{\tau_{\rm eff}}\right)=\frac{1}{\tau_{\rm
eff}\>\overline{\tau}_{\rm eff}}\>{\rm Im}\tau_{\rm eff}$ we recognize that
the truncated partition function is a modular function with (holomorphic and
anti-holomorphic) weight $(\frac{11}{20},\frac{11}{20})$ hence
$Z_{\varepsilon}({\mathbb{R}}^{4},\tau_{\rm eff})$ has a promissing behaviour
under $S$-duality transformations [18]. Of course to say something more
definitive on this topic (for instance what about the modular properties of
the full partition function with some meaningful $\tau_{\rm eff}$ and how
${\rm SU}(2)$ is replaced with its Langlands dual group ${\rm SO}(3)$, etc.)
one would need to calculate the complete partitition function
$Z({\mathbb{R}}^{4},\tau_{\rm eff})$ consisting of all instanton, etc.
corrections; this is however far beyond our technical skills at this stage of
the art.

Nevertheless we record here that essentially by verbatim repeating the
calculation below the partition function can also be computed in the vicinity
of an (anti-)instanton $\nabla^{k}$ with instanton number $k\in{\mathbb{Z}}$
as well. It takes the shape ${\rm e}^{-\sqrt{-1}\pi
k\tau}Z_{\varepsilon,k}({\mathbb{R}}^{4},\tau)$ if $k\geqq 0$ or ${\rm
e}^{\sqrt{-1}\pi k\overline{\tau}}Z_{\varepsilon,k}({\mathbb{R}}^{4},\tau)$ if
$k\leqq 0$ where $Z_{\varepsilon,k}({\mathbb{R}}^{4},\tau)$ is an expression
analogous to
$Z_{\varepsilon,0}({\mathbb{R}}^{4},\tau):=Z_{\varepsilon}({\mathbb{R}}^{4},\tau)$
in Theorem 1.1 such that the various ordinary Laplacians $\Delta_{i}={\rm
d}{\rm d}^{*}+{\rm d}^{*}{\rm d}$ and their corresponding functions
$\zeta_{\Delta_{i}}$ are to be replaced with the twisted ones
$\Delta^{k}_{i}:={\rm d}_{\nabla^{k}}{\rm d}^{*}_{\nabla^{k}}+{\rm
d}^{*}_{\nabla^{k}}{\rm d}_{\nabla^{k}}$ and $\zeta_{\Delta^{k}_{i}}$
respectively.

The paper is organized as follows. In Section 2 we recall the calculation of
the Gaussian and a certain non-Gaussian quartic integral in finite dimensions.
The calculation of the latter integral is due to Svensson [15]. The resulting
formulata allow formal generalizations to infinite dimensions. Then in Section
3 classical pure gauge theory with $\theta$-term is introduced in the standard
way and its truncated partition function is computed by evaluating these
infinite dimensional formal integrals using $\zeta$-function and heat kernel
techniques. Finally, Section 4 is an Appendix and consists a well-known no-go
result from infinite dimensional measure theory [5, 7]. This has been added to
gain a more comprehensive picture.

##  2 The Gaussian and a non-Gaussian integral

In this preliminary section we recall the evaluation of the well-known
Gaussian and a less-known certain non-Gaussian integral in finite dimensions;
these considerations then allow us to formally generalize these integrals to
infinite dimensions which is the relevant case for quantum field theory.

The Gaussian integral. Let $({\mathbb{R}}^{m},\eta)$ be the $m$ dimensional
Euclidean space and
$S:{\mathbb{R}}^{m}\times{\mathbb{R}}^{m}\rightarrow{\mathbb{R}}$ a positive
definite symmetric bilinear form on it given by $S(x,x):=\eta(x,Mx)$ where
$M:{\mathbb{R}}^{m}\rightarrow{\mathbb{R}}^{m}$ is a positive symmetric matrix
whose eigenvalues therefore are real and satisfy $0<\lambda_{i}<+\infty$ for
all $i=1,\dots,m$. Using a linear change of variables one can pass to a
principal axis basis of $S$ i.e. in which it looks like
$S(y,y)=\lambda_{1}y_{1}^{2}+\dots+\lambda_{m}y_{m}^{2}$ and then performing a
further change of variables $u_{i}:=\sqrt{\lambda_{i}}y_{i}$ we find that

$\lim\limits_{a_{i}\rightarrow+\infty}\int\limits_{-a_{i}}^{+a_{i}}{\rm
e}^{-\lambda_{i}y_{i}^{2}}{\rm
d}y_{i}=\lim\limits_{a_{i}\rightarrow+\infty}\frac{1}{\sqrt{\lambda_{i}}}\int\limits_{-\sqrt{\lambda_{i}}a_{i}}^{+\sqrt{\lambda_{i}}a_{i}}{\rm
e}^{-u_{i}^{2}}{\rm d}u_{i}=\frac{\sqrt{\pi}}{\sqrt{\lambda_{i}}}$

hence taking their product we come up with

$\int\limits_{{\mathbb{R}}^{m}}{\rm e}^{-S(x,x)}{\rm
d}x=\prod\limits_{i=1}^{m}\frac{\sqrt{\pi}}{\sqrt{\lambda_{i}}}=\frac{\pi^{\frac{m}{2}}}{\sqrt{\det
M}}$ (1)

giving rise to the well-known result. This integral has a truncated version,
too. Let $0<\delta<+\infty$ be a fixed number and using an orthonormal frame
$\\{{\bf e}_{1},\dots,{\bf e}_{m}\\}$ adapted to $S$ let

$C^{m}_{\delta}:=\left\\{y\in{\mathbb{R}}^{m}\>\left|\>y=\sum\limits_{i=1}^{m}y_{i}{\bf
e}_{i}\>,\>M{\bf e}_{i}=\lambda_{i}{\bf
e}_{i}\>,\>-\frac{\delta}{\sqrt{\lambda_{i}}}<y_{i}<+\frac{\delta}{\sqrt{\lambda_{i}}}\right.\right\\}$
(2)

denote the “principal hypercube” of $S$ more precisely an open rectangular
parallelepiped whose edges are parallel with the principal axes labeled by the
eigenvalues $\lambda_{i}$ of $S$ and having sizes
$\frac{2\delta}{\sqrt{\lambda_{i}}}$ respectively. Then introducing
$a_{i}:=\frac{\delta}{\sqrt{\lambda_{i}}}$ we can repeat the previous
calculation as follows:

$\int\limits_{-a_{i}}^{+a_{i}}{\rm e}^{-\lambda_{i}y_{i}^{2}}{\rm
d}y_{i}=\frac{1}{\sqrt{\lambda_{i}}}\int\limits_{-\delta}^{+\delta}{\rm
e}^{-u_{i}^{2}}{\rm d}u_{i}=\frac{\sqrt{K(\delta)}}{\sqrt{\lambda_{i}}}$

where $K(\delta)$, the square of the classical error function is defined as

$\sqrt{K(\delta)}:=\int\limits_{-\delta}^{+\delta}{\rm e}^{-u_{i}^{2}}{\rm
d}u_{i}=2\sum\limits_{j=0}^{+\infty}(-1)^{j}\frac{\delta^{2j+1}}{j!(2j+1)}=2\left(\delta-\frac{\delta^{3}}{3}+\frac{\delta^{5}}{10}-\frac{\delta^{7}}{42}+\dots\right)\>\>\>.$

It is independent of $S$ and is monotonly increasing in
$0\leqq\delta\leqq+\infty$ such that $0\leqq K(\delta)\leqq\pi$. Taking
product again we obtain an expression

$\int\limits_{C^{m}_{\delta}}{\rm e}^{-S(x,x)}{\rm
d}x=\prod_{i=1}^{m}\frac{\sqrt{K(\delta)}}{\sqrt{\lambda_{i}}}=\frac{K(\delta)^{\frac{m}{2}}}{\sqrt{\det
M}}\leqq\frac{\pi^{\frac{m}{2}}}{\sqrt{\det M}}$

for the integral over the principal axis hypercube, similar for the entire
integral above.

A non-Gaussian integral. Now let es compute a more general integral following
Svensson [15]. Namely, picking two positive definite bilinear forms
$S_{1},S_{2}$, we are interested in the quadratic integral

$\int\limits_{{\mathbb{R}}^{m}}{\rm e}^{-S_{1}(x,x)^{2}-S_{2}(x,x)}{\rm
d}x\>\>.$

Consider a curve $\Gamma_{c}(t):=t+\sqrt{-1}\>c\subset{\mathbb{C}}$ i.e. a
straight line in the complex plane which is parallel with the real axis
${\mathbb{R}}\subset{\mathbb{C}}$. It is easy to see that the integral
$\int_{\Gamma_{c}}{\rm
e}^{-\left(\frac{t}{2}-\sqrt{-1}\>S_{1}(x,x)\right)^{2}}{\rm d}t$ exists such
that it is independent of $c$ and its value is equal to $2\sqrt{\pi}$.
Referring to [15] we adjust our integral by inserting the Gaussian integral
$1=\frac{1}{2\sqrt{\pi}}\int_{\Gamma_{c}}{\rm
e}^{-\left(\frac{t}{2}-\sqrt{-1}\>S_{1}(x,x)\right)^{2}}{\rm d}t$ as follows:

$\displaystyle\int\limits_{{\mathbb{R}}^{m}}{\rm
e}^{-S_{1}(x,x)^{2}-S_{2}(x,x)}{\rm d}x$ $\displaystyle=$
$\displaystyle\int\limits_{{\mathbb{R}}^{m}}{\rm
e}^{-S_{1}(x,x)^{2}-S_{2}(x,x)}{\rm
d}x\left(\frac{1}{2\sqrt{\pi}}\int\limits_{\Gamma_{c}}{\rm
e}^{-\left(\frac{t}{2}-\sqrt{-1}\>S_{1}(x,x)\right)^{2}}{\rm d}t\right)$
$\displaystyle=$
$\displaystyle\frac{1}{2\sqrt{\pi}}\int\limits_{\Gamma_{c}}{\rm
e}^{-\frac{t^{2}}{4}}\left(\>\>\int\limits_{{\mathbb{R}}^{m}}{\rm
e}^{-S_{2}(x,x)+\sqrt{-1}\>tS_{1}(x,x)}{\rm d}x\right){\rm d}t$
$\displaystyle=$
$\displaystyle\frac{\pi^{\frac{m}{2}}}{2\sqrt{\pi}}\int\limits_{\Gamma}\frac{{\rm
e}^{-\frac{t^{2}}{4}}}{\sqrt{\det(M_{2}-\sqrt{-1}\>tM_{1})}}\>{\rm d}t$
$\displaystyle=$
$\displaystyle\frac{\sqrt{-1}^{\frac{m}{2}}\pi^{\frac{m}{2}}}{2\sqrt{\pi\det
M_{1}}}\int\limits_{\Gamma}\frac{{\rm e}^{-\frac{t^{2}}{4}}}{\sqrt{\det(t{\bf
1}+\sqrt{-1}M_{2}M_{1}^{-1})}}\>{\rm d}t$ $\displaystyle=$
$\displaystyle\frac{(\sqrt{-1}\pi)^{\frac{m}{2}}}{2\sqrt{\pi\det
M_{1}}}\int\limits_{\Gamma}\frac{{\rm
e}^{-\frac{t^{2}}{4}}}{\sqrt{(t-z_{1})\dots(t-z_{m})}}\>{\rm d}t$

where $z_{1},\dots,z_{m}\in{\mathbb{C}}$ are the eigenvalues of the matrix
$-\sqrt{-1}M_{2}M_{1}^{-1}$. We also have to specify the contour $\Gamma$ in
order to render the integral $\int_{\Gamma}\frac{{\rm
e}^{-\frac{t^{2}}{4}}}{\sqrt{(t-z_{1})\dots(t-z_{m})}}\>{\rm d}t$ well defined
which means that it must exist and be single-valued. Indeed, the eigenvalues
$z_{1},\dots,z_{m}\in{\mathbb{C}}$ are branching points of the integral and if
$m$ is even then these are the only branching points; if $m$ is odd then
beyond them the infinitely remote point $\infty$ is also a branching point.
Let us therefore perform branch cutting in the standard way as follows. If $m$
is even then take arbitrary pairings of the (not necessarily different)
eigenvalues $z_{1},\dots,z_{m}\in{\mathbb{C}}$ and connect these pairs with
segments such that these segments do not intersect. If $m$ is odd then do the
same within the collection
$z_{1},\dots,z_{m},\infty\in{\mathbb{C}}\cup\\{\infty\\}$ i.e. the set of
eigenvalues with the infinitely remote point added. Let $X$ be the space which
arises by cutting up ${\mathbb{C}}\cup\\{\infty\\}$ along these non-
intersecting segments and define $\Gamma\subset X$ to be any closed curve
which is homotopic within $X$ to any of the closed curves $\Gamma_{c}$ used
above i.e. satisfying $\frac{1}{2\sqrt{\pi}}\int_{\Gamma_{c}}{\rm
e}^{-\left(\frac{t}{2}-\sqrt{-1}\>S_{1}(x,x)\right)^{2}}{\rm d}t=1$ and having
the property $\Gamma_{c}\subset X$ (adjusting $c$ we can assume that
$\Gamma_{c}$ intersects the branch cutting segments at most in the infinite
remote point).

Let us specialize from now on to the case $S_{1}:=c_{1}S$ and $S_{2}:=c_{2}S$
with $c_{1},c_{2}>0$ real constants; this yields
$-\sqrt{-1}(c_{2}M)(c_{1}M)^{-1}=-\sqrt{-1}\frac{c_{2}}{c_{1}}{\bf 1}$ hence
$z_{1}=\dots=z_{m}=-\sqrt{-1}\frac{c_{2}}{c_{1}}\notin{\mathbb{R}}$. Moreover,
regardless the parity of $m$, we can put again $\Gamma$ to be for instance of
the form $\Gamma_{c}(t)=t+\sqrt{-1}\>c$ with a small positive constant $c>0$
(actually, if $m$ is even then we can put $c=0$ i.e., use
${\mathbb{R}}\subset{\mathbb{C}}$ for integration). We eventually come up with

$\int\limits_{{\mathbb{R}}^{m}}{\rm e}^{-c^{2}_{1}S(x,x)^{2}-c_{2}S(x,x)}{\rm
d}x=\frac{\pi^{\frac{m}{2}}}{2\sqrt{\pi\det
M}}\int\limits_{\Gamma}\left(c_{2}-\sqrt{-1}\>c_{1}t\right)^{-\frac{m}{2}}{\rm
e}^{-\frac{t^{2}}{4}}\>{\rm d}t$

together with the truncated integral

$\int\limits_{C^{m}_{\delta}}{\rm e}^{-c_{1}^{2}S(x,x)^{2}-c_{2}S(x,x)}{\rm
d}x=\frac{K(\delta)^{\frac{m}{2}}}{2\sqrt{\pi\det
M}}\int\limits_{\Gamma}\left(c_{2}-\sqrt{-1}\>c_{1}t\right)^{-\frac{m}{2}}{\rm
e}^{-\frac{t^{2}}{4}}\>{\rm d}t\>\>.$ (3)

It is easy to see that in the limit $c_{1}\rightarrow 0$ these integrals
reduce to the corresponding previous Gaussian ones.

Having warmed up with these rigorous but only finite dimensional results, let
us generalize them to infinite dimensions at least formally. Let $(M,g)$ be a
connected, compact, oriented Riemannian $4$-manifold without boundary and
consider the Laplacian $\Delta_{k}:C^{\infty}(M;\wedge^{k}M)\rightarrow
C^{\infty}(M;\wedge^{k}M)$ i.e. the second order linear, symmetric, elliptic
partial differential operator $\Delta_{k}={\rm d}{\rm d}^{*}+{\rm d}^{*}{\rm
d}$ naturally acting on the space of smooth $k$-forms. This space admits
Hilbert space completions like $L^{2}_{s}(M;\wedge^{k}M)$ for any
$s\in{\mathbb{R}}$ and one can demonstrate via elliptic regularity that
$\Delta_{k}$ extends to a densly defined, self-adjoint, unbounded linear
operator $\Delta_{k}:L^{2}(M;\wedge^{k}M)\rightarrow L^{2}(M;\wedge^{k}M)$. By
elliptic regularity the kernel of this map contains precisely the space
${\mathscr{H}}^{k}(M)\subset C^{\infty}(M;\wedge^{k}M)\subset
L^{2}(M;\wedge^{k}M)$ of smooth harmonic $k$-forms; by the Hodge decomposition
theorem this kernel is isomorphic to the de Rham cohomology group $H^{k}(M)$
hence is finite dimensional i.e. a closed subspace. Therefore $c\Delta_{k}$
with $c>0$ a real constant gives rise to a positive self-adjoint operator on
the orthogonal complement Hilbert space

${\mathscr{H}}^{k}(M)^{\perp}\subset L^{2}(M;\wedge^{k}M)\>\>.$

By the finite dimensional analogue (1) it is therefore natural to define a
Gaussian-like integral involving the Laplacian as

$\int\limits_{{\mathscr{H}}^{k}(M)^{\perp}}{\rm
e}^{-\left(a\>,\>c\Delta_{k}a\right)_{L^{2}(M)}}\>{\rm
D}a:=\frac{\pi^{\frac{1}{2}{\rm rk^{\prime}}(c\Delta_{k})}}{\sqrt{{\rm
det^{\prime}}\left(c\Delta_{k}\right)}}$ (4)

where the regularized rank ${\rm rk^{\prime}}$ and determinant ${\rm
det^{\prime}}$ is yet to be defined somehow.

Likewise, let $C_{\delta}\subset L^{2}(M;\wedge^{k}M)$ be the “principal axis
hypercube” for $\Delta_{k}$ defined as in the finite dimensional case (2) more
precisely as the corresponding finite linear combinations of the eigen-forms
of $\Delta_{k}$. Note that in spite of the fact that the eigen-forms of
$\Delta_{k}$ span a dense subspace of $L^{2}(M;\wedge^{k}M)$ the subset
$C_{\delta}$ is not open (unlike in finite dimensions) because the eigenvalues
of the Laplacian form an unbounded sequence i.e.
$\lambda_{i}\rightarrow+\infty$ hence the size of the edges of $C_{\delta}$
satisfy $2a_{i}\rightarrow 0$ as $i\rightarrow+\infty$. Keeping in mind this
subtlety and taking into account (3) nevertheless we put

$\int\limits_{C_{\delta}\cap{\mathscr{H}}^{k}(M)^{\perp}}{\rm
e}^{-(a\>,\>c_{1}\Delta_{k}a)^{2}_{L^{2}(M)}-\left(a\>,\>c_{2}\Delta_{k}a\right)_{L^{2}(M)}}\>{\rm
D}a:=\frac{1}{2\sqrt{\pi}}\frac{K(\delta)^{\frac{1}{2}{\rm
rk^{\prime}}(\Delta_{k})}}{\sqrt{{\rm
det^{\prime}}\Delta_{k}}}\int\limits_{\Gamma}\left(c_{2}-\sqrt{-1}\>c_{1}t\right)^{-\frac{1}{2}{\rm
rk^{\prime}}(\Delta_{k})}{\rm e}^{-\frac{t^{2}}{4}}\>{\rm d}t$ (5)

acknowledgeing that the integration contour $\Gamma$ might require further
specification in infinite dimensions.

We will also assume that the following “monotonicity principles” hold true for
these infinite dimensional formal integrals:

Monotonicity principles. If $\emptyset\subseteqq A,B\subseteqq
L^{2}(S^{4};\wedge^{1}S^{4})$ are two “measurable” subsets in the $L^{2}$
Hilbert space of $1$-forms over the $4$-sphere satisfying $A\subseteqq B$ and
$f:L^{2}(S^{4};\wedge^{1}S^{4})\rightarrow{\mathbb{R}}$ is a non-negative
“integrable” function then

$0\leqq\int\limits_{A}f(a){\rm D}a\leqq\int\limits_{B}f(a){\rm
D}a\leqq+\infty\>\>.$

Moreover, if $f,g:L^{2}(S^{4};\wedge^{1}S^{4})\rightarrow{\mathbb{R}}$ are two
“integrable” functions satisfying $0\leqq f\leqq g$ then

$0\leqq\int\limits_{A}f(a){\rm D}a\leqq\int\limits_{A}g(a){\rm
D}a\leqq+\infty$

is valid.

######  Remark.

1\. These monotonicity properties of integration are straighforward in finite
dimensions however are not easily accessable in infinite dimensions. But more
surprisingly, it seems these properties even may not hold over any
$4$-manifold. For instance, as we will see in Section 3, over the $4$-sphere
the regularized dimension of $L^{2}(S^{4};\wedge^{1}S^{4})$ with respect to
the Laplacian is positive (see Lemma 3.1) hence the above monotonicity
properties are expected to hold true. However, over the flat $4$-torus for
example, the regularized dimension of $L^{2}(T^{4};\wedge^{1}T^{4})$ with
respect to the Laplacian is negative hence one would expect that some sort of
converse of the above monotonicity might work in these case.

2\. As we also mentioned before the “principal axis hypercube”
$C_{\delta}\subset L^{2}(M;\wedge^{k}M)$ for the Laplacian is not open in
infinite dimensions. If nevertheless the formal integral (5) happens to attain
a non-zero value then this would imply that infinite dimensional integration
over very small (i.e. which do not contain any open ball) subsets might yield
non-trivial results.

All of these oddities of integration in infinite dimensions likely are
connected with the conflict between $\sigma$-additivity and infinite
dimensionlity (cf. the Appendix here).

##  3 The partition function

After these preliminaries we are ready to calculate the partition function.
Let us begin with recalling and introducing $4$ dimensional Euclidean non-
supersymmetric ${\rm SU}(2)$ gauge theory with $\theta$ term in the usual way.

Consider ${\mathbb{R}}^{4}$ with its standard flat Riemannian metric $\eta$.
Let $E\cong{\mathbb{R}}^{4}\times{\mathbb{C}}^{2}$ be the unique trivial
complex rank-two ${\rm SU}(2)$ vector bundle over ${\mathbb{R}}^{4}$ and take
a compatible (i.e. ${\rm SU}(2)$-valued) connection $\nabla$ on it. Denoting
by $\wedge^{k}{\mathbb{R}}^{4}\otimes{\mathfrak{s}}{\mathfrak{u}}(2)$ the
bundle of ${\mathfrak{s}}{\mathfrak{u}}(2)$-valued $k$-forms over
${\mathbb{R}}^{4}$, by the global triviality of $E$ we can globally write
$\nabla={\rm d}+A$ where the gauge potential $A$ is a section of
$\wedge^{1}{\mathbb{R}}^{4}\otimes{\mathfrak{s}}{\mathfrak{u}}(2)$ with the
corresponding field strength $F_{\nabla}={\rm d}A+A\wedge A$ giving rise to a
section of $\wedge^{2}{\mathbb{R}}^{4}\otimes{\mathfrak{s}}{\mathfrak{u}}(2)$.
Moreover let $e\in{\mathbb{R}}$ and $\theta\in{\mathbb{R}}$ denote the
coupling constant and the $\theta$-parameter of the theory respectively. The
non-supersymmetric $4$ dimensional Euclidean ${\rm SU}(2)$ gauge theory is
then defined by the action

$S(\nabla,e,\theta):=-\frac{1}{2e^{2}}\int\limits_{{\mathbb{R}}^{4}}{\rm
tr}(F_{\nabla}\wedge*F_{\nabla})+\frac{\sqrt{-1}\>\theta}{16\pi^{2}}\int\limits_{{\mathbb{R}}^{4}}{\rm
tr}(F_{\nabla}\wedge F_{\nabla})\>\>.$

The $\theta$-term is a characteristic class hence its variation is identically
zero consequently the Euler–Lagrange equations (togehter with the Bianchi
identity) of this theory are nothing but the usual vacuum Yang–Mills equations

$\left\\{\begin{array}[]{ll}{\rm d}_{\nabla}F_{\nabla}=0\\\ {\rm
d}^{*}_{\nabla}F_{\nabla}=0\>\>.\end{array}\right.$

Introducing the complex coupling constant

$\tau:=\frac{\theta}{2\pi}+\frac{4\pi}{e^{2}}\sqrt{-1}$ (6)

taking its values on the upper complex half-plane ${\mathbb{C}}^{+}$, and the
positive definite $L^{2}$ scalar product
$(\Phi,\Psi)_{L^{2}({\mathbb{R}}^{4})}:=-\int_{{\mathbb{R}}^{4}}{\rm
tr}(\Phi\wedge*\Psi)$ on the space of ${\mathfrak{s}}{\mathfrak{u}}(2)$-valued
$2$-forms, with induced norm therefore satisfying
$\|\Phi\|_{L^{2}({\mathbb{R}}^{4})}\geqq 0$, the action above can be re-
written as

$\displaystyle S(\nabla,\tau)$ $\displaystyle=$
$\displaystyle-\frac{\sqrt{-1}\pi}{2}\tau\left(\frac{1}{8\pi^{2}}\|F_{\nabla}\|^{2}_{L^{2}({\mathbb{R}}^{4})}+\frac{1}{8\pi^{2}}(F_{\nabla},*F_{\nabla})_{L^{2}({\mathbb{R}}^{4})}\right)$
(7)
$\displaystyle+\frac{\sqrt{-1}\pi}{2}\overline{\tau}\left(\frac{1}{8\pi^{2}}\|F_{\nabla}\|^{2}_{L^{2}({\mathbb{R}}^{4})}-\frac{1}{8\pi^{2}}(F_{\nabla},*F_{\nabla})_{L^{2}({\mathbb{R}}^{4})}\right)$

since $*^{2}={\rm Id}_{\wedge^{2}{\mathbb{R}}^{4}}$ hence the topological term
takes the shape $-\int_{{\mathbb{R}}^{4}}{\rm tr}(F_{\nabla}\wedge
F_{\nabla})=(F_{\nabla},*F_{\nabla})_{L^{2}({\mathbb{R}}^{4})}$ in this
notation.

The orientation and the flat Euclidean metric $\eta$ on ${\mathbb{R}}^{4}$ is
used to introduce various Sobolev spaces. Let $\nabla^{0}$ denote the trivial
flat connection on $E$ i.e. the unique connection which satisfies
$F_{\nabla^{0}}=0$. Then define

${\mathscr{A}}(\nabla^{0}):=\\{\mbox{$\nabla$ is an ${\rm SU}(2)$ connection
on $E$}\>|\>\mbox{$\nabla-\nabla^{0}\in
L^{2}_{1}({\mathbb{R}}^{4}\>;\>\wedge^{1}{\mathbb{R}}^{4}\otimes{\mathfrak{s}}{\mathfrak{u}}(2))$}\\}.$

This is the $L^{2}_{1}$ Sobolev space of ${\rm SU}(2)$ connections on $E$
relative to $\nabla^{0}$. Notice that this is a vector space (not an affine
space) and in fact

${\mathscr{A}}(\nabla^{0})\ni\nabla\longmapsto\nabla-\nabla^{0}=:a\in
L^{2}_{1}({\mathbb{R}}^{4}\>;\>\wedge^{1}{\mathbb{R}}^{4}\otimes{\mathfrak{s}}{\mathfrak{u}}(2))$

is a canonical isomorphism between ${\mathscr{A}}(\nabla^{0})$ and
$L^{2}_{1}({\mathbb{R}}^{4}\>;\>\wedge^{1}{\mathbb{R}}^{4}\otimes{\mathfrak{s}}{\mathfrak{u}}(2))$.
Furthermore write ${\mathscr{S}}{\mathscr{U}}(2)$ for the $L^{2}_{2}$
completion of

$\\{\mbox{$\gamma$ is an ${\rm SU}(2)$ gauge transformation on
$E$}\>|\>\mbox{$\gamma-{\rm Id}_{E}\in C^{\infty}_{0}({\mathbb{R}}^{4};{\rm
End}E)$, $\gamma\in C^{\infty}({\mathbb{R}}^{4};{\rm Aut}E)$ a.e.}\\}$

that is, the space of compactly supported smooth ${\rm SU}(2)$ gauge
transformations. Therefore $\gamma\in{\mathscr{S}}{\mathscr{U}}(2)$ means that
$\|\gamma-{\rm End}E\|_{L^{2}_{2}({\mathbb{R}}^{4})}<+\infty$. The space
${\mathscr{A}}(\nabla^{0})$ is acted upon by ${\mathscr{S}}{\mathscr{U}}(2)$
in the usual way as $\nabla\mapsto\gamma^{-1}\nabla\gamma$ and the
corresponding gauge equivalence class of $\nabla\in{\mathscr{A}}(\nabla^{0})$
is denoted by $[\nabla]$ and the orbit space
${\mathscr{A}}(\nabla^{0})/{\mathscr{S}}{\mathscr{U}}(2)$ of these equivalence
classes with its quotient topology by ${\mathscr{B}}([\nabla^{0}])$ as usual.
In the non-Abelian case ${\mathscr{B}}([\nabla^{0}])$ is not a linear space
however at least locally it can be modeled on various Banach spaces as we
shall see shortly. Also note that under our assumptions appropriate Sobolev
embedding and multiplication theorems valid in $4$ dimensions ensure us that
the curvature $2$-forms are always at least in $L^{2}$ that is, $F_{\nabla}\in
L^{2}({\mathbb{R}}^{4}\>;\>\wedge^{2}{\mathbb{R}}^{4}\otimes{\mathfrak{s}}{\mathfrak{u}}(2))$
for any $\nabla\in[\nabla]\in{\mathscr{B}}([\nabla^{0}])$.

Having now the classical non-supersymmetric Euclidean gauge theory at our
disposal, the partition function of the induced quantum theory is formally
defined by the integral

$Z({\mathbb{R}}^{4},\tau):=\frac{1}{{\rm
Vol}\left({{\mathscr{S}}{\mathscr{U}}(2)}\right)}\int\limits_{\nabla\in{\mathscr{A}}(\nabla^{0})}{\rm
e}^{-S(\nabla,\tau)}{\rm D}\nabla$

or formally equivalently

$Z({\mathbb{R}}^{4},\tau):=\int\limits_{[\nabla]\in{\mathscr{B}}([\nabla^{0}])}{\rm
e}^{-S(\nabla,\tau)}{\rm D}[\nabla]$

where ${\rm D}\nabla$ is the formal (probably never definable) measure on
${\mathscr{A}}(\nabla^{0})$ while ${\rm D}[\nabla]$ is the induced formal
measure (including the Faddeev–Popov determinant) on the orbit space
${\mathscr{B}}([\nabla^{0}])$. The ideal goal would be to calculate this
integral in its full glory however it is an extraordinary difficult task
because of the non-linearity of ${\mathscr{B}}([\nabla^{0}])$. Therefore we
will evaluate it in the vicinity of the vacuum
${\mathscr{B}}_{\varepsilon}([\nabla^{0}])$ only i.e. we are interested in an
appropriately truncated integral

$Z_{\varepsilon}({\mathbb{R}}^{4},\tau):=\int\limits_{[\nabla]\in{\mathscr{B}}_{\varepsilon}([\nabla^{0}])}{\rm
e}^{-S(\nabla,\tau)}{\rm D}[\nabla]$

where ${\mathscr{B}}_{\varepsilon}([\nabla^{0}])$ is a small open subset about
$[\nabla^{0}]$ defined by
$0\leqq|S(\nabla,\tau)|<\frac{|\tau|}{8\pi}\varepsilon^{2}$ possessing the
crucial property that, unlike the whole ${\mathscr{B}}([\nabla^{0}])$, it is
well approximated by (a quotient of) a small open ball in an appropriate
Hilbert space.

To make this picture more precise and in order to avoid several technical
difficulties we make a technical interlude and extend the ${\rm SU}(2)$
Yang–Mills theory from $({\mathbb{R}}^{4},\eta)$ to its one-point conformal
compactification $(S^{4},g_{R})$ where $g_{R}$ denotes the standard round
metric on $S^{4}={\mathbb{R}}^{4}\cup\\{\infty\\}$ such that it has radius
$0<R<+\infty$. From the physical viewpoint this conformal compactification is
justified at least classically by the conformal invariance of classical gauge
theory defined by (7) in four dimensions. From the mathematical or technical
viewpoint a further support is Uhlenbeck’s singularity removal theorem [16]
asserting that if $\nabla\in{\mathscr{A}}(\nabla^{0})$ is a connection on
${\mathbb{R}}^{4}$ (which by definition means that there exists an $L^{2}_{1}$
gauge relative to $\nabla^{0}$ moreover
$\|F_{\nabla}\|_{L^{2}({\mathbb{R}}^{4})}<+\infty$ as we mentioned above)
which solves the Yang–Mills equations then there exists an $L^{2}_{2}$ gauge
transformation around the asymptotic region of ${\mathbb{R}}^{4}$ such that
the gauge transformed connection $\nabla^{\prime}$ smoothly extends over
${\mathbb{R}}^{4}\cup\\{\infty\\}=S^{4}$. Since the vacuum $\nabla^{0}$ is
obviously a classical solution of Yang–Mills theory it extends over the
conformal compactification. Recalling that we wish to perform computations in
the vicinity of the vacuum, we will assume that a sufficiently small
neighbourhood in ${\mathscr{A}}(\nabla^{0})$ of the vacuum connection
$\nabla^{0}$ conformally extends as well. Therefore, from now on we consider
the classical Yang–Mills theory (7) over $(S^{4},g_{R})$ and correspondingly
we are interested in calculating the formal truncated Feynman integral
$Z_{\varepsilon}({\mathbb{R}}^{4},\tau)$ by working over $(S^{4},g_{R})$. It
is therefore understood that the action $S$, the Sobolev space
${\mathscr{A}}(\nabla^{0})$ consisting of our connections $\nabla$ and the
various differential operators like ${\rm d}^{*}$, $\Delta_{k}$, etc. are
defined over the round $4$-sphere $(S^{4},g_{R})$. In this compactified
setting Uhlenbeck’s gauge fixing theorem [17] can be formulated as follows
(cf. [2, Proposition 2.3.13]): There exists a constant $0<\varepsilon$ such
that if a connection $\nabla\in{\mathscr{A}}(\nabla^{0})$ on the trivial
bundle $E\cong S^{4}\times{\mathbb{C}}^{2}$ satisfies
$\|F_{\nabla}\|_{L^{2}(S^{4})}<\varepsilon$ then there exists an $L^{2}_{2}$
gauge transformation $\gamma$ and a constant $0<N<+\infty$ such that the gauge
transformed connection $\nabla^{\prime}=\gamma^{-1}\nabla\gamma$ with
corresponding decomposition $\nabla^{\prime}={\rm d}+A^{\prime}$ satisfies the
Coulomb gauge condition together with an estimate

$\left\\{\begin{array}[]{ll}{\rm d}^{*}A^{\prime}=0\\\
\|A^{\prime}\|_{L^{2}_{1}(S^{4})}\leqq
N\|F_{\nabla^{\prime}}\|_{L^{2}(S^{4})}\end{array}\right.$ (8)

implying $\|A^{\prime}\|_{L^{2}_{1}(S^{4})}\leqq N\varepsilon$ in Coulomb
gauge.

Now we are in a position to define the truncated partition function more
carefully. Fix a sufficiently small constant $0<\varepsilon<1$ and consider
those connections which satisfy $\|F_{\nabla}\|_{L^{2}(S^{4})}<\varepsilon$.
By conformal invariance of the norm this is equivalent to consider those
connections over the original space which satisfy
$\|F_{\nabla}\|_{L^{2}({\mathbb{R}}^{4})}<\varepsilon$. The action takes a
more clear shape in the compactified setting as follows. Regarding its
topological term $\frac{1}{8\pi^{2}}\int_{S^{4}}{\rm tr}(F_{\nabla}\wedge
F_{\nabla})$ we know that it is proportional to the second Chern number of the
extended ${\rm SU}(2)$ bundle over $S^{4}$ hence it assumes discrete values
only; however by the Cauchy–Schwarz inequality
$0\leqq\left|(F_{\nabla}\>,\>*F_{\nabla})_{L^{2}(S^{4})}\right|\leqq\|F_{\nabla}\|_{L^{2}(S^{4})}\|*F_{\nabla}\|_{L^{2}(S^{4})}=\|F_{\nabla}\|^{2}_{L^{2}(S^{4})}<\varepsilon^{2}<1$
the $\theta$-term simply vanishes over $S^{4}$ in the small energy regime.
This also implies that the connections we are interested in are realized in
the extended gauge theory on the trivial bundle $E\cong
S^{4}\times{\mathbb{C}}^{2}$ alone and Uhlenbeck’s gauge fixing theorem
applies. Consequently the action (7) about $\nabla^{0}$ looks like

$S(\nabla,\tau)=S(\nabla^{0}+a,\tau)=\frac{{\rm
Im}\tau}{8\pi}\|F_{\nabla^{0}+a}\|^{2}_{L^{2}(S^{4})}=\frac{{\rm
Im}\tau}{8\pi}\|{\rm d}a+a\wedge a\|^{2}_{L^{2}(S^{4})}$

which also shows by conformal invariance of the action that
$[\nabla]\in{\mathscr{B}}_{\varepsilon}([\nabla^{0}])$. The key technical
observation now is [2, Proposition 4.2.9] saying that for a sufficiently small
$\varepsilon$ there exists an $\eta>0$ such that
${\mathscr{B}}_{\varepsilon}([\nabla^{0}])$ is homeomorphic to
$(B_{\eta}(\nabla^{0})\cap{\rm ker}\>{\rm d}^{*})/G_{0}$ with
$B_{\eta}(\nabla^{0})\subset{\mathscr{A}}(\nabla^{0})$ being a small open ball
and $G_{0}\cong{\rm SU}(2)$ the gauge isotropy subgroup of the flat hence
reducible connection $\nabla^{0}$. Hence put

${\mathscr{A}}_{\varepsilon,N}(\nabla^{0}):=\left\\{a\in
L^{2}_{1}(S^{4};\wedge^{1}S^{4}\otimes{\mathfrak{s}}{\mathfrak{u}}(2))\>\left|\>\|a\|_{L^{2}_{1}(S^{4})}<\min(\eta,N\varepsilon)\right.\right\\}$
(9)

where $\varepsilon,N$ are the same constants as in (8). By the aid of the
homeomorphism

${\mathscr{B}}_{\varepsilon}([\nabla^{0}])\cong\frac{{\mathscr{A}}_{\varepsilon,N}(\nabla^{0})\cap{\rm
ker}\>{\rm d}^{*}}{G_{0}}$

we suppose that the “measure” ${\rm D}[\nabla]$ arises from a
$G_{0}$-invariant “measure” on
${\mathscr{A}}_{\varepsilon,N}(\nabla^{0})\cap{\rm ker}\>{\rm d}^{*}$ what we
denote ${\rm D}a$. The main advantage of this non-linear isomorphism is that
it locally “straightens” the gauge orbits hence its effect is analogous to
passing from a general curved coordinate system to the standard Descartes one.
Consequently the Faddeev–Popov determinant is locally transformed away i.e.
gives only a constant multiplyer (cf. Footnote 4). Our truncated Feynman
integral to be calculated is then defined more carefully as

$Z_{\varepsilon}({\mathbb{R}}^{4},\tau):=\int\limits_{{\mathscr{A}}_{\varepsilon,N}(\nabla^{0})\cap{\rm
ker}\>{\rm d}^{*}}{\rm e}^{-\frac{{\rm Im}\tau}{8\pi}\|{\rm d}a+a\wedge
a\|^{2}_{L^{2}(S^{4})}}\frac{{\rm D}a}{{\rm Vol}(G_{0})}$ (10)

where by compactness of $S^{4}$ the integration domain
${\mathscr{A}}_{\varepsilon,N}(\nabla^{0})\cap{\rm ker}\>{\rm d}^{*}$ is a
small open ball of radius $\min(\eta,N\varepsilon)$ in the closed hence
Hilbert subspace ${\rm ker}\>{\rm d}^{*}\subset
L^{2}_{1}(S^{4};\wedge^{1}S^{4}\otimes{\mathfrak{s}}{\mathfrak{u}}(2))$ which
therefore is independent of the radius $R$ of $S^{4}$ however depends on the
Uhlenbeck constant $N$ in (8). For notational simplicity we shall plug the
numerical factor $0<\frac{1}{{\rm Vol}(G_{0})}<+\infty$ into ${\rm D}a$.

Let us work out a two-sided estimate of the action along the integration
domain of (10) as follows. Regarding an estimate from below, it readily
follows from (8) that

$\|{\rm d}a\|_{L^{2}(S^{4})}\leqq N\|{\rm d}a+a\wedge a\|_{L^{2}(S^{4})}\>\>.$
(11)

Assume $N\geqq 1$. Now $\|a\wedge a\|_{L^{2}(S^{4})}$ is bounded by
$\|a\|^{2}_{L^{4}(S^{4})}$; but
$\|a\|_{L^{4}(S^{4})}\\!\leqq\\!c_{1}\|a\|_{L^{2}_{1}(S^{4})}$ by the Sobolev
embedding $L^{2}_{1}\subset L^{4}$ which is sharp in $4$ dimensions; moreover
elliptic regularity for ${\rm d}+{\rm d}^{*}$ gives
$\|a\|_{L^{2}_{1}(S^{4})}\leqq c_{2}\|({\rm d}+{\rm
d}^{*})a\|_{L^{2}(S^{4})}+c_{3}\|a\|_{L^{2}(S^{4})}=c_{2}\|{\rm
d}a\|_{L^{2}(S^{4})}$ since ${\rm d}^{*}a=0$ by the Coulomb gauge condition
and we can put $c_{3}=0$ because $H^{1}(S^{4})=0$ consequently ${\rm ker}({\rm
d}+{\rm d}^{*})={\rm ker}\>\Delta_{1}=\\{0\\}$. Combining these and
introducing $c:=(c_{1}c_{2})^{2}>0$ we get

$\|{\rm d}a\|_{L^{2}(S^{4})}\leqq\|{\rm d}a+a\wedge
a\|_{L^{2}(S^{4})}+\|a\wedge a\|_{L^{2}(S^{4})}\leqq\|{\rm d}a+a\wedge
a\|_{L^{2}(S^{4})}+c\|{\rm d}a\|^{2}_{L^{2}(S^{4})}\>\>.$

Adding the two inequalities for $\|{\rm d}a\|_{L^{2}(S^{4})}$ gives $\|{\rm
d}a\|_{L^{2}(S^{4})}(2-c\|{\rm d}a\|_{L^{2}(S^{4})})\leqq(N+1)\\!\|{\rm
d}a+a\wedge a\|_{L^{2}(S^{4})}$; the integration domain (9) is bounded and it
implies a particular higher bound $\|{\rm
d}a\|_{L^{2}(S^{4})}<(N+2)\varepsilon$ yielding $\|{\rm
d}a\|_{L^{2}(S^{4})}\left(2-c(N+2)\varepsilon\right)\leqq(N+1)\|{\rm
d}a+a\wedge a\|_{L^{2}(S^{4})}$. Provided $\varepsilon$ is small enough
compared with the initial value of $N$ in (8) hence (11), more precisely
$\varepsilon<\frac{2}{c(N+2)}$ we can replace $N$ in (11) with
$\frac{N+1}{2-c(N+2)\varepsilon}$ such that iterating this particular process
$N$ will converge to the particular lower fixed point

$N_{*}:=\frac{1-2c\varepsilon-\sqrt{(1-2c\varepsilon)^{2}-4c\varepsilon}}{2c\varepsilon}$
(12)

with the property $N_{*}<1$. In this way we come up with an estimate from
below

$\|{\rm d}a\|^{2}_{L^{2}(S^{4})}\leqq N_{*}^{2}\|{\rm d}a+a\wedge
a\|^{2}_{L^{2}(S^{4})}$ (13)

along the particularly resized ball
${\mathscr{A}}_{\varepsilon,N_{*}}(\nabla^{0})\cap{\rm ker}\>{\rm d}^{*}$
defined analogously to (9). Regarding an upper bound over this resized ball we
start with $\|{\rm d}a+a\wedge a\|^{2}_{L^{2}(S^{4})}\leqq 2\|{\rm
d}a\|^{2}_{L^{2}(S^{4})}+2\|a\wedge a\|^{2}_{L^{2}(S^{4})}$ and then proceed
along the same lines as above to get

$\|{\rm d}a+a\wedge a\|^{2}_{L^{2}(S^{4})}\leqq 2\|{\rm
d}a\|^{2}_{L^{2}(S^{4})}+2c^{2}\|{\rm d}a\|^{4}_{L^{2}(S^{4})}\>\>.$ (14)

Putting together (13) and (14), multiplying each term with $-\frac{{\rm
Im}\tau}{8\pi}<0$ and exponentiating we obtain

${\rm e}^{-\frac{{\rm Im}\tau}{8\pi N_{*}^{2}}\|{\rm
d}a\|^{2}_{L^{2}(S^{4})}}\geqq{\rm e}^{-\frac{{\rm Im}\tau}{8\pi}\|{\rm
d}a+a\wedge a\|^{2}_{L^{2}(S^{4})}}\geqq{\rm e}^{-\frac{{\rm
Im}\tau}{4\pi}\|{\rm d}a\|^{2}_{L^{2}(S^{4})}-\frac{{\rm
Im}\tau\>c^{2}}{4\pi}\|{\rm d}a\|^{4}_{L^{2}(S^{4})}}$

or equivalently, using ${\rm d}^{*}a=0$ again we end up with

${\rm e}^{-\left(a\>,\>\frac{{\rm Im}\tau}{8\pi
N_{*}^{2}}\Delta_{1}a\right)_{L^{2}(S^{4})}}\geqq{\rm e}^{-\frac{{\rm
Im}\tau}{8\pi}\|{\rm d}a+a\wedge a\|^{2}_{L^{2}(S^{4})}}\geqq{\rm
e}^{-\left(a\>,\>\sqrt{\frac{{\rm
Im}\tau}{4\pi}}c\>\Delta_{1}a\right)^{2}_{L^{2}(S^{4})}-\left(a\>,\>\frac{{\rm
Im}\tau}{4\pi}\Delta_{1}a\right)_{L^{2}(S^{4})}}\>\>.$ (15)

Having obtained these rigorous estimates, the time has come to apply the
formal integral expressions from Setion 2.

######  Definition 3.1.

Taking into account that $H^{1}(S^{4})=\\{0\\}$ hence
${\mathscr{H}}^{1}(S^{4})=\\{0\\}$ and
$\dim_{\mathbb{R}}{\mathfrak{s}}{\mathfrak{u}}(2)=3$, substituting
$c:=\frac{{\rm Im}\tau}{8\pi N_{*}^{2}}$ in (4) we define the Gaussian-like
integral as

$\int\limits_{{\rm ker}\>{\rm d}^{*}}{\rm e}^{-\left(a\>,\>\frac{{\rm
Im}\tau}{8\pi N_{*}^{2}}\Delta_{1}a\right)_{L^{2}(S^{4})}}\>{\rm
D}a:=\left(\frac{\pi^{\frac{1}{2}{\rm rk^{\prime}}\left(\frac{{\rm
Im}\tau}{8\pi N_{*}^{2}}\Delta_{1}|_{{\rm ker}\>{\rm
d}^{*}}\right)}}{\sqrt{{\rm det^{\prime}}\left(\frac{{\rm Im}\tau}{8\pi
N_{*}^{2}}\Delta_{1}|_{{\rm ker}\>{\rm d}^{*}}\right)}}\right)^{3}\>\>.$

Likewise, substituing $c_{1}:=\frac{{\rm Im}\tau}{4\pi}$ and
$c_{2}:=\sqrt{\frac{{\rm Im}\tau}{4\pi}}\>c$ in (5) we define a truncated
quartic integral to be

$\displaystyle\int\limits_{C_{\delta}\cap{\rm ker}\>{\rm
d}^{*}}\\!\\!\\!\\!\\!\\!\\!\\!{\rm e}^{-\left(a\>,\>\sqrt{\frac{{\rm
Im}\tau}{4\pi}}c\>\Delta_{1}a\right)^{2}_{L^{2}(S^{4})}-\left(a\>,\>\frac{{\rm
Im}\tau}{4\pi}\Delta_{1}a\right)_{L^{2}(S^{4})}}{\rm D}a$ $\displaystyle:=$
$\displaystyle\frac{1}{2\sqrt{\pi}}\left(\frac{K(\delta)^{\frac{1}{2}{\rm
rk^{\prime}}\left(\Delta_{1}|_{{\rm ker}\>{\rm d}^{*}}\right)}}{\sqrt{{\rm
det^{\prime}}\left(\Delta_{1}|_{{\rm ker}\>{\rm
d}^{*}}\right)}}\right)^{3}\times$
$\displaystyle\int\limits_{\Gamma}\left(\frac{{\rm
Im}\tau}{4\pi}-\>\sqrt{\frac{-{\rm Im}\tau}{4\pi}}ct\right)^{-\frac{3}{2}{\rm
rk^{\prime}}\left(\Delta_{1}|_{{\rm ker}\>{\rm d}^{*}}\right)}{\rm
e}^{-\frac{t^{2}}{4}}\>{\rm d}t$

where the contour $\Gamma$ still depends on whether or not the infinitely
distant point is a branching point of the integral.

A familiar way to make sense of ${\rm rk^{\prime}}$ and ${\rm det^{\prime}}$
in Definition 3.1 i.e. to regularize the dimension and the functional
determinant in infinite dimensions is an application of $\zeta$-function
regularization.

######  Lemma 3.1.

(cf. [4, Definition 3.1 and Lemma 3.1]) Using $\zeta$-function regularization
to define ${\rm rk^{\prime}}$ and ${\rm det^{\prime}}$ and then heat kernel
techniques to calculate the zero values of various resulting $\zeta$-functions
over $(S^{4},g_{R})$ we obtain from its definition above that the Gaussian
integral looks like

$\int\limits_{{\rm ker}\>{\rm d}^{*}}{\rm e}^{-\left(a\>,\>\frac{{\rm
Im}\tau}{8\pi N_{*}^{2}}\Delta_{1}a\right)_{L^{2}(S^{4})}}\>{\rm
D}a=\left(\frac{{\rm Im}\tau}{8\pi^{2}N_{*}^{2}}\right)^{-\frac{11}{20}}{\rm
e}^{\frac{3}{2}\zeta^{\prime}_{\Delta_{1}}(0)-3\zeta^{\prime}_{\Delta_{0}}(0)}$

over $(S^{4},g_{R})$. Likewise,

$\displaystyle\int\limits_{C_{\delta}\cap{\rm ker}\>{\rm
d}^{*}}\\!\\!\\!\\!\\!\\!\\!\\!\\!{\rm e}^{-\left(a\>,\>\sqrt{\frac{{\rm
Im}\tau}{4\pi}}c\>\Delta_{1}a\right)^{2}_{L^{2}(S^{4})}-\left(a\>,\>\frac{{\rm
Im}\tau}{4\pi}\Delta_{1}a\right)_{L^{2}(S^{4})}}{\rm D}a$ $\displaystyle=$
$\displaystyle\frac{K(\delta)^{\frac{11}{20}}}{2\sqrt{\pi}}\>{\rm
e}^{\frac{3}{2}\zeta^{\prime}_{\Delta_{1}}(0)-3\zeta^{\prime}_{\Delta_{0}}(0)}\times$
$\displaystyle\int\limits_{\mathbb{R}}\left(\frac{{\rm
Im}\tau}{4\pi}-\>\sqrt{\frac{-{\rm
Im}\tau}{4\pi}}ct\right)^{-\frac{11}{20}}\\!\\!\\!\\!\\!{\rm
e}^{-\frac{t^{2}}{4}}\>{\rm d}t$

is the shape of the truncated quartic integral over $(S^{4},g_{R})$.

######  Remark.

Before embarking upon the proof we note that in our opinion the particular
value $\frac{11}{20}>0$ of the exponents in these integral expressions is not
important because it is a just the consequence of one of the possible (namely
$\zeta$-function combined with heat kernel) regularization procedures over one
of the possible (namely $(S^{4},g_{R})$ i.e. the one-point conformal)
compactifications of $({\mathbb{R}}^{4},\eta)$. Only its sign, namely that it
is positive, bears relevance. Indeed, this exponent does not have to always
assume a positive value because of some a priori reason. For example in the
case of the flat torus $T^{4}$ the corresponding exponent turns out to be
$-\frac{9}{2}<0$ leading to a completely different situation; e.g. the
Monotonicity principles break down due to this negative scaling. These
oddities are related with lacking a good measure in infinite dimensions, see
the Appendix.

Proof. Since the spectrum of the Laplacian over a compact Riemannian manifold
$(M,g)$ is non-negative real and discrete, one sets

$\zeta_{\Delta_{k}}(s):=\sum\limits_{\lambda\in{\rm
Spec}\>\Delta_{k}\setminus\\{0\\}}\lambda^{-s},\>\>\>\>\>\mbox{with
$s\in{\mathbb{C}}$ and ${\rm Re}\>s>0$ sufficiently large}$

and observes that this function can be meromorphically continued over the
whole complex plane (cf. e.g. [13, Theroem 5.2]) having no pole at
$s=0\in{\mathbb{C}}$. A formal calculation then convinces us that the
regularized rank and the determinant of the Laplacian should be ${\rm
rk^{\prime}}\>\Delta_{k}:=\zeta_{\Delta_{k}}(0)$ and ${\rm
det^{\prime}}\Delta_{k}:={\rm e}^{-\zeta^{\prime}_{\Delta_{k}}(0)}$ yielding
${\rm rk^{\prime}}(c\Delta_{k})=\zeta_{\Delta_{k}}(0)$ and ${\rm
det^{\prime}}(c\Delta_{k})=c^{\zeta_{\Delta_{k}}(0)}{\rm
e}^{-\zeta^{\prime}_{\Delta_{k}}(0)}$.

Because of the Coulomb gauge condition we have to calculate restrictions of
these $\zeta$-functions over the round $4$-sphere $(S^{4},g_{R})$. Since
$H^{1}(S^{4})=\\{0\\}$ hence $\Delta_{1}$ has trivial kernel, the Hodge
decomposition theorem says that $L^{2}(S^{4};\wedge^{1}S^{4})\cong{\rm
im}\>{\rm d}_{0}\oplus{\rm im}\>{\rm d}^{*}_{2}$. Moreover ${\rm im}\>{\rm
d}_{0}\cap{\rm ker}\>{\rm d}^{*}_{1}=\\{0\\}$ and ${\rm im}\>{\rm
d}^{*}_{2}\subseteqq{\rm ker}\>{\rm d}^{*}_{1}$ hence

$L^{2}(S^{4};\wedge^{1}S^{4})\cong{\rm im}\>{\rm d}_{0}\oplus{\rm ker}\>{\rm
d}^{*}_{1}\>\>.$

Applying this decomposition we can write any element $a\in
L^{2}(S^{4};\wedge^{1}S^{4})$ uniquely as $a={\rm d}_{0}f+\alpha$ with $f\in
L^{2}_{1}(S^{4};\wedge^{0}S^{4})$ a function and $\alpha\in
L^{2}(S^{4};\wedge^{1}S^{4})$ satisfying ${\rm d}^{*}_{1}\alpha=0$. A simple
calculation ensures us that

$\left(a\>,\>\Delta_{1}a\right)_{L^{2}(S^{4})}=\left({\rm
d}_{0}f+\alpha\>,\>\Delta_{1}({\rm
d}_{0}f+\alpha)\right)_{L^{2}(S^{4})}=\left(f\>,\>\Delta^{2}_{0}f\right)_{L^{2}(S^{4})}+\left(\alpha\>,\>\Delta_{1}\alpha\right)_{L^{2}(S^{4})}$

where $\Delta^{2}_{0}$ is the square of the scalar Laplacian on
$(S^{4},g_{R})$. Taking into account these decompositions then we obtain that
${\rm Spec}\>\Delta_{1}={\rm Spec}\>\Delta^{2}_{0}\sqcup{\rm
Spec}\>\Delta_{1}|_{{\rm ker}\>{\rm d}^{*}_{1}}$. This decomposition together
with the proof of [13, Theorem 5.2] ensures us that
$\zeta_{\Delta_{1}}=\zeta_{\Delta^{2}_{0}}+\zeta_{\Delta_{1}|{\rm ker}\>{\rm
d}^{*}_{1}}$ consequently

$\zeta_{\Delta_{1}|{\rm ker}\>{\rm
d}^{*}}=\zeta_{\Delta_{1}}-\zeta_{\Delta^{2}_{0}}\>\>.$

Moreover $\zeta_{\Delta^{2}_{0}}(s)=\zeta_{\Delta_{0}}(2s)$ hence ${\rm
rk^{\prime}}(c\Delta^{2}_{0})=\zeta_{\Delta_{0}}(0)$ and
$\det^{\prime}(c\Delta^{2}_{0})=c^{\zeta_{\Delta_{0}}(0)}{\rm
e}^{-2\zeta^{\prime}_{\Delta_{0}}(0)}$. Therefore in the case of the first
integral of Definition 3.1 putting $c:=\frac{{\rm Im}\tau}{8\pi N_{*}^{2}}$ we
find

$\left\\{\begin{array}[]{ll}{\rm rk^{\prime}}\left(\frac{{\rm Im}\tau}{8\pi
N_{*}^{2}}\Delta_{1}|_{{\rm ker}\>{\rm
d}^{*}}\right)&=\zeta_{\Delta_{1}}(0)-\zeta_{\Delta_{0}}(0)\\\ {\rm
det^{\prime}}\left(\frac{{\rm Im}\tau}{8\pi N_{*}^{2}}\Delta_{1}|_{{\rm
ker}\>{\rm d}^{*}}\right))&=\left(\frac{{\rm Im}\tau}{8\pi
N_{*}^{2}}\right)^{\zeta_{\Delta_{1}}(0)-\zeta_{\Delta_{0}}(0)}{\rm
e}^{-\zeta^{\prime}_{\Delta_{1}}(0)+2\zeta^{\prime}_{\Delta_{0}}(0)}\>\>.\end{array}\right.$

We can easily calculate at least $\zeta_{\Delta_{1}}(0)-\zeta_{\Delta_{0}}(0)$
explicitly applying standard heat kernel techniques. Over a compact
$4$-manifold $(M,g)$ without boundary it is well-known [13, Theorem 5.2] that

$\zeta_{\Delta_{k}}(0)=-\dim_{\mathbb{R}}{\rm
ker}\Delta_{k}+\frac{1}{16\pi^{2}}\int\limits_{M}{\rm tr}(u^{4}_{k}){\rm
d}V_{g}$

where the sections $u^{p}_{k}\in C^{\infty}(M;{\rm End}(\wedge^{k}M))$ with
$p=0,1,\dots$ appear [13, Chapter 3] in the coefficients of the short time
asymptotic expansion of the heat kernel for the $k$-Laplacian

$\sum\limits_{\lambda\in{\rm Spec}\Delta_{k}}{\rm e}^{-\lambda
t}\>\sim\>\frac{1}{(4\pi
t)^{2}}\sum\limits_{p=0}^{+\infty}\left(\>\>\int\limits_{M}{\rm
tr}(u^{p}_{k}){\rm d}V_{g}\right)t^{\frac{p}{2}}\>\>\>\>\>\mbox{as
$t\rightarrow 0$}\>\>.$

These functions are expressible with the curvature of $(M,g)$ and one can
demonstrate [6, p. 340] that

$\left\\{\begin{array}[]{ll}u^{4}_{0}&=\frac{1}{360}\left(2|{\rm
Riem}|_{g}^{2}-2|{\rm Ric}|_{g}^{2}+5\>{\rm Scal}^{2}\right)\\\ &\\\ {\rm
tr}(u^{4}_{1})&=\frac{1}{360}\left(-22|{\rm Riem}|_{g}^{2}+172|{\rm
Ric}|_{g}^{2}-40\>{\rm Scal}^{2}\right)\end{array}\right.$

yielding together with $\dim_{\mathbb{R}}{\rm ker}\>\Delta_{0}=1$ and
$\dim_{\mathbb{R}}{\rm ker}\>\Delta_{1}=0$ over $(S^{4},g_{R})$ that

$\zeta_{\Delta_{1}}(0)-\zeta_{\Delta_{0}}(0)=1-\frac{1}{\pi^{2}}\int\limits_{S^{4}}\left(\frac{1}{120}|{\rm
Riem}|_{g}^{2}-\frac{87}{2880}|{\rm Ric}|_{g}^{2}+\frac{1}{128}{\rm
Scal}^{2}\right){\rm d}V_{R}\>\>.$

In addition we recall over $(S^{4},g_{R})$ the classical expressions

$\left\\{\begin{array}[]{ll}|{\rm Riem}|_{g_{R}}^{2}&=2|{\rm
Ric}|_{g_{R}}^{2}-\frac{1}{3}\>{\rm Scal}^{2}\\\ |{\rm
Ric}|_{g_{R}}^{2}&=\frac{1}{4}{\rm Scal}^{2}\\\ {\rm
Scal}&=\frac{12}{R^{2}}\end{array}\right.$

and plug them into the integral and also perform $\int_{S^{4}}{\rm
d}V_{R}=\frac{8\pi^{2}}{3}R^{4}$. We come up with

$\frac{3}{2}\left(\zeta_{\Delta_{1}}(0)-\zeta_{\Delta_{0}}(0)\right)=\frac{3}{2}\left(1-\frac{1}{\pi^{2}}\left(\frac{1}{120}\left(\frac{1}{2}-\frac{1}{3}\right)-\frac{87}{2880}\cdot\frac{1}{4}+\frac{1}{128}\right)\frac{144}{R^{4}}\cdot\frac{8\pi^{2}}{3}R^{4}\right)=\frac{11}{20}$

and find in particular that
$\frac{3}{2}\left(\zeta_{\Delta_{1}}(0)-\zeta_{\Delta_{0}}(0)\right)$ is
independent of $R$ offering a sort of justification for using the conformal
compactification $(S^{4},g_{R})$ in place of the original space
$({\mathbb{R}}^{4},\eta)$. Inserting all of these formulata into the right
hand side of the first integral of Definition 3.1 we obtain the first
expression of the lemma.444The Faddeev–Popov determinant is therefore formally
equal to ${\rm e}^{3\zeta^{\prime}_{\Delta_{0}}(0)}$ hence is indeed constant
in this picture.

Repeating the same with the truncated quartic integral, the corresponding
result also follows. The only remaining thing is to specify the contour
$\Gamma$ in the second integral. The branching points of the algebraic
function $t\mapsto(c_{2}-\sqrt{-1}\>c_{1}t)^{-\frac{11}{20}}$ are its zero
$z=-\sqrt{-1}\frac{c_{2}}{c_{1}}=-\sqrt{-1}\sqrt{\frac{4\pi}{{\rm
Im}\tau}}\>c$ along the imaginary axis and $\infty$ since $\frac{11}{20}$ is
not an integer and these are the only ones. Consequently a good branch cutting
$X$ of ${\mathbb{C}}\cup\\{\infty\\}$ arises by cutting it up along a segment
which connects $z$ and $\infty$. Since ${\rm Im}\>z<0$ we can define $\Gamma$
to be simply the real axis ${\mathbb{R}}\subset{\mathbb{C}}$. $\Diamond$

We are ready to formally integrate our key inequality (15). Consider the small
ball about the flat connection
${\mathscr{A}}_{\varepsilon,N_{*}}(\nabla^{0})\cap{\rm ker}\>{\rm d}^{*}$ i.e.
the vicinity of the vacuum in Coulomb gauge as in (9). Recall that $N$ is
replaced with $N_{*}$ in its radius (hence it depends on $N_{*}$). Take the
Laplacian $\Delta_{1}$ and the corresponding $C_{\delta}\subset
L^{2}_{1}(S^{4};\wedge^{1}S^{4})$ introduced as its finite dimensional
analogue (2). If $0<\lambda_{{\rm min}}<+\infty$ is the smallest eigenvalue of
$\Delta_{1}$ then picking any $0<\delta<\sqrt{\lambda_{{\rm
min}}}\>\min(\eta,N_{*}\varepsilon)$ we know by (9) that
${\mathscr{A}}_{\varepsilon,N_{*}}(\nabla^{0})\supset C_{\delta}$ yielding two
inclusions ${\rm ker}\>{\rm
d}^{*}\supset{\mathscr{A}}_{\varepsilon,N_{*}}(\nabla^{0})\cap{\rm ker}\>{\rm
d}^{*}\supset C_{\delta}\cap{\rm ker}\>{\rm d}^{*}$ for these subsets in
$L^{2}_{1}(S^{4};\wedge^{1}S^{4})$. Now let us integrate the left term of (15)
over ${\rm ker}\>{\rm d}^{*}$, the middle term of (15) over
${\mathscr{A}}_{\varepsilon,N_{*}}(\nabla^{0})\cap{\rm ker}\>{\rm d}^{*}$ and
finally the right term of (15) over $C_{\delta}\cap{\rm ker}\>{\rm d}^{*}$.
Referring at this step to our Monotonicity principles this procedure obeys the
ordering in (15) that is,

$\begin{array}[]{ll}\int\limits_{{\rm ker}\>{\rm d}^{*}}\\!\\!\\!\\!\\!\\!{\rm
e}^{-\left(a,\frac{{\rm Im}\tau}{8\pi
N_{*}^{2}}\Delta_{1}a\right)_{\\!\\!L^{2}(S^{4})}}\\!{\rm
D}a\geqq\\!\\!\int\limits_{{\mathscr{A}}_{\varepsilon,N_{*}}(\nabla^{0})\cap{\rm
ker}\>{\rm
d}^{*}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!{\rm
e}^{-\frac{{\rm Im}\tau}{8\pi}\|{\rm d}a+a\wedge
a\|^{2}_{L^{2}(S^{4})}}\\!{\rm D}a\geqq\int\limits_{C_{\delta}\cap{\rm
ker}\>{\rm d}^{*}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!{\rm
e}^{-\left(a,\sqrt{\frac{{\rm
Im}\tau}{4\pi}}c\>\Delta_{1}a\right)^{2}_{\\!\\!L^{2}(S^{4})}\\!\\!\\!\\!\\!\\!\\!\\!-\left(a,\>\frac{{\rm
Im}\tau}{4\pi}\Delta_{1}a\right)_{L^{2}(S^{4})}}{\rm D}a\end{array}$

continues to hold. By Lemma 3.1 and (10) we eventually arrive at the two-sided
estimate

$\begin{array}[]{ll}\left(\frac{{\rm
Im}\tau}{8\pi^{2}N_{*}^{2}}\right)^{-\frac{11}{20}}{\rm
e}^{\frac{3}{2}\zeta^{\prime}_{\Delta_{1}}(0)-3\zeta^{\prime}_{\Delta_{0}}(0)}\geqq
Z_{\varepsilon}({\mathbb{R}}^{4},\tau)\geqq\frac{K(\delta)^{\frac{11}{20}}}{2\sqrt{\pi}}{\rm
e}^{\frac{3}{2}\zeta^{\prime}_{\Delta_{1}}(0)-3\zeta^{\prime}_{\Delta_{0}}(0)}\int\limits_{-\infty}^{+\infty}\left(\frac{{\rm
Im}\tau}{4\pi}-\>\sqrt{\frac{-{\rm
Im}\tau}{4\pi}}ct\right)^{-\frac{11}{20}}\\!\\!\\!\\!\\!{\rm
e}^{-\frac{t^{2}}{4}}\>{\rm d}t\>.\end{array}$ (16)

######  Lemma 3.2.

For a fixed sufficiently small $0<\varepsilon<1$ of the vicinity parameter as
before and a fixed sufficiently large $0<{\rm Im}\tau<+\infty$ of the complex
coupling constant (6) one can choose the compactification radius $0<R<+\infty$
such that the left and right hand sides of (16) get equal yielding

$Z_{\varepsilon}({\mathbb{R}}^{4},\tau)=\left(\frac{{\rm
Im}\tau}{8\pi^{2}N_{*}^{2}}\right)^{-\frac{11}{20}}{\rm
e}^{\frac{3}{2}\zeta^{\prime}_{\Delta_{1}}(0)-3\zeta^{\prime}_{\Delta_{0}}(0)}\>\>.$

Moreover the truncated partition function
$Z_{\varepsilon}({\mathbb{R}}^{4},\tau)$ depends on $R$ only through its
determinant term ${\rm
e}^{\frac{3}{2}\zeta^{\prime}_{\Delta_{1}}(0)-3\zeta^{\prime}_{\Delta_{0}}(0)}$.
More precisely if $(S^{4},g_{R_{i}})$ with $i=1,2$ are two conformal one-point
compactifications of $({\mathbb{R}}^{4},\eta)$ then the corresponding
partition functions are related by
$Z^{2}_{\varepsilon}({\mathbb{R}}^{4},\tau)=\left(\frac{R_{2}}{R_{1}}\right)^{\frac{11}{10}}Z^{1}_{\varepsilon}({\mathbb{R}}^{4},\tau)$.

Proof. Since the determinant expressions on both sides of (16) are the same it
easily follows by multiplying the right hand side of (16) with
$\left(\frac{{\rm Im}\tau}{8\pi^{2}N_{*}^{2}}\right)^{\frac{11}{20}}$ that for
given values of $\varepsilon$ and $\tau$ the left and the right hand sides of
(16) get equal if we can fine-tune the remaining technical constants
$\delta,R$ so that it makes the expression

$\left(\frac{K(\delta)}{2\pi
N_{*}^{2}}\right)^{\frac{11}{20}}\frac{1}{2\sqrt{\pi}}\int\limits_{-\infty}^{+\infty}\frac{{\rm
e}^{-\frac{t^{2}}{4}}}{\left(1-\sqrt{-1}c\sqrt{\frac{4\pi}{{\rm
Im}\tau}}\>t\right)^{\frac{11}{20}}}\>{\rm d}t$ (17)

equal to $1$. Asymptotic expansion shows that as ${\rm Im}\tau$ ranges between
$0$ and $+\infty$ the integral from

$\lim\limits_{{\rm Im}\tau\rightarrow
0}\>\>\frac{1}{2\sqrt{\pi}}\int\limits_{-\infty}^{+\infty}\frac{{\rm
e}^{-\frac{t^{2}}{4}}}{\left(1-\sqrt{-1}c\sqrt{\frac{4\pi}{{\rm
Im}\tau}}\>t\right)^{\frac{11}{20}}}\>{\rm d}t=0$

monotonly increases to

$\lim\limits_{{\rm
Im}\tau\rightarrow+\infty}\>\>\frac{1}{2\sqrt{\pi}}\int\limits_{-\infty}^{+\infty}\frac{{\rm
e}^{-\frac{t^{2}}{4}}}{\left(1-\sqrt{-1}c\sqrt{\frac{4\pi}{{\rm
Im}\tau}}\>t\right)^{\frac{11}{20}}}\>{\rm d}t=1\>\>.$

Consequently this integral is a positive number less than $1$ such that by
increasing the value of ${\rm Im}\tau$ it approaches $1$ from below. Hence our
demand can be reformulated as $2\pi N_{*}^{2}<K(\delta)$ with some $\delta$.
Suppose first that $N_{*}<\frac{1}{\sqrt{2}}$ implying $2\pi N_{*}^{2}<\pi$.
Then, knowing that $K(\delta)\rightarrow\pi$ as $\delta\rightarrow+\infty$
there exists a sufficiently large $\delta_{0}$ such that $2\pi
N_{*}^{2}<K(\delta_{0})<\pi$. Moreover, by shrinking $R$, i.e. conformally
rescaling $(S^{4},g_{R})$ with a constant if necessary, we can scale up
$\lambda_{{\rm min}}$ to be arbitrary large without affecting the other
conformally invariant parameters $\varepsilon,N_{*},c$ of the theory.
Therefore we can make $0<\delta_{0}<\sqrt{\lambda_{\rm
min}}\>\min(\eta,N_{*}\varepsilon)$ as large as we please without breaking the
inclusion
$C_{\delta_{0}}\subset{\mathscr{A}}_{\varepsilon,N_{*}}(\nabla^{0})$. Having
found a suitable large $\delta_{0}$ (over a suitable small
$(S^{4},g_{R_{0}})$) we can now take a $\tau_{0}\in{\mathbb{C}}^{+}$ with
sufficiently large imaginary part such that with the choice
$(\varepsilon,N_{*},\tau_{0},\delta_{0},R_{0})$ the expression (17) is equal
to $1$ as required.

Replacing the inequalities (13) and (14) providing lower and upper estimates
for the action with

$\frac{1}{N_{*}^{2}}\|{\rm d}a\|^{2}_{L^{2}(S^{4})}\leqq\|{\rm d}a+a\wedge
a\|^{2}_{L^{2}(S^{4})}\leqq(1+p)\|{\rm
d}a\|^{2}_{L^{2}(S^{4})}+\left(1+\frac{1}{p}\right)c^{2}\|{\rm
d}a\|^{4}_{L^{2}(S^{4})}$

where $p>0$ is arbitrary and then repeating the whole analysis so far we can
replace the condition $N_{*}<\frac{1}{\sqrt{2}}$ with
$N_{*}<\frac{1}{\sqrt{1+p}}$ for all $p>0$ i.e. it is sufficient to demand
only $N_{*}<1$ which is satisfied with our choice (12) for it.

Regarding the $R$ dependence observe first that the constant $c$ in (12) is
independent of $R$ since it says $\|a\|_{L^{4}(S^{4})}\leqq\sqrt{c}\|{\rm
d}a\|_{L^{2}(S^{4})}$ and both norms are conformally invariant. Secondly if
$(S^{4},g_{R_{i}})$ with $i=1,2$ are two conformal one-point compactifications
of $({\mathbb{R}}^{4},\eta)$ then obviously
$g_{R_{2}}=\left(\frac{R_{2}}{R_{1}}\right)^{2}g_{R_{1}}$ which can be
regarded as a homothety applied on $g_{R_{1}}$. Therefore the eigenvalues of
$\Delta_{1}$ under this re-sizeing simply change as
$\lambda_{k}\mapsto\left(\frac{R_{1}}{R_{2}}\right)^{2}\lambda_{k}$ i.e.
coincide with that of the scaled Laplacian
$\left(\frac{R_{1}}{R_{2}}\right)^{2}\Delta_{1}$ hence
$\zeta_{\Delta_{1}}\mapsto\zeta_{\left(\frac{R_{1}}{R_{2}}\right)^{2}\Delta_{1}}$.
Consequently

${\rm
e}^{\frac{3}{2}\zeta^{\prime}_{\Delta_{1}}(0)-3\zeta^{\prime}_{\Delta_{0}}(0)}\longmapsto\left(\frac{R_{1}}{R_{2}}\right)^{2\left(-\frac{3}{2}\left(\zeta_{\Delta_{1}}(0)-\zeta_{\Delta_{0}}(0)\right)\right)}{\rm
e}^{\frac{3}{2}\zeta^{\prime}_{\Delta_{1}}(0)-3\zeta^{\prime}_{\Delta_{0}}(0)}$

but we already know that
$\frac{3}{2}\left(\zeta_{\Delta_{1}}(0)-\zeta_{\Delta_{0}}(0)\right)=\frac{11}{20}$
hence the asserted scaling of $Z_{\varepsilon}({\mathbb{R}}^{4},\tau)$
follows. $\Diamond$

Putting together the contents of Lemmata 3.1 and 3.2 we obtain Theorem 1.1.
$\Diamond$

##  4 Appendix: There is no good measure in infinite dimensions

For completeness we recall the following simple but important general fact
about measures in infinite dimensions. Perhaps this no-go result demonstrates
in the sharpest way the existence of a “demarcation line” between finite and
infinite dimensional integration. We also refer to the excellent survey book
[7] to gain a broader picture.

Let $(X,\mu)$ be any measure space. As a very basic demand in measure theory
the measure $\mu$ is always assumed to be $\sigma$-additive i.e.
$\mu(\sqcup_{i}A_{i})=\sum_{i}\mu(A_{i})$ to hold for all countable collection
of pairwise disjoint measurable subsets $A_{1},A_{2},.\>.\>.\subset X$. If $X$
admits further structures, further natural assumptions can be imposed on a
measure. If $X$ can be given the structure of a Banach space for instance,
then mimicing the properties of the Lebesgue measure in finite dimensions, one
can further demand $\mu$ to be (i) positive i.e. $0\leqq\mu(U)\leqq+\infty$
for every open subset $\emptyset\subseteqq U\subseteqq X$; (ii) locally finite
i.e. every point $x\in X$ has an open neighbourhood $N_{x}\subseteqq X$ such
that $-\infty<\mu(N_{x})<+\infty$; (iii) and finally translation invariant
that is for every measurable subset $\emptyset\subseteqq A\subseteqq X$ and
every vector $x\in X$ the translated set $x+A$ is measurable and
$\mu(x+A)=\mu(A)$ holds.

However, as it is well-known, these natural demands conflict each other in
infinite dimensions:

######  Theorem 4.1.

(cf. e.g. [5, Theorem 4, p. 359], or [7, Theorem 3.1.5]) Let
$(X,\|\>\cdot\>\|)$ be an infinite dimensional, separable Banach space. Then
the only locally finite and translation invariant Borel measure $\mu$ on $X$
is the trivial measure, with $\mu(A)=0$ for every measurable subset $A$.
Equivalently, every translation invariant measure that is not identically zero
assigns infinite measure to all open subsets of $X$.

Proof. Take a locally finite, translation invariant measure $\mu$ on an
infinite dimensional, separable Banach space $(X,\|\>\cdot\>\|)$. Using local
finiteness, suppose that, for some $\varepsilon>0$, the open ball
$B_{\varepsilon}(0)\subset X$ of radius $\varepsilon$ and centered at the
origin, has a finite $\mu$-measure. Since $X$ is infinite dimensional, there
is a countable infinite sequence of pairwise disjoint open balls
$B_{\frac{\varepsilon}{4}}(x_{i})$ of radius for instance
$\frac{\varepsilon}{4}$ and centers $x_{i}\in X$, with all the smaller balls
$B_{\frac{\varepsilon}{4}}(x_{i})$ with $i=1,2,\dots$ contained within the
larger ball $B_{\varepsilon}(0)$. By translation invariance, all of the
smaller balls have the same measure; since by $\sigma$-additivity the absolute
value of the sum of these measures is estimated from above by
$\mu(B_{\varepsilon}(0))<+\infty$ hence is finite, the smaller balls must all
have $\mu$-measure zero. Now, since $X$ is separable, it can be covered by a
countable collection of balls of radius $\frac{\varepsilon}{4}$; since each
such ball has $\mu$-measure zero, by $\sigma$-additivity again so must the
whole space $X$. Therefore $\mu$ is the trivial measure. $\Diamond$

This means that our ad hoc “measure” ${\rm D}a$ used for integration in a
Hilbert space throughout Sections 3 and 4 lacks at least one of the standard
properties listed above. In the long-lasting debate of what is then going to
be dropped, we have occupied a radical position: In our opinion the most
fundamental requirement, namely $\sigma$-additivity is to be given up in
infinite dimensions hence in quantum field theory.

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