# Four dimensional Riemannian general relativity as a spontaneous symmetry
breaking

Gábor Etesi
Department of Algebra and Geometry, Institute of Mathematics,
Budapest University of Technology and Economics,
Műegyetem rkp. 3., H-1111 Budapest, Hungary 111E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In this paper $4$ dimensional Riemannian (or Euclidean) vacuum general
relativity is recovered from a phase transition by spontaneous symmetry
breaking within a quantum field theory (all in the sense of the algebraic
approach [3, 14]) provided by the unique ${\rm II}_{1}$ hyperfinite factor von
Neumann algebra. The formal temperature of this phase transition is
$T=\frac{1}{2}T_{\rm Planck}\approx 7.06\times 10^{31}{\rm K}$.

More precisely, first we recall [12, 13] that making use of its smooth
structure only, out of a connected oriented smooth $4$-manifold $M$ a von
Neumann algebra ${\mathfrak{R}}$ can be constructed which is geometric in the
sense that it is generated by local operators and as a special four
dimensional phenomenon it contains all algebraic (i.e., formal or coming from
a metric) curvature tensors of the underlying $4$-manifold. The von Neumann
algebra ${\mathfrak{R}}$ itself is a hyperfinite ${\rm II}_{1}$ factor hence
is unique up to abstract isomorphisms of von Neumann algebras. Interpreting
${\mathfrak{R}}$ as the operator algebra of a (bosonic) quantum field theory,
on the one hand, the unique tracial state on ${\mathfrak{R}}$ gives rise to a
phase whose only state is the corresponding infinitely high temperature KMS
state of the theory with trivial Tomita–Takesaki or modular dynamics. On the
other hand, $M$ induces further states on ${\mathfrak{R}}$ having the property
that they are of finite temperature, are stationary and stable under a new
non-trivial dynamics on ${\mathfrak{R}}$ what we call the Hodge dynamics.
Hence it is reasonable to call these states as finite temperature equilibrium
states too (however they are not KMS states in the sense of [7, 3, 14]).
Furthermore, $M$ embeds into ${\mathfrak{R}}$ which induces a Riemannian
$4$-manifold $(M,g)$ and it is observed that $(M,g)$ satisfies the vacuum
Einstein equation if and only if both $M\subset{\mathfrak{R}}$ and its
Riemannian curvature tensor $R_{g}\in{\mathfrak{R}}$ are invariant under the
corresponding Hodge dynamics. Thus the other phase of the proposed quantum
field theory is built up from the huge moduli space of states provided by all
smooth $4$-manifold structures and the transition between these phases can be
considered as a spontaneous symmetry breaking by cooling (perhaps about the
Big Bang times).

AMS Classification: Primary: 46L10, 83C45, Secondary: 57R55

Keywords: Smooth $4$-manifold; Hyperfinite ${\rm II}_{1}$ factor; Symmetry
breaking; Quantum gravity

##  1 Introduction and summary

A longstanding problem of contemporary theoretical physics is how to unify the
obviously successful and mathematically consistent theory of general
relativity with the obviously successful but yet mathematically problematic
relativistic quantum field theory. It has been generally believed that these
two fundamental pillars of modern theoretical physics are in a clash not only
because of the different mathematical tools they use but are in tension even
at a deep conceptional level: classical notions of general relativity such as
the space-time event, the light cone or the event horizon of a black hole are
too “sharp” objects from a quantum theoretic viewpoint whereas relativistic
quantum field theory is not background independent from the aspect of general
relativity.

The demand by general relativity summarized as the principle of general
covariance is perhaps one of the two main obstacles why general relativity has
remained outside of the mainstream classical and quantum field theoretic
expansion in the $20^{\rm th}$ century. Indeed, an implementation of this
inherent principle of general relativity forces that a robust group, namely
the full diffeomorphism group of the underlying space-time manifold must
belong to the symmetry group of a field theory compatible with general
relativity. However an unexpected consequence of the vast diffeomorphism
symmetry is that it even allows one to transform time itself away from the
theory (known as the “problem of time” in general relativity, see e.g. [25,
Appendix E] for a technical presentation as well as e.g. [5, Chapter 2] and
[16, Subsection 2.1] for a broader philosophical survey on this problem)
making it problematic to apply standard canonical quantization methods—based
on Hamiltonian formulation hence on an essential explicit reference to an
”auxiliary time”—in case of general relativity. The other reason is the as
well in-built core idea, the equivalence principle which renders general
relativity a strongly self-interacting classical field theory in the sense
that precisely in four dimensions the “free” and the “interaction-with-itself”
modes of the gravitational field have energetically the same magnitudes,
obfuscating perturbative considerations. In fact the equivalence principle
says that there is no way to make a physical distinction between these two
modes of gravity. Heisenberg and Pauli were still optimistic concerning
canonical and perturbative quantization of gravity with respect to a fixed
time or more generally a reference or ambient space-time in their 1929 paper
[17]; however these initial hopes quickly evaporated already in the 1930’s by
recognizing the essential impossibility of quantizing general relativity via
canonical quantization and exhibit it as a perturbatively renormalizable
quantum field theory in a coherent way. This was clearly observed by Bronstein
[4] first; as he wrote in his 1936 paper: “[…] the elimination of the logical
inconsistencies [requires] rejection of our ordinary concepts of space and
time, modifying them by some much deeper and nonevident concepts.” (also cited
by Smolin [23, p. 85]).

Roughly the thinking about gravity has split into two main branches since the
1950-60’s [16]. The first older and more accepted direction postulates that
gravity should be quantized akin to other fundamental forces but with more
advanced methods including (super)string theoretic [15], Feynman integral,
loop quantum gravity or some further techniques—or at least one should
construct it as a low energy effective field theory of an unknown high energy
theory; the other newer and yet less-accepted attitude declares that gravity
is an emergent macroscopic phenomenon in the sense that it always involves a
huge amount of physical degrees-of-freedom (beyond the obvious astronomical
evidences, also supported by various theoretical discoveries during the
1970-80’s such as Hawking’s area theorem, black hole radiation, all resembling
thermodynamics) hence is not subject to quantization at all. Nevertheless, as
a matter of fact in the 2020’s, we have to admit that an overall accepted
quantum theory of gravity does not exist yet and even general relativity as a
classical field theory persists to keep its conceptionally isolated position
within current theoretical physics [16]. Perhaps it is worth mentioning here
that general relativity receives further challenges from low dimensional
differential topology too by recent discoveries which were unforeseenable
earlier, cf. e.g. [11, Section 1] for a brief summary.

Therefore, respecting its long resistance against quantization, we do not make
a new attempt here to quantize general relativity. Rather the purpose of this
paper is to occupy an in-between position in the battlefield and in part
motivated by [4, 8] to take a fresh look on precisely four dimensional and
Riemannian general relativity by identifying it as a spontaneously broken
classical phase of a proposed quantum field theory based upon the famous
unique hyperfinite factor von Neumann algebra of ${\rm II}_{1}$-type and its
rich representation theory (also cf. [6]). The remaining part of this
introduction is an informal trailer to this idea and we refer for the physical
details to Section 2 while for the mathematical subtleties to Section 3 below.

As it is known [1] the hyperfinite ${\rm II}_{1}$ factor von Neumann algebra
${\mathfrak{R}}$ is unique up to abstract isomorphisms but admits lot of
inequivalent representations on various Hilbert spaces. These representations
are classified by their so-called ${\mathfrak{R}}$-dimensions (or Murray–von
Neumann coupling constants) taking values in the non-negative real line
$[0,+\infty)$ and these are their appropriately defined dimensions as
(left-)modules over ${\mathfrak{R}}$. For instance the representation having
${\mathfrak{R}}$-dimension precisely $1\in[0,+\infty)$ is its standard
represention on itself by (left-)multiplications. This is the best-known
representation and within the framework of the GNS-construction it can be
obtained from the distinguished faithful state provided by the unique finite
trace on the hyperfinite ${\rm II}_{1}$ factor. However many other new
representations can be constructed using further non-faithful differential-
geometric states provided by connected oriented smooth $4$-manifolds as
observed in our earlier efforts [12, 13]. The ${\mathfrak{R}}$-dimensions of
these representations always belong to the interval $[0,1)\subset[0,+\infty)$
hence they are never equivalent to the standard representation. It was Haag’s
remarkable observation long time ago [14, Subsection II.1.1] that the
cardinality of the degrees-of-freedom in a quantum system can be characterized
by the existence of inequivalent representations of its algebra of
observables. Since ${\mathfrak{R}}$ admits lot of representations it is
reasonable to expect that the hyperfinite ${\rm II}_{1}$ factor has something
to do with quantum statistical mechanics of infinite systems.222As it is
known, the hyperfinite ${\rm II}_{1}$ factor was von Neumann’s favourite
operator algebra and he wanted to use it to bring quantum mechanics on an even
conceptionally acceptable basis as a statistical theory relative to an
unsorted or absolute statistical ensemble provided by this von Neumann algebra
(cf. e.g. [20]). Indeed, the common feature of the various states on
${\mathfrak{R}}$ to be introduced here (and giving rise to representations by
the GNS-construction) is that they represent thermal equlibrium states at
various temperatures. The temperature can be assigned to them in a uniform way
by computing the periodicity $\hbar\beta$ where $\beta=\frac{1}{k_{B}T}$ is
the inverse temperature (note that $[\hbar\beta]=$time) of the imaginary-time
dynamics they are related with. The connection between periodicity in
imaginary time and the existence of a temperature in real time is well-known
cf. e.g. [18].

First let us consider the well-known case of a KMS state. Recall that the KMS
condition is as follows. Take a $C^{*}$-algebra and consider a dynamics on
this algebra given by a $1$-real-parameter continuous family of its
$*$-automorphisms $t\mapsto\alpha_{t}$. A state $\omega$ on this
$C^{*}$-algebra satisfies the KMS condition if and only if for every operator
$A$ the function $t\mapsto\omega(\alpha_{t}(A))$ can be extended to
$z\mapsto\omega(\alpha_{z}(A))$ which is complex analytic in the interiors and
continuous up to the boundaries of two strips above and below the real line of
width $0\leqq\hbar\beta\leqq+\infty$ i.e. for ${\rm Re}\>z\in{\mathbb{R}}$ and
$0\leqq\pm{\rm Im}\>z\leqq\hbar\beta$ such that for every $t\in{\mathbb{R}}$
and every $A,B$ from the $C^{*}$-algebra

$\omega(\alpha_{t}(A)B)=\omega(B\alpha_{t\pm\sqrt{-1}\hbar\beta}(A))$

holds. However this implies that the corresponding “pseudo-dynamics” in the
imaginary-time direction333Actually this is not a dynamics in the strict sense
because in this case automorphisms fail to be $*$-automorphisms hence for
instance
$(\alpha_{\sqrt{-1}\hbar\beta}(A))^{\divideontimes}=\alpha_{-\sqrt{-1}\hbar\beta}(A^{\divideontimes})$
only. In general observe that on a $C^{*}$-algebra among the two automorphisms
$t\mapsto\alpha_{t}$ and $t\mapsto\alpha_{\sqrt{-1}t}$ with $t\in{\mathbb{R}}$
at most one can be a $*$-automorphism hence represent a dynamics. Moreover we
can term the one which is periodic as “imaginary-time (pseudo-)dynamics” and
the other one as “real-time (pseudo-)dynamics”. is periodic with period
$\hbar\beta$. Indeed, the KMS state is faithful [14, Subsection V.1.4]
consequently $\omega(TT^{\divideontimes})=0$ implies $T=0$; putting
$T:=A-\alpha_{\sqrt{-1}\hbar\beta}(A)$ and using the KMS condition couple of
times at $t=0$ with straightforward substitutions we compute

$\displaystyle\omega\big{(}(A-\alpha_{\sqrt{-1}\hbar\beta}(A))(A-\alpha_{\sqrt{-1}\hbar\beta}(A))^{\divideontimes}\big{)}$
$\displaystyle=$
$\displaystyle\omega\big{(}(A-\alpha_{\sqrt{-1}\hbar\beta}(A))(A^{\divideontimes}-\alpha_{-\sqrt{-1}\hbar\beta}(A^{\divideontimes}))\big{)}$
$\displaystyle=$
$\displaystyle\omega(AA^{\divideontimes})-\omega\big{(}A^{\divideontimes}A)-\omega(A^{\divideontimes}A\big{)}+\omega\big{(}AA^{\divideontimes}\big{)}$

demonstrating that if $A$ is self-adjoint then the right hand side is zero
hence so is the other side consequently $A=\alpha_{\sqrt{-1}\hbar\beta}(A)$;
since every element in a $C^{*}$-algebra is a sum (a ${\mathbb{C}}$-linear
combination) of two self-adjoint operators we find that given a “real-time
dynamics” the corresponding KMS condition indeed implies the periodicity of
the “imaginary-time pseudo-dynamics”. On the one hand we can apply this
observation on the hyperfinite ${\rm II}_{1}$ factor ${\mathfrak{R}}$ in its
standard representation by picking the positive self-adjoint modular operator
$\Delta$ and taking the Tomita–Takesaki or modular “real-time dynamics”
$t\mapsto{\rm Ad}_{\Delta^{\sqrt{-1}t}}$ as well as the corresponding
“imaginary-time pseudo-dynamics” $t\mapsto{\rm Ad}_{\Delta^{t}}$ (Footnote 3).
However in this case $\Delta=1$ hence the former dynamics is trivial likewise
the latter one yielding $\hbar\beta=0$ consequently the temperature of the
corresponding KMS state $\tau$, which is nothing but the unique trace on
${\mathfrak{R}}$, is infinite. Hence this tracial state (and its corresponding
standard representation $\pi$ on a Hilbert space ${\mathscr{H}}$ whose
${\mathfrak{R}}$-dimension therefore satisfies
$\dim_{\mathfrak{R}}{\mathscr{H}}=1$) describes an infinite high temperature
thermal equilibrium state on ${\mathfrak{R}}$.

On the other hand we can consider another dynamics on ${\mathfrak{R}}$ which
is introduced by an oriented Riemannian $4$-manifold $(M,g)$ through its
Hodge-operator acting on $2$-forms over $M$. This operator turns out to
represent a unitary element $*\in{\mathfrak{R}}$ hence gives rise to a
dynamics $t\mapsto{\rm Ad}_{*^{t}}$ on ${\mathfrak{R}}$. This dynamics is
already non-trivial however the well-known property $*^{2}=1$ makes it
periodic; therefore here this “Hodge dynamics” should be termed as the
“imaginary-time dynamics” while the corresponding $t\mapsto{\rm
Ad}_{*^{\sqrt{-1}t}}$ as the “real-time pseudo-dynamics” (cf. Footnote 3).
Nevertheless a corresponding temperature can be computed from
$\hbar\beta=2t_{\rm Planck}$ yielding $T=\frac{1}{2}T_{\rm Planck}$. It turns
out that picking a surface $\Sigma\subset M$ and a $2$-form $\omega$ on $M$
there exists a state $F_{\Sigma,\omega}$ on ${\mathfrak{R}}$ induced by
$(M,g)$ which is stationary under the Hodge dynamics i.e.
$F_{\Sigma,\omega}(*^{t}A*^{-t})=F_{\Sigma,\omega}(A)$ for every
$t\in{\mathbb{R}}$ and $A\in{\mathfrak{R}}$ moreover it is stable against
small perturbations. Consequently it is reasonable to call this state as a
finite temparture thermal equilibrium state on ${\mathfrak{R}}$ too obtained
from a connected oriented smooth $4$-manifold $M$. However this state cannot
be KMS because the corresponding representation $\rho_{M}$ of ${\mathfrak{R}}$
on a Hilbert space ${\mathscr{I}}(M)^{\perp}$ obtained by the GNS construction
is not faithful, actually its ${\mathfrak{R}}$-dimension satisfies
$0\leqq\dim_{\mathfrak{R}}{\mathscr{I}}(M)^{\perp}<1$.

Our key observation in this paper is that if $(M,g)$ is an oriented Riemannian
$4$-manifold which satisfies the vacuum Einstein equation then $M$ together
with its Riemannian curvature tensor $R_{g}$ embeds into ${\mathfrak{R}}$ as a
portion of the fixed-point-subalgebra of the corresponding “Hodge dynamics”
generated by the Hodge operator of $(M,g)$. Therefore, speaking expressively,
“starting” with the infinitely high temperature standard representation of
${\mathfrak{R}}$ whose corresponding dynamics is trivial hence its fixed-
point-set is the whole ${\mathfrak{R}}$ we can “pass” to one of its finite
temperature representation whose dynamics is not trivial anymore and its
fixed-point-set “cuts out” the vacuum space-time $(M,g)$ from ${\mathfrak{R}}$
we began with; and regard this whole story as a phase transition by a
spontaneous symmetry breaking because of cooling in the early Big Bang times
(this scheme is formally displayed in (1) below). An advantage of this pattern
is that it avoids the problem of the Beginning for it dissolves the classical
dynamical phase of the Universe in a dynamically trivial abstract quantum
phase “just before” reaching the “Big Bang event”. We also have to admit that
the whole machinery here works in Riemannian (i.e. Euclidean) signature only.
But at least the space-time dimension, which must be precisely four here,
matches.

The paper is organized as follows. Section 2 contains a detailed presentation
of this phase transition approach while Section 3 contains all the (supposed
to be complete) mathematical constructions.

##  2 Recovering Riemannian general relativity

In this section by recalling some results from [12, 13] we will introduce a
framework based on the algebraic quantum field theory approach [3, 14] in
which the immense class of classical space-times of general relativity is
replaced with a single universal “quantum space-time” allowing us to lay down
the foundations of a manifestly four dimensional, covariant, non-perturbative
and genuinely quantum theory related with gravitation in some way. This
construction is natural, simple and self-contained. More precisely here not
one particular Riemannian $4$-manifold—physically regarded as a particular
classical space-time—but the unique hyperfinte ${\rm II}_{1}$ factor von
Neumann algebra—physically viewed as the universal quantum space-time—is
declared to be the primarily given object and $4$ dimensional Riemannian
general relativity is derived by a phase transition within this theory from an
infinitely high temperature pure quantum phase to a finite temperature
classical mixed phase carried out by a spontaneous symmetry breaking. Let us
see how it works!

A quantum theory and its quantum phase. To begin with, let ${\mathscr{H}}$ be
an abstractly given infinite dimensional complex separable Hilbert space and
${\mathfrak{R}}\subset{\mathfrak{B}}({\mathscr{H}})$ be a type ${\rm II}_{1}$
hyperfinite factor hence tracial von Neumann algebra acting on ${\mathscr{H}}$
by the standard representation $\pi$. Its unique trace is denoted by $\tau$
and satisfies $\tau(1)=1$. We call ${\mathfrak{R}}$ the algebra of
observables, its tangent space $T_{1}{\mathfrak{R}}\supset{\mathfrak{R}}$
consisting of the Fréchet derivatives of $1$-parameter families of observables
at the unit $1\in{\mathfrak{R}}$ the algebra of fields, while ${\mathscr{H}}$
the state space. The subgroup ${\rm U}({\mathscr{H}})\cap{\mathfrak{R}}$ of
the unitary group of ${\mathscr{H}}$ operating as inner automorphisms on
${\mathfrak{R}}$ is the gauge group. Note that the gauge group acts on both
${\mathfrak{R}}$ and ${\mathscr{H}}$ but in a different way. If the unique
standard representation
$\pi:{\mathfrak{R}}\rightarrow{\mathfrak{B}}({\mathscr{H}})$ is constructed
through the GNS construction then it is obtained from the faithful state
$\tau$ on ${\mathfrak{R}}$ hence the resulting ${\mathscr{H}}$ possesses a
cyclic and separating vector; thus $\pi$ renders ${\mathfrak{R}}$ a dynamical
system under its own Tomita–Takesaki or modular automorphisms [3, 7, 8, 14]
which dynamics is however trivial—or rephrasing, the corresponding imaginary-
time pseudo-dynamics is periodic with period $\hbar\beta=0$—because in
addition $\tau$ is tracial [3, Vol. I, p. 90]. Nevertheless $\tau$ is the
corresponding KMS state which is therefore of infinitely high temperature as
well as stationary and stable in the sense of [14, Sections V.1-3]. The
cardinality of the physical degrees-of-freedom in a quantum system can be
characterized by the existence of inequivalent representations of its algebra
of observables cf. e.g. [14, Subsection II.1.1]. It is well-known [1, Chapter
8] that beyond its standard representation ${\mathfrak{R}}$ admits lot of
further inequivalent representations too hence this operator algebra is
capable to describe an infinite degree-of-freedom physical system e.g. a
field. Thus we can say that by default the theory exhibited by
${\mathfrak{R}}$ together with its state $\tau$ and corresponding standard
representation $\pi$ on ${\mathscr{H}}$ describes some infinite degree-of-
freedom quantum statistical system in a pure phase whose only state is an
infinitely high temperature thermal equilibrium state. We call from now on
this infinitely high temperature (physically rather equal to $T_{\rm
Planck}\approx 1.42\times 10^{32}$K) equilibrium phase as the quantum phase of
the proposed quantum field theory.

However before proceeding further we ask ourselves what kind of quantum theory
is the one in which ${\mathfrak{R}}$ plays the role of the algebra of
(bounded) physical observables? The starting point to answer this question in
particular and for a more detailed analysis of this quantum theory in general
is the following result taken from [13] whose proof for convenience (as well
as to fix notation to be used throughout) is elaborated here in Section 3
again:

######  Theorem 2.1.

(i) Let $M$ be a connected oriented smooth $4$-manifold. Making use of its
smooth structure only, a von Neumann algebra ${\mathfrak{R}}$ can be
constructed which is geometric in the sense that it is generated by local
operators including all bounded complexified algebraic (i.e., formal or
stemming from a metric) curvature tensors of $M$ and ${\mathfrak{R}}$ itself
is a hyperfinite factor of type ${\rm II}_{1}$ hence is unique up to abstract
isomorphism of von Neumann algebras.

(ii) Furthermore $M$ admits an embedding $M\subset{\mathfrak{R}}$ via
projections and this embedding induces a Riemannian structure $(M,g)$ whose
Riemannian curvature, if bounded, satisfies $R_{g}\in{\mathfrak{R}}$. Two
connected oriented smooth $4$-manifolds $M,N$ with corresponding embeddings
have abstractly isomorphic von Neumann algebras however not canonically.
Nevertheless different abstract isomorphisms between these von Neumann
algebras induce orientation-preserving diffeomorphisms of $M$ and $N$
respectively i.e. leave their embeddings unchanged. Hence up to
diffeomorphisms every connected oriented smooth $4$-manifold admits an
embedding into a commonly given abstract von Neumann algebra ${\mathfrak{R}}$.

(iii) Finally taking $M$ its von Neumann algebra ${\mathfrak{R}}$ admits a
non-faithful representation on a certain complex separable Hilbert space such
that the unitary equivalence class of this representation is invariant under
orientation-preserving diffeomorphisms of $M$ (hence gives rise to a smooth
$4$-manifold invariant).

We see by item (i) of Theorem 2.1 that the operator algebra ${\mathfrak{R}}$,
when regarded to operate on a differentiable manifold, contains all bounded
algebraic curvature tensors along it whenever the real dimension of this
manifold is precisely four. Moreover ${\mathfrak{R}}$ is generated by local
operators more precisely by algebraic operators $R$ acting like multiplication
operators and by operators $L_{X}$ acting like first order differential
operators. Picking canonically conjugate pairs i.e. necessarily unbounded
self-adjoint operators $R,\sqrt{-1}L_{X}\in T_{1}{\mathfrak{R}}$ satisfying
$[R,\sqrt{-1}L_{X}]=\hbar$ shows that $T_{1}{\mathfrak{R}}$ contains a CCR
algebra hence revealing the bosonic character of ${\mathfrak{R}}$. Recall that
in classical general relativity local gravitational phenomena are caused by
the curvature of space-time. Hence interpreting ${\mathfrak{R}}$ physically as
an operator algebra of local observables the corresponding quantum theory is
expected to be a four dimensional quantum theory of pure gravity (hence a
bosonic theory). In this way we fulfill the Heisenberg dictum that a quantum
theory should completely and unambigously be formulated and interpreted in
terms of its local physical observables (and not the other way round). In
modern understanding by a physical theory one means a two-level description of
a branch of natural phenomena which are interrelated phenomenologically: the
theory possesses a syntax provided by its mathematical core structure and a
semantics which is the meaning i.e. interpretation of the bare mathematical
model in terms of physical concepts. In this context our quantum theory is not
merely a mathematical theory anymore but a physical theory. This is because
the bare mathematical structure ${\mathfrak{R}}$ (together with a
representation $\pi$ on ${\mathscr{H}}$) is dressed up i.e., interpreted by
assigning a physical (in fact, gravitational) meaning to the experiments
consistently performable by the aid of this structure (i.e., the usual quantum
measurements of operators $A\in{\mathfrak{R}}$ in pure states
$v\in{\mathscr{H}}$ or in more general ones, see below). In our opinion it is
of particular interest that the geometrical dimension—equal to four—is fixed
at the semantical level only and it matches the known phenomenological
dimension of space-time. This is in sharp contrast to e.g. string theory where
the geometrical dimension of the theory is fixed already at its syntactical
level i.e., by its mathematical structure (namely, demaninding the theory to
be free of conformal anomaly) and turns out to be much higher than the
phenomenological dimension of space-time. As a further clarification it is
stressed that the aim here is not to quantize general relativity in one or
another way but rather conversely: being already a successful theory at the
classical level, general relativity should be somehow derived from this
abstractly given quantum theory by taking an appropriate “classical limit” as
naturally as possible.

Observables as the universal space of all space-times. Item (ii) of Theorem
2.1 says that any connected oriented smooth $4$-manifold $M$ embeds into
${\mathfrak{R}}$. This embedding is given by (8) and is “physical” for it maps
a point of $x\in M$ i.e. a classical elementary event to a projection
$P_{x}\in{\mathfrak{R}}$ i.e. a quantum elementary event. Moreover, as
emphasized in Theorem 2.1 too, due to these inclusions and the uniqueness of
the hyperfinite ${\rm II}_{1}$ factor, ${\mathfrak{R}}$ can be considered as
the collection of all classical space-time manifolds and we can interpret the
appearance of the gauge group here as the manifestation of the diffeomorphism
gauge symmetry of classical general relativity in this quantum theory. Indeed,
orientation-preserving diffeomorphisms of $M$ interchange its points as well
as act on the corresponding operators (projections) in ${\mathfrak{R}}$ by
unitary inner automorphisms i.e. ${\rm Diff}^{+}(M)$ embeds into ${\rm
Aut}\>{\mathfrak{R}}$. Reformulating this in a more geometric language we can
say that classical space-times within ${\mathfrak{R}}$ appear as special
orbits of its gauge group. The operators in ${\mathfrak{R}}$ representing
geometric points are the simplest possible operators namely projections.
Consequently the full non-commutative algebra ${\mathfrak{R}}$ is not
exhausted by operators representing points of space-time alone; it certainly
contains much more operators—e.g. various projections which are not of
geometric origin—therefore this “universal quantum space-time” is more than a
bunch of all classical space-times. As a comparison, in algebraic quantum
field theory [14] one starts with a particular smooth $m$-manifold $M$ and
considers an assignment $U\mapsto{\mathfrak{R}}(U)$ describing local algebras
of observables along all open subsets $\emptyset\subseteqq U\subseteqq M$.
However in our case, quite conversely, space-times are secondary structures
only and all of them are injected into the unique observable algebra
${\mathfrak{R}}$ which is considered to be primary.

More examples of observables and fields. Let us take a closer look of the
elements of ${\mathfrak{R}}$ and of
$T_{1}{\mathfrak{R}}\supset{\mathfrak{R}}$. Referring to item (ii) of Theorem
2.1 again we see that the embedding of a connected oriented smooth
$4$-manifold $M$ into ${\mathfrak{R}}$ canonically improves it to a Riemannian
manifold $(M,g)$. The metric arises simply by pulling back the standard scalar
product on ${\mathfrak{R}}$ provided by $\tau$ (see Section 3 for details and
concerning the notation below). Remember that given a connected oriented
Riemannian $4$-manifold $(M,g)$ one can introduce the Hodge operator $*$ which
satisfies $*^{2}=1$ on $2$-forms hence induces a splitting

$\Omega^{2}_{c}(M)=\Omega^{+}_{c}(M)\oplus\Omega^{-}_{c}(M)$

into (anti)self-dual or $\pm 1$-eigenspaces. Thus the Hodge star as an
operator on $\Omega^{2}_{c}(M)$ with respect to this splitting has the form
$*=\left(\begin{smallmatrix}1&0\\\ 0&-1\end{smallmatrix}\right)$ hence is
symmetric; likewise the curvature tensor $R_{g}$ of $(M,g)$ looks like (5)
i.e., is also a pointwise symmetric operator. We summarize these facts by
writing that as pointwise algebraic operators $*,R_{g}\in
C^{\infty}(M;S^{2}\wedge^{2}T^{*}M)\subset
C^{\infty}(M;\operatorname{End}(\wedge^{2}T^{*}M))$. The complex-linear
extension of $*$ whose trace satisfies $\tau(*)=0$ i.e. is finite, (by
considerations in Section 3) always gives an element $*\in{\mathfrak{R}}$
which is self-adjoint. Likewise if $R_{g}$ is a bounded operator (in the sense
explained in Section 3) then its complex-linear extension gives rise to
$R_{g}\in{\mathfrak{R}}$ which is moreover self-adjoint i.e.,
$R_{g}^{\divideontimes}=R_{g}$ because $R_{g}$ is symmetric. Therefore its
trace is also real. Proceeding further $R_{g}$ satisfies the algebraic Bianchi
identity. It has the form $b(R_{g})=0$ where $b$ is a certain fiberwise
averaging operator on symmetric bundle morphisms satisfying $b^{2}=b$ and
$b^{*}=b$ with respect to the metric. This induces a $g$-orthogonal
decomposition $S^{2}\wedge^{2}T^{*}_{x}M\otimes{\mathbb{C}}={\rm
ker}\>b_{x}\oplus{\rm im}\>b_{x}$ at every point $x\in M$. A subtlety of four
dimensions is that ${\rm im}\>b_{x}\subset
S^{2}\wedge^{2}T_{x}^{*}M\otimes{\mathbb{C}}$ is spanned by the symmetric
operator $*\in S^{2}\wedge^{2}T_{x}^{*}M\otimes{\mathbb{C}}$ alone cf. [22].
Consequently $b(R_{g})=0$ is equivalent to saying that $g(R_{g},*)=0$ i.e.,
$R_{g},*\in S^{2}\wedge^{2}T_{x}^{*}M\otimes{\mathbb{C}}$ are such that
$R_{g}$ is $g$-orthogonal to $*$ at every $x\in M$. By exploiting that $*$ is
a pointwise symmetric operator we can use the pointwise equality
$g(R_{g},*)={\rm tr}(R_{g}*)$ and then (7) to compute the trace of the product
of $R_{g}$ and $*$ in ${\mathfrak{R}}$ as follows:

$\tau(R_{g}*)=\frac{1}{6}\int\limits_{M}{\rm tr}(R_{g}*)\mu_{g}=0$

which geometrically means that $(\hat{R}_{g},\hat{*})=0$ i.e., $\hat{R}_{g}$
is perpendicular to $\hat{*}$ as vectors within ${\mathscr{H}}$. Hence the
Bianchi identity.

The vacuum Einstein equation with cosmological constant
$\Lambda\in{\mathbb{R}}$ reads as ${\rm Ric}=\Lambda g$ therefore it is
equivalent to the vanishing of the traceless Ricci part ${\rm Ric}_{0}$ of
$R_{g}$ in (5). From these it readily follows, as noticed in [22], that
$(M,g)$ is Einstein i.e., satisfies the vacuum Einstein equation with
cosmological constant if and only if $*R_{g}=R_{g}*$ or equivalently
$*R_{g}*^{-1}=R_{g}$ within ${\mathfrak{R}}$. Moreover taking into account
that ${\rm Weyl}^{\pm}\in
C^{\infty}(M;S^{2}\wedge^{2}T^{*}M\otimes{\mathbb{C}})$ in (5) are traceless
their pointwise scalar products with the identity of
$C^{\infty}(M;S^{2}\wedge^{2}T^{*}M\otimes{\mathbb{C}})$ look like $g({\rm
Weyl}^{\pm}\>,1)={\rm tr}({\rm Weyl}^{\pm})=0$. Using (7) this shows again
that the corresponding vectors $\widehat{\rm Weyl}^{\pm}\in{\mathscr{H}}$ are
perpendicular to $\hat{1}\in{\mathscr{H}}$ i.e., $(\widehat{\rm
Weyl}^{\pm}\>,\>\hat{1})=0$. Moreover ${\rm Scal}=4\Lambda$ consequently via
(5) we get $\tau(R_{g})=(\hat{R}_{g}\>,\>\hat{1})=\frac{1}{12}(\widehat{\rm
Scal},\hat{1})=\frac{4\Lambda}{12}(\hat{1},\hat{1})=\frac{\Lambda}{3}$. All of
these serve as a motivation for the following operator-algebraic reformulation
or generalization of the classical vacuum Einstein equation with cosmological
constant [13, Definition 3.1]:

######  Definition 2.1.

Let $M$ be a connected oriented smooth $4$-manifold and ${\mathfrak{R}}$ its
${\rm II}_{1}$-type hyperfinite factor von Neumann algebra as before.

(i) A refinement of ${\mathfrak{R}}$ is a pair $({\mathfrak{R}},*)$ where
$1\not=*\in{\mathfrak{R}}$ and satisfies $*^{2}=1$;

(ii) An operator $Q\in{\mathfrak{R}}$ solves the quantum vacuum Einstein
equation with respect to $({\mathfrak{R}},*)$ if

$\left\\{\begin{array}[]{lll}Q^{\divideontimes}&=Q&\mbox{(self-
adjointness)}\\\ \tau(Q*)&=0&\mbox{(algebraic Bianchi identity)}\\\
Q*^{-1}&=Q&\mbox{(Einstein condition);}\end{array}\right.$

(iii) The trace $\tau(Q)=:\frac{\Lambda}{3}\in{\mathbb{R}}$ is called the
corresponding quantum cosmological constant.

Note that the curvature tensor $R_{g}$ of an Einstein manifold $(M,g)$ if
bounded as an operator always solves the quantum vacuum Einstein equation with
respect to the canonical refinement provided by metric (anti)self-duality.
However in sharp contrast to the classical Einstein equation whose solution is
a metric therefore is non-linear, its quantum generalization is linear hence
easily solvable. Of course this is beacuse in the generalization we do not
request the solution (which can be any non-local operator) to originate from a
metric. Indeed, given $B\in{\mathfrak{R}}$ then
$S=\frac{1}{2}(B+B^{\divideontimes})$ is self-adjoint and then picking an
arbitrary refinement $({\mathfrak{R}},*)$ and taking into account that $*$ is
always self-adjoint, the averaged operator
$Q:=\frac{1}{2}\big{(}S+*S*\big{)}-\tau(S*)*\in{\mathfrak{R}}$ is
automatically a solution of the quantum vacuum Einstein equation; moreover
using $\tau(*)=0$ its trace $\frac{\Lambda}{3}$ is equal to $\tau(B)$ hence is
independent of the particular refinement. It is important to note that,
contrary to smooth solutions, many singular solutions of classical general
relativity theory cannot be interpreted as observables because their
curvatures lack being bounded operators hence do not belong to
${\mathfrak{R}}$. As a result, we expect that the classical (Riemannian)
Schwarzschild or Kerr black hole solutions and more generally gravitatioal
fields of isolated bodies (cf. [9, 10]) give rise not to observables in
${\mathfrak{R}}$ but rather fields in
$T_{1}{\mathfrak{R}}\supset{\mathfrak{R}}$.

The classical phase of the quantum theory. Now we are in a position to
introduce another phase of the proposed quantum field theoretic interpretation
of ${\mathfrak{R}}$ which—unlike its pure quantum phase above consisting of a
single infinitely high temperature thermal equilibrium state $\tau$ with the
corresponding unique standard representation $\pi$ of ${\mathfrak{R}}$ only—is
built up from a huge moduli of finite temperature thermal equilibrium states
$F_{\Sigma,\omega}$ with corresponding new representations $\rho_{M}$ of
${\mathfrak{R}}$. Moreover we want to understand these states as classical
(Riemannian) vacuum space-times $(M,g)$. These will be achieved by introducing
a new dynamics on ${\mathfrak{R}}$.

######  Definition 2.2.

Let $({\mathfrak{R}},*)$ be a refinement of ${\mathfrak{R}}$ as in Definition
2.1. The condition $*\not=1$ and $*^{2}=1$ implies that there exists a basis
in which $*=\left(\begin{smallmatrix}1&0\\\ 0&-1\end{smallmatrix}\right)$
therefore introducig the skew-Hermitian operator
$\log*:=\left(\begin{smallmatrix}0&0\\\
0&\sqrt{-1}\pi\end{smallmatrix}\right)$ for every $t\in{\mathbb{R}}$ we can
set $*^{t}:={\rm e}^{t\log*}$ which is a unitary within ${\mathfrak{R}}$. The
corresponding $1$-parameter family of unitary inner automorphisms on
${\mathfrak{R}}$ given by

$A\longmapsto*^{t}A*^{-t}$

for all $A\in{\mathfrak{R}}$ and $t\in{\mathbb{R}}$ introduces a dynamics on
${\mathfrak{R}}$ what we call the Hodge dynamics. Accordingly
$\big{(}{\mathfrak{R}},\\{{\rm Ad}_{*^{t}}\\}_{t\in{\mathbb{R}}}\big{)}$ is a
Hodge dynamical system on the hyperfinite ${\rm II}_{1}$ factor.

How these dynamics on ${\mathfrak{R}}$ look like? Observe first that contrary
to the unique and trivial Tomita–Takesaki dynamics these highly non-unique
Hodge dynamics are already non-trivial however $*^{2}=1$ says that they are
uniformly periodic with physical duration $\hbar\beta=2t_{\rm Planck}$
consequently also give rise to a uniform temperature equal to
$\frac{\hbar}{2k_{B}t_{\rm Planck}}=\frac{1}{2}T_{\rm Planck}$. In particular
an embedding $M\subset{\mathfrak{R}}$ induces an oriented Riemannian
$4$-manifold $(M,g)$ hence thanks to its Hodge operator $*\in{\mathfrak{R}}$
it generates a Hodge dynamics on ${\mathfrak{R}}$ too. What kind of operators
are its (non-trivial) fixed points?

######  Lemma 2.1.

Let $M$ be a connected oriented smooth $4$-manifold and consider its embedding
$M\subset{\mathfrak{R}}$ given by $x\mapsto P_{x}$ and the induced oriented
Riemannian $4$-manifold $(M,g)$ as in Theorem 2.1. Take its Hodge and
curvature operators $*,R_{g}$ respectively. We know that $*\in{\mathfrak{R}}$
and assume that $R_{g}\in{\mathfrak{R}}$ too.

Then $M\subset{\mathfrak{R}}$ is invariant under the induced Hodge dynamics on
${\mathfrak{R}}$ i.e. the corresponding projections satisfy
$*^{t}P_{x}*^{-t}=P_{x}$ for every $x\in M$ and $t\in{\mathbb{R}}$; moreover
$(M,g)$ is Einstein if and only if its Riemannian curvature operator is also
invariant i.e. $*^{t}R_{g}*^{-t}=R_{g}$ for all $t\in{\mathbb{R}}$.

###### Proof.

Concerning $M\subset{\mathfrak{R}}$ given by $x\mapsto P_{x}$ where
$P_{x}\in{\mathfrak{R}}$ is a projection, it follows from its construction
(see the construction of the map (8) in Section 3) that $P_{x}$ commutes with
$*$ hence is invariant under the corresponding Hodge dynamics. Consequently
the dynamics on ${\mathfrak{R}}$ generated by $M\subset{\mathfrak{R}}$ leaves
this embedding unchanged.

Concerning $R_{g}$ the assertion is straightforward as by recalling [22] we
already noticed that $(M,g)$ is Einstein if and only if $*$ commutes with
$R_{g}$ (see Definition 2.1). ∎

Therefore classical Riemannian general relativity in vacuum i.e. an oriented
Riemannian $4$-manifold $(M,g)$ satisfying the vacuum Einstein equation (with
cosmological constant) $R_{g}=\Lambda g$ can be characterized quantum
mechanically as a subset $\\{P_{x}\\}_{x\in
M}\cup\\{R_{g}\\}\subset{\mathfrak{R}}$ which belongs to the fixed-point-set
of a natural non-trivial dynamics on this operator algebra generated by this
subset.444Actually the full fixed-point-set of any dynamics is a von Neumann
subalgebra which always contains the center of ${\mathfrak{R}}$. It is a
normal subalgebra if and only if the dynamics is inner and periodic [24]. We
conjecture that in our case this normal von Neumann subalgebra is generated by
the self-adjoint subset $\\{1\\}\cup\\{P_{x}\\}_{x\in
M}\cup\\{*,R_{g}\\}\subset{\mathfrak{R}}$. It seems there is a connection
between $4$ dimensional Einstein structures and normal von Neumann subalgebras
of the hyperfinte ${\rm II}_{1}$ factor.

Having introduced non-trivial dynamics on ${\mathfrak{R}}$ then following the
KMS analysis of [3, 14] now we would like to find states on ${\mathfrak{R}}$
which are stationary and stable under these finite temperature dynamics hence
could be termed as “thermal equilibrium states”. This is possible if we recall
now item (iii) of Theorem 2.1 which is based on Lemma 9 in Section 3. Consider
again $M$ and take a pair $(\Sigma,\omega)$ consisting of an (immersed) closed
orientable surface $\Sigma\looparrowright M$ with induced orientation and
$\omega\in\Omega^{2}_{c}(M;{\mathbb{C}})$ which is also closed i.e. ${\rm
d}\omega=0$. Consider the differential geometric ${\mathbb{C}}$-linear
functional $A\mapsto\int_{\Sigma}A\omega$ and continuously extend it from
$\operatorname{End}(\Omega^{2}_{c}(M;{\mathbb{C}}))\cap{\mathfrak{R}}$ to a
state $F_{\Sigma,\omega}:{\mathfrak{R}}\rightarrow{\mathbb{C}}$. This
extension is unique because
$\operatorname{End}(\Omega^{2}_{c}(M;{\mathbb{C}}))\cap{\mathfrak{R}}$ is
norm-dense in ${\mathfrak{R}}$.

######  Lemma 2.2.

Consider an embedding $M\subset{\mathfrak{R}}$ with induced oriented
Riemannian $4$-manifold $(M,g)$ and corresponding Hodge dynamical system
$\big{(}{\mathfrak{R}},\\{{\rm Ad}_{*^{t}}\\}_{t\in{\mathbb{R}}}\big{)}$.
Assume that both $\Sigma\looparrowright M$ and
$\omega\in\Omega^{2}_{c}(M;{\mathbb{C}})$ are self-dual in the sense that the
(real and not necessarily compactly supported) Poincaré-dual $2$-form
$\eta_{\Sigma}\in\Omega^{2}(M)\subset\Omega^{2}(M;{\mathbb{C}})$ of $\Sigma$
is self-dual i.e. $*\eta_{\Sigma}=\eta_{\Sigma}$ and likewise
$\omega\in\Omega^{2}_{c}(M;{\mathbb{C}})$ satisfies $*\omega=\omega$.

Then the corresponding state
$F_{\Sigma,\omega}:{\mathfrak{R}}\rightarrow{\mathbb{C}}$ is stationary under
the induced Hodge dynamics i.e.
$F_{\Sigma,\omega}(*^{t}A*^{-t})=F_{\Sigma,\omega}(A)$ for every
$A\in{\mathfrak{R}}$ and $t\in{\mathbb{R}}$.

Moreover take
$*^{\prime}\in\operatorname{End}(\Omega^{2}_{c}(M;{\mathbb{C}}))\cap{\mathfrak{R}}$
and let $\big{(}{\mathfrak{R}},\\{{\rm
Ad}_{(*^{\prime})^{t}}\\}_{t\in{\mathbb{R}}}\big{)}$ be a “nearby” dynamical
system in the sense that $*^{\prime}$ commutes with $*$ (therefore preserves
$M\subset{\mathfrak{R}}$ too by Lemma 2.1) and
$(*^{\prime})^{2\pm\varepsilon}=1$ (i.e. having slightly disturbed
periodicity). Then there exist $\Sigma^{\prime}\looparrowright M$ and
$\omega^{\prime}\in\Omega_{c}^{2}(M;{\mathbb{C}})$ such that the “nearby”
state $F_{\Sigma^{\prime},\omega^{\prime}}$ on ${\mathfrak{R}}$ is stationary
under the “nearby” dynamics.

###### Proof.

Assume that
$A\in\operatorname{End}(\Omega^{2}_{c}(M;{\mathbb{C}}))\cap{\mathfrak{R}}$. By
its definition and the requested properties of the Poincaré-dual the
restricted functional can be re-written as a unitary $L^{2}$ scalar product on
the space $\Omega^{2}_{c}(M;{\mathbb{C}})\times\Omega^{2}(M;{\mathbb{C}})$
over $(M,g)$ as follows:

$F_{\Sigma,\omega}(A)=\int\limits_{\Sigma}A\omega=\int\limits_{M}A\omega\wedge\eta_{\Sigma}=\int\limits_{M}A\omega\wedge*\overline{\eta}_{\Sigma}=\big{(}A\omega\>,\>\eta_{\Sigma}\big{)}_{L^{2}(M,g)}\>\>.$

Now let $t\in{\mathbb{R}}$ be arbitrary and taking into account that $*^{t}$
is unitary with respect to this $L^{2}$ scalar product we can rearrange things
as
$\big{(}*^{t}A*^{-t}\omega,\eta_{\Sigma}\big{)}_{L^{2}(M,g)}=\big{(}A*^{-t}\omega,*^{-t}\eta_{\Sigma}\big{)}_{L^{2}(M,g)}$.
However

$\frac{{\rm d}}{{\rm
d}t}F_{\Sigma,\omega}(*^{t}A*^{-t})=-\big{(}A*^{-t}(\log*)\omega\>,\>*^{-t}\eta_{\Sigma}\big{)}_{L^{2}(M,g)}-\big{(}A*^{-t}\omega\>,\>*^{-t}(\log*)\eta_{\Sigma}\big{)}_{L^{2}(M,g)}=0$

because we know that in the (anti)self-dual basis
$\log*=\left(\begin{smallmatrix}0&0\\\ 0&\sqrt{-1}\pi\end{smallmatrix}\right)$
hence its action on the self-dual elements $\binom{\omega}{0}$ and
$\binom{\eta_{\Sigma}}{0}$ vanishes. This implies by continuity the
stationarity of $F_{\Sigma,\omega}$ for all $A\in{\mathfrak{R}}$.

Concerning stability since by assumption
$A\mapsto*^{\prime}A(*^{\prime})^{-1}$ is a $*$-automorhism of
${\mathfrak{R}}$ we necessarily find that
$*^{\prime}\in\operatorname{End}(\Omega^{2}_{c}(M;{\mathbb{C}}))\cap{\mathfrak{R}}$
is unitary and maps $\Omega^{2}_{c}(M;{\mathbb{C}})$ into itself like $*$
does; thus there exists a unitary operator
$U\in\operatorname{End}(\Omega^{2}_{c}(M;{\mathbb{C}}))\cap{\mathfrak{R}}$
such that $*^{\prime}=*U$. It readily follows from $*^{\prime}*=*U*$ and
$**^{\prime}=U$ that the assumption that $*$ and $*^{\prime}$ commute implies
$*$ also commutes with $U$ hence the perturbed dynamics decomposes as
$(*^{\prime})^{t}=*^{t}U^{t}$. The further condition
$(*^{\prime})^{2\pm\varepsilon}=1$ eventually yields that in the (anti)self-
dual basis we can write $U=z\left(\begin{smallmatrix}1&0\\\ 0&\pm
1\end{smallmatrix}\right)$ where the phase $z=z(\varepsilon)\in{\mathbb{C}}$
satisfies $z(0)=1$ but $|z(\varepsilon)|=1$ in general. Consequently the two
dynamical systems differ by a simple inner automorphism ${\rm Ad}_{U}$ of
${\mathfrak{R}}$ only. Therefore putting simply $\Sigma^{\prime}:=\Sigma$
hence $\eta_{\Sigma^{\prime}}:=\eta_{\Sigma}$ and $\omega^{\prime}:=\omega$ we
can introduce a new state which simply satisfies
$F_{\Sigma^{\prime},\omega^{\prime}}(B)=F_{\Sigma,\omega}(B)$ i.e. coincides
with the old state. Then we compute

$\displaystyle
F_{\Sigma^{\prime},\omega^{\prime}}((*^{\prime})^{t}A(*^{\prime})^{-t})$
$\displaystyle=$ $\displaystyle F_{\Sigma,\omega}(*^{t}U^{t}AU^{-t}*^{-t})$
$\displaystyle=$
$\displaystyle\big{(}AU^{-t}*^{-t}\omega\>,\>U^{-t}*^{-t}\eta_{\Sigma}\big{)}_{L^{2}(M,g)}$
$\displaystyle=$
$\displaystyle|z|^{-t}\big{(}Az^{t}U^{-t}*^{-t}\omega\>,\>z^{t}U^{-t}*^{-t}\eta_{\Sigma}\big{)}_{L^{2}(M,g)}$
$\displaystyle=$
$\displaystyle\big{(}A(z^{-1}U)^{-t}*^{-t}\omega\>,\>(z^{-1}U)^{-t}*^{-t}\eta_{\Sigma}\big{)}_{L^{2}(M,g)}$

hence using again that $(\log*)\omega=0=(\log*)\eta_{\Sigma}$ and noticing
that $\log(z^{-1}U)=\left(\begin{smallmatrix}0&0\\\ 0&{\rm
something}\end{smallmatrix}\right)$ hence likewise
$\log(z^{-1}U)\omega=0=\log(z^{-1}U)\eta_{\Sigma}$ we can differentiate the
last expression and obtain

$\frac{{\rm d}}{{\rm
d}t}F_{\Sigma^{\prime},\omega^{\prime}}((*^{\prime})^{t}A(*^{\prime})^{-t})=0$

demonstrating that $F_{\Sigma^{\prime},\omega^{\prime}}$ is stationary as
stated. ∎

Thus summarizing: in addition to its infinitely high temperature pure quantum
phase above (consisting of the single KMS state $\tau$ and corresponding
standard representation $\pi$ on ${\mathscr{H}}$), we hereby introduce the so-
called classical phase of the quantum field theory based on ${\mathfrak{R}}$
to be the moduli space comprising all finite temperature thermal equilibrium
states like $F_{\Sigma,\omega}$ on ${\mathfrak{R}}$ provided by all possible
embeddings $M\subset{\mathfrak{R}}$ of connected oriented smooth $4$-manifolds
into this algebra such that the induced Riemannian $4$-manifold $(M,g)$ is
Einstein. The uniform temperature of these states is $\frac{1}{2}T_{\rm
Planck}$. Application of the GNS construction to states in the classical phase
gives further representations of ${\mathfrak{R}}$. More precisely it is proved
in Lemma 9 below that given an embedding $M\subset{\mathfrak{R}}$ and a
functional $F_{\Sigma,\omega}:{\mathfrak{R}}\rightarrow{\mathbb{C}}$ the
isomorphism class of the corresponding representation $\rho_{M}$ on a complex
separable Hilbert space ${\mathscr{I}}(M)^{\perp}$ is independent of the
particular $\Sigma\looparrowright M$ and
$\omega\in\Omega^{2}_{c}(M;{\mathbb{C}})$ i.e. these representations depend
only on $M$. Moreover (unlike the standard representation $\pi$) all of these
representations $\rho_{M}$ are not faithful. Consequently the states
$F_{\Sigma,\omega}$ (unlike the tracial state $\tau$) are not KMS states.

Vacuum general relativity from a spontaneous symmetry breaking. The modern
physical and mathematical basis for thinking about space, time, gravity and
the structure and history of the Universe (cosmology) has been provided by
classical general relativity in the form of (pseudo-)Riemannian $4$-manifolds
$(M,g)$ subject to Einstein’s field equation. As we have summarized in Theorem
2.1 on the one hand out of $M$ alone an operator algebra ${\mathfrak{R}}$ and
a Hilbert space ${\mathscr{I}}(M)^{\perp}$ carrying a representation
$\rho_{M}$ of ${\mathfrak{R}}$ can be constructed; on the other hand
$M\subset{\mathfrak{R}}$ holds which automatically induces a Riemannian
structure $(M,g)$ which is moreover Einstein if and only if this embedding
together with its Riemannian tensor $R_{g}$ is invariant with respect to the
Hodge dynamics on ${\mathfrak{R}}$ also induced by the embedding. Remember the
plain technicality that a curvature tensor of $(M,g)$ must be bounded in order
to belong to the operator algebra. However it has been known for a long time
that in general relativity the gravitational field of an isolated massive
configuration cannot be regular everywhere [9, 10]. Thus we are forced to move
towards a global non-singular cosmological context when trying to work out a
physical interpretation of the present quite circular interaction between
${\mathfrak{R}}$ and $M$. Therefore hereby we make a working hypothesis: the
unique abstract triple $({\mathfrak{R}},{\mathscr{H}},\pi)$ as a quantum
theory describes the Universe from the Big Bang till about $2t_{\rm
Planck}\approx 1.07\times 10^{-43}$ sec while a highly non-unique triple
$({\mathfrak{R}},{\mathscr{I}}(M)^{\perp},\rho_{M})$ giving rise to a $4$
dimensional vacuum space-time $(M,g)$ in classical general relativity
describes its evolution from $2t_{\rm Planck}$ onwards. A usual choice for
$(M,g)$ is the FLRW solution with or without cosmological constant. The
“moment” $2t_{\rm Planck}$ would therefore formally or symbolically label the
emergence or onset of a space-time structure in the course of the history of
the Universe555We are aware of, but cannot do better, how nonsense it is,
strictly speaking, to talk about the emergence of space and time at the moment
$2t_{\rm Planck}$; this assertion requires the contradictory pre-supposition
of time before $t_{\rm Planck}$. establishing a physical correspondence
between the two representations of ${\mathfrak{R}}$. We display symbolically
this passage as

$\begin{matrix}({\mathfrak{R}},{\mathscr{H}},\pi)\Longrightarrow&\big{(}\mbox{A
Hodge-invariant subset $\\{P_{x}\\}_{x\in
M}\cup\\{R_{g}\\}$}\subset{\mathfrak{R}},{\mathscr{I}}(M)^{\perp}\subset{\mathscr{H}},\rho_{M}=\pi|_{{\mathscr{I}}(M)^{\perp}}\big{)}\\\
\Updownarrow&\Updownarrow\\\ \mbox{The ``quantum space-time''}&\mbox{A
particular $4$ dimensional Riemannian vacuum space-time $(M,g)$}\\\ \mbox{at
infinite temperature}&\mbox{at temperature $\frac{1}{2}T_{\rm
Planck}$}\end{matrix}$ (1)

having in mind a sort of phase transition at $2t_{\rm Planck}$ from the
quantum to the classical phase of the theory via spontaneous symmetry breaking
driven by cooling (or by a spontaneous jump from the unique Tomita–Takesaki to
a particular Hodge dynamics on ${\mathfrak{R}}$). Note that this transition
from the unique quantum regime $({\mathfrak{R}},{\mathscr{H}},\pi)$ to a
particular $4$ dimensional classical vacuum regime $(M,g)$ given by another
representation $({\mathfrak{R}},{\mathscr{I}}(M)^{\perp},\rho_{M})$ has been
captured in the framework of algebraic quantum field theory [3, 14] as
switching from the unique representation $\pi$ to a different particular
representation $\rho_{M}$ of the same algebra ${\mathfrak{R}}$. One can also
formally label the transition with $\frac{1}{2}T_{\rm Planck}\approx
7.06\times 10^{31}$ K; this high temperature is reasonable if we make a
further technical observation that $\pi$ is induced by the unique tracial
state $\tau$ on ${\mathfrak{R}}$ hence one can imagine that
$({\mathfrak{R}},{\mathscr{H}},\pi)$ describes a quantum statistical
mechanical system in an infinitely high temperature KMS equilibrium state
$\tau$ (hence having trivial modular dynamics in the sense of [8]).

Finally let us mention that although this picture is unphysical because uses
imaginary time it avoids the problem of “initial singularity” in a natural way
for it dissolves the dynamically non-trivial classical evolutionary phase of
the Universe in a dynamically trivial quantum phase about $2t_{\rm Planck}$
i.e. “just before the initial singularity”.

A comment on the measurement problem. It is dictated by the mathematical
structure of the ${\rm II}_{1}$ hyperfinite factor that if
$A\in{\mathfrak{R}}$ represents a (bounded) observable while
$B\in{\mathfrak{R}}$ a state (regarded as a “density matrix”) describing a
physical situation in this theory then without further specifications on $A,B$
the expectation value of $A$ in $B$ should be given by
$\tau(AB)\in{\mathbb{C}}$ i.e. a complex number in general. We may then ask
ourselves: after performing the physical experiment to measure the quantity
represented by $A$ in the state represented by $B$ should we expect that the
state will necessarily “collapse” to an eigenstate $B_{\lambda}$ of $A$? In
our opinion no and this is a fundmental difference between gravity and quantum
mechanics [4]. Namely, in quantum mechanics an ideal observer compared to the
physical object to be observed is infinitely large hence the immense physical
interaction accompanying the measurement procedure drastically disturbs the
entity leading to the collapse of its state. However, in sharp contrast to
this, in gravity an ideal observer is infinitely small hence it is reasonable
to expect that measurements might not alter gravitational states. This is in
accordance with our old experience concerning measurements in astronomy.

As an example we can compute the energy content of a classical space-time
$(M,g)$ occupied by pure gravitation only i.e. without matter. As we have seen
in the classical phase the valid dynamics on ${\mathfrak{R}}$ is the Hodge
dynamics generarted by $(M,g)$. It is therefore reasonable to suppose that the
energy should be captured by measuring the Hamilton operator $H(M,g)$ of the
Hodge dynamics in the state represented by the curvature operator $R_{g}$
satisfying the vacuum Einstein equation. Let us suppose that this latter
operator belongs to ${\mathfrak{R}}$ then it is self-adjoint. Moreover
recalling that $*^{t}={\rm e}^{t\log*}$ we can introduce the corresponding
self-adjoint Hamilton operator $H(M,g):=\frac{1}{\sqrt{-1}}\log*$ which looks
like $H(M,g)=\left(\begin{smallmatrix}0&0\\\ 0&\pi\end{smallmatrix}\right)$ in
the (anti)self-dual basis hence belongs to ${\mathfrak{R}}$ (cf. Definition
2.2). In addition the curvature in this basis has the shape (5). Taking into
account that both $R_{g}$ and $H(M,g)$ act pointwise on $M$ more precisely are
bundle morphisms over $\wedge^{2}T^{*}M\otimes{\mathbb{C}}$ hence so is their
product $H(M,g)R_{g}\in{\mathfrak{R}}$ we can use (7) to compute the trace.
Recalling that ${\rm Scal}=4\Lambda$ and using the normalized measure
$\mu_{g}$ (which surely exists if $M$ is compact) we obtain

$E(M,g)=\tau(H(M,g)R_{g})=\frac{1}{6}\int\limits_{M}{\rm
tr}(H(M,g)R_{g})\mu_{g}=\frac{1}{6}\cdot
3\pi\cdot\frac{4\Lambda}{12}=\frac{\pi}{6}\Lambda$

consequently the quantum mechanical energy content of a vacuum space-time is
proportional to the cosmological constant as one would expect on physical
grounds. This also helps to recover the resulting cosmological constant from
“inside” the spontaneously broken state.

This is a good point to exit our yet admittedly incomplete trial to set up
some physical picture behind the unique ${\rm II}_{1}$ hyperfinite factor.

##  3 Appendix: mathematical constructions

In this section for completeness and the Reader’s convenience we recall and
partly extend further the mathematical exposition in [13] and give a detailed
proof of Theorem 2.1. First we shall exhibit a simple self-contained two-step
construction of a von Neumann algebra attached to any oriented smooth
$4$-manifold. Then the structure of this algebra will be explored in some
detail. Finally we exhibit a new (i.e. not the standard) representation of
this von Neumann algebra induced by the whole contruction. For clarity we
emphasize that the forthcoming constructions are rigorous in the sense that no
physical ideas, considerations, steps, etc. to be used.

Construction of an algebra. Take the isomorphism class of a connected oriented
smooth $4$-manifold (without boundary) and from now on let $M$ be a once and
for all fixed representative in it carrying the action of its own orientation-
preserving group of diffeomorphisms ${\rm Diff}^{+}(M)$. Among all tensor
bundles $T^{(p,q)}M$ over $M$ the $2^{\rm nd}$ exterior power
$\wedge^{2}T^{*}M\subset T^{(0,2)}M$ is the only one which can be endowed with
a pairing in a natural way i.e., with a pairing extracted from the smooth
structure (and the orientation) of $M$ alone. Indeed, consider its associated
vector space $\Omega^{2}_{c}(M):=C^{\infty}_{c}(M;\wedge^{2}T^{*}M)$ of
compactly supported smooth $2$-forms on $M$. Define a pairing
$\langle\>\cdot\>,\>\cdot\>\rangle_{L^{2}(M)}:\Omega^{2}_{c}(M)\times\Omega^{2}_{c}(M)\rightarrow{\mathbb{R}}$
via integration:

$\langle\alpha,\beta\rangle_{L^{2}(M)}:=\int\limits_{M}\alpha\wedge\beta$ (2)

and observe that this pairing is non-degenerate however is indefinite in
general thus can be regarded as an indefinite scalar product on
$\Omega^{2}_{c}(M)$. It therefore induces an indefinite real quadratic form
$Q$ on $\Omega^{2}_{c}(M)$ given by
$Q(\alpha):=\langle\alpha,\alpha\rangle_{L^{2}(M)}$. Let $C(M)$ denote the
complexification of the infinite dimensional real Clifford algebra associated
with $(\Omega^{2}_{c}(M),Q)$. Because Clifford algebras are usually
constructed out of definite quadratic forms, we summarize this construction
[19, Section I.§3] to make sure that the resulting object $C(M)$ is well-
defined i.e. is not sensitive for the indefiniteness of (2). To begin with,
let $V_{m}\subset\Omega^{2}_{c}(M)$ be an $m$ dimensional real subspace and
assume that $Q_{r,s}:=Q|_{V_{m}}$ has signature $(r,s)$ on $V_{m}$ that is,
the maximal positive definite subspace of $V_{m}$ with respect to $Q_{r,s}$
has dimension $r$ while the dimension of the maximal negative definite
subspace is $s$ such that $r+s=m$ by the non-degeneracy of $Q_{r,s}$. Then out
of the input data $(V_{m},Q_{r,s})$ one constructs in the standard way a
finite dimensional real Clifford algebra $C_{r,s}(M)$ with unit $1\in
C_{r,s}(M)$ and an embedding $V_{m}\subset C_{r,s}(M)$ with the property
$\alpha^{2}=Q_{r,s}(\alpha)1$ for every element $\alpha\in V_{m}$. This real
algebra depends on the signature $(r,s)$ however fortunately its
complexification $C_{m}(M):=C_{r,s}(M)\otimes{\mathbb{C}}$ is already
independent of it. In fact, if ${\mathfrak{M}}_{k}({\mathbb{C}})$ denotes the
algebra of $k\times k$ complex matrices, then it is well-known [19, Section
I.§3] that $C_{0}(M)\cong{\mathfrak{M}}_{1}({\mathbb{C}})$ while
$C_{1}(M)\cong{\mathfrak{M}}_{1}({\mathbb{C}})\oplus{\mathfrak{M}}_{1}({\mathbb{C}})$
and the higher dimensional cases follow from the complex periodicity
$C_{m+2}(M)\cong C_{m}(M)\otimes{\mathfrak{M}}_{2}({\mathbb{C}})$.
Consequently depending on the parity $C_{m}(M)$ is isomorphic to either
${\mathfrak{M}}_{2^{\frac{m}{2}}}({\mathbb{C}})$ or
${\mathfrak{M}}_{2^{\frac{m-1}{2}}}({\mathbb{C}})\oplus{\mathfrak{M}}_{2^{\frac{m-1}{2}}}({\mathbb{C}})$.
These imply that $2$-step-chains of successive embeddings of real subspaces
$V_{m}\subset V_{m+1}\subset V_{m+2}\subset\Omega^{2}_{c}(M)$ starting with
$V_{0}=\\{0\\}$ and given by iterating $\omega\mapsto\binom{\omega}{0}$
provide us with injective algebra homomorphisms
${\mathfrak{M}}_{2^{\frac{m}{2}}}({\mathbb{C}})\hookrightarrow{\mathfrak{M}}_{2^{\frac{m}{2}+1}}({\mathbb{C}})$
having the shape $A\mapsto\left(\begin{smallmatrix}A&0\\\
0&A\end{smallmatrix}\right)$. Therefore $C(M)$ is isomorphic to the injective
limit of this directed system, that is there exists a linear-algebraic
isomorphism

$C(M)\cong\bigcup\limits_{n=0}^{+\infty}{\mathfrak{M}}_{2^{n}}({\mathbb{C}})$
(3)

or equivalently

$C(M)\cong{\mathfrak{M}}_{2}({\mathbb{C}})\otimes{\mathfrak{M}}_{2}({\mathbb{C}})\otimes\dots$

because this injective limit is also isomorphic to the infinite tensor product
of ${\mathfrak{M}}_{2}({\mathbb{C}})$’s. For clarity note that being (2) a
non-local operation, $C(M)$ is a genuine global infinite dimensional object.

It is well-known (cf. [7, Section I.3]) that any complexified infinite
Clifford algebra like $C(M)$ above generates the ${\rm II}_{1}$-type
hyperfinite factor von Neumann algebra. Let us summarize this procedure too
(cf. [1, Section 1.1.6]). It readily follows that $C(M)$ possesses a unit
$1\in C(M)$ and its center comprises the scalar multiples of the unit only.
Moreover $C(M)$ continues to admit a canonical embedding
$\Omega^{2}_{c}(M;{\mathbb{C}})\subset C(M)$ satisfying
$\omega^{2}=Q(\omega)1$ where now $Q$ denotes the quadratic form induced by
the complex-bilinear extension of (2). We also see via (3) already that $C(M)$
is a complex $*$-algebra whose $*$-operation (provided by taking Hermitian
matrix transpose, a non-local operation) is written as $A\mapsto
A^{\divideontimes}$. The isomorphism (3) also shows that if $A\in C(M)$ then
one can pick the smallest $n\in{\mathbb{N}}$ such that
$A\in{\mathfrak{M}}_{2^{n}}({\mathbb{C}})$ consequently $A$ has a finite trace
defined by $\tau(A):=2^{-n}{\rm Trace}(A)$ i.e., taking the usual normalized
trace of the corresponding $2^{n}\times 2^{n}$ complex matrix. It is
straightforward that $\tau(A)\in{\mathbb{C}}$ does not depend on $n$. We can
then define a sesquilinear inner product on $C(M)$ by
$(A,B):=\tau(AB^{\divideontimes})$ which is non-degenerate thus the completion
of $C(M)$ with respect to the norm $\|\>\cdot\>\|$ induced by
$(\>\cdot\>,\>\cdot\>)$ renders $C(M)$ a complex Hilbert space what we shall
write as ${\mathscr{H}}$ and its Banach algebra of all bounded linear
operators as ${\mathfrak{B}}({\mathscr{H}})$. Multiplication in $C(M)$ from
the left on itself is continuous hence gives rise to a representation
$\pi:C(M)\rightarrow{\mathfrak{B}}({\mathscr{H}})$. Finally our central object
effortlessly emerges as the weak closure of the image of $C(M)$ under $\pi$
within ${\mathfrak{B}}({\mathscr{H}})$ or equivalently, by referring to von
Neumann’s bicommutant theorem [1, Theorem 2.1.3] we put

${\mathfrak{R}}:=(\pi(C(M)))^{\prime\prime}\subset{\mathfrak{B}}({\mathscr{H}})\>\>.$

This von Neumann algebra of course admits a unit $1\in{\mathfrak{R}}$ moreover
continues to have trivial center i.e., is a factor. Moreover by construction
it is hyperfinite. The trace $\tau$ as defined extends from $C(M)$ to
${\mathfrak{R}}$ and satisfies $\tau(1)=1$. Moreover [1, Proposition 4.1.4]
this trace is unique on ${\mathfrak{R}}$. Likewise we obtain by extension a
representation $\pi:{\mathfrak{R}}\rightarrow{\mathfrak{B}}({\mathscr{H}})$.
The canonical inclusion $\Omega^{2}_{c}(M;{\mathbb{C}})\subset C(M)$ recorded
above extends to both $\Omega^{2}_{c}(M;{\mathbb{C}})\subset{\mathscr{H}}$ and
$\Omega^{2}_{c}(M;{\mathbb{C}})\subset{\mathfrak{R}}$. Thus in order to
carefully distinguish the two different completions ${\mathfrak{R}}$ and
${\mathscr{H}}$ of one and the same object $C(M)$ we shall write
$A\in{\mathfrak{R}}$ but $\hat{B}\in{\mathscr{H}}$ from now on as usual. This
is necessary since ${\mathfrak{R}}$ and ${\mathscr{H}}$ are very different for
example as ${\rm U}({\mathscr{H}})$-modules: given a unitary operator
$V\in{\rm U}({\mathscr{H}})$ then $A\in{\mathfrak{R}}$ is acted upon as
$A\mapsto VAV^{-1}$ but $\hat{B}\in{\mathscr{H}}$ transforms as
$\hat{B}\mapsto V\hat{B}$. Using this notation and introducing
$\hat{A}:=\pi(A)\hat{1}$ the trace always can be written as a scalar product
with the image of the unit in ${\mathscr{H}}$ that is, for every
$A\in{\mathfrak{R}}$ we have

$\tau(A)=(\hat{A},\hat{1})$

yielding a general and geometric expression for the trace.

Exploring the algebra ${\mathfrak{R}}$. Before proceeding further let us make
a digression here to gain a better picture. This is desirable because taking
the weak closure like ${\mathfrak{R}}$ of some explicitly known structure like
$C(M)$ often involves a sort of loosing control over the latter. Nevertheless
we already know promisingly that ${\mathfrak{R}}$ is a hyperfinite factor von
Neumann algebra of ${\rm II}_{1}$-type. Let us now exhibit some of its
elements.

1\. Our first examples are the $2$-forms themselves as it follows from the
already mentioned canonical embedding $\Omega^{2}_{c}(M;{\mathbb{C}})\subset
C(M)$ combined with $C(M)\subset{\mathfrak{R}}$. This also implies that in
fact ${\mathfrak{R}}\supset C(M)$ is weakly generated by $1\in C(M)$ and all
finite products of $2$-forms $\omega_{1}\omega_{2}\dots\omega_{n}$ within the
associative algebra $C(M)$(and likewise, ${\mathscr{H}}\supset C(M)$ is the
closure of the unit and all finite products too). We might call this as the
first picture on ${\mathfrak{R}}$ provided by the embedding
$\Omega^{2}_{c}(M;{\mathbb{C}})\subset C(M)$ however this description is not
very informative.

2\. To see more examples, let us return to the Clifford algebra in (3) for a
moment. We already know that there exists a canonical embedding
$\Omega^{2}_{c}(M;{\mathbb{C}})\subset C(M)$. In addition to this let us find
a Clifford module for $C(M)$. Consider again any finite even dimensional
approximation $C_{m}(M)=C_{r,s}(M)\otimes{\mathbb{C}}$ constructed from
$(V_{m},Q_{r,s})$ where now $V_{m}\subset\Omega^{2}_{c}(M)$ is a real even
$m=r+s$ dimensional subspace. Choose any $2^{\frac{m}{2}}$ dimensional complex
vector subspace $S_{m}$ within $\Omega^{2}_{c}(M;{\mathbb{C}})$. If
$\operatorname{End}(\Omega^{2}_{c}(M;{\mathbb{C}}))$ denotes the associative
algebra of all ${\mathbb{C}}$-linear transformations of
$\Omega^{2}_{c}(M;{\mathbb{C}})$ then
$S_{m}\subset\Omega^{2}_{c}(M;{\mathbb{C}})$ induces an embedding
$\operatorname{End}S_{m}\subset\operatorname{End}(\Omega^{2}_{c}(M;{\mathbb{C}}))$
moreover we know that
$\operatorname{End}S_{m}\cong{\mathfrak{M}}_{2^{\frac{m}{2}}}({\mathbb{C}})\cong
C_{m}(M)$. Therefore we obtain a non-canonical inclusion
$C_{m}(M)\subset\operatorname{End}(\Omega^{2}_{c}(M;{\mathbb{C}}))$ for every
fixed $m\in 2{\mathbb{N}}$. Furthermore $S_{m}\subset
S_{m+2}\subset\Omega^{2}_{c}(M;{\mathbb{C}})$ given by
$\omega\mapsto\binom{\omega}{0}$ induces a sequence $C_{m}(M)\subset
C_{m+2}(M)\subset\operatorname{End}(\Omega^{2}_{c}(M;{\mathbb{C}}))$ for
Clifford algebras which is compatible with the previous ascending chain of
their matrix algebra realizations. Consequently taking the limit
$m\rightarrow+\infty$ we come up with a non-canonical injective linear-
algebraic homomorphism

$C(M)\subset\operatorname{End}(\Omega^{2}_{c}(M;{\mathbb{C}}))$ (4)

and this embedding gives rise to the second picture on ${\mathfrak{R}}$. Of
course, unlike the first picture above, this second one does not exist in the
finite dimensional case.

Although the algebra $\operatorname{End}(\Omega^{2}_{c}(M;{\mathbb{C}}))$ is
yet too huge, we can at least exhibit some of its elements. The simplest ones
are the $2$-forms themselves because $\Omega^{2}_{c}(M;{\mathbb{C}})\subset
C(M)\subset\operatorname{End}(\Omega^{2}_{c}(M;{\mathbb{C}}))$ holds as we
already know. Furthermore orientation-preserving diffeomorphisms act
${\mathbb{C}}$-linearly on $\Omega^{2}_{c}(M;{\mathbb{C}})$ via pullbacks thus
we conclude that ${\rm
Diff}^{+}(M)\subset\operatorname{End}(\Omega^{2}_{c}(M;{\mathbb{C}}))$.
Likewise ${\rm Lie}({\rm Diff}^{+}(M))\cong C^{\infty}_{c}(M;TM)$ consisting
of compactly supported real vector fields acts ${\mathbb{C}}$-linearly on
$\Omega^{2}_{c}(M;{\mathbb{C}})$ through Lie derivatives hence we also find
that ${\rm Lie}({\rm
Diff}^{+}(M))\subset\operatorname{End}(\Omega^{2}_{c}(M;{\mathbb{C}}))$.

Moreover
$C^{\infty}(M;\operatorname{End}(\wedge^{2}T^{*}M\otimes{\mathbb{C}}))\subset\operatorname{End}(\Omega^{2}_{c}(M;{\mathbb{C}}))$
i.e., bundle morphisms are also included. These are local algebraic operators
but are important because they allow to make a contact with local four
dimensional differential geometry.666In fact all the constructions so far work
for an arbitrary oriented and smooth $4k$-manifold with $k=0,1,2,\dots$ (note
that in $4k+2$ dimensions the indefinite pairing (2) gives rise to a
symplectic structure on $2k+1$-forms). A peculiarity of four dimensions is
that the space
$C^{\infty}(M;\operatorname{End}(\wedge^{2}T^{*}M\otimes{\mathbb{C}}))$
contains curvature tensors (more precisely their complex linear extensions) on
$M$. If $(M,g)$ is an oriented Riemannian $4$-manifold then its Riemannian
curvature tensor $R_{g}$ is indeed a member of this subalgebra: with respect
to the splitting of complexified $2$-forms into their (anti)self-dual parts
its complex linear extension looks like (cf. [22])

$R_{g}=\left(\begin{matrix}\frac{1}{12}{\rm Scal}+{\rm Weyl}^{+}&{\rm
Ric}_{0}\\\ {\rm Ric}_{0}^{*}&\frac{1}{12}{\rm Scal}+{\rm
Weyl}^{-}\end{matrix}\right)\>\>\>\>\>:\>\>\>\>\>\begin{matrix}\Omega^{+}_{c}(M;{\mathbb{C}})\\\
\bigoplus\\\
\Omega^{-}_{c}(M;{\mathbb{C}})\end{matrix}\>\>\>\longrightarrow\>\>\>\begin{matrix}\Omega^{+}_{c}(M;{\mathbb{C}})\\\
\bigoplus\\\ \Omega^{-}_{c}(M;{\mathbb{C}})\end{matrix}$ (5)

and more generally,
$C^{\infty}(M;\operatorname{End}(\wedge^{2}T^{*}M\otimes{\mathbb{C}}))$
contains the complex linear extensions of all algebraic (i.e. formal only, not
stemming from a metric) curvature tensors $R$ over $M$.

How to decide whether or not these elements of
$\operatorname{End}(\Omega^{2}_{c}(M;{\mathbb{C}}))$ belong to
${\mathfrak{R}}$? The key concept here is the trace. Compared with the above
trace expression $\tau(A)=(\hat{A},\hat{1})$ generally valid on
${\mathfrak{R}}$, more specific trace formulata are obtained if $M$ is endowed
with a normalized Riemannian metric $g$ i.e., the corresponding volume form
$\mu_{g}=*1$ satisfies $\int_{M}\mu_{g}=1$. The unique sesquilinear extension
of $g$ induces a positive definite sesquilinear $L^{2}$-scalar product

$(\varphi,\psi)_{L^{2}(M,g)}:=\int\limits_{x\in
M}g(\varphi(x),\psi(x))\mu_{g}(x)=\int\limits_{M}\varphi\wedge*\overline{\psi}$

on $\Omega^{2}_{c}(M;{\mathbb{C}})$. If $\\{\varphi_{1},\varphi_{2},\dots\\}$
is a smooth orthonormal frame in $\Omega^{2}_{c}(M;{\mathbb{C}})$ then it
readily follows that the trace of any
$B\in\operatorname{End}(\Omega^{2}_{c}(M;{\mathbb{C}}))$ formally looks like

$\tau(B)=\lim\limits_{n\rightarrow+\infty}\frac{1}{2^{n}}\sum\limits_{i=1}^{2^{n}}(B\varphi_{i}\>,\>\varphi_{i})_{L^{2}(M,g)}$
(6)

and, if exists, is independent of the frame used. Obviously
$B\in\operatorname{End}(\Omega^{2}_{c}(M;{\mathbb{C}}))\cap C(M)$ if and only
if the sum on the right hand side is constant after finitely many terms; and
an inspection of this trace expression at finite stages shows that in general
$B\in\operatorname{End}(\Omega^{2}_{c}(M;{\mathbb{C}}))\cap{\mathfrak{R}}$ if
and only if $\tau(B)$ exists.777By definition
$\operatorname{End}(\Omega^{2}_{c}(M;{\mathbb{C}}))\cap{\mathfrak{R}}$
contains those operators in
${\mathfrak{R}}\subset{\mathfrak{B}}({\mathscr{H}})$ which are defined on the
whole ${\mathscr{H}}$ but map
$\Omega^{2}_{c}(M;{\mathbb{C}})\subset{\mathscr{H}}$ into itself; obviously
such operators have finite trace. Conversely, given
$B\in\operatorname{End}(\Omega^{2}_{c}(M;{\mathbb{C}}))$ we may try to extend
it from $\Omega^{2}_{c}(M;{\mathbb{C}})$ to ${\mathscr{H}}$ step-by-step as
follows. For a fixed $n$ take the sub-basis
$\\{\varphi_{1},\dots,\varphi_{2^{n}}\\}$ in $\Omega^{2}_{c}(M;{\mathbb{C}})$
and consider the restriction $B_{n}$ to the corresponding $2^{n}$ dimensional
complex subspace; then iterating
$B_{n}\mapsto\left(\begin{smallmatrix}B_{n}&0\\\
0&B_{n}\end{smallmatrix}\right)$ embed it into $C(M)$ and define the action of
$B_{n}$ on ${\mathscr{H}}$ by continuously extending over ${\mathscr{H}}$ the
multiplication from the left on $C(M)\subset{\mathscr{H}}$ with this image; it
is clear that $B_{n}\rightarrow B$ weakly as $n\rightarrow+\infty$ yielding a
well-defined action of $B$ on ${\mathscr{H}}$ such that $B\in{\mathfrak{R}}$
if $\tau(B)<+\infty$ i.e. in this case
$B\in\operatorname{End}(\Omega^{2}_{c}(M;{\mathbb{C}}))\cap{\mathfrak{R}}$
indeed. As a consequence note that
$\operatorname{End}(\Omega^{2}_{c}(M;{\mathbb{C}}))\cap{\mathfrak{R}}$ is
already independent of the particular inclusion (4).

An example: $\Phi\in{\rm Diff}^{+}(M)$ acts on
$\Omega^{2}_{c}(M;{\mathbb{C}})$ via $(\Phi^{-1})^{*}$ however this action
induces a conjugate action on
$\operatorname{End}(\Omega^{2}_{c}(M;{\mathbb{C}}))$ too consequently
$(\Phi^{-1})^{*}$ is unitary yielding $|\tau((\Phi^{-1})^{*})|=1$; thus it
extends to ${\mathscr{H}}$ (cf. Footnote 7) as a unitary operator
demonstrating ${\rm
Diff}^{+}(M)\subset\operatorname{End}(\Omega^{2}_{c}(M;{\mathbb{C}}))\cap{\mathfrak{R}}$.
Another example: likewise if $X\in C^{\infty}_{c}(M;TM)$ and its corresponding
Lie derivative $L_{X}$ operating on $2$-forms satisfies $\tau(L_{X})<+\infty$
then it extends to ${\mathscr{H}}$ (cf. again Footnote 7) such that
$L_{X}\in\operatorname{End}(\Omega^{2}_{c}(M;{\mathbb{C}}))\cap{\mathfrak{R}}$.

Furthermore the curvature $R_{g}$ of $(M,g)$ as an operator in (5) acts on
$\Omega^{2}_{c}(M;{\mathbb{C}})$. If in addition it is bounded which means
that

$\sup\limits_{\|\omega\|_{L^{2}(M,g)}=1}\|R_{g}\omega\|_{L^{2}(M,g)}\leqq
K<+\infty$

then

$0\leqq|\tau(R_{g})|\leqq\lim\limits_{n\rightarrow+\infty}\frac{1}{2^{n}}\sum\limits_{i=1}^{2^{n}}|(R_{g}\varphi_{i}\>,\>\varphi_{i})_{L^{2}(M,g)}|\leqq\lim\limits_{n\rightarrow+\infty}\frac{1}{2^{n}}\sum\limits_{i=1}^{2^{n}}\|R_{g}\varphi_{i}\|_{L^{2}(M,g)}\leqq
K$

thus extends over ${\mathscr{H}}$ (cf. again Footnote 7) and satisfies
$R_{g}\in
C^{\infty}(M;\operatorname{End}(\wedge^{2}T^{*}M\otimes{\mathbb{C}}))\cap{\mathfrak{R}}$
and more generally any $R\in
C^{\infty}(M;\operatorname{End}(\wedge^{2}T^{*}M\otimes{\mathbb{C}}))\cap{\mathfrak{R}}$
if it is bounded.

Actually when $R\in
C^{\infty}(M;\operatorname{End}(\wedge^{2}T^{*}M\otimes{\mathbb{C}}))$ the
previous trace formula can be further specified because one can compare the
global trace $\tau(R)$ and the local trace function $x\mapsto{\rm tr}(R(x))$
given by the pointwise traces of the local operators
$R(x):\wedge^{2}T^{*}_{x}M\otimes{\mathbb{C}}\rightarrow\wedge^{2}T^{*}_{x}M\otimes{\mathbb{C}}$
at every $x\in M$. Recall that ${\mathfrak{R}}$ has been constructed as the
weak closure of the Clifford algebra (3). In fact [1, Section 1.1.6] the
universality of ${\mathfrak{R}}$ permits to obtain it from other matrix
algebras too, like for instance from
$\bigcup\limits_{n=0}^{+\infty}{\mathfrak{M}}_{6^{n}}({\mathbb{C}})$ whose
weak closure therefore is again ${\mathfrak{R}}$. By the aid of this altered
construction we can formally start with

$\tau(R)=\lim\limits_{n\rightarrow+\infty}\frac{1}{6^{n}}\sum\limits_{i=1}^{6^{n}}(R\varphi_{i}\>,\>\varphi_{i})_{L^{2}(M,g)}\>\>.$

Fix $n\in{\mathbb{N}}$, write $M_{n}:=\bigcap\limits_{i=1}^{6^{n}}{\rm
supp}\varphi_{i}\subseteqq M$ and take a point $x\in M_{n}$. Since
$\dim_{\mathbb{C}}(\wedge^{2}T^{*}_{x}M\otimes{\mathbb{C}})=\binom{4}{2}=6$
the maximal number of completely disjoint linearly independent sub-$6$-tuples
in $\\{\varphi_{1}(x),\varphi_{2}(x),\dots,\varphi_{6^{n}}(x)\\}$ is equal to
$\frac{6^{n}}{6}=6^{n-1}$. Moreover it follows from Sard’s lemma that with a
generic smooth choice for $\\{\varphi_{1},\varphi_{2},\dots\\}$ the subset of
those points $y\in M_{n}$ where this number is less than $6^{n-1}$ has measure
zero in $M_{n}$ with respect to the measure $\mu_{g}$. Consequently

$\sum\limits_{i=1}^{6^{n}}(R\varphi_{i},\varphi_{i})_{L^{2}(M_{n},g)}=\int\limits_{x\in
M_{n}}\sum\limits_{i=1}^{6^{n}}g(R(x)\varphi_{i}(x),\varphi_{i}(x))\>\mu_{g}(x)=6^{n-1}\int\limits_{x\in
M_{n}}{\rm tr}(R(x))\mu_{g}(x)\>\>.$

Since $\\{\varphi_{1},\varphi_{2},\dots\\}$ is a basis in
$\Omega^{2}_{c}(M;{\mathbb{C}})$ therefore
$M\setminus\bigcup\limits_{n=0}^{+\infty}M_{n}$ has measure zero as well we
can let $n\rightarrow+\infty$ to end up with

$\tau(R)=\frac{1}{6}\int\limits_{M}{\rm tr}(R)\mu_{g}$ (7)

and observe that $\tau$ in this form is nothing else than the generalization
of the total scalar curvature of a Riemannian manifold. Moreover if and only
if (7) exists $R$ extends to ${\mathscr{H}}$ (cf. Footnote (7) as usual) and
gives $R\in
C^{\infty}(M;\operatorname{End}(\wedge^{2}T^{*}M\otimes{\mathbb{C}}))\cap{\mathfrak{R}}$.
So if we start with (4) i.e. the second picture we can use several useful
tracial criteria for checking whether or not an operator in
$\operatorname{End}(\Omega^{2}_{c}(M;{\mathbb{C}}))$ extends to an operator in
${\mathfrak{R}}$.

3\. Now we are ready to exhibit an especially important class of elements in
${\mathfrak{R}}$ through a natural regular embedding

$i_{M}:M\longrightarrow{\mathfrak{R}}$ (8)

of any connected oriented smooth $4$-manifold $M$ into its ${\mathfrak{R}}$ by
the aid of the first picture as follows. To every sufficiently nice closed
subset $\emptyset\subseteqq X\subseteqq M$ there exists an associated linear
subspace
$\Omega^{2}_{c}(M,X;{\mathbb{C}})\subset\Omega^{2}_{c}(M;{\mathbb{C}})\subset
C(M)\subset{\mathscr{H}}$ consisting of compactly supported smooth $2$-forms
vanishing at least along $X$. In this way to every point $x\in M$ one can
attach a subspace $V_{x}\subset{\mathscr{H}}$ defined by using the
multiplication in $C(M)$ and then taking the closure of the
${\mathbb{C}}$-linear subspace

${\mathbb{C}}1+\Omega^{2}_{c}(M,x;{\mathbb{C}})+\Omega^{2}_{c}(M,x;{\mathbb{C}})\Omega^{2}_{c}(M;{\mathbb{C}})+\Omega^{2}_{c}(M,x;{\mathbb{C}})\Omega^{2}_{c}(M;{\mathbb{C}})\Omega^{2}_{c}(M;{\mathbb{C}})+\dots$

within ${\mathscr{H}}$. Let $P_{x}:{\mathscr{H}}\rightarrow V_{x}$ be the
corresponding orthogonal projection. Observe that a priori
$P_{x}\in{\mathfrak{B}}({\mathscr{H}})$ however in fact
$P_{x}\in{\mathfrak{R}}$. This is because $V_{x}$ arises as the image of an
operator $A_{x}\in{\mathfrak{R}}$ which for instance simply looks like
$\omega_{1}\dots\omega_{n}\mapsto\frac{1}{Q(\psi_{x})}\psi_{x}\omega_{1}\dots\omega_{n}$
on the dense subset $C(M)\subset{\mathscr{H}}$ i.e. multiplication from the
left within $C(M)$ by a fixed $2$-form
$\psi_{x}\in\Omega^{2}_{c}(M,x;{\mathbb{C}})$ satisfying $Q(\psi_{x})\not=0$.
Indeed, obviously $A_{x}C(M)\subseteqq V_{x}$ however
$A_{x}(\psi_{x}(V_{x}\cap C(M)))=V_{x}\cap C(M)$ hence
$A_{x}{\mathscr{H}}=V_{x}$. It readily follows that the resulting map in (8)
defined to be $x\mapsto P_{x}$ is injective and continuous in the norm
topology hence gives rise to a continuous embedding of $M$ into
${\mathfrak{R}}$ via projections.

Proceeding further take a connected oriented smooth $4$-manifold $M$ and
consider its embedding into ${\mathfrak{R}}$ via (8). Orientation-preserving
diffeomorphisms act transitively on $M$ thus for any two points $x,y\in M$
there exists $\Phi\in{\rm Diff}^{+}(M)$ such that $\Phi(x)=y$ which implies
$(\Phi^{-1})^{*}V_{x}=V_{y}$ consequently for their corresponding projections
$P_{y}=(\Phi^{-1})^{*}P_{x}\Phi^{*}$ holds hence they are unitary equivalent
(yielding $\tau$ is constant along $i_{M}M\subset{\mathfrak{R}}$). Let
$\\{\gamma(t)\\}_{t\in(-\varepsilon,+\varepsilon)}$ be a smooth curve in $M$
satisfying $\gamma(0)=x\in M$ and $\dot{\gamma}(0)=X\in T_{x}M$; then putting
$P(t):=i_{M}\gamma(t)$ to be the corresponding family of projections in
${\mathfrak{R}}$ we find $P(0)=P_{x}$ and there exists a $1$-parameter family
of diffeomorphisms such that $P(t)=\Phi^{*}_{-t}P_{x}\Phi^{*}_{t}$. Therefore
formally $\dot{P}(0)=[P_{x}\>,\>L_{Z}]$ where
$L_{Z}\in\operatorname{End}(\Omega^{2}_{c}(M;{\mathbb{C}}))$ is the
corresponding Lie derivative representing the infinitesimal generator of
$\\{\Phi_{t}\\}_{t\in(-\varepsilon,+\varepsilon)}$. Without loss of generality
we can assume $\Phi_{t}={\rm id}_{M}$ outside a small neighbourhood of $x\in
M$ hence the support of $Z\in C^{\infty}_{c}(M;TM)$ is also small. Therefore
we can suppose $\tau(L_{Z})<+\infty$ allowing a unique extension
$L_{Z}\in\operatorname{End}(\Omega^{2}_{c}(M;{\mathbb{C}}))\cap{\mathfrak{R}}$
which implies that the commutator exists and
$\dot{P}(0)=[P_{x},L_{Z}]\in{\mathfrak{R}}$. Consequently
$T_{P_{x}}(i_{M}M)\subset{\mathfrak{R}}$ moreover in a manner such that,
taking into account (as a special four dimensional phenomenon) that in a von
Neumann algebra every element is a sum of at most four unitary operators, the
whole tangent space at $x\in M$ is actually spanned over ${\mathbb{R}}$ by
four fixed unitary operators from ${\mathfrak{R}}$ i.e. which depend only on
$x$. Thus having $X\in T_{x}M$ define $i_{M*}X:=\dot{P}(0)\in
T_{P_{x}}(i_{M}M)\subset{\mathfrak{R}}$. Assume that $X\not=0$ hence
$L_{Z}\not=0$ however $\dot{P}(0)=[P_{x},L_{Z}]=0$. This would imply that the
image $V_{x}$ of $P_{x}$ is invariant under $L_{Z}$ however this is not
possible for if e.g. $\varphi_{x}$ is any $2$-form vanishing at $x\in M$ hence
belongs to $V_{x}$ then in general $L_{Z}\varphi_{x}$ does not vanish there
hence is not in $V_{x}$. All of these finally allow us to use the non-
degenerate scalar product $(A,B):=\tau(AB^{\divideontimes})$ on
${\mathfrak{R}}$ which completes it to ${\mathscr{H}}$ as above to obtain a
non-degenerate Riemannian metric $g$ on $M$ by the pullback formula

$g(X,Y):={\rm Re}\big{(}i_{M*}X\>,\>i_{M*}Y\big{)}\>\>.$

Therefore embedding $M$ via (8) into ${\mathfrak{R}}$ canonically enhances it
an oriented Riemannian $4$-manifold $(M,g)$. Observe that if $*\in
C^{\infty}(M;{\rm End}(\wedge^{2}T^{*}M\otimes{\mathbb{C}}))$ denotes the
induced complexified Hodge operator on $(M,g)$ then $\tau(*)=0$ implies
$*\in{\mathfrak{R}}$ and $*^{2}=1$ that it is self-adjoint. Hence $*$ commutes
with the projection $P_{x}\in{\mathfrak{R}}$ for every $x\in M$. To see this,
take ${\mathscr{H}}=V_{x}\oplus V_{x}^{\perp}$ with respect to $P_{x}$; then
$*$ obviously preserves
$\Omega^{2}_{c}(M,x;{\mathbb{C}})\subset\Omega^{2}_{c}(M;{\mathbb{C}})$ thus
if $A_{x}\in{\mathfrak{R}}$ is as before the multiplication from the left by
the element $\psi_{x}\in\Omega^{2}_{c}(M,x;{\mathbb{C}})$ hence
$A_{x}{\mathscr{H}}=V_{x}$ then $B_{x}:=*A_{x}\in{\mathfrak{R}}$ is
multiplication from the left with
$*\psi_{x}\in\Omega^{2}_{c}(M,x;{\mathbb{C}})$ hence
$B_{x}{\mathscr{H}}=V_{x}$ too consequently
$*V_{x}=*A_{x}{\mathscr{H}}=B_{x}{\mathscr{H}}=V_{x}$; likewise
$0=(V_{x},V_{x}^{\perp})=(*^{2}V_{x},V_{x}^{\perp})=(*V_{x},*V_{x}^{\perp})=(V_{x},*V_{x}^{\perp})$
consequently $*V_{x}^{\perp}=V_{x}^{\perp}$ yielding $[P_{x},*]=0$. Finally we
remark that (8) is analogous to embedding Riemannian manifolds into Hilbert
spaces via heat kernel techniques [2].

Having understood $M\subset{\mathfrak{R}}$ for a given $4$-manifold let us
compare these embeddings for different spaces. So let $M,N$ be two connected
oriented smooth $4$-manifolds and consider their corresponding embeddings into
their von Neumann algebras via (8) respectively. Regardless what $M$ or $N$
are, their abstractly given algebras are both hyperfinite factors of ${\rm
II}_{1}$-type, therefore these latter structures are isomorphic [1, Theorem
11.2.2] however not in a canonical fashion. Indeed, if
$F^{\prime}:{\mathfrak{R}}\rightarrow{\mathfrak{R}}$ is an isomorphism between
the ${\mathfrak{R}}$’s for $M$ and $N$ respectively then any other isomorphism
between them has the form $F^{\prime\prime}=\beta^{-1}F^{\prime}\alpha$ where
$\alpha$ and $\beta$ are inner automorphisms of the abstractly given
${\mathfrak{R}}$ for $M$ and the abstractly given ${\mathfrak{R}}$ for $N$
respectively.888If one wants to consider ${\mathfrak{R}}$ not abstractly i.e.
in itself but spatially i.e. together with
${\mathfrak{R}}\subset{\mathfrak{B}}({\mathscr{H}})$ provided by its
construction then in this case isomorphisms are to be parameterized not only
by inner automorhisms of ${\mathfrak{R}}$ but by all automorphisms of
${\mathfrak{R}}\subset{\mathfrak{B}}({\mathscr{H}})$. However one can prove
[21] that inner automorphisms are dense in the ${\rm strong}^{*}$-topology in
this larger group hence in this sense all outer automorphisms of
${\mathfrak{R}}\subset{\mathfrak{B}}({\mathscr{H}})$ can be approximated by
inner ones of ${\mathfrak{R}}$. For further characterization of inner
automorphisms cf. [24]. Therefore to understand the freedom how operator
algebras for different $4$-manifolds are identified we need a description of
the inner automorphism group of ${\mathfrak{R}}$.

Since $C(M)$ is weakly dense in ${\mathfrak{R}}$ by construction and the
former is invariant against the inner automorphisms of the latter, it is
enough to understand how the inner automorphisms of $C(M)$ look like. However
these are simply conjugations with unitaries from $C(M)$ and referring to (4)
they belong to ${\rm Aut}(\Omega^{2}_{c}(M;{\mathbb{C}}))\cap C(M)$ where the
former component is the group of all invertible ${\mathbb{C}}$-linear
transformations of $\Omega^{2}_{c}(M;{\mathbb{C}})$. If $x\in M$ recall that
$\Omega^{2}_{c}(M,x;{\mathbb{C}})\subset\Omega^{2}_{c}(M;{\mathbb{C}})$ is a
complex subspace and it is easy to show by invertability that subspaces of
this kind are permuted by elements of ${\rm
Aut}(\Omega^{2}_{c}(M;{\mathbb{C}}))$. Consequently any member of ${\rm
Aut}(\Omega^{2}_{c}(M;{\mathbb{C}}))$ induces an orientation-preserving
diffeomorphism $\Phi$ of $M$ and a fiberwise ${\mathbb{C}}$-linear
diffeomorphism $f$ of $\wedge^{2}T^{*}M\otimes{\mathbb{C}}$ such that

$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\wedge^{2}T^{*}M\otimes{\mathbb{C}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\scriptstyle{f}$$\textstyle{\wedge^{2}T^{*}M\otimes{\mathbb{C}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi}$$\textstyle{M}$

is commutative i.e. yields a bundle isomorphism of
$\wedge^{2}T^{*}M\otimes{\mathbb{C}}$. Conversely it is straightforward that
every bundle isomorphism of $\wedge^{2}T^{*}M\otimes{\mathbb{C}}$ gives rise
to an element of ${\rm Aut}(\Omega^{2}_{c}(M;{\mathbb{C}}))$. Thus there
exists an isomorphism of groups ${\rm
Aut}(\Omega^{2}_{c}(M;{\mathbb{C}}))\cong{\rm
Iso}(\wedge^{2}T^{*}M\otimes{\mathbb{C}})$. Consider the short exact sequence
$1\rightarrow{\mathscr{G}}({\wedge^{2}T^{*}M\otimes{\mathbb{C}}})\rightarrow{\rm
Iso}(\wedge^{2}T^{*}M\otimes{\mathbb{C}}){\rightarrow}{\rm
Diff}^{+}(M)\rightarrow 1$ involving the fiberwise automorphism group (the
gauge group) ${\mathscr{G}}({\wedge^{2}T^{*}M\otimes{\mathbb{C}}})\cong
C^{\infty}(M;{\rm Aut}(\wedge^{2}T^{*}M\otimes{\mathbb{C}}))$ the global
automorphism group ${\rm Iso}(\wedge^{2}T^{*}M\otimes{\mathbb{C}})\cong{\rm
Aut}(C^{\infty}(M;\wedge^{2}T^{*}M\otimes{\mathbb{C}}))={\rm
Aut}(\Omega^{2}_{c}(M;{\mathbb{C}}))$, both of the vector bundle
$\wedge^{2}T^{*}M\otimes{\mathbb{C}}$, and the diffeomorphism group ${\rm
Diff}^{+}(M)$ of the underlying space $M$ respectively. This short exact
sequence can therefore be re-written as

$1\longrightarrow C^{\infty}(M;{\rm
Aut}(\wedge^{2}T^{*}M\otimes{\mathbb{C}}))\longrightarrow{\rm
Aut}(\Omega^{2}_{c}(M;{\mathbb{C}}))\longrightarrow{\rm
Diff}^{+}(M)\longrightarrow 1\>\>.$ (9)

In addition the map $\Phi\mapsto(\Phi^{-1})^{*}$ gives rise to group injection
${\rm Diff}^{+}(M)\rightarrow{\rm Aut}(\Omega^{2}_{c}(M;{\mathbb{C}}))$
consequently (9) can be supplemeted to

$1\longrightarrow C^{\infty}(M;{\rm
Aut}(\wedge^{2}T^{*}M\otimes{\mathbb{C}}))\longrightarrow{\rm
Aut}(\Omega^{2}_{c}(M;{\mathbb{C}}))\>\>^{\longrightarrow}_{\longleftarrow}\>\>{\rm
Diff}^{+}(M)\longrightarrow 1$

inducing a set-theoretical splitting which means that at least as a set there
exists a decomposition ${\rm
Aut}(\Omega^{2}_{c}(M;{\mathbb{C}}))=C^{\infty}(M;{\rm
Aut}(\wedge^{2}T^{*}M\otimes{\mathbb{C}}))\>{\rm Diff}^{+}(M)$. Therefore an
inner automotphism $\alpha$ of $C(M)$ as a conjugation with an element of
${\rm Aut}(\Omega^{2}_{c}(M;{\mathbb{C}}))$ admits a unique splitting
$\alpha={\rm Ad}_{g}{\rm Ad}_{(\Phi^{-1})^{*}}$ where $g$ is a gauge
transformation of $\wedge^{2}T^{*}M\otimes{\mathbb{C}}$ hence leaves $M$
pointwise fixed and $\Phi$ is an orientation-preserving diffeomorphism of $M$.
Likewise, $\beta={\rm Ad}_{h}{\rm Ad}_{(\Psi^{-1})^{*}}$ is the shape of an
inner automorphism of $C(N)$. These make sure that given an isomorphism
$F^{\prime}:{\mathfrak{R}}\rightarrow{\mathfrak{R}}$ between the von Neumann
algebras for $M$ and $N$ respectively then any other isomorphism has the form
$F^{\prime\prime}={\rm Ad}_{\Psi^{*}}({\rm Ad}_{h^{-1}}F^{\prime}{\rm
Ad}_{g}){\rm Ad}_{(\Phi^{-1})^{*}}$ between them. Consequently isomorphisms
between a pair of abstractly given von Neumann algebras differ only by inner
automorphisms which preserve their underlying $4$-manifolds as embedded within
their algebras respectively; i.e. differences between identifications are
inessential in this sense. Our overall conclusion therefore is that up to
diffeomorphisms every connected oriented smooth $4$-manifold $M$ admits an
embedding into a commonly given abstract von Neumann algebra ${\mathfrak{R}}$
via (8) and these embeddings induce Riemannian structures $(M,g)$ on them.

4\. We close the partial comprehension of ${\mathfrak{R}}$ with an observation
regarding its general structure. The shape of (9) at the Lie algebra level
looks like

$0\longrightarrow C^{\infty}(M;{\rm
End}(\wedge^{2}T^{*}M\otimes{\mathbb{C}}))\longrightarrow{\rm
End}(\Omega^{2}_{c}(M;{\mathbb{C}}))\longrightarrow{\rm Lie}({\rm
Diff}^{+}(M))\longrightarrow 0\>\>.$

We already have an embedding (4). In addition to this there exists an
isomorphism of Lie algebras $L:C^{\infty}_{c}(M;TM)\rightarrow{\rm Lie}({\rm
Diff}^{+}(M))$ such that $X\mapsto L_{X}$ is nothing but taking Lie derivative
with respect to a compactly supported real vector field where the first-order
${\mathbb{C}}$-linear differential operator $L_{X}$ is supposed to act on
$2$-forms hence ${\rm Lie}({\rm
Diff}^{+}(M))\subset\operatorname{End}(\Omega^{2}_{c}(M;{\mathbb{C}}))$ as we
know already too. Therefore the intersection of this sequence with
$C(M)\subset{\rm End}(\Omega^{2}_{c}(M;{\mathbb{C}}))$ is meaningful and gives

$0\longrightarrow C^{\infty}(M;{\rm
End}(\wedge^{2}T^{*}M\otimes{\mathbb{C}}))\cap C(M)\longrightarrow
C(M)\longrightarrow{\rm Lie}({\rm Diff}^{+}(M))\cap C(M)\longrightarrow
0\>\>.$

The second term consists of fiberwise algebraic hence local operators having
finite trace (7) and likewise the fourth term consists Lie derivatives having
finite trace via (6) thus belongs to the class of local operators too. Since
the vector spaces underlying $C(M)$ considered either as an associative or a
Lie algebra are isomorphic we conclude, as an important structural
observation, that the overall construction here is geometric in the sense that
the algebra ${\mathfrak{R}}$ is generated by local operators.

Summing up all of our findings so far: ${\mathfrak{R}}$ is a hyperfinite
factor von Neumann algebra of ${\rm II}_{1}$-type associated to $M$ such that
the solely input in its construction has been the pairing (2). Hence
${\mathfrak{R}}$ depends only on the orientation and the smooth structure of
$M$ in a functorial way. It contains, certainly among many other non-local
operators, the space $M$ itself as projections, its orientation-preseving
diffeomorphisms as well as its space of bounded algebraic curvature tensors.
Nevertheless ${\mathfrak{R}}$ is geometric in the sense that it is generated
by $M$’s local operators alone. It is remarkable that despite the plethora of
smooth $4$-manifolds detected since the 1980’s their associated von Neumann
algebras here are unique offering a sort of justification terming
${\mathfrak{R}}$ as “universal”. Moreover one is allowed to say that every
connected oriented smooth $4$-manifold $M$ togeter with its curvature tensor
embeds into a common ${\mathfrak{R}}$ and to look upon this von Neumann
algebra as a natural common non-commutative space generalization of all
oriented smooth $4$-manifolds or all $4$-geometries. This universality also
justifies the simple notation ${\mathfrak{R}}$ used throughout the text.

A new representation of the algebra ${\mathfrak{R}}$. After these preliminary
considerations we are in a position to exhibit lot of new representations of
the hyperfinite ${\rm II}_{1}$ factor. The following lemma is repeated
verbatim from [13, Lemma 2.1]:

######  Lemma 3.1.

Let $M$ be a connected oriented smooth $4$-manifold and ${\mathfrak{R}}$ its
von Neumann algebra with trace $\tau$ as before. Then there exists a complex
separable Hilbert space ${\mathscr{I}}(M)^{\perp}$ and a representation
$\rho_{M}:{\mathfrak{R}}\rightarrow{\mathfrak{B}}({\mathscr{I}}(M)^{\perp})$
with the following properties. If
$\pi:{\mathfrak{R}}\rightarrow{\mathfrak{B}}({\mathscr{H}})$ is the standard
representation constructed above then
$\\{0\\}\subseteqq{\mathscr{I}}(M)^{\perp}\subsetneqq{\mathscr{H}}$ and
$\rho_{M}=\pi|_{{\mathscr{I}}(M)^{\perp}}$ holds; therefore, although
$\rho_{M}$ can be the trivial representation, it is surely not unitary
equivalent to the standard representation. Moreover the unitary equivalence
class of $\rho_{M}$ is invariant under orientation-preserving diffeomorphisms
of $M$.

Thus the Murray–von Neumann coupling constant999Also called the
${\mathfrak{R}}$-dimension of a left ${\mathfrak{R}}$-module hence denoted
$\dim_{{\mathfrak{R}}}$, cf. [1, Chapter 8]. of $\rho_{M}$ is invariant under
orientation-preserving diffeomorphisms. Writing
$P_{M}:{\mathscr{H}}\rightarrow{\mathscr{I}}(M)^{\perp}$ for the orthogonal
projection the coupling constant is equal to
$\tau(P_{M})\in[0,1)\subset{\mathbb{R}}_{+}$ consequently
$\gamma(M):=\tau(P_{M})$ is a smooth $4$-manifold invariant.

###### Proof.

First let us exhibit a representation of ${\mathfrak{R}}$; this construction
is inspired by the general Gelfand–Naimark–Segal technique however exploits
the special features of our construction so far as well. Pick a pair
$(\Sigma,\omega)$ consisting of an (immersed) closed orientable surface
$\Sigma\looparrowright M$ with induced oriantation and
$\omega\in\Omega^{2}_{c}(M;{\mathbb{C}})$ which is also closed i.e., ${\rm
d}\omega=0$. Consider the differential geometric ${\mathbb{C}}$-linear
functional $F_{\Sigma,\omega}:{\mathfrak{R}}\rightarrow{\mathbb{C}}$ by
continuously extending

$A\longmapsto\int\limits_{\Sigma}A\omega$

from $\operatorname{End}(\Omega^{2}_{c}(M;{\mathbb{C}}))\cap{\mathfrak{R}}$.
This extension is unique because
$\operatorname{End}(\Omega^{2}_{c}(M;{\mathbb{C}}))\cap{\mathfrak{R}}$ is
norm-dense in ${\mathfrak{R}}$. In case of $F_{\Sigma,\omega}(1)\not=0$ let
$\\{0\\}\subseteqq I(M)\subseteqq{\mathfrak{R}}$ be the subset of elements
$A\in{\mathfrak{R}}$ satisfying $F_{\Sigma,\omega}(A^{\divideontimes}A)=0$. In
fact for all pairs $(\Sigma,\omega)$ we obviously find $\\{0\\}\subsetneqq
I(M)$ and $I(M)\cap{\mathbb{C}}1=\\{0\\}$ hence
$I(M)\subsetneqq{\mathfrak{R}}$ too. We assert that $I(M)$ is a multiplicative
left-ideal in ${\mathfrak{R}}$ which is independent of $(\Sigma,\omega)$. In
the case of $F_{\Sigma,\omega}(1)=0$ we put $I(M)={\mathfrak{R}}$ hence it is
again independent of $(\Sigma,\omega)$ but trivially in this way.

Consider the case when $F_{\Sigma,\omega}(1)\not=0$. Then we can assume that
$F_{\Sigma,\omega}(1)=1$ hence $F_{\Sigma,\omega}$ is a positive functional;
applications of the standard inequality
$|F_{\Sigma,\omega}(A^{\divideontimes}B)|^{2}\leqq
F_{\Sigma,\omega}(A^{\divideontimes}A)F_{\Sigma,\omega}(B^{\divideontimes}B)$
show that $I_{\Sigma,\omega}(M)$ defined by the elements satisfying
$F_{\Sigma,\omega}(A^{\divideontimes}A)=0$ is a multiplicative left-ideal in
${\mathfrak{R}}$. Concerning its $\omega$-dependence, let
$\omega^{\prime}\in\Omega^{2}_{c}(M;{\mathbb{C}})$ be another closed $2$-form
having the property $F_{\Sigma,\omega^{\prime}}(1)=1$; since neither $\omega$
nor $\omega^{\prime}$ are identically zero, we can pick an invertible operator
$T\in{\rm Aut}(\Omega^{2}_{c}(M;{\mathbb{C}}))$ satisfying
$\omega^{\prime}|_{\Sigma}=T\omega|_{\Sigma}$. Then by
$F_{\Sigma,\omega^{\prime}}(A^{\divideontimes}A)=F_{\Sigma,\omega}(A^{\divideontimes}AT)$
and applying the above inequality in the form
$|F_{\Sigma,\omega^{\prime}}(A^{\divideontimes}AT)|^{2}\leqq
F_{\Sigma,\omega}(A^{\divideontimes}A)F_{\Sigma,\omega}((AT)^{\divideontimes}(AT))$
we find $I_{\Sigma,\omega^{\prime}}(M)\subseteqq I_{\Sigma,\omega}(M)$.
Likewise, making use of
$F_{\Sigma,\omega}(A^{\divideontimes}A)=F_{\Sigma,\omega^{\prime}}(A^{\divideontimes}AT^{-1})$
and $|F_{\Sigma,\omega}(A^{\divideontimes}AT^{-1})|^{2}\leqq
F_{\Sigma,\omega^{\prime}}(A^{\divideontimes}A)F_{\Sigma,\omega^{\prime}}((AT^{-1})^{\divideontimes}(AT^{-1}))$
imply the converse inequality $I_{\Sigma,\omega^{\prime}}(M)\supseteqq
I_{\Sigma,\omega}(M)$. Consequently
$I_{\Sigma,\omega^{\prime}}(M)=I_{\Sigma,\omega}(M)$.

Concerning the general $(\Sigma,\omega)$-dependence of $I_{\Sigma,\omega}$ we
argue as follows. Let $\eta_{\Sigma}\in\Omega^{2}(M;{\mathbb{R}})$ be a closed
real $2$-form representing the Poincaré-dual $[\eta_{\Sigma}]\in
H^{2}(M;{\mathbb{R}})$ of $\Sigma\looparrowright M$. Equip $M$ with an
arbitrary Riemannian metric $g$; since the corresponding Hodge operator $*$ is
an isomorphism on $\wedge^{2}T^{*}M$ we can take a $2$-form $\varphi_{\Sigma}$
such that $\eta_{\Sigma}=*\varphi_{\Sigma}$. Then via
$\int_{\Sigma}\omega=\int_{M}\omega\wedge\overline{\eta}_{\Sigma}=\int_{M}\omega\wedge*\overline{\varphi}_{\Sigma}=(\omega,\varphi_{\Sigma})_{L^{2}(M,g)}$
the functional can be re-expressed as
$F_{\Sigma,\omega}(A^{\divideontimes}A)=(A^{\divideontimes}A\omega,\varphi_{\Sigma})_{L^{2}(M,g)}$
in terms of the corresponding definite sesquilinear $L^{2}$-scalar product on
$(M,g)$. Let $\Sigma^{\prime}\looparrowright M$ be another closed surface and
$\omega^{\prime}$ another closed $2$-form such that
$F_{\Sigma^{\prime},\omega^{\prime}}(1)=1$. Altering $\omega$ and
$\omega^{\prime}$ along $\Sigma$ and $\Sigma^{\prime}$ respectively if
necessary (which has no effect on $I(M)$ as we have seen) we can pick some
compactly supported $2$-form $\Omega$ on $M$ such that
$\Omega|_{\Sigma}=\omega$ and $\Omega|_{\Sigma^{\prime}}=\omega^{\prime}$.
Moreover as before take a representative
$*\varphi_{\Sigma^{\prime}}\in\Omega^{2}(M;{\mathbb{R}})$ for the Poincaré-
dual of $\Sigma^{\prime}$. We can use again $T\in{\rm
Aut}(\Omega^{2}_{c}(M;{\mathbb{C}}))$ satisfying
$\varphi_{\Sigma^{\prime}}=T\varphi_{\Sigma}$. Thus

$F_{\Sigma^{\prime},\omega^{\prime}}(A^{\divideontimes}A)=(A^{\divideontimes}A\Omega,\varphi_{\Sigma^{\prime}})_{L^{2}(M,g)}=(A^{\divideontimes}A\Omega,T\varphi_{\Sigma})_{L^{2}(M,g)}=(T^{\divideontimes}A^{\divideontimes}A\Omega,\varphi_{\Sigma})_{L^{2}(M,g)}=F_{\Sigma,\omega}(T^{\divideontimes}A^{\divideontimes}A)$

together with
$|F_{\Sigma,\omega}(T^{\divideontimes}A^{\divideontimes}A)|^{2}\leqq
F_{\Sigma,\omega}\big{(}(AT)^{\divideontimes}AT\big{)}F_{\Sigma,\omega}(A^{\divideontimes}A)$
demonstrate that $I_{\Sigma^{\prime},\omega^{\prime}}(M)\subseteqq
I_{\Sigma,\omega}(M)$. In the same fashion
$F_{\Sigma,\omega}(A^{\divideontimes}A)=F_{\Sigma^{\prime},\omega^{\prime}}((T^{-1})^{\divideontimes}A^{\divideontimes}A)$
together with the corresponding inequality convinces us that
$I_{\Sigma^{\prime},\omega^{\prime}}(M)\supseteqq I_{\Sigma,\omega}(M)$. Thus
$I_{\Sigma^{\prime},\omega^{\prime}}(M)=I_{\Sigma,\omega}(M)$.

Secondly if $(\Sigma,\omega)$ is such that $F_{\Sigma,\omega}(1)=0$ then by
definition $I_{\Sigma,\omega}(M)={\mathfrak{R}}$. Therefore if
$(\Sigma^{\prime},\omega^{\prime})$ is another pair with
$F_{\Sigma^{\prime},\omega^{\prime}}(1)=0$ then
$I_{\Sigma,\omega}(M)=I_{\Sigma^{\prime},\omega^{\prime}}(M)$ (and equal to
${\mathfrak{R}}$). We are now convinced that it is correct to write
$I_{\Sigma,\omega}(M)$ as $I(M)$. In summary it satisfies $\\{0\\}\subsetneqq
I(M)\subseteqq{\mathfrak{R}}$.

Let us proceed further by exploiting now the observation made during the
construction of ${\mathfrak{R}}$ that it acts on a Hilbert space
${\mathscr{H}}$ with scalar product $(\>\cdot\>,\>\cdot\>)$ by the
representation $\pi$ i.e., multiplication from the left. In fact, since
$(\hat{A},\hat{B})=\tau(AB^{\divideontimes})$ we see that $\pi$ is the
standard representation. Consider the space $\\{0\\}\subsetneqq
I(M)I(M)^{\divideontimes}\subseteqq{\mathfrak{R}}$ consisting of all finite
sums $A_{1}B_{1}+\dots+A_{k}B_{k}\in{\mathfrak{R}}$ where $A_{i}\in I(M)$ and
similarly $B_{j}\in I(M)^{\divideontimes}$. It gives rise to a closed linear
subspace $\\{0\\}\subsetneqq{\mathscr{I}}(M)\subseteqq{\mathscr{H}}$ by taking
the closure of $C(M)\cap I(M)I(M)^{\divideontimes}$ within
${\mathscr{H}}\supset C(M)$. Therefore ${\mathscr{I}}(M)$ is a well-defined
closed subspace of ${\mathscr{H}}$ which is non-trivial if
$F_{\Sigma,\omega}(1)\not=0$ and coincides with ${\mathscr{H}}$ whenever
$F_{\Sigma,\omega}(1)=0$. Take its orthogonal complementum
$\\{0\\}\subseteqq{\mathscr{I}}(M)^{\perp}\subsetneqq{\mathscr{H}}$. Note that
${\mathscr{I}}(M)^{\perp}$ is isomorphic to ${\mathscr{H}}/{\mathscr{I}}(M)$.
Taking into account that the subset $I(M)$ is a multiplicative left-ideal
$\pi:{\mathfrak{R}}\rightarrow{\mathfrak{B}}({\mathscr{H}})$ given by left-
multiplication restricts to ${\mathscr{I}}(M)$ but even more, since the scalar
product on ${\mathscr{H}}$ satisfies the identity
$(\widehat{AB},\widehat{C}\>)=(\hat{B},\widehat{A^{\divideontimes}C})$ the
standard representation restricts to ${\mathscr{I}}(M)^{\perp}$ as well. This
latter representation is either a unique non-trivial representation if
${\mathscr{I}}(M)^{\perp}\not=\\{0\\}$ (provided by a functional over $M$ with
$F_{\Sigma,\omega}(1)\not=0$ if exists), or the trivial one if
${\mathscr{I}}(M)^{\perp}=\\{0\\}$ (provided by a functional with
$F_{\Sigma,\omega}(1)=0$ which always exists). Keeping these in mind, for a
given $M$ we define

$\rho_{M}:{\mathfrak{R}}\rightarrow{\mathfrak{B}}({\mathscr{I}}(M)^{\perp})\>\>\>\mbox{to
be}\>\>\>\left\\{\begin{array}[]{ll}\mbox{$\pi|_{{\mathscr{I}}(M)^{\perp}}$ on
${\mathscr{I}}(M)^{\perp}\not=\\{0\\}$ if possible,}\\\
\mbox{$\pi|_{{\mathscr{I}}(M)^{\perp}}$ on ${\mathscr{I}}(M)^{\perp}=\\{0\\}$
otherwise.}\end{array}\right.$

The choice is unambigously determined by the topology of $M$ (see the Remark
below).

From the general theory [1, Chapter 8] we know that if
$P_{M}:{\mathscr{H}}\rightarrow{\mathscr{I}}(M)^{\perp}$ is the orthogonal
projection then $P_{M}\in{\mathfrak{R}}$ because ${\mathscr{H}}$ is the
standard ${\mathfrak{R}}$-module. The Murray–von Neumann coupling constant of
$\rho_{M}$ depends only on the unitary equivalence class of $\rho_{M}$ and is
equal to $\tau(P_{M})\in[0,1]$. However observing that $\rho_{M}$ is surely
not isomorphic to $\pi$ since ${\mathscr{I}}(M)$ is never trivial the case
$\tau(P_{M})=1$ is excluded i.e., $\tau(P_{M})\in[0,1)$. Let
$\Phi:M\rightarrow M$ be an orientation-preserving diffeomorphism. It induces
an inner automorphism $A\mapsto(\Phi^{-1})^{*}A\Phi^{*}$ of ${\mathfrak{R}}$.
Taking into account that the scalar product on ${\mathscr{H}}$ is induced by
the trace which is invariant under cyclic permutations this inner automorphism
is unitary. Moreover it transforms $I_{\Sigma,\omega}(M)$ into
$I_{\Sigma^{\prime},\omega^{\prime}}(M)=I_{\Phi(\Sigma),(\Phi^{-1})^{*}\omega}(M)$
hence $F_{\Sigma,\omega}(1)=0$ if and only if
$F_{\Sigma^{\prime},\omega^{\prime}}(1)=0$ consequently the Hilbert space
${\mathscr{I}}(M)^{\perp}$ is invariant under $\Phi$. Thus $\Phi$ transforms
$\rho_{M}$ into a new representation $(\Phi^{-1})^{*}\rho_{M}\Phi^{*}$ on
${\mathscr{I}}(M)^{\perp}$ which is unitary equivalent to $\rho_{M}$.

We conclude that $\gamma(M):=\tau(P_{M})\in[0,1)$ is a smooth invariant of $M$
as stated. ∎

######  Remark.

Note that $\gamma(M)=0$ corresponds to the situation when $\rho_{M}$ is the
trivial representation on ${\mathscr{I}}(M)^{\perp}=\\{0\\}$. To avoid this we
have to demand $F_{\Sigma,\omega}(1)\not=0$ which by the closedness
assumptions on $\Sigma\looparrowright M$ and
$\omega\in\Omega^{2}_{c}(M;{\mathbb{C}})$ is in fact a topological condition:
it is equivalent that

$\frac{1}{2\pi\sqrt{-1}}F_{\Sigma,\omega}(1)=\frac{1}{2\pi\sqrt{-1}}\int\limits_{\Sigma}\omega=\langle[\Sigma],[\omega]\rangle\in{\mathbb{C}}$

as a pairing of $[\Sigma]\in H_{2}(M;{\mathbb{Z}})$ and $[\omega]\in
H^{2}(M;{\mathbb{C}})$ in homology is not trivial. Hence $\gamma(M)=0$ iff
$H_{2}(M;{\mathbb{C}})=H_{2}(M;{\mathbb{Z}})\otimes{\mathbb{C}}=\\{0\\}$ (or
equivalently,
$H^{2}(M;{\mathbb{C}})=H^{2}(M;{\mathbb{Z}})\otimes{\mathbb{C}}=\\{0\\}$).
Thus unfortunately $\gamma(M)=0$ for all acyclic or aspherical manifolds
(including homology $4$-spheres). Examples in the simply connected case are
$M=S^{4},{\mathbb{R}}^{4},R^{4}$ (this latter is any exotic or fake
${\mathbb{R}}^{4}$) while $M=S^{3}\times S^{1}$ is a non-simply connected
example. Nevertheless $\gamma(M)$ is not always trivial and some of the
topological properties of this new smooth $4$-manifold invariant have been
investigated in [13, Theorems 1.1-3].

Proof of Theorem 2.1. Parts (i) and (ii) follow from the material presented in
Section 3 while part (iii) from Lemma 9. $\square$

Acknowledgements. All the not-referenced results in this work are fully the
author’s own contribution. There are no conflict of interest to declare that
are relevant to the content of this article. The work meets all ethical
standards applicable here. No funds, grants, or other financial supports were
received. Data sharing is not applicable to this article as no datasets were
generated or analysed during the underlying study.

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