Application 01/04 to the Addendum Nu. 1 of the Protocol on Scientific Cooperation between the Austrian Academy of Sciences and the National Academy of Sciences of Ukraine concluded on February 7, 1996 Multilateral research project Quantum Gravity, Cosmology and Categorification Contents 1 Status of Research 1 1.1 2D dilaton quantum gravity ......................... 1 1.2 Non-commutative geometry ......................... 4 1.3 Cosmology ................................... 5 1.4 Topological quantum field theories ...................... 6 1.5 Categorification of quantum gravity ..................... 11 2 Aim of the Project and Work Plan 16 3 Personnel, Rearch Institutes and Funds 17 i
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At the moment, cosmology is one of the fastest changing fields in physics. This fact
might, on the one hand, be ascribed to the vast amount of new observational data
(cf [1–3] e.g.), on the other hand there are still fundamental open questions within what
is nowadays called the cosmological standard model [4,5]: Where does the inflaton field
come from? Is there something like the cosmological constant Λ which contributes to
the dark energy etc.?
From a theoretical viewpoint, one might divide efforts today within cosmology into
two broad subclasses.
Firstly, we have models which extend the standard model to a certain amount, inflation
[6, 7], e.g., can be viewed as an add-on for the classical Friedman-Lemaıtre-Robertson-
Walker (FLRW) model. All these models have in common that they do not affect the
structure of space-time itself, i.e., they are still bound to a four-dimensional Riemannian
space-time and, in addition, do not modify the underlying gravity theory, i.e., General
Relativity (GR).
Secondly, we have models which are no longer tied to Riemannian space-time and might
also modify the underlying gravity theory.
1.1 2D dilaton quantum gravity
One example for the latter is the generalised gravity theories in two dimensions. Models
of gravity in two dimensions have been largely studied in recent years as toy models
addressing issues that are too complex to be faced directly in four dimensions. However,
the dynamics of two-dimensional gravity is rather different from its four-dimensional
counterpart, since the Hilbert-Einstein action is a topological invariant in two dimensions
and hence gives rise to trivial field equations. In order to derive the field equations
from an action principle it is then necessary to introduce an auxiliary scalar field η
(which in the following will be called dilaton) [8]. Recent astrophysical data suggest
that the Hilbert-Einstein action may need to be modified by rather non-standard terms
1
(1/R and even ln R were considered as possibilities, see [9]). Dilaton theories provide
a natural framework for such modifications, and two dimensional models are, as usual,
a convenient test ground. Such models, the most prominent being the dilaton gravity
of Jackiw and Teitelboim (JT) [8, 10–13], represent linear gauge theories. An excellent
summary (containing also a more comprehensive list of references on literature before
1988) is contained in the textbook of Brown [14]. Among those models spherically
reduced gravity (SRG), the truncation of D = 4 gravity to its s-wave part, possesses
perhaps the most direct physical motivation. One can either treat this system directly
in D = 4 and impose spherical symmetry in the equations of motion (e.o.m.-s) [15]
or impose spherical symmetry already in the action [15–25], thus obtaining a dilaton
theory. The rekindled interest in generalised dilaton theories in D = 2 (henceforth
GDTs) started in the early 1990-s, triggered by the string inspired [26–33] dilaton black
hole model1, studied in the influential paper of Callan, Giddings, Harvey and Strominger
(CGHS) [35]. At approximately the same time it was realized that 2D dilaton gravity
can be treated as a non-linear gauge-theory [37,38]. As already suggested by earlier work,
all GDTs considered so far could be extracted from a second order dilaton action [39,40].
A common feature of these classical treatments of models with and without torsion is
the almost exclusive use2 of the gauge-fixing for the D = 2 metric familiar from string
theory, namely the conformal gauge. Then the e.o.m.-s become complicated partial
differential equations. The determination of the solutions, which turns out to be always
possible in the matterless case, for nontrivial dilaton field dependence usually requires
considerable mathematical effort. The same had been true for the first papers on theories
with torsion [42, 43]. However, in that context it was realized soon that gauge-fixing is
not necessary, because the invariant quantities R and T aTa themselves may be taken
as variables in the Katanaev-Volovich (KV)-model [44–47]. This approach has been
extended to general theories with torsion3. As a matter of fact, in GR many other gauge-
fixings for the metric have been well-known for a long time: the Eddington-Finkelstein
1A textbook-like discussion of this model can be found in refs. [34, 36].2A notable exception is Polyakov [41].3A recent review of this approach is provided by Hehl and Obukhov [48].
2
(EF) gauge, the Painleve-Gullstrand gauge, the Lemaitre gauge etc. . As compared to
the “diagonal” gauges like the conformal and the Schwarzschild type gauge, they possess
the advantage that coordinate singularities can be avoided, i.e. the singularities in those
metrics are essentially related to the “physical” ones in the curvature. It was shown for
the first time in [49] that the use of a temporal gauge for the Cartan variables in the
(matterless) KV-model made the solution extremely simple. This gauge corresponds
to the EF gauge for the metric. Soon afterwards it was realized that the solution
could be obtained even without previous gauge-fixing, either by guessing the Darboux
coordinates [50] or by direct solution of the e.o.m.-s [51]. Then the temporal gauge of [49]
merely represents the most natural gauge fixing within this gauge-independent setting.
The basis of these results had been a first order formulation of D = 2 covariant theories
by means of a covariant Hamiltonian action in terms of the Cartan variables and further
auxiliary fields Xa which (beside the dilaton field X) take the role of canonical momenta.
They cover a very general class of theories comprising not only the KV-model, but also
more general theories with torsion4. The most attractive feature of such theories is that
an important subclass of them is in a one-to-one correspondence with the GDT-s. This
dynamical equivalence, including the essential feature that also the global properties are
exactly identical, seems to have been noticed first in [52] and used extensively in studies
of the corresponding quantum theory [53–55]. In the latter the temporal gauge again
prevaricates complications from Faddeev-Popov ghosts [56] which are present otherwise.
Generalizing the first order formulation to the much more comprehensive class of
“Poisson-Sigma models” [57, 58] on the one hand helped to explain the deeper reasons
of the advantages from the use of the first oder version, on the other hand led to very
interesting applications in other fields [59], including especially also string theory [60,
61]. Recently this approach was shown to represent a very direct route to 2D dilaton
supergravity [62] without auxiliary fields. For more technical and historical details on
dilaton gravity the review [63] may be consulted.
4In that case there is the restriction that it must be possible to eliminate all auxiliary fields Xa and
X .
3
There are also purely two-dimensional reasons to look for generalisations of the dilaton
gravities. It has been demonstrated [64] that the exact string black hole [29] cannot be
embedded in generalised dilaton gravities. Later Strobl [65] suggested a very general
framework for topological gravity theories. However, the relation between these theories
and metric theories of gravity is not clear. The methods of non-perturbative treatment
of classical and quantum dilaton gravities have been developed for pseudo-Euclidean
spaces. The Euclidean regime differs considerably from the pseudo-Euclidean one since,
for example, different asymptotic conditions and different gauge conditions have to be
used. Another example is non-commutative gravity in two dimensions which leads to
the second main topic.
1.2 Non-commutative geometry
Over recent years, non-commutative geometry interacts fruitfully with theoretical physics.
We want to mention the Seiberg-Witten approach to non-commutative field theory
[61, 66, 67], especially. There, matter and gauge fields are replaced by Seiberg-Witten
maps of the commutative fields and variables, the pointwise product by the Weyl-Moyal
product. The Seiberg-Witten approach provides a systematic way to introduce Lorentz
violating operators into the Lagrangian. It also enables one to use arbitrary gauge
groups. The action can be expanded in the non-commutativity parameter. The ze-
roth order term resembles the commutative action. The additional Lorentz violating
terms are not put in by hand, but they represent the effect of the non-commutative
space-time structure [66–71]. Therefore, also the standard model with gauge group
SU(3)C × SU(2)F × U(1)Y can be attacked by these means [68]. However, also an al-
ternative approach to the standard model exists [69]; additional degrees of freedom are
introduced, which they have to get rid of at the end. Quantisation in the θ-expanded
theory, as presented here, seems to be straight forward. Feynman rules can be extracted
from the Lagrangian directly. No problems with unitarity are expected to be encoun-
tered. However, problems with unitarity occur in the non-expanded theory. These prob-
lems can be solved by a consequent analysis of perturbation theory in a Hamiltonian
4
approach, cf. [72–76] for scalar field theory.
The Seiberg-Witten approach to non-commutative field theory does not only work
for constant non-commutativity parameter θ, but can also be generalised to space-time
dependent θ(x) [77–81].
Of special interest is the so called κ-deformation. Similar to q-deformation, space-time
acquires a quantum group symmetry. Poincare covariance is not broken but deformed
to κ-Poincare covariance [78, 79, 82–84]. The connection of κ-deformed field theory, or
deformed field theories in general, to quantum gravity has to be explored thoroughly.
First steps have been done in [85].
Since no satisfactory non-commutative gravity exists so far, two-dimensional theories
(cf. sect. 1.1) may be again a good starting point. Indeed, some progress in this direc-
tion already exists (see [86, 87] where a non-commutative version of the JT model was
constructed).
1.3 Cosmology
One of the main topics in the theory of gravitation is the study of cosmological models.
In [88] have been studied two-dimensional cosmologies in the context of the JT-model,
in the case of minimally coupled and conformally coupled matter. On the one hand, the
main reason to consider more general structures within cosmology is the idea that new
geometrical quantities might shed light on the problems of the cosmological standard
model, e.g. provide an explanation for the rather artificial introduction of an additional
scalar field, like the inflaton. Especially the inclusion of torsion, and possibly non-
metricity, may be a good starting point for extending the standard model of cosmology
by means of new geometrical quantities. First promising results by including torsion are
in [89]. The new quantities couple to the anisotropic space-time [90, 91], spin, shear,
and dilaton current of matter, which are supposed to come into play at high energy
densities, i.e., at early stages of the universe. An example for the latter is the assertion
that quantum effects of the electromagnetic field (EMF) in the external gravitational
field in the anisotropic Bianchi I model give a contribution to the degree of polarisa-
5
tion of the EMF in the quadrupole harmonics. It is known that the size of this effect
parametrically depends on the moment of time starting from which the vacuum of EMF
became unstable. On the assumption of the observational limits on the quantity of the
degree of polarisation of the cosmic microwave background (CMB) one can determine
the limits on the amount of red shift beyond which quantum effects started to play a
role. According to the results of the papers [90,91], the moment of time when the quan-
tum effects of photons switch on can correspond to the rising of the anisotropy on the
background of the initially isotropic matter.
1.4 Topological quantum field theories
The topics of 1.1-1.3, although connected by the use of special (2D) models, seem quite
separated. Part of the difficulty of combining general relativity and quantum theory is
that they use different sorts of mathematics: one is based on objects such as manifolds,
the other on objects such as Hilbert spaces. As “sets equipped with extra structure”,
these look like very different things, so combining them in a single theory has always led
to difficulties. However, work on topological quantum field theory has uncovered a deep
analogy between the two. Moreover, this analogy operates at the level of categories and
here the modern methods of category theory may provide further relations [92–96]. In
refs. [97–100] is has been attempted to formulate the method of additional structures as
a set of axioms for a category, which would be sufficient for an abstract expression of
the basic concepts of the theory of structures on objects of a category. Then all main
properties of a structure are properties of its forgetful functor. Additional (external)
structures on objects of a category provide the possibility to construct new categories
for physics [101].
In physics, interest in categories was sparked by developments relating topology and
quantum field theory. In 1985, Jones [102] came across an invariant of knots, which
could be systematically derived from quantum groups, invented in exactly soluble 2-
dimensional field theories. In a next step, 3-dimensional gravity was introduced into
modern physics by Deser, Jackiw and ’t Hooft [103, 104], then Witten arrived at a
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manifestly 3-dimensional approach to the new knot invariants, deriving them from a
quantum field theory in 3-dimensional space-time (Chern-Simons theory) [105]. This
approach also gave invariants of 3-dimensional manifolds.
Atiyah formulated in 1989 an axiomatic setup for topological quantum field theories
(TQFTs) [106]. Independently and at about the same time G. Segal gave a mathematical
definition of conformal field theories (CFTs) [107], which is very similarly based on
categories and functors. In 1993 J. Frohlich and T. Kerler demonstrated that tensor
categories play a central role in mathematical formulation of quantum groups and TQFT
[108].
We shall focus on two categories in this project. One is the category CKS whose
objects are Cayley-Klein spaces (CKS) [109–111] and whose morphisms are linear op-
erators between these. The other is the category nCKG whose objects are (n − 1)-
dimensional Cayley-Klein geometries (CKG) [112–116] and whose morphisms are n-
dimensional Cayley-Klein geometry going between these. This plays an important role
in relativistic theories where spacetime is assumed to be n-dimensional: in these theo-
ries the objects of nCKG represents possible choices of “space geometries”, while the
morphisms – called “cobordism” – represent possible choices of “space-time geometries”.
While an individual manifold does not resembles very much like a Cayley-Klein space,
the category nCKG turns out to have many structural similarities to the category CKS.
The goal of this project is to explain these similarities and show that the most puzzling
features of quantum theory all arise from ways in which CKS resembles nCKG more
than the category Set, whose objects are sets and whose morphisms are functions. In
quantum field theory on curved space-time, space and space-time are not just mani-
folds: they come with fixed “Cayley-Klein metrics” that allow us to measure distances
and times. In this context, S and S ′ are Cayley-Klein manifolds, and M : S → S ′ is a
Cayley-Klein cobordism from S to S ′. A topological quantum field theory then consists
of a map Z assigning a Hilbert space of states Z(S) to any (n−1)-manifold S and a linear
operator Z(M) : Z(S) → Z(S ′) to any cobordism between such manifolds. This map
cannot be arbitrary, though: it must be a functor from the category of n-dimensional
7
cobordisms to the category of Hilbert spaces. A functor between categories is a map
sending objects to objects and morphisms to morphisms, preserving composition and
identities. Our main point is that treating a TQFT as a functor from the category of
n-dimensional cobordisms to the category of Hilbert spaces is a way of making very
precise some of the analogies between general relavity and quantum theory. However,
we can go further! A TQFT is more than just a functor. It must also be compatible with
the “monoidal category” structure of the category of n-dimensional cobordisms and the
category of Hilbert spaces.
So, a n-dimensional TQFT is defined as a monoidal functor from the category Cob(n+1)
of oriented (n+1)-cobordism with disjoint union as tensor product to the category Vect
of complex finite dimensional vector spaces with the usual tensor product of vector
spaces.
In recent years, there has been also an increasing interest in algebraic structures on
a modular category motivated by coherence problems arising from TQFT [117, 118].
The categories of representations of Cayley-Klein quantum groups are braided monoidal
Cayley-Klein categories [119, 120]. Another motivation comes from developments in
homotopy theory, in particular, models for the stable homotopy category. Monoidal
categories correspond to loop spaces, and the group completion of the classifying space
of a braided monoidal category is a two-fold loop space [121]. Modular categories are
monoidal categories with additional structure (braiding, twist, duality, a finite set of
dominating simple objects satisfying a non-degeneracy axiom). If we remove the last
axiom, we get a pre-modular category. A pre-modular category provides invariants of
links, tangles, and sometimes of 3-manifolds. Any modular category yields a TQFT
[122–124].
There are modifications of the definition of TQFTs by modifying the cobordism cate-
gory by using additional structures on manifolds. For instance, we can specify a framing
of the tangent bundle of the cobordisms and of a formal neighborhood of the closed
manifolds. Another possibility is to include insertions of submanifolds in the manifolds
and matching insertions in the cobordisms. Tensor and duality preserving functors from
8
such modified cobordism categories to Vect are called TQFTs too.
Any modification in the cobordism category may lead to a modification in TQFT.
This modification can be thought of as an extended version of TQFT. For example in
Chern-Simons TQFT, cobordisms are supplied with some additional structures.
The role of higher-dimensional algebras is clear from the various constructions of
extended TQFTs. Baez and Dolan [125] outline a program in which n-dimensional
TQFTs are described as n-category representations. They described a n-dimensional
extended TQFT as a weak n-functor from the free stable weak n-category with duals
of one objects to n-Hilb , the category of n-Hilbert spaces, which preserve all levels of
duality. Homotopy theory methods were used to build examples of TQFTs. Homotopy
quantum field theories (HQFT) are defined as topological quantum field theories for
manifolds endowed with additional structure in the form of a map into some background
space X, it is a theory of objects over X. All these theories do is to fix a background
space X and to compute a weighted sum over homotopy classes of maps f : M → X for
a closed manifold M .
There are the important theorems by Reshetikhin and Turaev saying that HQFTs
only depended on the n-homotopy type of X [123, 126, 127]. We suggest that TQFTs
can be considered as a first approximation to full-blown quantum gravity, HQFTs are a
first approximation to gravity coupled with matter.
In n-categorical set up, one of the examples of monoidal 2-categories is the category
nCob , which has 0-manifolds as 0-cells, 1-manifolds with corners, i.e., cobordism between
0-manifolds as 1-cells, and 2-manifolds with corners as 2-cells.
Instead of taking 0-cells as 0-manifolds, one can also start with objects as 1-manifolds
with or without corners to get Atiyah-Segal-style TQFT.
A 2-dimensional TQFT is a particular case of the construction. Here the category
Cob 1+1 or Cob 2 has compact oriented 1-manifolds as objects and compact oriented
cobordism between them as morphisms.
Extended TQFTs constructed by Kerler and Lyubashenko [128] involves higher cat-
egory theory, namely double categories and double functors. Their construction of ex-
9
tended version of TQFTs is quite different from the n-categorical version of extended
TQFTs proposed by Baez and Dolan. It is not a generalized version of Turaev’s construc-
tion of TQFT functor, actually both constructions are different because of the different
base categories.
Baez’ and Dolan’s hypothesis for extended TFQTs [129] shows that the TQFT func-
tor which produce 2-dimensional extended TQFTs cannot be generalised easily to a
3-dimensional extended TQFT functor. Either they do not have a nice structure in
higher dimensions or their structure is very complicated, e.g. the enriched n-categorical
version of Vect is not very clear in dimension n ≥ 2.
For n = 2, one can think 2-vector spaces as a vector space over the category Vect k of
vector spaces over k.
Thus, we have a monoidal category M with tensor product ⊗ and a functor ⊗ :
Vect k ×M → M satisfying various conditions. One needs to construct different TQFT
functors at different dimensional levels. This suggests that in most of the higher dimen-
sional cases these TQFTs functors will be independent of each other.
For the higher dimensional extended TQFTs, one needs to generalise internal categor-
ical structures for higher dimensions in such a way that existing base category structures
remain preserved, e.g. as in the case of 2Vect , which contains ordinary vector spaces as
objects. If we consider 3Vect to be the category having objects as internal categories of
2Vect and arrows are internal functors, then under suitable conditions 3Vect can give
a higher category version of 2Vect which also contains 2-vector spaces as objects.
The n-category structure is a result of iteration using the ordinary categorical structure
with weaker modified coherence conditions. Our deformation of the category structure is
similar. We modify diagonal comultiplication, but save all diagrams from the categorical
axioms.
We describe a deformation of categories which gives new structures. But their theory
is similar to the category theory because we deform only comultiplications which are in
compositions on all levels in n-category. Our deformation can be applied to n-categories
on different levels. Such a deformation of j-level induces a deformation of the structure
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on all higher levels.
On the other hand, weak n-categories (as far as the notion is properly developed)
have been put to use in extended TQFTs, pointing towards applications in the field of
quantum gravity. In a n-dimensional TQFT, a functor is given from the n-dimensional