# Research Overview

On this page I will give an overview of my research with an eye towards key techniques, results, and possible future work. For more technical details (and a list of my publications) you can see my Resources page, and if you want a more extensive overview with some background information check out my Blog.

My main research interest is the geometry and topology of physical models, specifically those related to gravitational and high energy physics. The close interaction between theoretical physics and mathematics has lead to many deep and important results in both fields, and I think that expanding this relationship to include recent developments will only continue to do so. It is the classification of smooth manifolds that has occupied most of my research efforts. I started by studying inequivalent smooth structures in semiclassical gravity, and found that these apparently exotic mathematical constructs can effect physical observables. Then I worked on a formulation of loop quantum gravity (LQG) in which one can completely specify both the geometry and the topology of the spatial sections. This means the classification is not an issue, and the exotic smooth structures are automatically included. This also provides a framework for studying topology in LQG, which is not possible in the usual approach. This work was expanded into a similar construction which uses the fact that any four-manifold as a branched cover of the sphere, which allows for the semiclassical partition function to be completely specified. This avoids the exotic smoothness problem, and also naturally includes cosmic strings. This suggests an interesting connection between symmetry breaking in the early universe and semiclassical gravity. I now will discuss each of these projects in more detail.

## Exotic Smoothness and Quantum Gravity

The first chapter in my thesis focused on understanding the way we construct the geometric structure upon which we base all our theoretical models – that is, a smooth manifold. It has long been known that the transition from a topological manifold to a smooth manifold is not a trivial process in all dimensions, particularly in the physically important dimension four. In other words, there need not exist a diffeomorphism between homeomorphic manifolds. Two manifolds with inequivalent smooth structures are said to posses exotic smooth structures. Since diffeomorphisms are the gauge transformations of general relativity, these exotic smooth structures represent inequivalent classical solutions to the Einstein field equations.

The question that needed to be answered was does it matter that we choose one smooth structure to use out of the set of inequivalent ones? In Duston (2011) I showed that yes, in some cases the choice of smooth structure can have a physically meaningful effect. To show this we calculated the semiclassical expectation value of volume for a specific set of exotic spaces, which were presented as iterated branched covers of $\mathbb{CP}^2$. This made the geometry particularly simple, so we could explicitly calculate the semiclassical partition function restricted to a family of homeomorphic but non-diffeomorphic manifolds. We were able to show that in the 1-loop approximation, the expectation value of volume was strongly dependent on the conformal scale factor. It occurs over a small range in the conformal scale factor, and is thus much like a phase transition between the spaces with the largest and smallest volumes in the homeomorphic family. This calculation was the first result of its kind of dimension four.

### Future Work

While my thesis focuses on exotic smoothness in quantum gravity, it has always been in classical gravity where exotic smoothness presents solutions to some of the more interesting puzzles in physics. For instance, Carl Brans and Torsten Asselmayer-Maluga have written extensively on the idea that small exotic $\mathbb{R}^4$ (Akbulut corks) can explain gravitational lensing in galaxy clusters, and remove the need for the inclusion of dark matter to match theory with these observations. However, it has not yet been possible to explicitly calculate this effect because there are not yet any presentations of exotic $\mathbb{R}^4$ which include a sufficiently explicit metric.

I have considered several approaches to this problem which do not require finding metrics over the entire exotic $\mathbb{R}^4$. There is an observation by Laurence Taylor that suggests that distance functions on exotic $\mathbb{R}^4$ do not have bounded critical values (Taylor, 2005). Thus, perhaps studying such functions can give some insight into what the exotic structure might look like by focusing on these unbounded critical values. One idea that I would like to try is to study the critical values of distance functions under generalized Morse theory (generalized since distance functions are Lipschitz rather than smooth). It may be possible to find a local form for the distance functions near these critical points and see what effect there would be on geodesics. This might also shed some light on the Casson handle presentations of exotic $\mathbb{R}^4$, of which there are many explicit ones.

## Loop Quantum Gravity

In Duston (2012) I used Topspin networks (first presented by Denicola, Marcolli, and al-Yasry (2012)) to construct a form of LQG in which the spin networks encode both the topological and geometric data of the spatial sections. This is based on a theorem by Alexander (1920), which states that any closed, orientable $n$-dimensional manifold can be presented as a branched covering of an $n$-sphere branched along an $(n-2)$ subcomplex. For $n=3$, if we identify the spin networks with the one dimensional branch locus we can track the topology of the spatial section (now the branched cover) by adding topological labels to the spin network which keep track of how the sheets of the covers are stitched together. Such upgraded spin networks are called topspin networks. Since a topspin network tells us the topology of the spatial sections and the geometry of the gravitational field (from the original spin network), it provides us with an appropriate structure to study both the geometry and topology of LQG.

As it turns out, formulating LQG on topspin networks does in fact change the details of the theory. Specifically, there is an extra symmetry which results from interchanging sheets of the cover (a deck transformation). Since some of these sheets are identified at the branch locus, the metric at these locations must be invariant under the deck transformation. This symmetry can be carried through the usual construction of LQG, and it turns out that the effect of different topologies can be seen from the form of the area operator. In essence, the area operator gives the same eigenvalues, but the explicit form is now different in a way that tracks some of the topological information. We were also able to show that the action of the Hamiltonian on topspin networks is distinctly different than on spin networks, due to the presence of certain covering moves which implement an equivalence relation on the covers.

Turning to the question of exotic smoothness in LQG, it was already known that Alexander’s theorem can be upgraded to four manifolds branched over 2-complexes in the 4-sphere. In turn, we can upgrade the topspin networks to topspin foams and track the topology and geometry of four manifolds as well. However, since the “exotic smoothness problem” is a lack of information regarding the smooth structure of a manifold, there is no exotic smoothness when one constructs LQG in this way; the topspin foams tell us everything we need to know about both the topology and the geometry of spacetime.

### Future Work

The next natural step to take in this direction is to look for topology-changing processes in LQG. The approach would be to look at very simple knots and loops to try to determine if one could specify what characterizes topology change. This would probably require representing the topspin network as a graph and using tools from graph theory to find an appropriate notion of irreducibility. This result might be able to be extended to dipole spinfoam amplitudes, so that one could work out the topology-changing processes present in the early universe.

## Semiclassical Gravity and Cosmic Strings

The last section of my thesis was a combination of the previous two ideas; using Alexander’s theorem and what we learned from its application to LQG, write down a partition function for semiclassical quantum gravity which a) explicitly includes contributions from inequivalent smooth structures, and b) can be used to calculate observables.

The partition function for semiclassical quantum gravity involves a sum over 4-manifolds, and a result of Piergallini (1995) lets us upgrade these to embedded surfaces in $\mathbb{S}^4$. The generalized Weierstrass representation of embedded surfaces as solutions to a pair of Dirac equations can be used to represent these in a generic way. By using these surfaces to furnish codimension 2 foliations of the sphere, one can calculate the action through the Codazzi equation.

These surfaces can generically be “smoothed out” (Troyanov, 1991) so that they are flat everywhere with a finite number of conical points. In fact, the metric around such conical points is exactly that which are found around cosmic strings, so this approach naturally introduces cosmic strings into the semiclassical partition function.

I have looked at several specific examples of such a construction, and found that a key physical quantity is the area of the string worldsheet. Then the action can be written as a finite sum of string worldsheets, with some possible spinor terms coming from free solutions to the Dirac equations of the Weierstrass representations. Since any manifold is uniquely determined by a branch locus and some topological labels, the resulting partition function is formally complete; that is, no exotic smoothness.

### Future Work

To further this approach, I would like to explore the connections to cosmic strings. Specifically, when dealing with open strings, can the area of the string worldsheet be expressed in some way that would be connected with physical observables? Then the partition functions could be used to actually confirm predictions from cosmological observations. We also considered only non-interacting strings, and it would be interesting to work on the interacting case to look for characteristic differences. In fact, since the interacting strings would geometrically mean strings which crossed the branch locus, it might be possible to replace “all branch loci” with “all interacting strings” and formulate semiclassical gravity entirely with cosmic strings.

### Noncommutative Geometry

The recent incorporation of noncommutative geometry into physics provides the starting point for many interesting investigations. For the past several years I have participated in an informal seminar on noncommutative geometry and physics, held jointly by graduate students in the physics and math department at Florida State University. Using the spectral action principle, one can study the Lagrangians on spectral triples derived from various kinds of physical spacetimes. It would be very interesting to choose a spectral triple associated with loop quantum gravity (using spin or topspin networks) and calculate the spectral action. This might lead to a different way of formulating the classical limit for LQG, or provide a different approach to the sum-over-states models.

The language of noncommutative geometry has also lead to an interesting notion of emergent geometry, the construction of spacetime from spontaneous symmetry breaking. One can construct dynamics on an algebra and find the low temperature equilibrium (KMS) states. This has been done for the area operator in loop quantum gravity, but many generalizations are possible. For instance, the Hamiltonian operator on topspin networks generates a category; it would be very interesting to try and calculate the KMS states associated with this category. This would represent a different approach to solving the Hamiltonian constraint.

### Topological Quantum Field Theory

The introduction of the deck transformations over topspin networks suggests that some tools from topological quantum field theory (TQFT) might be applicable. This would isolate the topological properties of the gravitational field in LQG, and it might be interesting to see what kind of invariants would result from such considerations.

Specifically, the Dijkgraaf-Witten type topological field theory can be constructed using the finite action of the fundamental group on the branched covering space associated to a topspin network or a topspin foam. At least in the topspin network case, the topological action is Chern-Simons and we can classify the possible TQFTs with $H^4(BG,\mathbb{Z})$. The same construction is possible for the topspin foam, although TQFTs are less well-understood in four dimensions. In that case one could use the work of Baez (1996) for BF theory as the topological field theory. Either of these approaches would clarify the interaction between topological field theories and the spinfoam models, which has recently been getting considerable attention.