My research is in mathematical physics, especially classical and quantum gravity, and applications of geometry, topology, and n-category theory to physics. There's a lot of overlap in the different areas of my research, but roughly, I could break up my work into these categories:
This page is currently under construction (October 2009). Some of the descriptions below are incomplete...
Much of my research is motivated by the problem of quantum gravity.
"Topologically massive gravity" is interesting because, while it is in some ways simpler than 4d general relativity, it has all of the same conceptual difficulties for quantization. In particular, both theories are diffeomorphism-invariant, and unlike ordinary 2+1 GR, have local degrees of freedom.
The action for topologically massive gravity is just a linear combination of two topological actions: the Einstein-Hilbert action in 2+1 dimensions, with an optional cosmological constant, and a Chern-Simons action for the spin connection:
where R is the curvature of ω, e is the coframe field, and μ is called the topological mass. Very important, however, is that e and ω are not allowed to vary independently: ω is constrained to be the torsion free spin connection, so the action depends only on e. This constraint makes topologically massive gravity considerably more complex than its μ→∞ limit, namely 3d general relativity. In particular, topologically massive gravity is not topological, in the sense that not all solutions of TMG are locally gauge equivalent.
Here are two papers I've written on this subject:
For the full nonlinear theory, we also also found a set of chiral pp-wave-like solutions, and a formulation of topologically massive gravity as the difference of two constrained SL(2,R) Chern-Simons theories.
The following paper explains the geometry behind the MacDowell-Mansouri approach to gravity:
This paper was described by John Baez in Week 243 of his column.Abstract: The geometric content of the MacDowell-Mansouri formulation of general relativity is best understood in terms of Cartan geometry. In particular, Cartan geometry gives clear geometric meaning to the MacDowell-Mansouri trick of combining the Levi-Civita connection and coframe field, or soldering form, into a single physical field. The Cartan perspective allows us to view physical spacetime as tangentially approximated by an arbitrary homogeneous "model spacetime", including not only the flat Minkowski model, as is implicitly used in standard general relativity, but also de Sitter, anti de Sitter, or other models. A "Cartan connection" gives a prescription for parallel transport from one "tangent model spacetime" to another, along any path, giving a natural interpretation of the MacDowell-Mansouri connection as "rolling" the model spacetime along physical spacetime. I explain Cartan geometry, and "Cartan gauge theory", in which the gauge field is replaced by a Cartan connection. In particular, I discuss MacDowell-Mansouri gravity, as well as its recent reformulation in terms of BF theory, in the context of Cartan geometry.
You can also read summaries of parts of this work in Week 232 of John Baez's column, and in a blog entry by Urs Schreiber. In work with John Baez and Alissa Crans, I have studied string-like excitations in 4d topological gravity, giving a concrete example of a 4d theory with matter obeying statistics that are neither fermionic nor bosonic:
The reason exotic statistics arise in this setting is that strings in 4 dimensions can pass though each other, entangling their worldsheets in infinitely many topologically distinct ways. This can be illustrated using a higher-dimensional sort of "braid diagram" for the worldsheets, or by drawing frames for a "movie" of strings in 3 dimensional space:
The group that describes the motion of n strings in 3d space is the loop braid group LBn. We used an action of LBn on the moduli space of flat G-bundles to construct unitary representations of LBn on the Hilbert space of states for 4d BF theory, whenever the gauge group G admits a conjugation-invariant measure.
Here are the slides from a talk on this subject, in which also I discussed in more detail how this generalizes to other "motion groups"—groups describing how one manifold ("matter") can move around another manifold ("spacetime"):
On a related topic, I have been working with Jeffrey Morton on extended topological quantum field theory. This is much like ordinary topological field theory, but includes "matter" as submanifolds cut out of spacetime, much like the "strings" above, or like particles in 3d quantum gravity. Doing this properly uses tools from n-category theory.
I am interested in higher category theory and its applications to physics, topology, and geometry.
Here are two papers in which we study representations of "2-groups", and discuss applications of this representation theory to quantum topology and to physics:
A "2-group", sometimes called a "categorical group", is a sort of hybrid of a category and a group: it has invertible multiplication operations for both its objects and morphisms. As I explained in this talk, the representation theory we study in these papers is a step toward representations of 2-groups on infinite-dimensional "2-Hilbert spaces", certain categories analogous to Hilbert spaces. However, infinite-dimensional 2-Hilbert spaces have yet to be rigorously defined; this is another project I'm currently working on, with Jeff Morton.
Abstract: We investigate p-form electromagnetism—with the Maxwell and Kalb-Ramond fields as lowest-order cases—on discrete spacetimes, including the regular lattices commonly used in lattice gauge theory, but also more general examples. After constructing a maximally general model of discrete spacetime suitable for our purpose—a chain complex equipped with an inner product on (p+1)-cochains—we study both the classical and quantum versions of the theory, with either R or U(1) as gauge group. We find results—such as a `p-form Bohm–Aharonov effect'—that depend in interesting ways on the cohomology of spacetime. We quantize the theory via the Euclidean path integral formalism, where the natural kernels in the U(1) theory are not Gaussians but theta functions. As a special case of the general theory, we show p-form electromagnetism in p+1 dimensions has an exact solution which reduces when p = 1 to the abelian case of 2d Yang-Mills theory as studied by Migdal and Witten. Our main result describes p-form electromagnetism as a `chain field theory'—a theory analogous to a topological quantum field theory, but with chain complexes replacing manifolds. This makes precise a notion of time evolution in the context of discrete spacetimes of arbitrary topology.
Here are the abstract and slides for my talk at Loops '05.So far, my published work on geometry is on applications to gravity, so you can read about that above. Besides this, I have done other work, particularly with James Dolan, on Cartan geometry, which I hope to write up soon.
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