2012 SUMMER UNDERGRADUATE RESEARCH PROJECTS TITLES AND DESCRIPTIONS

• Project 1: Asymptotic safety of unimodular quantum gravity, Astrid Eichhorn
• Project 2: Eternal Inflation in the Cosmic Microwave Background Radiation, Matt Johnson
• Project 3: Glueballs from Gravity, Lilia Anguelova
• Project 4: Mass Estimates for Dwarf Galaxies without Spherical Symmetry, Adrienne Erickcek and Niayesh Afshordi
• Project 5: Shape Dynamics Observables, Tim Koslowski
• Project 6: Statistical mechanics on random lattices via random tensor models, Valentin Bonzom
• Project 7: Typical properties of quantum states with a given amount of entanglement, Markus Mueller
• Project 8: Quantum field theories in 3+1 dimensions and 1+1 dimensional spin chains, Amit Sever

Project 1: Asymptotic safety of unimodular quantum gravity, Astrid Eichhorn

The search for a quantum theory of gravity is a very exciting area of research, where many proposals coexist. One idea is the concept of asymptotic safety, that allows to construct a quantum field theory of the metric which is valid up to arbitrarily high momentum scales. Very similar to asymptotic freedom, where a theory is UV-complete, as the running couplings approach a non-interacting fixed point at high energies, asymptotic safety yields UV-complete theories with the help of an interacting fixed point.

With the help of functional Renormalisation Group techniques, a lot of evidence has been accumulated during the last decade, that Einstein gravity may be asymptotically safe.

In this project, we will study a formulation of gravity that is classically equivalent to Einstein gravity, namely unimodular gravity. Here the determinant of the metric is held fixed, which allows for a solution to the cosmological constant problem, namely why the cosmological constant is not of order of the Planck mass squared due to quantum corrections.

Holding the determinant of the metric fixed alters the quantum theory, and it is thus interesting to study whether unimodular quantum gravity is also asymptotically safe. Such a theory would give a proposal for a quantum theory of gravity solving the cosmological constant problem.

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Project 2: Eternal Inflation in the Cosmic Microwave Background Radiation, Matt Johnson

Inflation, a postulated epoch of accelerated expansion in the early universe, is a key component of the ‘standard model’ of cosmology. However, in many specific realizations of this idea, inflation ceases only locally in ‘pockets’ or ‘bubbles’ that may become radiation- or matter-dominated. One of these could contain our observable universe. If the rate at which pockets form is slower than the rate at which space is expanding, inflation is always happening somewhere; this is eternal inflation. Surprisingly, models of eternal inflation make testable predictions for the pattern of fluctuations in the Cosmic Microwave Background (CMB) radiation. The fluctuations from the phase our pocket was formed from can influence the evolution inside of our bubble, and be imprinted on the CMB. This project will focus on constructing a theoretical model of eternal inflation where all contributions to the fluctuations in the CMB can be calculated explicitly, and explore correlations between different statistical properties of the fluctuations in the temperature and polarization of the CMB radiation. Along the way, the student will be exposed to various topics in field theory and general relativity. Previous experience in these areas is not necessary, although a basic understanding of differential equations and fourier analysis is important.

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Project 3: Glueballs from Gravity, Lilia Anguelova

Gauge/Gravity Duality is a new area of research in high energy physics. It gives a powerful and systematic method to study non-perturbative phenomena that have long been difficult to address with standard Quantum Field Theory methods. The basic idea is that the strongly coupled regime of some gauge theories is encoded in a different (called dual) description, provided by a weakly coupled gravitational background in a different number of dimensions. Thus, nonperturbative field-theoretic problems are transformed into (almost) classical gravitational computations. This powerful method has already begun to provide interesting insights into a wide variety of physical phenomena, among which high-temperature superconductors, quark-gluon plasma and dynamical electroweak symmetry breaking.

The goal of this research project is to contribute to the study of a model of dynamical electroweak symmetry breaking via its dual gravitational description. The gravitational dual of this model has already been established in the recent literature (in a couple of papers by me and collaborators). Nevertheless, there are still various interesting observables to be computed, like for example the spectrum of glueball states in the strongly coupled sector responsible for the breaking of the electroweak symmetry. More concretely, a students' contribution would be in working on solving a second order differential equation, which determines the glueball spectrum. It is likely (although not guaranteed) that this may require the use of numerical methods.

To recapitulate, in working on this project, a student will acquire both familiarity with interesting new physics concepts as well as technical experience regarding solving differential equations. In addition, and perhaps most importantly, the above model may be put to the test by the Large Hadron Collider at CERN. So participating in this project might turn out to be rather exciting.

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Project 4: Mass Estimates for Dwarf Galaxies without Spherical Symmetry, Adrienne Erickcek and Niayesh Afshordi

Numerical simulations of galaxy-sized dark matter halos predict that the Milky Way dark matter halo contains several subhalos (e.g. Diemand et al. 2008; Springel et al. 2008). The dwarf spheroidal galaxies that surround the Milky Way are thought to dwell in the largest of these subhalos. However, recent comparisons between the largest subhalos in simulated galactic halos and the Milky Way dwarf spheroidals have revealed a surprising discrepancy: the simulations predict that the Milky Way’s largest subhalos should be denser than the Milky Way dwarf spheroidals (Boylan-Kolchin, Bullock & Kaplinghat 2011a, b). This conclusion is based on a technique that uses the observed line-of-sight velocity dispersion of the stars within the dwarf spheroidals to deduce the mass enclosed in their half-light radii (Wolfe et al. 2011). The resulting points on the dwarf spheroidals’ mass profiles lie below the mass profiles of the most massive subhalos in simulations. If these masses are accurate, then this discrepancy poses a difficult challenge for the cold-dark-matter paradigm: either galaxies fail to form in the largest subhalos, or there is some mechanism that reduces the central densities of these objects.

Alternatively, it is possible that the inferred masses for the dwarf spheroidals galaxies are incorrect. The procedure that Wolf et al. (2011) uses to obtain these masses from the measured line-of-sight velocity dispersions assumes that the dwarf spheroidals are spherically symmetric. However, simulations predict that the central regions of dark matter subhalos are triaxial (Kuhlen, Diemand & Madau 2007; Knebe et al. 2010). Furthermore, observations of the Milky Way dwarf spheroidals indicate that their stellar populations are not spherically symmetric (Lokas et al. 2011). In this project, the student will investigate how deviations from spherical symmetry affect the relationship between line-of-sight stellar velocities and the galaxy’s mass profile; the goal of this project is to determine if the apparent discrepancy between simulated subhalo populations and the Milky Way satellites may result from inaccurate mass estimates. The student will gain extensive knowledge of the theory of galactic dynamics; the only prerequisites are courses in multivariable calculus, differential equations, and classical mechanics.

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Project 5: Shape Dynamics Observables, Tim Koslowski

Shape Dynamics is a novel Hamiltonian formulation of General Relativity where refoliation symmetry is traded for local spatial conformal symmetry. One can in particular show that the observable algebras of General Relativity and Shape Dynamics are equivalent. This is very important, because the constraints that Shape Dynamics observables have to satisfy are considerably simpler than the constraints that observables in General Relativity have to satisfy. A second aspect of the relation between Shape Dynamics and General Relativity is a very simple relation between Shape Dynamics and General Relativity at leading order in a large volume expansion.

The idea for this project is to consider Shape Dynamics observables (functions on the phase space of metric gravity that are invariant under spatial diffeomorphisms and spatial conformal transformations) and to use this large volume relation to relate these (at least at the level of perturbation theory) to observables in General Relativity. Understanding this classical relation is a very important first step that is needed to interpret Quantum Shape Dynamics as a candidate for Quantum Gravity.

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Project 6: Statistical mechanics on random lattices via random tensor models, Valentin Bonzom

The project is part of a larger research program of the author which consists in understanding the world of random lattices and random geometries in dimensions higher than two, and their physical consequences on statistical mechanics.

I expect some parts of the project to be quite straightforward. Once they are understood, there are several possible directions, depending on the student’s interests and available time. Some directions are challenging issues which could be interesting for mid or long-term research.

Although the project is part of a research program which is related to statistical field theory and quantum gravity, no prior knowledge in those fields is required (but that would obviously be helpful to get the global picture). Some field theory tools we use are Feynman expansions and Schwinger-Dyson equations, but they can be easily understood directly in the context of our models (as the Feynman amplitudes are much simpler than those in usual field theory courses), and offer a flavor of field theory methods.

From the point of view of the student, this project will be a nice introduction to several key ideas and tools that are useful in several fields of modern theoretical physics (statistical and condensed matter physics, strings, quantum gravity). Among the cross-disciplinary tools, the student will learn large N expansions, find situations where resummations of Feynman expansions are possible, random matrices and tensors. The models we study are generalizations of random matrix models to higher dimensions. We also try as most as possible to follow the steps and ideas that made random matrix models so successful. That means that, if interested, the student can have first a walk in matrix model reviews (like [1]), i.e. in the early ideas and results on non-critical strings and Liouville theory coupled to matter.

The study of random tensor models has been revived in the past years thanks to people starting with the loop quantum gravity point of view (more precisely, group field theory). While it is not part of the project itself, the student can take advantage of it to see how contact with loop quantum gravity can be reached and hence get a taste at those ideas. However, we stress that the project is self-contained and is independent of loop quantum gravity interpretations.

It is remarkable that in two dimensions, some statistical mechanical models are exactly solvable on random lattices, thanks to reparametrization invariance (like the Ising model in a non-zero magnetic field [2]). Using a dictionary (the KPZ correspondence), it is then sometimes possible to freeze the dynamics of the lattice to get results of direct relevance in statistical mechanics. This is the method of coupling to 2d quantum gravity (see [3] for a recent application). One aim of the tensor models program is to achieve similar results in higher dimensions. As preliminary steps in this direction, the project I propose is to couple such matter systems to random lattices via random tensor models.

THE PROJECT

The same way random matrix models generate sums over random lattices of two-dimensional surfaces [1], random (rank D) tensor models generate sums over random lattices on D-dimensional spaces. As such, they were proposed quite early to probe random geometries in dimensions higher than two [4-6].

In statistical mechanical terms, microstates are lattices, the canonical partition function is defined for macrostates characterized by a fixed number of lattice sites, and the grand-canonical partition function is obtained by summing over all possible numbers of lattice sites with a fixed chemical potential1. The thermodynamic (or continuum) limit is the limit of large number of sites, and it is reached for the grand-canonical partition function when the latter is dominated by lattices with a very large number of sites (so that it develops a singular behavior).

There has been only few results until recently since one of the main ingredient of random matrix models, the large N expansion, was not known to exist with tensors. That implies that it was impossible to define a continuum limit for the lattices. For some years, numerical studies were performed, known as dynamical triangulations [7], but these simulations had no analytic counter-part. The situation has now changed as Gurau introduced a class of tensor models (known as colored models) and subsequently shown that they do possess a large N expansion [8].

Like in two dimensions, taking this limit selects random lattices with the spherical topology. We have then described their specific combinatorial structure and performed explicitly the sum over those lattices, coined melons, to get the thermodynamic limit [9]. One interesting exponent is the entropy exponent, which characterizes the proliferation of microstates at fixed number of lattice sites. It indicates how the thermodynamic limit is reached and is a first hint of universality classes in random tensor models. What we have found so far is very close to the results of the older numerical studies in the regime of the so-called branched polymer phase.

Several papers have followed and we now have a better knowledge of those melons [10-15]. We know how to couple them to some models from statistical mechanics, can handle nodes of different valence (composite nodes, or bubbles whose internal structures are melonic) and have observed multi-critical behaviors for instance. The project is about the following points.

• Couple random lattices to Ising spins. This is something we have done in a simple case [11], and shown there is no phase transition at finite temperature. However, universality has not been checked, and it should be done using the effective nodes which corresponds to putting the spins on bubbles.
• Couple random lattices to different statistical mechanical models, like the Potts model, loop models, etc.
• If a phase transition is observed, one should compute the whole set of critical exponents.
• As there is a one-to-one correspondence between melons and trees (branched polymers), they share some features (like the entropy exponent). It would be useful to compare with the same systems on random trees.
We stress that the framework is now well-established. Once the method and the physical ideas are understood, it is straightforward to study a given system. Most of the calculations can be done analytically. In the end, it may happen that we get rational functions of several variables, in which case, a numerical study leads to the conclusion.

There are numerous ideas to extend the project, mainly related to symmetries and to the description of the continuum limit. In particular, it is important that in the future we can keep contributions which are not in the melonic family. That could be investigated in the toy model of [12] which is a matrix model in disguise, for which some powerful tools are available.

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Project 7: Typical properties of quantum states with a given amount of entanglement, Markus Mueller

In recent years, typicality or "measure concentration" has been used as an important tool in quantum information theory. The idea is that in high dimensions, "almost all" pure quantum states behave very similarly: drawing a state randomly, expectation values of observables have "typical" values with overwhelming probability.

Usually, the random quantum states are drawn from subspaces. As a consequence, in the case of bipartite quantum systems AB, random states are typically almost maximally entangled. However, in some cases, it would be interesting to have statements of typicality for quantum states which are not maximally entangled -- but, say, have a fixed amount of purity in the local reduced density matrix. The ultimate goal of the proposed project is to prove a "concentration of measure" result for quantum states with a fixed amount of entanglement, possibly with an additional restriction to a subspace. This is a challenging task from mathematical physics, resting mainly on tools developed in Ref. [1]. Clearly, it cannot be expected that this ultimate goal will be achieved in a few weeks. However, there are some clear intermediate steps that can be taken by a student:
• Review known measure concentration results, and learn an example how they are applied in quantum information theory.
• Do numerical experiments to discover how "sharp" distributions become, if states with fixed local purity are drawn randomly by the computer, according to some distribution.
• Possibly things can easily be done analytically if one of the two systems is a single qubit.
• Try to generalize results from [1] to the new situation.

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Project 8: Quantum field theories in 3+1 dimensions and 1+1 dimensional spin chains, Amit Sever

2d spin chains are a well established tool in describing condensed matter systems. A cute miracle in 2d is that in some cases the 2d spin chain can be solve exactly. Such cases and named "integrable". These include for example the spectrum on spinons in KCuF3 and the Hubbard model. Recent development in quantum field theory in four dimensions uses exactly the same techniques. For example, the most advanced computation of scattering amplitude in a 4d gauge theory rely on a 2d spin chain description. No Feynman diagrams are involved! In another example, the conformal dimensions of local operators in the 4d theory are solved by mapping them to energies of an integrable spin chain. The project will be centered around integrability in scattering amplitudes.

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