University of Massachusetts
Mathematical Physics Seminar
Fall 2008

Various authors such as J. Lidsey, T. Christodoulakis, T. Grammenos, C. Helias, P. Kevrekidis, G. Papadopoulos and F. Williams are known to have formulated equivalent versions of the 3+1-dimensional Einstein's field equations in terms of a so-called generalized Ermakov-Milne-Pinney (EMP) differential equation. This reformulation provides an alternate method for acquiring exact solutions to the field equations, and has been accomplished within the frameworks of FRLW and some Bianchi universe models. Further inspired by an EMP-Schrodinger correspondence as noted by J. Lidsey, the author has recently published a linear Schrodinger version of the Bianchi I scalar field cosmology. This model has now been extended to an arbitrary number of spatial dimensions, and will be presented here. We will additionally comment on this cosmological modelÕs connection with Bose-Einstein condensates.

1. J. D'Ambroise, F.L. Williams, A non-linear Schršdinger type formulation of FRLW scalar field cosmology, International Journal of Pure and Applied Mathematics, Vol. 34, No. 1, p.117
2. J. D'Ambroise, A Schrodinger formulation of Bianchi I scalar field cosmology, Interational Journal of Pure and Applied Mathematics, Vol. 42, No. 3, p. 405
3. R. Hawkins, J. Lidsey, The Ermakov-Pinney equation in scalar field cosmologies, Physical Review D66 (2002), 023523-023531
4. G.Hohn,Universitat Bonn PhD Thesis,ArXiv:07060236
5. P. Kevrekidis, F. Williams, On 2+1-dimensional Friedmann-Robertson-Walker universes: an Ermakov-Pinney equation approach, Classical and Quantum Gravity, 20 (2003) L177-L184
6. J. Lidsey, Cosmic dynamics of Bose-Einstein Condensates, Classical and Quantum Grav., 21 (2004), 777-785
7. J. Lidsey, Multiple and anisotropic inflation with exponential potentials, Class. Quantum Grav. 9 (1992), 1239-1253
8. F. Williams, An EMP Model of Bianchi I Cosmology, Proceedings of the 11th Marcel Grossman Meeting on Relativity and Gravitation, Berlin, Germany, 2006

We use the theory of Hecke operators to give an explicit computation of E. Witten's extremal partition function $Z\_k(z)$ of level $k$, ,$k=1,2,3,...$, which he proposes to correspond to a conformal field theory (CFT) of central charge $24k$. In particular we correct an error in Witten's result (with A. Maloney) in case k=4. We also find the asymptotics of the Fourier coefficients of $Z\_k(z)$, whose sub-leading terms provide for corrections to Bekenstein-Hawking black hole entropy in AdS 3 gravity. Our approach to asymptotics is based on some recent number-theoretic results of N. Brisebarre and G. Philibert. In case $k=1$, Witten's proposal is known to be true, due to work of I. Frenkel, J. Lepowsky, A. Meurman, B. Fischer, R. Griess, and others. In this case $Z\_\{k=1\}(z)$ reduces to the well-known elliptic modular function J(z), and moreover the CFT states form a representation of a special finite group G called the "monster", since G has an enourmous number of elements - approximately 10 to the 54th power. Thus we hope to tie these various magnificant, seemingly unrelated, ideas together.

1. T.Apostol, Modular Functions and Dirichlet Series in Number Theory, 2nd edition, Grad.Texts in Math.41, Springer-Verlag(1989)
2. M.Banados, C.Teitelboim,and J.Zanelli, Black hole in three-dimensional spacetime, Phys.Rev.Letters69, no.13, (1992), 1849-1851
3. N.Brisebarre and G.Philibert, Effective lower and upper bounds for the Fourier coefficients of powers of the modular invariant j, Journal of the Ramanujan Math.Soc.20(2005), 255-282
4. I.Frenkel, J.Lepowsky, and A.Meurman, A moonshine module for the monster, from Vertex Operators in Mathematical Physics, MSRI Publication, (1984), 231-273
5. A.Maloney and E.Witten, Quantum gravity partition functions in three dimensions, arXiv:0712.0155
6. Panel on Group Theory(F.Williams and others), Group Theory:The Language of Symmetry in Science and Technology, National Academy Press(1996)
7. F.Williams, Note on quantum corrections to BTZ instanton entropy, Proceedings of Science, electronic journal:POS(IC 2006), 006
8. E.Witten, Three dimensional gravity revisited, arXiv:0706.3359

Smectic liquid crystals are materials with periodic order in only one direction. They are characterized as a series of equally-spaced fluid layers. In some materials, the ground states exhibit an ordered array of topological defects. I will describe one such structure in smectic liquid crystals, the twist-grain boundary phase. Finally, I will discuss some aspects of two-dimensional smectic order on curved surfaces, where the background geometry frustrates the local smectic ordering. This is analogous to a type of geometrical frustration that occurs in three dimensional smectics in Euclidean space.

A Quantum Field Theory on a manifold is said to be topological if it only depend on topoogy of the manifold, but not on the metric. This is the opposite from the standard case where QFT is studied on the Minkowski space only, so there is no topology in the game. The most famous examples of topological QFTs are thso called A and B models which appear as drastic simplifications of supersymmetric sigma models. TQFTs are more accessible mathematically. They were hugely influential in mathetics (Mirror Symmetry, Geometric Langlands) and in physics they are of lower importance as simpler approximations of deeper theories.