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1. CJM 2012 (vol 65 pp. 1164)

Vitagliano, Luca
Partial Differential Hamiltonian Systems
We define partial differential (PD in the following), i.e., field theoretic analogues of Hamiltonian systems on abstract symplectic manifolds and study their main properties, namely, PD Hamilton equations, PD Noether theorem, PD Poisson bracket, etc.. Unlike in standard multisymplectic approach to Hamiltonian field theory, in our formalism, the geometric structure (kinematics) and the dynamical information on the ``phase space'' appear as just different components of one single geometric object.

Keywords:field theory, fiber bundles, multisymplectic geometry, Hamiltonian systems
Categories:70S05, 70S10, 53C80

2. CJM 2000 (vol 52 pp. 503)

Gannon, Terry
The Level 2 and 3 Modular Invariants for the Orthogonal Algebras
The `1-loop partition function' of a rational conformal field theory is a sesquilinear combination of characters, invariant under a natural action of $\SL_2(\bbZ)$, and obeying an integrality condition. Classifying these is a clearly defined mathematical problem, and at least for the affine Kac-Moody algebras tends to have interesting solutions. This paper finds for each affine algebra $B_r^{(1)}$ and $D_r^{(1)}$ all of these at level $k\le 3$. Previously, only those at level 1 were classified. An extraordinary number of exceptionals appear at level 2---the $B_r^{(1)}$, $D_r^{(1)}$ level 2 classification is easily the most anomalous one known and this uniqueness is the primary motivation for this paper. The only level 3 exceptionals occur for $B_2^{(1)} \cong C_2^{(1)}$ and $D_7^{(1)}$. The $B_{2,3}$ and $D_{7,3}$ exceptionals are cousins of the ${\cal E}_6$-exceptional and $\E_8$-exceptional, respectively, in the A-D-E classification for $A_1^{(1)}$, while the level 2 exceptionals are related to the lattice invariants of affine~$u(1)$.

Keywords:Kac-Moody algebra, conformal field theory, modular invariants
Categories:17B67, 81T40

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