ar X iv : m at h / 04 10 55 1 v 1 [ m at h . D G ] 2 6 O ct 2 00 4 CLASSICAL FIELD THEORY ON LIE ALGEBROIDS : VARIATIONAL ASPECTS

ar X iv : m at h / 04 10 55 1 v 2 [ m at h . D G ] 1 6 N ov 2 00 4 CLASSICAL FIELD THEORY ON LIE ALGEBROIDS : VARIATIONAL ASPECTS

The variational formalism for classical field theories is extended to the setting of Lie algebroids. Given a Lagrangian function we study the problem of finding critical points of the action functional when we restrict the fields to be morphisms of Lie algebroids. In addition to the standard case, our formalism includes as particular examples the case of systems with symmetry (covariant Euler-P...

ar X iv : m at h / 04 10 17 9 v 1 [ m at h . Q A ] 6 O ct 2 00 4 ABELIAN HOMOTOPY DIJKGRAAF – WITTEN THEORY

We construct a version of Dijkgraaf–Witten theory based on a compact abelian Lie group within the formalism of Turaev's homotopy quantum field theory. As an application we show that the 2+1–dimensional theory based on U (1) classifies lens spaces up to homotopy type.

ar X iv : m at h / 04 10 61 1 v 1 [ m at h . A G ] 2 8 O ct 2 00 4 ON RATIONAL CUSPIDAL PROJECTIVE PLANE CURVES

ar X iv : m at h / 04 10 51 6 v 1 [ m at h . G R ] 2 4 O ct 2 00 4 DIMENSION FILTRATION ON LOOPS

We show that the graded group associated to the dimension filtration on a loop acquires the structure of a Sabinin algebra after being tensored with a field of characteristic zero. The key to the proof is the interpretation of the primitive operations of Umirbaev and Shestakov in terms of the operations on a loop that measure the failure of the associator to be a homomorphism.

ar X iv : m at h / 04 10 14 1 v 1 [ m at h . A P ] 6 O ct 2 00 4 Existence of conformal metrics with constant Q

Given a compact four dimensional manifold, we prove existence of conformal metrics with constant Q-curvature under generic assumptions. The problem amounts to solving a fourth-order nonlinear elliptic equation with variational structure. Since the corresponding Euler functional is in general unbounded from above and from below, we employ topological methods and minimax schemes, jointly with the...