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Locally variational invariant field equations and global currents: Chern-Simons theories

Mauro Francaviglia, M. Palese, E. Winterroth (2012)

Communications in Mathematics

We introduce the concept of conserved current variationally associated with locally variational invariant field equations. The invariance of the variation of the corresponding local presentation is a sufficient condition for the current beeing variationally equivalent to a global one. The case of a Chern-Simons theory is worked out and a global current is variationally associated with a Chern-Simons local Lagrangian.

Logarithmic structure of the generalized bifurcation set

S. Janeczko (1996)

Annales Polonici Mathematici

Let G : ℂ n × ℂ r → ℂ be a holomorphic family of functions. If Λ ⊂ ℂ n × ℂ r , π r : ℂ n × ℂ r → ℂ r is an analytic variety then    Q Λ ( G ) = ( x , u ) ∈ ℂ n × ℂ r : G ( · , u ) h a s a c r i t i c a l p o i n t i n Λ ∩ π r - 1 ( u ) is a natural generalization of the bifurcation variety of G. We investigate the local structure of Q Λ ( G ) for locally trivial deformations of Λ ₀ = π r - 1 ( 0 ) . In particular, we construct an algorithm for determining logarithmic stratifications provided G is versal.

Loop spaces and Riemann-Hilbert problems

G. Khimshiashvili (2007)

Banach Center Publications

We present a survey of recent results concerned with generalizations of the classical Riemann-Hilbert transmission problem in the context of loop spaces. Specifically, we present a general formulation of a Riemann-Hilbert problem with values in an almost complex manifold and illustrate it by discussing two particular cases in more detail. First, using the generalized Birkhoff factorization theorem of A. Pressley and G. Segal we give a criterion of solvability for generalized Riemann-Hilbert problems...

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