Novikov Ring
Various choices of coefficient ring for the quantum cohomology of X are possible. Usually a ring is chosen that encodes information about the second homology of X. This allows the quantum cup product, defined below, to record information about pseudoholomorphic curves in X. For example, let
be the second homology modulo its torsion. Let R be any commutative ring with unit and Λ the ring of formal power series of the form
where
- the coefficients come from R,
- the are formal variables subject to the relation ,
- for every real number C, only finitely many A with ω(A) less than or equal to C have nonzero coefficients .
The variable is considered to be of degree, where is the first Chern class of the tangent bundle TX, regarded as a complex vector bundle by choosing any almost complex structure compatible with ω. Thus Λ is a graded ring, called the Novikov ring for ω. (Alternative definitions are common.)
Read more about this topic: Quantum Cohomology
Famous quotes containing the word ring:
“There is no magic decoding ring that will help us read our young adolescent’s feelings. Rather, what we need to do is hold out our antennae in the hope that we’ll pick up the right signals.”
—The Lions Clubs International and the Quest Nation. The Surprising Years, III, ch.4 (1985)