Про потенциалы

[posic]

Есть наука про матричные факторизации. В ней выбирается потенциал -- многочлен от нескольких коммутирующих переменных и рассматривается (Z/2Z-градуированная) CDG-алгебра с нулевым дифференциалом и элементом кривизны, равным потенциалу.

Есть наука про колчаны. В ней выбирается потенциал -- формальный степенной ряд из циклических слов, составленных из стрелок колчана (без соотношений). Рассматривается DG-алгебра Гинзбурга, связанная с таким потенциалом.

Это два совершенно разных потенциала и две совершенно разные науки? Или есть какая-то связь?

Comments

[potap.livejournal.com]

Матричные факторизации знаю. А какое отношение они имеют к CDG-алгебрам и при чем тут кривизна? (Да, кстати, что такое C в слове CDG?)

[posic.livejournal.com]

Let me comment in English, as I have no Russian keyboard here and using www.translit.ru is just getting too complicated.

The C in "CDG-algebras" stands for "curved". A CDG-algebra is a triple (B,d,h) where B is a graded (or Z/2Z-graded) algebra, d is an odd derivation of B of degree 1, and h is an element of B_2, satisfying the equations d^2(x)=[h,x] for all x in B and d(h)=0. So d^2 is not zero (as usual in homological algebra) but the commutator with the "curvature element" h.

The typical examples of CDG-algebras are de Rham algebras corresponding to vector bundles with nonflat connections. Let M be an (affine algebraic or smooth) manifold, E be a vector bundle on M, and \nabla be a connection in E. Let \Omega be the algebra of differential forms on M with values in the vector bundle End(E) of endomorphisms of E. Then the curvature of \nabla is an element of \Omega_2, and the connection on End(E) induced by \nabla provides a de Rham differential on \Omega with a nonzero square. This is a CDG-algebra, hence the term "curvature element" for h.

A CDG-module M over a CDG-algebra B is a graded B-module endowed with an odd derivation d (satisfying the Leibniz rule together with the given derivation of B) satisfying the equation d^2(y) = hy for all y in M. CDG-modules form a DG-category, i.e., B-linear morphisms over B between two CDG-modules form a complex.

CDG-modules over the de Rham CDG-algebra corresponding to (M,E) are related via Koszul duality to complexes of modules over the ring D_{M,E} of differential operators acting on sections of E.

Now if h is a polynomial in several variables, then one can construct a Z/2-graded CDG-algebra by taking B to be the algebra of polynomials, d=0, and h as the curvature element. If the variables are even, one presumably takes B_1=0, too. I don't really know what matrix factorizations are, but my understanding is that they are Z/2-graded CDG-modules over this CDG-algebra for which the underlying graded B-module is free an finitely generated.

[posic.livejournal.com]

"between two CDG-modules over B"
"and finitely generated"