Hochschild Cohomology Tethers to Closed String Algebra by way of Cyclicity.

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When we have an open and closed Topological Field Theory (TFT) each element ξ of the closed algebra C defines an endomorphism ξa = ia(ξ) ∈ Oaa of each object a of B, and η ◦ ξa = ξb ◦ η for each morphism η ∈ Oba from a to b. The family {ξa} thus constitutes a natural transformation from the identity functor 1B : B → B to itself.

For any C-linear category B we can consider the ring E of natural transformations of 1B. It is automatically commutative, for if {ξa}, {ηa} ∈ E then ξa ◦ ηa = ηa ◦ ξa by the definition of naturality. (A natural transformation from 1B to 1B is a collection of elements {ξa ∈ Oaa} such that ξa ◦ f = f ◦ ξb for each morphism f ∈ Oab from b to a. But we can take a = b and f = ηa.) If B is a Frobenius category then there is a map πab : Obb → Oaa for each pair of objects a, b, and we can define jb : Obb → E by jb(η)a = πab(η) for η ∈ Obb. In other words, jb is defined so that the Cardy condition ιa ◦ jb = πab holds. But the question arises whether we can define a trace θ : E → C to make E into a Frobenius algebra, and with the property that

θaa(ξ)η) = θ(ξja(η)) —– (1)

∀ ξ ∈ E and η ∈ Oaa. This is certainly true if B is a semisimple Frobenius category with finitely many simple objects, for then E is just the ring of complex-valued functions on the set of classes of these simple elements, and we can readily define θ : E → C by θ(εa) = θa(1a)2, where a is an irreducible object, and εa ∈ E is the characteristic function of the point a in the spectrum of E. Nevertheless, a Frobenius category need not be semisimple, and we cannot, unfortunately, take E as the closed string algebra in the general case. If, for example, B has just one object a, and Oaa is a commutative local ring of dimension greater than 1, then E = Oaa, and so ιa : E → Oaa is an isomorphism, and its adjoint map ja ought to be an isomorphism too. But that contradicts the Cardy condition, as πaa is multiplication by ∑ψiψi, which must be nilpotent.

The commutative algebra E of natural endomorphisms of the identity functor of a linear category B is called the Hochschild cohomology HH0(B) of B in degree 0. The groups HHp(B) for p > 0, vanish if B is semisimple, but in the general case they appear to be relevant to the construction of a closed string algebra from B. For any Frobenius category B there is a natural homomorphism K(B) → HH0(B) from the Grothendieck group of B, which assigns to an object a the transformation whose value on b is πba(1a) ∈ Obb. In the semisimple case this homomorphism induces an isomorphism K(B) ⊗ C → HH0(B).

For any additive category B the Hochschild cohomology is defined as the cohomology of the cochain complex in which a k-cochain F is a rule that to each composable k-tuple of morphisms

Y0φ1 Y1φ2 ··· →φk Yk —– (2)

assigns F(φ1,…,φk) ∈ Hom(Y0,Yk). The differential in the complex is defined by

(dF)(φ1,…,φk+1) = F(φ2,…,φk+1) ◦ φ1 + ∑i=1k(−1)i F(φ1,…,φi+1 ◦ φi,…,φk+1) + (−1)k+1φk+1 ◦ F(φ1,…,φk) —– (3)

(Notice, in particular, that a 0-cochain assigns an endomorphism FY to each object Y, and is a cocycle if the endomorphisms form a natural transformation. Similarly, a 2-cochain F gives a possible infinitesimal deformation F(φ1, φ2) of the composition law (φ1, φ2) ↦ φ2 ◦ φ1 of the category, and the deformation preserves the associativity of composition iff F is a cocycle.)

In the case of a category B with a single object whose algebra of endomorphisms is O the cohomology just described is usually called the Hochschild cohomology of the algebra O with coefficients in O regarded as a O-bimodule. This must be carefully distinguished from the Hochschild cohomology with coefficients in the dual O-bimodule O. But if O is a Frobenius algebra it is isomorphic as a bimodule to O, and the two notions of Hochschild cohomology need not be distinguished. The same applies to a Frobenius category B: because Hom(Yk, Y0) is the dual space of Hom(Y0, Yk) we can think of a k-cochain as a rule which associates to each composable k-tuple of morphisms a linear function of an element φ0 ∈ Hom(Yk, Y0). In other words, a k-cochain is a rule which to each “circle” of k + 1 morphisms

···→φ0 Y0φ1 Y1 →φ2···→φk Ykφ0··· —– (4)

assigns a complex number F(φ01,…,φk).

If in this description we restrict ourselves to cochains which are cyclically invariant under rotating the circle of morphisms (φ01,…,φk) then we obtain a sub-cochain complex of the Hochschild complex whose cohomology is called the cyclic cohomology HC(B) of the category B. The cyclic cohomology, which evidently maps to the Hochschild cohomology is a more natural candidate for the closed string algebra associated to B than is the Hochschild cohomology. A very natural Frobenius category on which to test these ideas is the category of holomorphic vector bundles on a compact Calabi-Yau manifold.