By E. Ramirez De Arellano

ISBN-10: 0387521755

ISBN-13: 9780387521756

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**Example text**

We shall now survey a few kinds of examples by way of illustration, sometimes giving only a sketch of the details. The most general target category we can consider is a symmetric tensor category: clearly we need a tensor product, and the axiom HY1 ⊔Y2 ∼ = H Y1 ⊗ HY2 only makes sense if there is an involutory canonical isomorphism HY1 ⊗ H Y2 ∼ = H Y2 ⊗ H Y1 . , vector spaces V with a mod 2 grading V = V 0 ⊕V 1 , where the canonical isomorphism V ⊗ W ∼ = W ⊗ V is v ⊗ w → (−1)deg v deg w w ⊗ v. One can also consider the category of Z-graded vector spaces, with the same sign convention for the tensor product.

Thus when d = 2 the objects of the geometric category are disjoint unions of circles and oriented intervals with labelled ends. A functor from this category to complex vector spaces which takes disjoint unions to tensor products will be called an open and closed topological field theory: such theories will give us a “baby” model of the theory of D-branes. , so that it is a cobordism from the point b to the point a, and NOT the other way round). a a b c b c Figure 1. Basic cobordism on open strings.

Associativity, commutativity, and unit constraints in the closed case. The unit constraint requires the natural assumption that the cylinder correspond to the identity map C → C. boundary circle being the incoming closed circle, while the other boundary circle is subdivided into an outgoing interval and an interval of constrained boundary. 4. Sewing theorem. Geometrically, any oriented surface can be decomposed into a composition of morphisms corresponding to the basic data defining the Frobenius structure.

### Algebraic Geometry and Complex Analysis by E. Ramirez De Arellano

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Categories: Algebraic Geometry