Conformal field theory and elliptic cohomology

What is the Jacobian of a Riemann Surface with Boundary?

“…We begin by the same considerations as in [4], starting on p. 351. Consider an even lattice L and a bilinear form…”

“…(The papers [4,5] used the word "lax" instead of "pseudo", but the first author [1] discovered that "pseudo" conforms more with existing terminology of higher category theory.) 1 In the present paper, the meaning of the target of the map (1), which is defined in [5], plays only a marginal role.…”

“…Of course, one has to start out by being precise about what exact abstract structure captures the notion of gluing, and then generalize the notion of conformal field theory to be defined on such abstract structures. Following ideas of Segal [9], this was done in [1,4,5], with a correction in [3]. The desired structure is called stack of pseudo commutative monoids with cancellation (SPCMC -see [3] for a correct definition) and a CFT is a pseudo morphism of certain SPCMC's…”

“…When L is an even lattice (together with a Z/2-valued bilinear form b satisfying a suitable condition), one has a notion of conformal field theory associated with L ( [9,4]). It could be argued that the definition only uses additive data, so the lattice conformal theories should "factor through open abelian varieties".…”

“…Those are 2-dimensional real-analytic manifolds with boundary which have a complex structure and real-analytically parametrized boundary components. The notion of SPCMC, which is defined in [4,5], is designed to capture the operations of disjoint union and gluing in C, along with the fact that C is a groupoid (under holomorphic maps compatible with the boundary parametrizations), and in fact a stack over the Grothendieck topology of complex-analytic manifolds and open covers. In particular, gluing in C is defined by noticing that the parametrized boundary components of a worldsheet can have two possible orientations with respect to the complex structure -one usually calls them inbound and outbound.…”