The n-Category Café

Bergman on Infinity-Vector Bundles Coupled to Topological Strings

Posted by Urs Schreiber

As Aaron Bergman kindly pointed out here, he has an interesting article

Aaron Bergman
New Boundaries for the B-Model
arXiv:0808.0168

which gives a direct interpretation of the derived categories of coherent sheaves appearing in topological strings in terms of connections on $\infty$-(Chan-Paton)-vector bundles, using a crucial insight by Jonathan Block.

Formerly, the way to see that the branes (= boundary conditions) of the topological B-model string are objects in a derived category of coherent sheaves was somewhat involved and a bit mysterious. Once upon a time I had tried to summarize the main steps involved as reviewed by Aspinwall here

Now, Jonathan Block showed that these derived categories of coherent sheaves are actually equivalent to homotpy categories of representations of holomorphic tangent Lie algebroids on chain complexes – but these are special cases of flat linear (as opposed to principal) $\infty$-connections.

Aaron Bergman takes this theorem at face value and concludes that therefore it should be true that there is a direct way to see that the boundary conditions of the topological B-string come from such flat $\infty$-vector bundles with connection, pretty much analogous to how an ordinary conformal string (the “physical string”) couples on D-branes to ordinary (maybe $\mathbb{Z}_2$-graded) vector bundles with connection.

As an ansatz, he considers boundary insertions in the path integral that should correspond to the generalization of the familiar holonomy along the String’s boundary of the pulled back connection to the $\infty$ setup. He derives that this satisfies the B-string’s topological invariance precisely if the flat $\infty$-connection satisfies the appropriate axioms it should satisfy. He then derives that, similarly, imposing the required topological invariance on the boundary field insertions interpolating between two such flat $\infty$-connections yields the right notion of morphism in the category of flat $\infty$-connections that Jonathan Block considered.