August 17, 2008

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.

Posted at August 17, 2008 12:46 PM UTC

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Re: Bergman on Infinity-Vector Bundles Coupled to Topological Strings

I should say that it turns out that the coupling had already been derived in a different way in Herbst, Hori and Page.

Posted by: Aaron Bergman on August 17, 2008 3:07 PM | Permalink | Reply to this

Re: Bergman on Infinity-Vector Bundles Coupled to Topological Strings

these derived categories of coherent sheaves are actually equivalent to homotopy categories of representations of holomorphic tangent Lie algebroids on chain complexes

Wow that sounds very elegant.

Posted by: Bruce Bartlett on August 17, 2008 5:33 PM | Permalink | Reply to this

Re: Bergman on Infinity-Vector Bundles Coupled to Topological Strings

these derived categories of coherent sheaves are actually equivalent to homotopy categories of representations of holomorphic tangent Lie algebroids on chain complexes

Wow that sounds very elegant.

I guess in concrete applications it may depend which of the two points of view is more useful, but the interpretation in terms of reps of $L_\infty$-algebroids seems to clarify a bit more what is really going on here.

By the way, do you rememebr when we were all sitting in the chinese restaurant in Toronto during a break at the Fields institute workshop on higher categories and their applications with Aaron Lauda and a couple of other guys? We were talking about algebras assigned to boundaries in 2d TFT and how these can be interpreted as placeholders for 2-vector spaces. I think I said that we should try to understand how $\infty$-vector spaces come into this picture, because these are the ones that appear in the phyiscally interesting models. I still remember how you nodded. So here is Aaron Bergman making a bit of progress towards this goal.

Posted by: Urs Schreiber on August 17, 2008 8:13 PM | Permalink | Reply to this

Re: Bergman on Infinity-Vector Bundles Coupled to Topological Strings

Indeed. I can see there is some serious progress here; it’s just that I have to pump some serious iron in the mathematical gym before I can get my technical level up to the point where I can understand it.

Posted by: Bruce Bartlett on August 17, 2008 9:50 PM | Permalink | Reply to this

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