What is the Gromov-Witten potential associated to String Topology? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T11:19:53Z http://mathoverflow.net/feeds/question/27697 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/27697/what-is-the-gromov-witten-potential-associated-to-string-topology What is the Gromov-Witten potential associated to String Topology? skupers 2010-06-10T14:10:41Z 2010-07-29T20:59:57Z <p>Kevin Costello's article on <code>the Gromov-Witten potential associated to a TCFT</code> constructs for each TCFT, i.e. a functor from chains on Riemann surfaces with boundary to chain complexes satisfying certain conditions, a canonical formal power series $D$ with coefficients in a certain Fock space $\mathcal{F}$, which is constructed from the chain complex $V$ that a TCFT associates to the circle. </p> <p>When one can construct the Gromov-Witten invariants for a manifold, we get a TCFT from Gromov-Witten theory. In that case a certain choice of polarization allows us to identify this potential $D$ with the Gromov-Witten potential. This potential encodes the intersection numbers of $\Psi$-classes and the fundamental class.</p> <p>According to Costello's earlier article on <code>TCFT's and $A_\infty$ Calabi-Yau categories</code> the constructions of string topology allow us to define a TCFT for each oriented closed manifold $M$ (basically the chain level operations of Godin's operations in homology). One also gets a Gromov-Witten potential in this case. Is there an easier expression known for this potential? What geometric information does it encode?</p> http://mathoverflow.net/questions/27697/what-is-the-gromov-witten-potential-associated-to-string-topology/33855#33855 Answer by Daniel Pomerleano for What is the Gromov-Witten potential associated to String Topology? Daniel Pomerleano 2010-07-29T20:59:57Z 2010-07-29T20:59:57Z <p>Here's how I understand the situation(I'm not very deep and might well be wrong) --- Costello's paper explains how to construct the Gromov Witten potential from a compact A(infinity) Calabi Yau category given two little additional conditions 1) the Hodge to de-Rham spectral sequence degenerates and 2) the induced pairing as defined in his first paper on <code>$HH_*$</code> is non degenerate. This is explained for example on the bottom of page 9... </p> <p>In the case of an affine (Z- graded) dg-category--- e.g. C= the category of perfect modules over a dg-algebra A, these conditions are guaranteed by A being homologically smooth. The reference for the first being implied is a famous paper of D. Kaledin and the second is a paper of D. Shklyarov <a href="http://arxiv.org/abs/math/0702590" rel="nofollow">http://arxiv.org/abs/math/0702590</a> (I think anyways). </p> <p>None of these conditions are satisfied for string topology of a manifold(probably never somehow). The smoothness for example breaks pretty clearly --- think about $S^n$, C*(X) is $Q[x]/x^2$ with no higher operations.) The calculation of the HH to cyclic spectral sequence is a tad easier for odd spheres--- it is the case of a free commutative algebra on an odd variable which you can find in say Loday's book. HH is differential forms on the superspace $R(0,1)$ and normally the Connes operator acts by de Rham d extended to superdifferential forms(I haven't checked this little part but it's the only thing that makes any sense...)</p> <p>What you do have from degenerate theories like this is an ideal in a certain Weyl algebra (see page 10 of the paper) I am pretty confident you could compute it at least for the case of spheres without too many problems, but I tend to be optimistic about these sort of things so... Hopefully some of this is right.</p>