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Thomas Riepe
Thomas Riepe
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Here are Tom Coates' notes.

A try to summarize the seminar talks mentioned below: Only small quantum cohomology was discussed. The basic idea is to do the same as with normal motives, i.e. to map the cat. of varieties into something that looks like a Tannakian category. Whereas usually, one seeks to classify and identify motives by their corresponding representations of Gal(Q), here on looks for representations of a bigger "quantum Galois group of Q" found within known representations coming from geometry. That "quantum Galois group" was not described, maybe this relates to it?

A try to summarize the seminar talks mentioned below: Only small quantum cohomology was discussed. The basic idea is to do the same as with normal motives, i.e. to map the cat. of varieties into something that looks like a Tannakian category. Whereas usually, one seeks to classify and identify motives by their corresponding representations of Gal(Q), here on looks for representations of a bigger "quantum Galois group of Q" found within known representations coming from geometry. That "quantum Galois group" was not described, maybe this relates to it?

Here are Tom Coates' notes.

A try to summarize the seminar talks mentioned below: Only small quantum cohomology was discussed. The basic idea is to do the same as with normal motives, i.e. to map the cat. of varieties into something that looks like a Tannakian category. Whereas usually, one seeks to classify and identify motives by their corresponding representations of Gal(Q), here on looks for representations of a bigger "quantum Galois group of Q" found within known representations coming from geometry. That "quantum Galois group" was not described, maybe this relates to it?

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Thomas Riepe
Thomas Riepe
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Edit (Thomas):

A try to summarize the seminar talks mentioned below: Only small quantum cohomology was discussed. The basic idea is to do the same as with normal motives, i.e. to map the cat. of varieties into something that looks like a Tannakian category. Whereas usually, one seeks to classify and identify motives by their corresponding representations of Gal(Q), here on looks for representations of a bigger "quantum Galois group of Q" found within known representations coming from geometry. That "quantum Galois group" was not described, maybe this relates to it?

Golyshev mentioned as motivating analogy an article by Deligne, where D. proves the Weil-conjectures for K3-surfaces by embedding motives from them in a product of motives from abelian varieties, for whom the Weil-conj.s were known to be true. That embedding of motives was done by looking how the representations fit together, and by identifying whose rep.s come from abelian varieties. In Golyshev's analogy, even dim. quadrics should play the role of K3-surfaces, the orth. group the role of a construction by Kuga-Satake in Deligne's paper on the repr.s of ab. var.s

A search method for such representations which would led to "quantum motives" is made with help of mirror symmetry. That idea seems to go back to Golyshev.

Coates' method seems to be: The q.-cohomology of Fanos is known, that of toric varieties is not; On the other hand, the Laurent polynomials of toric varieties are known, but that of Fanos are not. Mirror symmetry connects q.-cohomology with such polynomials and Golyshev has a conjecture conc. the Laurent-poly.s of Fanos. Coates' computations try to bridge between Fanos and toric varieties by deformation theory and look if polynomials result which look like coming from pieces of Fanos, which then could be "quantum motives".

Maniel's idea is to look at the quantum cohomology of Grassmanians etc., (if I understood him correctly) producing other interesting representations through a quantum Satake correspondence. Unfortunately he mentioned that with the quantum Satake so shortly at the beginning of his talk, that I am unsure if my idea on it's use is correct.

A very fascinating talk was by Gorbounov, he plays around with Landau-Ginzberg potentials and finds quantum cohomology as special case of equivariant coh. However that may be, as far as I know, Landau-Ginzberg potentials are only used in Grassmanians etc., q.-coh. is not restricted to them.

One talk I understood nothing from was by Katzarkov, probably similar to his Oberwolfach talk earlier this year. The connection to quantum motives is unclear to me.

Edit (Thomas):

A try to summarize the seminar talks mentioned below: Only small quantum cohomology was discussed. The basic idea is to do the same as with normal motives, i.e. to map the cat. of varieties into something that looks like a Tannakian category. Whereas usually, one seeks to classify and identify motives by their corresponding representations of Gal(Q), here on looks for representations of a bigger "quantum Galois group of Q" found within known representations coming from geometry. That "quantum Galois group" was not described, maybe this relates to it?

Golyshev mentioned as motivating analogy an article by Deligne, where D. proves the Weil-conjectures for K3-surfaces by embedding motives from them in a product of motives from abelian varieties, for whom the Weil-conj.s were known to be true. That embedding of motives was done by looking how the representations fit together, and by identifying whose rep.s come from abelian varieties. In Golyshev's analogy, even dim. quadrics should play the role of K3-surfaces, the orth. group the role of a construction by Kuga-Satake in Deligne's paper on the repr.s of ab. var.s

A search method for such representations which would led to "quantum motives" is made with help of mirror symmetry. That idea seems to go back to Golyshev.

Coates' method seems to be: The q.-cohomology of Fanos is known, that of toric varieties is not; On the other hand, the Laurent polynomials of toric varieties are known, but that of Fanos are not. Mirror symmetry connects q.-cohomology with such polynomials and Golyshev has a conjecture conc. the Laurent-poly.s of Fanos. Coates' computations try to bridge between Fanos and toric varieties by deformation theory and look if polynomials result which look like coming from pieces of Fanos, which then could be "quantum motives".

Maniel's idea is to look at the quantum cohomology of Grassmanians etc., (if I understood him correctly) producing other interesting representations through a quantum Satake correspondence. Unfortunately he mentioned that with the quantum Satake so shortly at the beginning of his talk, that I am unsure if my idea on it's use is correct.

A very fascinating talk was by Gorbounov, he plays around with Landau-Ginzberg potentials and finds quantum cohomology as special case of equivariant coh. However that may be, as far as I know, Landau-Ginzberg potentials are only used in Grassmanians etc., q.-coh. is not restricted to them.

One talk I understood nothing from was by Katzarkov, probably similar to his Oberwolfach talk earlier this year. The connection to quantum motives is unclear to me.

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Peter Arndt
Peter Arndt
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"I'd like to know more" is the vaguest possible question :-) The best you can do is to watch the corresponding talk here - it is fantastic ! - and probably come back with more questions, since it is quite dense.

The background for the statement you are asking about is roughly the following (I am no expert and writing from memory - the following may contain nonsense and should not be taken literally):

Consider the moduli stacks $\overline{\mathcal{M}_{g,n}}$ of stable curves of genus g with n marked points. These are interconnected by a bunch of maps corresponding to glueing together stable curves, forgetting marked points, collapsing components to restore stability etc. (these are all operations with an intuitive geometric meaning and are wonderfully explained in Kock/Vainsencher's "Invitation to Quantum Cohomology").

If we map these into the category of motives, we can for each $n$ form their direct sum over $g$, thus getting a sequence of motives, and then assemble the above maps to form the structure maps of an operad, i.e. mainly "composition maps" $\overline{\mathcal{M}_{*,n_1}} \times \ldots \times \overline{\mathcal{M}_{*,n_k}} \rightarrow \overline{\mathcal{M}_{*,n_1 + \ldots + n_k}}$.

The word "motives" here means Chow motives and to make sense of this for stacks, one has to extend the definition of Chow groups to suitable stacks and then apply the usual constructions. This can be done in at least two different ways (Vistoli's and Toen's, see the talk for references).

Then the statement in the talk is that the motive of any smooth projective variety ("projective scheme" is a bit too bold because we are using Chow motives) is acted upon by this operad object in motives. To construct the action morphisms one uses cycles (and thus morphisms in motives) $I_{g,n}(V)$ corresponding to certain quantum cohomology classes, which are present in the quantum cohomology of any smooth projective variety...

Of course if you apply now any monoidal functor to all of this (e.g. a cohomology) you get an operad, and an object with an action from it, in the target category...