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have added to the Idea-section at Schubert calculus the following paragraph:
Schubert calculus is concerned with the ring structure on the cohomology of flag varieties and Schubert varieties. Traditionally this was considered for ordinary cohomology (see References – traditional) later also for generalized cohomology theories (see References – In generalized cohomology), notably in complex oriented cohomology theory such as K-theory, elliptic cohomology and algebraic cobordism.
And have added references on Schubert calculus for generalized cohomology.
have added to Schubert calculus (very) brief pointers to pull-push through Schubert correspondences and Schubert product formulas
One of these well-kept secrets that should be advertized in the evening news:
The “generalized Penrose transform” is called Radon transform in standard books on harmonic analysis (in about the same generality, see books by Helgason, for example), and sometimes Penrose-Radon transform. The treatment of intersection theory via pull-push has been one of the trade marks of Grothendieck school (K-theory, Grothendecik-Riemann Roch, Grothendieck duality…) and modern avatars like the study of Fourier-Mukai transforms and more general kernels. I would not say that it is correct however to say that the Schubert calculus is the same thing. Schubert calculus is the additional question on how the specific subvarieties, Schubert classes concretely behave under the intersection products, rather than the theory what intersection products are. It is like saying that graph theory is the same as set theory, as graphs are defined using set of vertices and set of arrows.
How generalized is ’generalized’? Is it all ultimately part of the ’push-pull through slices on spans’ story?
@Zoran, the point is not the general idea of pull-push in intersection theory here, but the specific case of that involving specifically correspondences of the form $G/P_1 \leftarrow G/(P_1 \cap P_2) \longrightarrow G/P_2$ for $P_i \hookrightarrow G$ parabolic subgroups of a suitable semisimple group.
@David: for Schubert calculus “generalized” means (by Ganter-Ram): pull-push not just in ordinary cohomology, but in generalized cohomology theories. for the Penrose transform “generalized” means: pull-push through correspondence as above more general than the original $SL(4)/SL(3) \leftarrow SL(4)/SL(2) \longrightarrow SL(4)/SL(2)\times SL(2)$.
Recently people are playing with this for the case of the 6d (2,0)-superconformal QFT in which case one expects to see pull-push in elliptic cohomology or the like. That’s where the two generalized theories effectively merge.
Shouldn’t Borel-Weil theorem link up with Schubert calculus at least as ’related’ but perhaps more profoundly? These notes see the Borel-Weil theorem as concern with the geometry and cohomology of Schubert varieties.
Yes, certainly, and orbit method. Just yesterday I said this to somebody, but didn’t get around to. Will add some quick cross-links now. But eventually this should be merged more, yes.
That reminds me of something I raised a while ago.
If, quoting you,
in 4d topological Yang-Mills, this is the way Wess-Zumino-Witten theory and and Wilson loop actions appears as a codimension-2 corner theory and as codimension-3 defects, respectively,
and
Wilson loops in an ambient gauge theory are 1d topological prequantum systems that mathematically are the content of Kirillov’s orbit method,
then shouldn’t there be ’methods’ like the orbit method but corresponding to different dimensions? E.g., shouldn’t there be a method relating to Wess-Zumino-Witten theory?
I gave Borel-Weil theorem a lead-in paragraph (it was missing one anyway) and added a pointer to your reference .
re #9:
yes, just this moment Joost Nuiten is working on this. So once geometric quantization is understood as just a special case of motivic quantization given by push-forwad in equivariant K-theory, we can ask all the previous questions again, but in a more general context.
The orbit method is really about pull-push of K-theory classes to Schubert/twistor-like correspondences. We understand well from Poisson holography what this means physically…
… and the corresponding higher analog for quantization of the string is pull-push through similar correspondences of equivariant elliptic cohomology. It is sort of known that the $G$-equivariant elliptic cohomology of the point knows about the loop group representations in direct analogy to how the $G$-equivariant K-theory of the point is the representation ring of the group itself. Hence pull-push here will give a “loop group orbit method”.
And then there’s quantization of higher branes as pull-push through correspondences of some higher chromatic equivariant cohomology?
That’s the idea, yes.
OK, so you say that if one takes generalized cohomology theory then one is beyond the case of Penrose-Radon transforms (which are themselves classically well beyond the simplest “twistor” case of $SL(4)$ etc.). I accept that, thanx. (On the other hand, this does not yet capture the Schubert calculus, as one has to single out the very specific subvarieties of consideration, and this is not implied by the very transform. There are also refinements like the positive Schubert cells and alike, which we encountered in the discussion of amplituhedron a couple of weeks ago)
What I said was that both Schubert calculus and twistor calculus are concerned with cohomology not of random spaces, but of certain flag varieties and certain maps between these.
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