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--- blog:2023-06-27 [2023/06/27 16:54] – created pzhou
+++ blog:2023-06-27 [2023/06/27 18:32] (current) – pzhou
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 ===== BFN approach of matter =====
+==== diversion on topology ====
 Let's tell the story just one more time. We have  $GL_1(\C)$  acting on  $\C$  in the standard way. We have two stacks, one is  $\C(K) / GL_1(K)$ , one is  $\C(O) / GL_1(O)$ . It is a set with some group action.
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 one thing that I noticed is that, if we have a quotient map by group action  $X \to X/G$ , then usually the natural topology on  $X/G$  (say it is discrete) and the one on  $X$  are different. For example, consider  $B$  acting on  $G/B$ . So, the topology on  $GL_1(K)$  and that on $GL_1(K)/GL_1(O)$ is not compatible. That is a bit confusing.
-What's the topology on  $GL_1(K)$  and  $K$ ? We have inclusion  $GL_1(K) = K \RM \{0\}$ .  $K$  is an infinite dimensional vector space
+What's the topology on  $GL_1(K)$  and  $K$ ? We have inclusion  $GL_1(K) = K \RM \{0\}$ . We can ask, what is the topology of  $K$ .  $K = \C( (t) )$ . The abstract metric completion of a metrized space equal to the induced closure after isometrically embedded into a complete metric space. We define the  $t$ -adic valuation on  $K$ , which is not the usual one induced from polynomial.
+The subset  $c_0 + t \C[ [t] ]$  is itself both open and closed,  since it is is the open and closed ball of radius  $r_0$  centered at  $c_0$ , where  $1 > r_0 > e^{-1}$ .
+So, the space  $K$  is disconnected. Or it is totally disconnected. wiki says, all non-archimedian division rings are toally disconnected. fine.
+So, why does affine Grassmannian have interesting topology? What topology are we talking about even? The analytic topology over  $\C$ ? Or some other ones? The naive picture that  $LG$  is a  $L^+G$  torsor over  $Gr_G$  seems to be correct.
+If  $G$  is  $GL_1$ , then there is no contradiction. But if  $G=GL_2$ , then  $Gr_G$  is not a totally disconnected space, take a connected component, take two points, connect them by a (real segment), lift it up (using some local trivialization of  $LG \to Gr_G$ ),  then it means that  $GL_2(K)$  is not totally disconnected. Even  $GL_2(O)$  may not be totally disconnected. The case for  $GL_1(O)$  is a coincidence.
+What is the topology on  $GL_2(O)$ ? We have the matrix  $M_2(O)$ . It is a nice enough vector space.
+OK, the t-adic metric on  $F( (t) )$  is viewing  $F$  as a totally disconnected space, like  $\F_q$ . Even treating  $\C$  as a disconnected space. And that is too fine.
+==== OK then, how about the weird convolution euler index ====
+slow down. let's work over  $\C$ , as real two dimensional space.
+Consider a rank  $r$  vector space over  $\C$ , with  $\C^*$ -diagonal action. First, consider the non-equivariant BM homology of the space. by definition, this is the relative homology of the one point compactification, relative to the unique point at infinity. So here, it is $H_*(S^{2r}, pt)$, which is generated by the fundamental class.
+Now, we turn on  $\C^*$  or  $S^1$  action. There are many invariant subspaces. We also have  $S^1$  action on  $S^{2r}$ , coming from the suspension of  $S^1$  acting on  $S^{2r-1}$ .
+What can we say then? If we want to compute the equivariant cohomology (which is equal to equivariant homology up to shift) of  $\C$  acted by  $S^1$ , then we pass to the Borel model, we form  $EG \times_G \C$ ,  $G=S^1$ , and we get a line bundle over  $BG=\C \P^\infty$ . The top dimensional 'homology', or 0-degree (degree = codim) cohomology is defined. That is  $[\C_G]$ , the fundamental (homology) class of the universal associated bundle.  Then, we can ask, what is the homology of  $[0_G]$ , the universal zero-section's fundamental class? By Thom isomorphism, we know **it is equivalent to something in the base, times the full fiber of  $[\C_G] \to [0_G]$ **, by perturbation and stretching. This is equal to computing the self-intersection of the zero-section.
+ $[0_G] = [\C_G] \cdot u$
+where  $u$  is cohomological degree  $2$ .  $[0_G] = [\C^r_G] \cdot u^r$ . This make sense at least degreewise.
+So, the actual non-compact space fundamental class is
+ $[\C^r_G]= \frac{[0_G]}{u^r}$
+OK, this summarizes my understanding of equivariant localization between a vector space and the zero section.
+For the abelian case, there should be several approaches, all leading to the correct consistent answers.
+BFN and VV both defined some convolution, maybe even BDG.
+==== BFN convolution space ====
+Recall what is matrix multiplication: given matrices of sizes  $n_1 \times n_2$ ,  $n_2 \times n_3$ , called  $M_{12}$  and  $M_{23}$ , we can form their product, which is given by
+ $(M_{13})_{ij} = \sum_k (M_{12})_{ik} (M_{12})_{kj}$
+This can be considered as composing correspondences. We need a way to do composition (multiplication), and a way to do summation (over all possible  $k$ , intermediate state).
+If we package this in terms of BM homology, we