Peng Zhou

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====== 2026-02-24 ======
I am recording this past discussion with Ben G. Here (n+1)-dim TFT will be given by some n-category, we use $nA$ or $nB$ as the corresponding category. 

Let $G$ and $G^\vee$ be Langlands dual groups. Assume relative Langlangds says there are some equivalence of 3-cats
$$ 3A_G \cong 3B_{G^{\vee}}, \quad 3IndPerv(BG) \cong 3IndCoh(BG^\vee). $$
where objects in $3A_G$ comes from Ham G-space $M = T^*X$, where $X$ are $G$-spherical varieties, which means Borel acts with finitely many objects, same for $3B_{G^L}$. In the following, we will denote objects by $X$ instead of $T^*X$. 

Example: $G = G_m, G^L = G_m^\vee$. As dual objects
$$ G_m \sim pt $$ 
$$ A^1 \sim  (A^1)^\vee $$ 
Then we have
$$ 3A(G_m, G_m) = 2Loc(G_m) = 2Coh(B G_m^\vee) = Rep(G^\vee)-mod $$

In general, $G$ acts on $pt$ is 4d dual to $G^\vee$ acts $T^*G / /_\chi N$ twisted Whittaker reduction. Quick sanity check
$$ (T^*pt/G, T^*pt/G)_G = T^*(pt/G), \quad (T^*G / /_\chi N, T^*G / /_\chi N)_{G^\vee} = T^*(N \RM G / N) $$
If we do $2B$ on the left, we $Rep(G)-Mod$, if we do $2A$ on the right, we get $2A$ on the multiplicative pure Coulomb branches. 

What the heck is the $Rep(G)-mod$? For $G=SL_2$, we know $Rep(G)$ is monoidally generated by the fundamental rep $F$, with and $F$ is self-dual. so we have $id \to F^R F$ and $F F^R \to id$. We also have $F^R F \to FF^R$ (some braiding functor?). We have a 2-category (of B-type)
$$ Rep(SL_2)-Mod = \{F: C \leftrightarrow C: F^R \mid [id_C \to F^R F \to F F^R \to id_C] = 2 \} $$
Where does such cat $C$ some from? Say $C = Coh(X/SL_2)$.