In <http://arxiv.org/pdf/1410.6240.pdf> M. McBreen and N. Proudfoot conjectured a precise relationship between the quantum cohomology of a symplectic resolution and the intersection cohomology of the cone which it is resolving. However, I could not find the motivation for this conjecture in the above article. Is there a heuristic reason why the two sides should be related? 



Why should intersection cohomology and quantum cohomology be related for a symplectic resolution?
I'll try to explain the genesis of the paper, then some intuition we developed later.


**The original motivation was a coincidence**. For $X$ hypertoric, Tom Braden and Nick Proudfoot had defined a ring structure on $IH^\*(X^{aff})$ whereas Daniel Shenfeld and I found a generators & relations presentation of $QH\_{\mathbb{C}^\*}(X)$. When the quantum parameter $q$ is specialized to the 'most singular value' $q\_0$, our two rings coincide.


We ended up finding something much stronger. Namely, the decomposition theorem tells us $H^\*(X)= IH^\*(X^{aff}) \oplus K$ for some perverse sheaf $K$, and hence gives an inclusion $IH^\*(X^{aff}) \to H^\*(X)$. The latter maps to a `polynomial subbring' $QH^\*\_{\mathbb{C}^\*}(X)\_{pol} \subset QH^\*\_{\mathbb{C}^\*}(X)$, which admits a specialization $QH^\*\_{\mathbb{C}^\*}(X)\_{pol} \to R'$ at $q=q\_0$.


In our paper, we show that for $X$ hypertoric, the composition
$$ IH^\*(X^{aff}) \to H^\*(X) \to QH^\*\_{\mathbb{C}^\*}(X)\_{pol} \to R' $$


is an isomorphism. This is (to me) still rather surprising! Neither surjectivity nor injectivity are given to you for free, and the fact that they hold for all hypertorics is very suggestive.


**There are nonetheless some general heuristics**. For $X$ a symplectic resolution, the Steinberg algebra $A = H^{top}(X \times\_{X^{aff}} X)$ acts on $H^\*(X)$, and the trivial isotypical component of this module is exactly $IH^\*(X^{aff})$. In fact, the Steinberg algebra decomposes as $\mathbb{C} id \oplus A^+$, where $id$ is the diagonal correspondence, and in the decomposition
$$ H^\*(X) = IH^\*(X^{aff}) \oplus K $$
we have $K = image(A^+)$.


The conjecture can be roughly reformulated as `specializing $q=q\_0$ kills the image of $A^+$'. 


Now, let $u \in QH^2\_{\mathbb{C}^\*}(X)$. Then [BravermanMaulikOkounkov] tells us that the operator of quantum product by $u$, viewed as a function of the quantum parameter $q \in H^2(X,\mathbb{C}^\*)$, has poles along certain divisors $D\_{\alpha} \subset H^2(X,\mathbb{C}^\*)$. The residues are Steinberg operators $Z\_{\alpha} \in A^+$.


For many symplectic resolutions, the $D\_{\alpha}$ intersect in a 'maximally singular point $q\_0$' (when this is not the case, our conjecture runs into some trouble and needs to be corrected). 


One would like to say: specializing $q$ to $D\_{\alpha}$ kills the image of $Z\_{\alpha}$ in $H^\*(X)$; specializing to $q\_0$ kills the image of all $Z\_{\alpha}$. In fact, modulo some technical difficulties it seems likely one can prove such a statement.


Hence, very roughly, our conjecture ressembles the claim that the $Z\_{\alpha}$ generate $A^+ \subset A$, or at least the set of idempotents in $A^+$. I do not know of a general reason why the $Z\_{\alpha}$ should generate, but perhaps you prefer this mystery to the one we started with.


**There are also some motivations coming from representation theory**, which I won't go into here as they require a lot of set-up.


**More speculatively,** the limit $q \to q\_0$ should correspond to a singular degeneration $Y \to Y\_0$ of the mirror variety of $X$. Since symplectic resolutions are almost 'self-mirror', one may hope that the cone singularity of $X^{aff}$ appears in $Y\_0$. Perhaps one can directly identify $R'$ with the intersection cohomology of this singularity. Of course, for now there are too many difficulties to make this precise.


