We prove that the functor of noncommutative deformations of every flipping or flopping irreducible rational curve in a $3$-fold is representable, and hence, we associate to every such curve a noncommutative deformation algebra ${\mathrm{A}}_{\mathrm{con}}$. This new invariant extends and unifies known invariants for flopping curves in $3$-folds, such as the width of Reid and the bidegree of the normal bundle. It also applies in the settings of flips and singular schemes. We show that the noncommutative deformation algebra ${\mathrm{A}}_{\mathrm{con}}$ is finite-dimensional, and give a new way of obtaining the commutative deformations of the curve, allowing us to make explicit calculations of these deformations for certain $(-3,1)$-curves.

We then show how our new invariant ${\mathrm{A}}_{\mathrm{con}}$ also controls the homological algebra of flops. For any flopping curve in a projective $3$-fold with only Gorenstein terminal singularities, we construct an autoequivalence of the derived category of the $3$-fold by twisting around a universal family over the noncommutative deformation algebra ${\mathrm{A}}_{\mathrm{con}}$, and prove that this autoequivalence is an inverse of Bridgeland’s flop-flop functor. This demonstrates that it is strictly necessary to consider noncommutative deformations of curves in order to understand the derived autoequivalences of a $3$-fold and, thus, the Bridgeland stability manifold.

The two Rogers–Ramanujan $q$-series

$${\sum }_{n=0}^{\infty }\frac{q^{n(n+\sigma )}}{(1-q)\cdots (1-q^n)},$$ where $\sigma =0,1$, play many roles in mathematics and physics. By the Rogers–Ramanujan identities, they are essentially modular functions. Their quotient, the Rogers–Ramanujan continued fraction, has the special property that its singular values are algebraic integral units. We find a framework which extends the Rogers–Ramanujan identities to doubly infinite families of $q$-series identities. If $a\in \{1,2\}$ and $m,n\ge 1$, then we have

$${\sum }_{\lambda ,{\lambda }_1\le m}{q}^{a\left|\lambda \right|}{P}_{2\lambda }(1,q,q^2,\dots ;q^n)=\left[\mathrm{infinite}\mathrm{product}\mathrm{modular}\mathrm{function}\right],$$where the $P_{\lambda }(x_1,x_2,\dots ;q)$ are Hall–Littlewood polynomials. These $q$-series are specialized characters of affine Kac–Moody algebras. Generalizing the Rogers–Ramanujan continued fraction, we prove in the case of ${\mathrm{A}}_{2n}^{\left(2\right)}$ that the relevant $q$-series quotients are integral units.

Let $\mathfrak{o}$ be the ring of integers in a finite extension $K$ of ${\mathbb{Q}}_p$, and let $k$ be its residue field. Let $G$ be a split reductive group over ${\mathbb{Q}}_p$, and let $T$ be a maximal split torus in $G$. Let $\mathcal{H}(G,I_0)$ be the pro-$p$ Iwahori–Hecke $\mathfrak{o}$-algebra. Given a semi-infinite reduced chamber gallery (alcove walk) $C^{(•)}$ in the $T$-stable apartment, a period $\varphi \in N\left(T\right)$ of $C^{(•)}$ of length $r$, and a homomorphism $\tau :{\mathbb{Z}}_p^{\times }\to T$ compatible with $\varphi$, we construct a functor from the category ${Mod}^{\mathrm{fin}}\left(\mathcal{H}\right(G,I_0\left)\right)$ of finite-length $\mathcal{H}(G,I_0)$-modules to étale $({\phi }^r,\Gamma )$-modules over Fontaine’s ring ${\mathcal{O}}_{\mathcal{E}}$. If $G={GL}_{d+1}\left({\mathbb{Q}}_p\right)$, then there are essentially two choices of $(C^{(•)},\varphi ,\tau )$ with $r=1$, both leading to a functor from ${Mod}^{\mathrm{fin}}\left(\mathcal{H}\right(G,I_0\left)\right)$ to étale $(\phi ,\Gamma )$-modules and hence to ${\mathrm{Gal}}_{{\mathbb{Q}}_p}$-representations. Both induce a bijection between the set of absolutely simple supersingular $\mathcal{H}(G,I_0){\otimes }_{\mathfrak{o}}k$-modules of dimension $d+1$ and the set of irreducible representations of ${\mathrm{Gal}}_{{\mathbb{Q}}_p}$ over $k$ of dimension $d+1$. We also compute these functors on modular reductions of tamely ramified locally unitary principal series representations of $G$ over $K$. For $d=1$, we recover Colmez’s functor (when restricted to $\mathfrak{o}$-torsion ${GL}_2\left({\mathbb{Q}}_p\right)$-representations generated by their pro-$p$ Iwahori invariants).