The paper gives an elegant construction of categorified Jones-Wenzl projectors as a limit of (suitably normalized) Khovanov chain complexes of torus braids. In the context of representation theory of the quantum group \(U_q(\mathfrak{sl}_2)\), the Temperley-Lieb algebra \(TL_n\) is the endomorphism algebra of the \(n\)-th tensor power of the natural \(2\)-dimensional representation, and the Jones-Wenzl projector \(P_n\) is the idempotent corresponding to the \((n+1)\)-dimensional irreducible direct summand. These projectors play a central role in constructions of \(SU(2)\) quantum invariants of \(3\)-manifolds.
When expressed in terms of Temperley-Lieb (planar) tangles, the Jones-Wenzl projectors have coefficients which are rational (rather than polynomial) functions of the parameter \(q\). Therefore their categorification takes the form of (semi)-infinite chain complexes. The main result is that the chain complex associated to the \((m,n)\)-torus braid (viewed as \(n\) strands with \(m\) full twists) in the Bar-Natan category [D. Bar-Natan, Geom. Topol. 9, 1443–1499 (2005; Zbl 1084.57011)] stabilizes as \(m\longrightarrow \infty\). This limit is the categorification of the Jones-Wenzl projector \(P_n\).
Alternative constructions of categorified Jones-Wenzl projectors appeared at about the same time in [B. Cooper and V. Krushkal, Quantum Topol. 3, No. 2, 139–180 (2012; Zbl 1362.57015)] and [I. B. Frenkel et al., ibid. 181–253 (2012; Zbl 1256.17006)].
The infinite torus braid construction, introduced in this paper, has already been used by a number of authors in follow-up works. In particular, it was applied to the \(\mathfrak{sl}_3\) case in [D. E. V. Rose, Quantum Topol. 5, No. 1, 1–59 (2014; Zbl 1294.57009)], and more generally to \(\mathfrak{sl}_n\) tangle invariants in [S. Cautis, Math. Ann. 363, No. 3–4, 1053–1115 (2015; Zbl 1356.57013)].