Lower central series and free resolutions of hyperplane arrangements

If $M$ is the complement of a hyperplane arrangement, and A=H^*(M,\k) is the cohomology ring of $M$ over a field of characteristic 0, then the ranks, $\phi_k$, of the lower central series quotients of $\pi_1(M)$ can be computed from the Betti numbers, b_{ii}=\dim_{\k} \Tor^A_i(\k,\k)_i, of the linear strand in a (minimal) free resolution of \k over $A$. We use the Cartan-Eilenberg change of rings spectral sequence to relate these numbers to the graded Betti numbers, b'_{ij}=\dim_{\k} \Tor^E_i(A,\k)_j, of a (minimal) resolution of $A$ over the exterior algebra $E$. From this analysis, we recover a formula of Falk for $\phi_3$, and obtain a new formula for $\phi_4$. The exact sequence of low degree terms in the spectral sequence allows us to answer a question of Falk on graphic arrangements, and also shows that for these arrangements, the algebra $A$ is Koszul iff the arrangement is supersolvable. We also give combinatorial lower bounds on the Betti numbers, $b'_{i,i+1}$, of the linear strand of the free resolution of $A$ over $E$; if the lower bound is attained for $i = 2$, then it is attained for all $i \ge 2$. For such arrangements, we compute the entire linear strand of the resolution, and we prove that all components of the first resonance variety of $A$ are local. For graphic arrangements (which do not attain the lower bound, unless they have no braid sub-arrangements), we show that $b'_{i,i+1}$ is determined by the number of triangles and $K_4$ subgraphs in the graph.Comment: 25 pages, to appear in Trans. Amer. Math. So

Resonance, linear syzygies, Chen groups, and the Bernstein-Gelfand-Gelfand correspondence

If \A is a complex hyperplane arrangement, with complement X, we show that the Chen ranks of G=\pi_1(X) are equal to the graded Betti numbers of the linear strand in a minimal, free resolution of the cohomology ring A=H^*(X,\k), viewed as a module over the exterior algebra E on \A: \theta_k(G) = \dim_\k Tor^E_{k-1}(A,\k)_k, where \k is a field of characteristic 0, and k\ge 2. The Chen ranks conjecture asserts that, for k sufficiently large, \theta_k(G) =(k-1) \sum_{r\ge 1} h_r \binom{r+k-1}{k}, where h_r is the number of r-dimensional components of the projective resonance variety R^1(\A). Our earlier work on the resolution of A over E and the above equality yield a proof of the conjecture for graphic arrangements. Using results on the geometry of R^1(\A) and a localization argument, we establish the conjectured lower bound for the Chen ranks of an arbitrary arrangement \A. Finally, we show that there is a polynomial P(t) of degree equal to the dimension of R^1(\A), such that \theta_k(G) = P(k), for k sufficiently large.Comment: 21 pages; final versio

A new handle on three-point coefficients: OPE asymptotics from genus two modular invariance

We derive an asymptotic formula for operator product expansion coefficients of heavy operators in two dimensional conformal field theory. This follows from modular invariance of the genus two partition function, and generalises the asymptotic formula for the density of states from torus modular invariance. The resulting formula is universal, depending only on the central charge, but involves the asymptotic behaviour of genus two conformal blocks. We use monodromy techniques to compute the asymptotics of the relevant blocks at large central charge to determine the behaviour explicitly.Comment: 32 pages, 2 figures, 1 appendix, 2 moose, a bear and an o