12,390 research outputs found
The characters of two new dipterous genera, with indications of some generic subdivisions and several undescribed species of Dolichopodidae
Lower central series and free resolutions of hyperplane arrangements
If is the complement of a hyperplane arrangement, and A=H^*(M,\k) is
the cohomology ring of over a field of characteristic 0, then the ranks,
, of the lower central series quotients of 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 . 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 over the exterior algebra .
From this analysis, we recover a formula of Falk for , and obtain a
new formula for . 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 is
Koszul iff the arrangement is supersolvable. We also give combinatorial lower
bounds on the Betti numbers, , of the linear strand of the free
resolution of over ; if the lower bound is attained for , then it
is attained for all . For such arrangements, we compute the entire
linear strand of the resolution, and we prove that all components of the first
resonance variety of are local. For graphic arrangements (which do not
attain the lower bound, unless they have no braid sub-arrangements), we show
that is determined by the number of triangles and 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
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