50 research outputs found
Computing a pyramid partition generating function with dimer shuffling
We verify a recent conjecture of Kenyon/Szendroi, arXiv:0705.3419, by
computing the generating function for pyramid partitions. Pyramid partitions
are closely related to Aztec Diamonds; their generating function turns out to
be the partition function for the Donaldson--Thomas theory of a non-commutative
resolution of the conifold singularity {x1x2 -x3x4 = 0}. The proof does not
require algebraic geometry; it uses a modified version of the domino shuffling
algorithm of Elkies, Kuperberg, Larsen and Propp.Comment: 19 pages, 13 figures. v2: fixed minor typos, updated references and
future work; added some definitions to Section
Wall Crossing and M-theory
We study BPS bound states of D0 and D2 branes on a single D6 brane wrapping a
Calabi-Yau 3-fold X. When X has no compact 4-cyles, the BPS bound states are
organized into a free field Fock space, whose generators correspond to BPS
states of spinning M2 branes in M-theory compactified down to 5 dimensions by a
Calabi-Yau 3-fold X. The generating function of the D-brane bound states is
expressed as a reduction of the square of the topological string partition
function, in all chambers of the Kahler moduli space.Comment: 19 pages, 1 figur
Open BPS Wall Crossing and M-theory
Consider the degeneracies of BPS bound states of one D6 brane wrapping
Calabi-Yau X with D0 branes and D2 branes. When we include D4-branes wrapping
Lagrangian cycle L in addition, D2-branes can end on them. These give rise to
new bound states in the d=2, N=(2,2) theory of the D4 branes. We call these
"open" BPS states, in contrast to closed BPS states that arise from D-branes
without boundaries. Lifting this to M-theory, we show that the generating
function is captured by free Fock space spanned by M2-brane particles ending on
M5 branes wrapping L. This implies that the open BPS bound states are counted
by the square of the open topological string partition function on X, reduced
to the corresponding chamber. Our results give new predictions for open BPS
invariants and their wall crossing phenomena when we change the open and closed
string moduli. We relate our results to the work of Cecotti and Vafa on wall
crossing in the two dimensional N=(2,2) theories. The findings from the crystal
melting model for the open BPS invariants proposed recently fit well with the
M-theory predictions.Comment: 18 pages, 1 figur