Alejandro H. Morales

E7(7) constraints on counterterms in N=8 supergravity

Abstract We prove by explicit computation that 6-point matrix elements of D 4 R 4 and D 6 R 4 in N = 8 supergravity have non-vanishing single-soft scalar limits, and therefore these operators violate the continuous E 7 ( 7 ) symmetry. The soft limits precisely match automorphism constraints. Together with previous results for R 4 , this provides a direct proof that no E 7 ( 7 ) -invariant candidate counterterm exists below 7-loop order. At 7-loops, we characterize the infinite tower of independent supersymmetric operators D 4 R 6 , R 8 , φ 2 R 8 , … with n > 4 fields and prove that they all violate E 7 ( 7 ) symmetry. This means that the 4-graviton amplitude determines whether or not the theory is finite at 7-loop order. We show that the corresponding candidate counterterm D 8 R 4 has a non-linear supersymmetrization such that its single- and double-soft scalar limits are compatible with E 7 ( 7 ) up to and including 6-points. At loop orders 7 , 8 , 9 we provide an exhaustive account of all independent candidate counterterms with up to 16 , 14 , 12 fields, respectively, together with their potential single-soft scalar limits.

Observation of halos in rotationally inelastic scattering of Na2 from Ar

We have measured level to level differential cross sections for rotational level changing collisions in ground state Na2 with Ar: Na2 (v\=0, ji=7)+Ar→Na2 (v\=0, jf)+Ar. Measurements range in Δj(jf−ji) from −4 to 80 and in Θ to π. A seeded supersonic beam source produces rotationally cold Na2 with a narrow velocity distribution; the levels are isolated and the differential cross sections are measured with cw dye lasers, using a new technique, angular distributions using the Doppler shift. The differential cross sections display a sharp rise and slow decline, reaching a maximum at an angle which is roughly proportional to Δj. We term these features ’’halos’’. (AIP)

Flow Polytopes of Signed Graphs and the Kostant Partition Function

We establish the relationship between volumes of flow polytopes associated to signed graphs and the Kostant partition function. A special case of this relationship, namely, when the graphs are signless, has been studied in detail by Baldoni and Vergne using techniques of residues. In contrast with their approach, we provide entirely combinatorial proofs inspired by the work of Postnikov and Stanley on flow polytopes. As a fascinating special family of flow polytopes, we study the Chan-Robbins-Yuen polytopes. Motivated by the beautiful volume formula

The polytope of Tesler matrices

\prod_{k=1}^{n-2} Cat(k)

Counting matrices over finite fields with support on skew Young diagrams and complements of Rothe diagrams

for the type

Matrices with restricted entries and

A_n

q

version, where

-analogues of permutations

Cat(k)

Hook formulas for skew shapes I. q-analogues and bijections

is the

Asymptotics of the number of standard Young tableaux of skew shape

k