Motivic amplitudes and cluster coordinates Golden, J. K.; Goncharov, A. B.; Spradlin, M. ...
The journal of high energy physics,
2014, Letnik:
2014, Številka:
1
Journal Article
Recenzirano
Odprti dostop
A
bstract
In this paper we study
motivic amplitudes
— objects which contain all of the essential mathematical content of scattering amplitudes in planar SYM theory in a completely canonical way, free ...from the ambiguities inherent in any attempt to choose particular functional representatives. We find that the cluster structure on the kinematic configuration space Conf
n
(ℙ
3
) underlies the structure of motivic amplitudes. Specifically, we compute explicitly the coproduct of the two-loop seven-particle MHV motivic amplitude
and find that like the previously known six-particle amplitude, it depends only on certain preferred coordinates known in the mathematics literature as
cluster
-
coordinates
on Conf
n
(ℙ
3
). We also find intriguing relations between motivic amplitudes and the geometry of generalized associahedrons, to which cluster coordinates have a natural combinatoric connection. For example, the obstruction to
being expressible in terms of classical
polylogarithms is most naturally represented by certain quadrilateral faces of the appropriate associahedron. We also find and prove the first known functional equation for the trilogarithm in which all 40 arguments are cluster
-coordinates of a single algebra. In this respect it is similar to Abel’s 5-term dilogarithm identity.
We present a compact analytic formula for the two-loop six-particle maximally helicity violating remainder function (equivalently, the two-loop lightlike hexagon Wilson loop) in N=4 supersymmetric ...Yang-Mills theory in terms of the classical polylogarithm functions Lik with cross ratios of momentum twistor invariants as their arguments. In deriving our formula we rely on results from the theory of motives.
ABSTRACT
The main goal of pulsar timing array experiments is to detect correlated signals such as nanohertz-frequency gravitational waves. Pulsar timing data collected in dense monitoring campaigns ...can also be used to study the stars themselves, their binary companions, and the intervening ionized interstellar medium. Timing observations are extraordinarily sensitive to changes in path-length between the pulsar and the Earth, enabling precise measurements of the pulsar positions, distances and velocities, and the shapes of their orbits. Here we present a timing analysis of 25 pulsars observed as part of the Parkes Pulsar Timing Array (PPTA) project over time spans of up to 24 yr. The data are from the second data release of the PPTA, which we have extended by including legacy data. We make the first detection of Shapiro delay in four Southern pulsars (PSRs J1017−7156, J1125−6014, J1545−4550, and J1732−5049), and of parallax in six pulsars. The prominent Shapiro delay of PSR J1125−6014 implies a neutron star mass of Mp = 1.5 ± 0.2 M⊙ (68 per cent credibility interval). Measurements of both Shapiro delay and relativistic periastron advance in PSR J1600−3053 yield a large but uncertain pulsar mass of $M_p = 2.06^{+0.44}_{-0.41}$ M⊙ (68 per cent credibility interval). We measure the distance to PSR J1909−3744 to a precision of 10 lyr, indicating that for gravitational wave periods over a decade, the pulsar provides a coherent baseline for pulsar timing array experiments.
Let
G
be a split, simple, simply connected, algebraic group over
Q
. The degree 4, weight 2 motivic cohomology group of the classifying space
BG
of
G
is identified with
Z
. We construct cocycles ...representing the generator, known as the second universal motivic Chern class. If
G
=
SL
(
m
)
, there is a canonical cocycle, defined by Goncharov (Explicit construction of characteristic classes. Advances in Soviet mathematics, 16, vol 1. Special volume dedicated to I.M.Gelfand’s 80th birthday, pp 169–210, 1993). For any group G, we define a collection of cocycles parametrised by cluster coordinate systems on the space of
G
-orbits on the cube of the principal affine space
G
/
U
. Cocycles for different clusters are related by explicit coboundaries, constructed using cluster transformations relating the clusters. The cocycle has three components. The construction of the last one is canonical and elementary; it does not use clusters, and provides the motivic generator of
H
3
(
G
(
C
)
,
Z
(
2
)
)
. However to lift it to the whole cocycle we need cluster coordinates: construction of the first two components uses crucially the cluster structure of the moduli spaces
A
(
G
,
S
)
related to the moduli space of
G
-local systems on
S
. In retrospect, it partially explains why cluster coordinates on the space
A
(
G
,
S
)
should exist. The construction has numerous applications, including explicit constructions of the universal extension of the group
G
by
K
2
, the line bundle on
Bun
(
G
)
generating its Picard group, Kac–Moody groups, etc. Another application is an explicit combinatorial construction of the second motivic Chern class of a
G
-bundle. It is a motivic analog of the work of Gabrielov et al. (1974), for any
G
. We show that the cluster construction of the measurable group 3-cocycle for the group
G
(
C
)
, provided by our motivic cocycle, gives rise to the quantum deformation of its exponent.
Inuit living in the Arctic are exposed to elevated levels of environmental contaminants primarily due to long-range atmospheric transport. Blood sampling and contaminant biomonitoring was conducted ...as part of the International Polar Year Inuit Health Survey in 2007-2008. The body burden of metals (e.g. Cd, Pb) and persistent organic pollutants (e.g. PCBs, DDT & DDE, toxaphene, chlordane, PBDEs) were measured for Inuit participants (n=2172) from 36 communities in Nunavut, Nunatsiavut, and the Inuvialuit Settlement Region, in Canada. The geometric mean of blood concentrations for Cd, Pb, PCBs, DDE & DDT, toxaphene, and chlordane were higher than those in the Canadian general population. A total of 9% of study participants exceeded the intervention guideline of 100μgL−1 for Pb, 11% of participants exceeded the trigger guideline of 5μgL−1 for Cd, and 1% exceeded the intervention guideline of 100μgL−1 for PCBs. Also, 3% of women of child-bearing age exceeded blood Pb of 100μgL−1 while 28% of women of child-bearing age exceeded 5μgL−1 of PCBs. This work showed that most Inuit Health Survey participants were below blood contaminant guidelines set by Health Canada but that metal and POP body burdens commonly exceed exposures observed in the general population of Canada.
•Blood was collected from 2172 Inuit adults for contaminant biomonitoring.•Levels of Cd, Pb, Chlordane, Toxaphene, PCB, PBDE were measured.•Contaminant levels were often higher than seen in the general population of Canada.•Average blood levels tended to be below screening and intervention guidelines.
Magnetron sputtering is used for preparing thin NbN films. The films are deposited on sapphire substrates at temperatures from 20 to 300°C. The superconducting transition temperature for various ...samples is in the range of 8–14 K depending on the substrate temperature during deposition. The critical current density
j
c
is in the range of 0.8–8 MA/cm
2
, which makes it possible to use these films to create multilayer structures due to the absence of anneals, which each underlying layer of structures is subjected to during the deposition of each subsequent layer.
A method is described that allows one to reduce the critical current and switching power of a short nanowire from a superconducting state to a normal state by embedding a section of a normal metal ...into the nanowire. This effect results from the local heating of the superconducting wire due to the heat released in the normal metal. An integrated resistor is created from the original NbN using ion irradiation through a mask. The obtained values of switching powers allow designing multilayer logical elements without galvanic coupling for classical cryo-computers.
K-decompositions and 3d gauge theories Dimofte, Tudor; Gabella, Maxime; Goncharov, Alexander B.
The journal of high energy physics,
11/2016, Letnik:
2016, Številka:
11
Journal Article
Recenzirano
Odprti dostop
A
bstract
This paper combines several new constructions in mathematics and physics. Mathematically, we study
framed
flat PGL(
K,
ℂ)-connections on a large class of 3-manifolds
M
with boundary. We ...introduce a moduli space ℒ
K
(
M
) of framed flat connections on the boundary ∂
M
that extend to
M
. Our goal is to understand an open part of ℒ
K
(
M
) as a Lagrangian subvariety in the symplectic moduli space
X
K
u
n
∂
M
of framed flat connections on the boundary — and more so, as a “K
2
-Lagrangian,” meaning that the K
2
-avatar of the symplectic form restricts to zero. We construct an open part of ℒ
K
(
M
) from elementary data associated with the
hypersimplicial K-decomposition
of an ideal triangulation of
M
, in a way that generalizes (and combines) both Thurston’s gluing equations in 3d hyperbolic geometry and the cluster coordinates for framed flat PGL(
K,
ℂ)-connections on surfaces. By using a canonical map from the complex of configurations of decorated flags to the Bloch complex, we prove that any generic component of ℒ
K
(
M
) is K
2
-isotropic as long as ∂
M
satisfies certain topological constraints (theorem 4.2). In some cases this easily implies that ℒ
K
(
M
) is K
2
-Lagrangian. For general
M
, we extend a classic result of Neumann and Zagier on symplectic properties of PGL(2) gluing equations to reduce the K
2
-Lagrangian property to a combinatorial statement.
Physically, we translate the
K
-decomposition of an ideal triangulation of
M
and its symplectic properties to produce an explicit construction of 3d
N
=
2
superconformal field theories
T
K
M
resulting (conjecturally) from the compactification of
K
M5-branes on
M
. This extends known constructions for
K
= 2. Just as for
K
= 2, the theories
T
K
M
are described as IR fixed points of abelian Chern-Simons-matter theories. Changes of triangulation (2-3 moves) lead to abelian mirror symmetries that are all generated by the elementary duality between
N
f
= 1 SQED and the XYZ model. In the large
K
limit, we find evidence that the degrees of freedom of
T
K
M
grow cubically in
K
.
A possibility of contactless switching of an NbN nanowire from superconducting to normal state by passing a current through a gate located at a certain distance from the nanowire is demonstrated. The ...gate, being isolated from the nanowire by an Al
2
O
3
layer, contains an integrated resistance formed by ion irradiation. Dependences of the minimum power released in the gate that is sufficient for nanowire to pass to the normal state on the dc current through the nanowire are experimentally obtained. A signal inverter containing three successive cascades is developed based on this principle. This design shows that the proposed approach can be used to form a logic element base for cryogenic computing.
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