75 research outputs found
Tree-level scattering amplitudes from the amplituhedron
7 pages, 2 figures, to be published in the Journal of Physics: Conference Series. Proceedings for the "7th Young Researcher Meeting", Torino, 2016A central problem in quantum field theory is the computation of scattering amplitudes. However, traditional methods are impractical to calculate high order phenomenologically relevant observables. Building on a few decades of astonishing progress in developing non-standard computational techniques, it has been recently conjectured that amplitudes in planar N=4 super Yang-Mills are given by the volume of the (dual) amplituhedron. After providing an introduction to the subject at tree-level, we discuss a special class of differential equations obeyed by the corresponding volume forms. In particular, we show how they fix completely the amplituhedron volume for next-to-maximally helicity violating scattering amplitudes.Peer reviewe
A system of difference equations with elliptic coefficients and Bethe vectors
An elliptic analogue of the deformed Knizhnik-Zamolodchikov equations is
introduced. A solution is given in the form of a Jackson-type integral of Bethe
vectors of the XYZ-type spin chains.Comment: 20 pages, AMS-LaTeX ver.1.1 (amssymb), 15 figures in LaTeX picture
environment
Asymptotics and Dimensional Dependence of the Number of Critical Points of Random Holomorphic Sections
We prove two conjectures from [M. R. Douglas, B. Shiffman and S. Zelditch,
Critical points and supersymmetric vacua, II: Asymptotics and extremal metrics.
J. Differential Geom. 72 (2006), no. 3, 381-427] concerning the expected number
of critical points of random holomorphic sections of a positive line bundle. We
show that, on average, the critical points of minimal Morse index are the most
plentiful for holomorphic sections of {\mathcal O}(N) \to \CP^m and, in an
asymptotic sense, for those of line bundles over general K\"ahler manifolds. We
calculate the expected number of these critical points for the respective cases
and use these to obtain growth rates and asymptotic bounds for the total
expected number of critical points in these cases. This line of research was
motivated by landscape problems in string theory and spin glasses.Comment: 14 pages, corrected typo
Weighted sums with two parameters of multiple zeta values and their formulas
A typical formula of multiple zeta values is the sum formula which expresses
a Riemann zeta value as a sum of all multiple zeta values of fixed weight and
depth. Recently weighted sum formulas, which are weighted analogues of the sum
formula, have been studied by many people. In this paper, we give two formulas
of weighted sums with two parameters of multiple zeta values. As applications
of the formulas, we find some linear combinations of multiple zeta values which
can be expressed as polynomials of usual zeta values with coeffcients in the
rational polynomial ring generated by the two parameters, and obtain some
identities for weighted sums of multiple zeta values of small depths.Comment: 14 page
A Selberg integral for the Lie algebra A_n
A new q-binomial theorem for Macdonald polynomials is employed to prove an
A_n analogue of the celebrated Selberg integral. This confirms the g=A_n case
of a conjecture by Mukhin and Varchenko concerning the existence of a Selberg
integral for every simple Lie algebra g.Comment: 32 page
Towards the Amplituhedron Volume
21 pages; v2: version published in JHEPIt has been recently conjectured that scattering amplitudes in planar N=4 super Yang-Mills are given by the volume of the (dual) amplituhedron. In this paper we show some interesting connections between the tree-level amplituhedron and a special class of differential equations. In particular we demonstrate how the amplituhedron volume for NMHV amplitudes is determined by these differential equations. The new formulation allows for a straightforward geometric description, without any reference to triangulations. Finally we discuss possible implications for volumes related to generic N^kMHV amplitudes.Peer reviewe
Combinatorial expression for universal Vassiliev link invariant
The most general R-matrix type state sum model for link invariants is
constructed. It contains in itself all R-matrix invariants and is a generating
function for "universal" Vassiliev link invariants. This expression is more
simple than Kontsevich's expression for the same quantity, because it is
defined combinatorially and does not contain any integrals, except for an
expression for "the universal Drinfeld's associator".Comment: 20 page
Geometry of q-Hypergeometric Functions as a Bridge between Yangians and Quantum Affine Algebras
The rational quantized Knizhnik-Zamolodchikov equation (qKZ equation)
associated with the Lie algebra is a system of linear difference
equations with values in a tensor product of Verma modules. We solve the
equation in terms of multidimensional -hypergeometric functions and define a
natural isomorphism between the space of solutions and the tensor product of
the corresponding quantum group Verma modules, where the parameter
is related to the step of the qKZ equation via .
We construct asymptotic solutions associated with suitable asymptotic zones
and compute the transition functions between the asymptotic solutions in terms
of the trigonometric -matrices. This description of the transition functions
gives a new connection between representation theories of Yangians and quantum
loop algebras and is analogous to the Kohno-Drinfeld theorem on the monodromy
group of the differential Knizhnik-Zamolodchikov equation.
In order to establish these results we construct a discrete Gauss-Manin
connection, in particular, a suitable discrete local system, discrete homology
and cohomology groups with coefficients in this local system, and identify an
associated difference equation with the qKZ equation.Comment: 66 pages, amstex.tex (ver. 2.1) and amssym.tex are required;
misprints are correcte
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