213 research outputs found
The notion of dimension in geometry and algebra
This talk reviews some mathematical and physical ideas related to the notion
of dimension. After a brief historical introduction, various modern
constructions from fractal geometry, noncommutative geometry, and theoretical
physics are invoked and compared.Comment: 29 pages, a revie
Looking backward: From Euler to Riemann
We survey the main ideas in the early history of the subjects on which
Riemann worked and that led to some of his most important discoveries. The
subjects discussed include the theory of functions of a complex variable,
elliptic and Abelian integrals, the hypergeometric series, the zeta function,
topology, differential geometry, integration, and the notion of space. We shall
see that among Riemann's predecessors in all these fields, one name occupies a
prominent place, this is Leonhard Euler. The final version of this paper will
appear in the book \emph{From Riemann to differential geometry and relativity}
(L. Ji, A. Papadopoulos and S. Yamada, ed.) Berlin: Springer, 2017
A mathematical formalism for the Kondo effect in WZW branes
In this paper, we show how to adapt our rigorous mathematical formalism for
closed/open conformal field theory so that it captures the known physical
theory of branes in the WZW model. This includes a mathematically precise
approach to the Kondo effect, which is an example of evolution of one
conformally invariant boundary condition into another through boundary
conditions which can break conformal invariance, and a proposed mathematical
statement of the Kondo effect conjecture. We also review some of the known
physical results on WZW boundary conditions from a mathematical perspective.Comment: Added explanations of the settings and main result
Scattering Forms and the Positive Geometry of Kinematics, Color and the Worldsheet
The search for a theory of the S-Matrix has revealed surprising geometric
structures underlying amplitudes ranging from the worldsheet to the
amplituhedron, but these are all geometries in auxiliary spaces as opposed to
kinematic space where amplitudes live. In this paper, we propose a novel
geometric understanding of amplitudes for a large class of theories. The key is
to think of amplitudes as differential forms directly on kinematic space. We
explore this picture for a wide range of massless theories in general spacetime
dimensions. For the bi-adjoint cubic scalar, we establish a direct connection
between its "scattering form" and a classic polytope--the associahedron--known
to mathematicians since the 1960's. We find an associahedron living naturally
in kinematic space, and the tree amplitude is simply the "canonical form"
associated with this "positive geometry". Basic physical properties such as
locality, unitarity and novel "soft" limits are fully determined by the
geometry. Furthermore, the moduli space for the open string worldsheet has also
long been recognized as an associahedron. We show that the scattering equations
act as a diffeomorphism between this old "worldsheet associahedron" and the new
"kinematic associahedron", providing a geometric interpretation and novel
derivation of the bi-adjoint CHY formula. We also find "scattering forms" on
kinematic space for Yang-Mills and the Non-linear Sigma Model, which are dual
to the color-dressed amplitudes despite having no explicit color factors. This
is possible due to a remarkable fact--"Color is Kinematics"--whereby kinematic
wedge products in the scattering forms satisfy the same Jacobi relations as
color factors. Finally, our scattering forms are well-defined on the
projectivized kinematic space, a property that provides a geometric origin for
color-kinematics duality.Comment: 77 pages, 25 figures; v2, corrected discussion of worldsheet
associahedron canonical for
Dynamical generation of wormholes with charged fluids in quadratic Palatini gravity
The dynamical generation of wormholes within an extension of General
Relativity (GR) containing (Planck's scale-suppressed) Ricci-squared terms is
considered. The theory is formulated assuming the metric and connection to be
independent (Palatini formalism) and is probed using a charged null fluid as a
matter source. This has the following effect: starting from Minkowski space,
when the flux is active the metric becomes a charged Vaidya-type one, and once
the flux is switched off the metric settles down into a static configuration
such that far from the Planck scale the geometry is virtually indistinguishable
from that of the standard Reissner-Nordstr\"om solution of GR. However, the
innermost region undergoes significant changes, as the GR singularity is
generically replaced by a wormhole structure. Such a structure becomes
completely regular for a certain charge-to-mass ratio. Moreover, the nontrivial
topology of the wormhole allows to define a charge in terms of lines of force
trapped in the topology such that the density of lines flowing across the
wormhole throat becomes a universal constant. To the light of our results we
comment on the physical significance of curvature divergences in this theory
and the topology change issue, which support the view that space-time could
have a foam-like microstructure pervaded by wormholes generated by quantum
gravitational effects.Comment: 14 pages, 3 figures, revtex4-1 style. New content added on section
VI. Other minor corrections introduced. Final version to appear in Phys. Rev.
Braiding Knots with Topological Strings
For an arbitrary knot in a three-sphere, the Ooguri-Vafa conjecture associates to it a unique stack of branes in type A topological string on the resolved conifold, and relates the colored HOMFLY invariants of the knot to the free energies on the branes. For torus knots, we use a modified version of the topological recursion developed by Eynard and Orantin to compute the free energies on the branes from the Aganagic-Vafa spectral curves of the branes, and find they are consistent with the known colored HOMFLY knot invariants Ă la the Ooguri-Vafa conjecture. In addition our modified topological recursion can reproduce the correct closed string free energies, which encode the information of the background geometry. We conjecture the modified topological recursion is applicable for branes associated to hyperbolic knots as well, encouraged by the observation that the modified topological recursion yields the correct planar closed string free energy from the Aganagic-Vafa spectral curves of hyperbolic knots. This has implications for the knot theory concerning distinguishing mutant knots with colored HOMFLY invariants. Furthermore, for hyperbolic knots, we present methods to compute colored HOMFLY invariants in nonsymmetric representations of U(N). The key step in this computation is computing quantum 6j-symbols in the quantum group U_q(sl_N)
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