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Register a new userWilson Lines from Representations of NQ-Manifolds
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An NQ-manifold is a non-negatively graded supermanifold with a degree 1 homological vector field. The focus of this paper is to define the Wilson loops/lines in the context of NQ-manifolds and to study their properties. The Wilson loops/lines, which give the holonomy or parallel transport, are familiar objects in usual differential geometry, we analyze the subtleties in the generalization to the NQ-setting and we also sketch some possible applications of our construction.
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We study the cusped Wilson line operators and Bremsstrahlung functions associated to particles transforming in the rank-$k$ symmetric representation of the gauge group $U(N)$ for ${cal N} = 4$ super Yang-Mills. We find the holographic D3-brane description for Wilson loops with internal cusps in two different limits: small cusp angle and $ksqrt{lambda}gg N$. This allows for a non-trivial check of a conjectured relation between the Bremsstrahlung function and the expectation value of the 1/2 BPS circular loop in the case of a representation other than the fundamental. Moreover, we observe that in the limit of $kgg N$, the cusped Wilson line expectation value is simply given by the exponential of the 1-loop diagram. Using group theory arguments, this eikonal exponentiation is conjectured to take place for all Wilson loop operators in symmetric representations with large $k$, independently of the contour on which they are supported.
This article is an overview of some of the remarkable progress that has been made in Sasaki-Einstein geometry over the last decade, which includes a number of new methods of constructing Sasaki-Einstein manifolds and obstructions.
We study the limit geometry of complete projective special real manifolds. By limit geometry we mean the limit of the evolution of the defining polynomial and the centro-affine fundamental form along certain curves that leave every compact subset of the initial complete projective special real manifold. We obtain a list of possible limit geometries, which are themselves complete projective special real manifolds, and find a lower limit for the dimension of their respective symmetry groups. We further show that if the initial manifold has regular boundary behaviour, every possible limit geometry is isomorphic to $mathbb{R}_{>0}ltimesmathbb{R}^{n-1}$.
Sasakian manifolds are odd-dimensional counterpart to Kahler manifolds. They can be defined as contact manifolds equipped with an invariant Kahler structure on their symplectic cone. The quotient of this cone by the homothety action is a complex manifold called Vaisman. We study harmonic forms and Hodge decomposition on Vaisman and Sasakian manifolds. We construct a Lie superalgebra associated to a Sasakian manifold in the same way as the Kahler supersymmetry algebra is associated to a Kahler manifold. We use this construction to produce a self-contained, coordinate-free proof of the results by Tachibana, Kashiwada and Sato on the decomposition of harmonic forms and cohomology of Sasakian and Vaisman manifolds. In the last section, we compute the supersymmetry algebra of Sasakian manifolds explicitly.
The Abelian dominance for the string tension was shown for the fundamental sources in MA gauge in the lattice simulations. For higher representations, however, it is also known that the naive Abelian Wilson loop, which is defined by using the diagonal part of the gauge field, does not reproduce the correct behavior. To solve this problem, for an arbitrary representation of an arbitrary compact gauge group, we propose to redefine the Abelian Wilson loop. By using this redefined operator, we demonstrate the Abelian dominance for sources in the adjoint representation and the sextet representation of $SU(3)$ gauge group in lattice simulations.
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