Abstract
We consider the AdS bulk dual to an external massive quark in SYM following an arbitrary trajectory on Minkowski background. While a purely outgoing boundary condition on the gluonic field allows one to express the corresponding string worldsheet in a closed form, the setup has curious consequences. In particular, we argue that any quark whose trajectory on flat spacetime approaches that of a light ray in the remote past (as happens e.g. in the case of uniform acceleration) must necessarily be accompanied by an anti-quark. This is puzzling from the field theory standpoint, since one would expect that a sole quark following any timelike trajectory should be allowed. We explain the resolution in terms of boundary and initial conditions. We analyze the configuration in global AdS, which naturally suggests a modification to the boundary conditions allowing for a single accelerated quark without accompanying anti-quark. We contrast this resolution with earlier proposals.
Highlights
Since in the field theory this corresponds to purely outgoing gluonic field configuration, we will refer to this boundary condition as ‘outgoing’
In global AdS5, whose boundary is the Einstein Static Universe (ESU) S3 × R1, we see the entire string; the field theory lives on a compact space, so that it must satisfy the Gauss law constraint: it cannot admit a quark without an anti-quark
Conformally symmetric Yang-Mills theory is in a deconfined phase, so individual quarks should be allowed, and one might expect that the absence of Gauss law constraint on the non-compact space should allow color charged states, including a quark which follows any timelike trajectory, without the presence of anti-quark
Summary
One can specify the quark’s worldline by xμ(τ ), parameterized by proper time τ. This worldline can be arbitrary provided it is timelike everywhere, and the parameterization by τ ensures that the four-velocity xμ(τ ). Given a quark following xμ(τ ) with outgoing boundary conditions, let us specify its gravity dual. The quark and its gluonic field is described by a fundamental string in (2.2) which ends on the boundary quark position xμ(τ ) at u = 0. Solved the worldsheet equations of motion for the string with outgoing boundary conditions, parameterized by the bulk radial coordinate u and a time coordinate τ , and found a remarkably simple expression for the embedding coordinates XM (τ, u) = (Xμ(τ, u), u): Xμ(τ, u) = u xμ(τ ) + xμ(τ ). In a snapshot at a constant x0 slice of AdS, the string described by (2.3) typically just ends in ‘mid-air’
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