Contact Term, its Holographic Description in QCD and Dark Energy
  In this work we study the well known contact term, which is the key element
in resolving the so-called $U(1)_A$ problem in QCD. We study this term using
the dual Holographic Description. We argue that in the dual picture the contact
term is saturated by the D2 branes which can be interpreted as the tunnelling
events in Minkowski space-time. We quote a number of direct lattice results
supporting this identification. We also argue that the contact term receives a
Casimir -like correction $\sim (\Lqcd R)^{-1}$ rather than naively expected
$\exp(-\Lqcd R)$ when the Minkowski space-time ${\cal R}_{3,1}$ is replaced by
a large but finite manifold with a size $R$. Such a behaviour is consistent
with other QFT-based computations when power like corrections are due to
nontrivial properties of topological sectors of the theory. In holographic
description such a behaviour is due to massless Ramond-Ramond (RR) field living
in the bulk of multidimensional space when power like corrections is a natural
outcome of massless RR field. In many respects the phenomenon is similar to the
Aharonov -Casher effect when the "modular electric field" can penetrate into a
superconductor where the electric field is exponentially screened. The role of
"modular operator" from Aharonov -Casher effect is played by large gauge
transformation operator $\cal{T}$ in 4d QCD, resulting the transparency of the
system to topologically nontrivial pure gauge configurations. We discuss some
profound consequences of our findings. In particular, we speculate that a slow
variation of the contact term in expanding universe might be the main source of
the observed Dark Energy.
