1202.2735 (Partha Mukhopadhyay)

On a semi-classical limit of loop space quantum mechanics    [PDF]

Partha Mukhopadhyay

Following earlier work, we view two dimensional non-linear sigma model with
target space $M$ as a single particle quantum mechanics whose extended
configuration space is given by the corresponding free loop space $LM$. In a
natural semi-classical limit ($\hbar=\alpha' \to 0$) of this model the
wavefunction localizes on the submanifold of vanishing loops which is
isomorphic to $M$. In such a vacuum one would expect that the semi-classical
expansion should be related to the tubular expansion of the theory around the
submanifold and an effective dynamics on the submanifold is obtainable using
Born-Oppenheimer approximation. Motivated by this picture, we first study a
finite dimensional analogue of the loop space quantum mechanics where we
discuss its tubular expansion and how that is related to a semi-classical
expansion of the Hamiltonian. Then we study an explicit construction of the
relevant tubular neighborhood in $LM$ using exponential maps. Such a tubular
geometry is obtained from a Riemannian structure on the tangent bundle of $M$
which views the zero-section as a submanifold admitting a tubular neighborhood.
Using this result and exploiting the analogy with the finite dimensional model
we show how the linearized tachyon effective equation at leading order in
$\alpha'$-expansion is correctly reproduced up to divergent terms all
proportional to the Ricci scalar of $M$.

View original: http://arxiv.org/abs/1202.2735