Physics – High Energy Physics – High Energy Physics - Theory
Scientific paper
2012-02-13
Physics
High Energy Physics
High Energy Physics - Theory
35 pages, 2 figures
Scientific paper
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 $M$.
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