2012/02/13 by Partha Mukhopadhyay, Mukhopadhyay, Partha
Physics and Astronomy · #Black Holes and Theoretical Physics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Noncommutative and Quantum Gravity Theories #Quantum Electrodynamics and Casimir Effect
paper · pdf · doi:10.48550/arxiv.1202.2735
openalex publication_date 2012/02/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Following earlier work, we view two dimensional non-linear sigma model with target space \cM as a single particle relativistic quantum mechanics in the corresponding free loop space \cLM. In a natural semi-classical limit (ℏ=α' → 0) of this model the wavefunction localizes on the submanifold of vanishing loops which is isomorphic to \cM. One would expect that the relevant 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. In this work we develop a framework to carry out such an analysis at the leading order in α'-expansion. In particular, we show that the linearized tachyon effective equation is correctly reproduced up to divergent terms all proportional to the Ricci scalar of \cM. The steps leading to this result are as follows: first we define a finite dimensional analogue of the loop space quantum mechanics (LSQM) 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 \cLM using exponential maps. Such a tubular geometry is obtained from a Riemannian structure on the tangent bundle of \cM which views the zero-section as a submanifold admitting a tubular neighborhood. Using this result and exploiting an analogy with the toy model we arrive at the final result for LSQM.