We consider the problem of locally describing tubular geometry around a submanifold embedded in a (pseudo)Riemannian manifold in its general form. Given the geometry of ambient space in an arbitrary coordinate system and equations determining the sub
manifold in the same system, we compute the tubular expansion coefficients in terms of this {it a priori data}. This is done by using an indirect method that crucially applies the tubular expansion theorem for vielbein previously derived. With an explicit construction involving the relevant coordinate and non-coordinate frames we verify consistency of the whole method up to quadratic order in vielbein expansion. Furthermore, we perform certain (long and tedious) higher order computation which verifies the first non-trivial spin connection term in the expansion for the first time. Earlier a similar method was used to compute tubular geometry in loop space. We explain this work in the light of our general construction.
Motivated by the computation of loop space quantum mechanics as indicated in [7], here we seek a better understanding of the tubular geometry of loop space ${cal L}{cal M}$ corresponding to a Riemannian manifold ${cal M}$ around the submanifold of va
nishing loops. Our approach is to first compute the tubular metric of $({cal M}^{2N+1})_{C}$ around the diagonal submanifold, where $({cal M}^N)_{C}$ is the Cartesian product of $N$ copies of ${cal M}$ with a cyclic ordering. This gives an infinite sequence of tubular metrics such that the one relevant to ${cal L}{cal M}$ can be obtained by taking the limit $Nto infty$. Such metrics are computed by adopting an indirect method where the general tubular expansion theorem of [12] is crucially used. We discuss how the complete reparametrization isometry of loop space arises in the large-$N$ limit and verify that the corresponding Killing equation is satisfied to all orders in tubular expansion. These tubular metrics can alternatively be interpreted as some natural Riemannian metrics on certain bundles of tangent spaces of ${cal M}$ which, for ${cal M} times {cal M}$, is the tangent bundle $T{cal M}$.
We consider tubular neighborhood of an arbitrary submanifold embedded in a (pseudo-)Riemannian manifold. This can be described by Fermi normal coordinates (FNC) satisfying certain conditions as described by Florides and Synge in cite{FS}. By generali
zing the work of Muller {it et al} in cite{muller} on Riemann normal coordinate expansion, we derive all order FNC expansion of vielbein in this neighborhood with closed form expressions for the curvature expansion coefficients. Our result is shown to be consistent with certain integral theorem for the metric proved in cite{FS}.
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 ($hbar=alpha to 0$) of thi
s 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 $alpha$-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.
We study a particular approach for analyzing worldsheet conformal invariance for bosonic string propagating in a curved background using hamiltonian formalism. We work in the Schrodinger picture of a single particle description of the problem where t
he particle moves in an infinite-dimensional space. Background independence is maintained in this approach by adopting DeWitts (Phys.Rev.85:653-661,1952) coordinate independent formulation of quantum mechanics. This enables us to construct certain background independent notion of Virasoro generators, called DeWitt-Virasoro (DWV) generators, and invariant matrix elements of an arbitrary operator constructed out of them in spin-zero representation. We show that the DWV algebra is given by the Witt algebra with additional anomalous terms that vanish for Ricci-flat backgrounds. The actual quantum Virasoro generators should be obtained by first introducing the vacuum state and then normal ordering the DWV generators with respect to that. We demonstrate the procedure in the simple cases of flat and pp-wave backgrounds. This is a shorter version of arXiv:0912.3987 [hep-th] with many technical derivations omitted.
We generalize the DeWitt-Virasoro (DWV) construction of arXiv:0912.3987 [hep-th] to tensor representations of higher ranks. A rank-$n$ tensor state, which is by itself coordinate invariant, is expanded in terms of position eigenstates that transform
as tensors of the same rank. The representation of the momentum operator in these basis states is then obtained by generalizing DeWitts argument in Phys.Rev.85:653-661,1952. Such a representation is written in terms of certain bi-vector of parallel displacement and its covariant derivatives. With this machinery at hand we find tensor representations of the DWV generators defined in the previous work. The results differ from those in spin-zero representation by additional terms involving the spin connection. However, we show that the DWV algebra found earlier as a scalar expectation value remains the same, as required by consistency, as all the additional contributions conspire to cancel in various ways. In particular, vanishing of the anomaly term requires the same condition of Ricci-flatness for the background.
We analyze exact conformal invariance of string worldsheet theory in non-trivial backgrounds using hamiltonian framework. In the first part of this talk we consider the example of type IIB superstrings in Ramond-Ramond pp-wave background. In particul
ar, we discuss the quantum definition of energy-momentum (EM) tensor and two methods of computing Virasoro algebra. One of the methods uses dynamical supersymmetries and indirectly establishes (partially) conformal invariance when the background is on-shell. We discuss the problem of operator ordering involved in the other method which attempts to compute the algebra directly. This method is supposed to work for off-shell backgrounds and therefore is more useful. In order to understand this method better we attempt a background independent formulation of the problem which is discussed in the second half of the talk. For a bosonic string moving in an arbitrary metric-background such a framework is obtained by following DeWitts work (Phys.Rev.85:653-661,1952) in the context of particle quantum mechanics. In particular, we construct certain background independent analogue of quantum Virasoro generators and show that in spin-zero representation they satisfy the Witt algebra with additional anomalous terms that vanish for Ricci-flat backgrounds. We also report on a new result which states that the same algebra holds true in arbitrary tensor representations as well.
We study worldsheet conformal invariance for bosonic string propagating in a curved background using the hamiltonian formalism. In order to formulate the problem in a background independent manner we first rewrite the worldsheet theory in a language
where it describes a single particle moving in an infinite-dimensional curved spacetime. This language is developed at a formal level without regularizing the infinite-dimensional traces. Then we adopt DeWitts (Phys.Rev.85:653-661,1952) coordinate independent formulation of quantum mechanics in the present context. Given the expressions for the classical Virasoro generators, this procedure enables us to define the coordinate invariant quantum analogues which we call DeWitt-Virasoro generators. This framework also enables us to calculate the invariant matrix elements of an arbitrary operator constructed out of the DeWitt-Virasoro generators between two arbitrary scalar states. Using these tools we further calculate the DeWitt-Virasoro algebra in spin-zero representation. The result is given by the Witt algebra with additional anomalous terms that vanish for Ricci-flat backgrounds. Further analysis need to be performed in order to precisely relate this with the beta function computation of Friedan and others. Finally, we explain how this analysis improves the understanding of showing conformal invariance for certain pp-wave that has been recently discussed using hamiltonian framework.
In a previous work (arXiv:0902.3750 [hep-th]) we studied the world-sheet conformal invariance for superstrings in type IIB R-R plane-wave in semi-light-cone gauge. Here we give further justification to the results found in that work through alternati
ve arguments using dynamical supersymmetries. We show that by using the susy algebra the same quantum definition of the energy-momentum (EM) tensor can be derived. Furthermore, using certain Jacobi identities we indirectly compute the Virasoro anomaly terms by calculating second order susy variation of the EM tensor. Certain integrated form of all such terms are shown to vanish. In order to deal with various divergences that appear in such computations we take a point-split definition of the same EM tensor. The final results are shown not to suffer from the ordering ambiguity as noticed in the previous work provided the coincidence limit is taken before sending the regularization parameter to zero at the end of the computation.
We reconsider the analysis done by Kazama and Yokoi in arXiv:0801.1561 (hep-th). We find that although the right vacuum of the theory is the one associated to massless normal ordering (MNO), phase space normal ordering (PNO) plays crucial role in the
analysis in the following way. While defining the quantum energy-momentum (EM) tensor one needs to take into account the field redefinition relating the space-time field and the corresponding world-sheet coupling. We argue that for a simple off-shell ansatz for the background this field redefinition can be taken to be identity if the interaction term is ordered according to PNO. This definition reproduces the correct physical spectrum when the background is on-shell. We further show that the right way to extract the effective equation of motion from the Virasoro anomaly is to first order the anomaly terms according to PNO at a finite regularization parameter $eps$ and then take the $eps to 0$ limit. This prescription fixes an ambiguity in taking the limit for certain bosonic and fermionic contributions to the Virasoro anomaly and is the natural one to consider given the above definition of the EM tensor.
