On a semi-classical limit of loop space quantum mechanics

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 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 $\alpha'$-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.


Introduction and summary
Strings in curved background is a well studied problem [1]. Usually the semi-classical expansion is formulated using the background field method of quantum field theory (QFT) in Lagrangian framework [2,3,4]. An attractive feature of this formulation is the use of Riemann normal coordinate (RNC) [5] expansion. This enables one to keep the Riemannian structure of the target manifold M manifest.
Although it is not usually used for QFT computations, Hamiltonian framework, on the other hand, is conceptually appealing. It is natural to view the two dimensional nonlinear sigma model (NLSM) under consideration as a single particle relativistic quantum mechanics in the infinite dimensional free loop space LM corresponding to M [6,7,8,9].
In [10] we discussed a framework of describing this quantum mechanics, hereafter called loop space quantum mechanics (LSQM), for the bosonic sigma model in terms of general coordinates in LM keeping the infinite dimensional Riemannian structure manifest. This may be viewed as a formal -deformation of the classical theory as divergences are present in the form of infinite dimensional traces. The problem of regularizing these divergences was emphasized earlier in [6,11,7]. As a first step towards this direction, in this article we discuss a semi-classical limit of LSQM and motivate the use of Fermi normal coordinate (FNC) [12,13] expansion describing the tubular neighborhood of M when it is viewed as the submanifold of vanishing loops embedded in LM. 1 We now roughly describe the general idea. One expects that in = α → 0 limit the worldsheet theory should reduce to a theory of particles in M. One also notices that LSQM has a potential which minimizes to zero on the submanifold of zero loops. Therefore a natural semi-classical limit is given by the situation where the wavefunction localizes on this submanifold. The general idea is to use Born-Oppenheimer type approximation to adiabatically decouple the longitudinal (slow) and transverse (fast) degrees of freedom 2 and finally to compute the effective theory on M → LM order by order in .
A complete understanding of the above procedure requires several technical questions to be answered. Some of them are as follows.
1. How to perform tubular expansion of tensors around a submanifold embedded in a higher dimensional ambient space? 2. Given a suitable quantum mechanical problem in the ambient space, how to set up the relevant semi-classical expansion of the Hamiltonian which relates to the above tubular expansion? How to get an effective theory on the submanifold? 3. Given the understanding of the above questions in a finite dimensional case, how to apply them to our present context of loop space?
We will discuss all the above three topics successively and our final goal will be to derive the linearized effective equation for the tachyon fluctuation at leading order in αexpansion. We will show that our analysis correctly reproduces the known result up to divergent terms all proportional to the Ricci scalar of M. Below we briefly discuss these topics to indicate how this result will be arrived at. This will also clarify relation to other works in the literature.
The first question is discussed in §2 (and in appendix A). Here we explain our basic set up for a finite dimensional submanifold embedding, introduce FNC and review the results of [14]. In [14], by generalizing the techniques of [15], we find all order FNC-expansion of vielbein components in the neighborhhood of a submanifold (say M ) embedded in a pseudo-Riemannian ambient space (say L) 3 . The expansion coefficients are given by certain tensors of L, all evaluated at M → L. For vielbein these tensors are given by combinations of various powers of the curvature, their covariant derivatives and spin connection. For the rest of our analysis the FNC-expansion of the metric tensor up to quadratic order, as given in eq.(2.3), will be crucially used.
To address the second question we consider a finite dimensional analogue of LSQM in §3. The analysis in this section is along the line of what is usually known as constrained quantum system in the literature. A partial list of references is [16,17,18,19,20]. Here one considers a non-relativistic classical system in an ambient space with a potential that tries to confine the motion into a submanifold. The idea is to realize this constraint at the quantum mechanical level through localization of wavefunction. This is done by rescaling the model with certain tunable parameter (e.g. representing the strength of the restoring force) in such a way that makes the transverse directions fast in the Born-Oppenheimer sense when the parameter is small. In our case the tunable parameter is the scale and therefore the procedure gives a semi-classical expansion of the theory. In §3 we give precise definition of the potential of our model and the procedure leading to semi-classical expansion of the Hamiltonian. This shows how the contribution at a given order in is related to tubular expansion of various geometric quantities at different orders. Finally, we define and compute an analogue of linearized tachyon effective equation at leading order in -expansion within this toy model.
The usefulness of this study lies in the fact that it is free of divergences. Moreover, as hinted in the next paragraph, there exists an analogy which can be exploited to translate the end results of the toy model to the case of LSQM. Once this is done it exhibits the pattern of divergences that are expected in the actual LSQM computations. This is how we arrive at the final result for the tachyon effective equation as mentioned earlier.
We now turn to the question of how to translate the results of finite dimensional model 3 M and L are our finite dimensional analogues of M and LM respectively.
to the case of LSQM which is the content of the third question. Given that the theory is being expanded around a submanifold, such results are in general expressed in terms of various tubular expansion coefficients which are tensors of the ambient space evaluated on the submanifold. Since the Riemannian structure of LM is induced from that of M, one would expect that all the relevant tubular expansion coefficients should be related to certain intrinsic geometric data of M. Finding such relations for the metric-expansion coefficients up to quadratic order will be the precise quantitative question addressed in §4. There are several technical steps to be followed in order to arrive at the final result which we explain in a self-contained manner in §4 4 . Once these relations are known, one can use the precise analogy between the toy model and LSQM to translate results of §3.
This will be discussed in §5. We conclude in §6 with some future directions. A brief note on loop space and LSQM that will be relevant for our discussion has been given in appendix C. Appendix B and D contain some technical details.

Tubular expansion of metric up to quadratic order
Here we describe the basic set up for submanifold embedding that will be used throughout the paper. We consider a D-dimensional subspace M embedded in a higher dimensional (pseudo) Riemannian space L of dimension d. We adopt the following notations. Greek indices (α, β, · · ·) run over D dimensions, capital Latin indices (A, B, · · ·) run over (d − D) dimensions and small Latin indices (a, b, · · ·), over all d dimensions. The coordinates of L will be denoted by z a = (x α , y A ) where x α is a general coordinate system in M . Indices kept inside parenthesis will refer to non-coordinate basis, η (ab) being the diagonal matrix with the indictors as diagonal elements.
In [12] Florides and Synge (FS) proved existence of certain submanifold based coordinate system, called FNC in modern literature [13], which satisfies special coordinate conditions. In the special case where M is a point, FNC reduces to RNC. The FS coordinate conditions can be described as follows. Equation for the submanifold is given by, The metric components of L, denoted by g ab (z), satisfy the following equations, whereḡ ab (x) = g ab (x, y = 0). As a general rule, we use the lower case symbols to denote the geometric quantities of L and the same symbols with bars to denote the same quantities restricted to the submanifold. With this convention in mind we will refrain from explicitly writing down the arguments of such quantities most of the time.
The results for the expansion of the metric components away from the submanifold that will be relevant for us later are given by, where ω a (b) (c) are components of the connection one-form of L (non-coordinate indices are converted to coordinate indices with the use of vielbein as usual), r acdb is the covariant Riemann curvature tensor 5 ands αβC =ω αβC +ω βαC , (2.4) is the second fundamental form of the submanifold embedding [22]. We obtain (2.3) from a closed form expression for the expansion of vielbein which is derived in [14]. Although the details of this result will not be directly used in this work, there will be some relevance in the discussion of §4. We therefore summarize the main results of [14] in appendix A. 5 We follow the same convention for the curvature as in [21].

Finite dimensional analogue of loop space quantum mechanics
In this section we will consider a finite dimensional analogue of LSQM in the framework discussed in [10]. In §3.1 we will define the model and its semi-classical expansion. The analogue of linearized effective equation for tachyon fluctuation at leading order will be derived in §3.2. Our discussion below will be done without any reference to LSQM. We will come back to the analogy later in §5.

Definition of model and semi-classical expansion of Hamiltonian
All our notations used in the previous section will be valid in this section. We consider a non-relativistic quantum mechanical system whose configuration space is given by L.
Hence it is assumed (only in this section) to have Euclidean signature. The Hamiltonian of the system is given by the standard expression, where dw = dz √ g is the invariant measure, D 2 is the Laplacian of L and V is a potential.
We will consider V to be confining to the submanifold M → L (a more precise definition will follow). We must define what we mean by performing a semi-classical expansion such that in the semi-classical limit the wavefunction collapses on the submanifold. This is a procedure given by the following steps 6 ,

Submanifold based description:
Given H pre as in (3.5), we first move to a submanifold based description where the natural measure is given by dydx √ G instead of dw √ g 7 . As discussed in appendix B, this is done by performing certain rescaling of the wavefunction so that the same matrix element in (3.5) is given in terms of the transformed Hamiltonian, which we 6 Various other, but similar, procedures have been discussed in the literature indicated earlier. Our procedure is adopted to suit LSQM. 7 Recalling our notations introduced in §2, dx √ G is the invariant measure on the submanifold with respect to the induced metric.
call H sub , and transformed wavefunctions (unprimed) as, where the expression for H sub can be found in eq.(B.68).

Tubular expansion:
Next we tubular expand H sub to write, where we adopt the following notation:

Definition of V :
The potential V is confining to the submanifold M whose embedding satisfies the following property,ω As a result the second fundamental form in (2.4) vanishes and therefore M is totally geodesic [22]. Furthermore, the transverse profile of V is given by the following where A is positive definite 9 .

Definition of semi-classical expansion:
Define a rescaled Hamiltonian, (3.10) 8 We will display the summation over indices explicitly, as we have done in eq.(3.9), whenever Einstein summation convention will not be valid. 9 Notice that due to the presence of the -factors in the potential, general covariance of the ambient manifold is broken down to that of the submanifold even at the classical level. Without such factors V will be a scalar under the full diffeomorphism of L, but will not have a non-trivial tubular expansion due to the coordinate condition in (2.2). As will be explained in §5, general covariance gets broken in LSQM only due to the semi-classical limit.
Semi-classical expansion of H is given by rescaling the transverse coordinates as, (3.11) in the tubular expansion of H. This gives, where we have used the following notation, The expression for H (n) can be found in eq.(B.73). For the rest of our analysis we will restrict the expansion in (3.12) up to O( ). Explicit computation using the metricexpansion in (2.3) yields the following results, where in the first equation we have defined annihilation and creation operators, As noted in the literature, Λ AB is the angular momentum operator in the transverse space andω αAB is analogous to a non-abelian (SO(d − D)) Berry connection [23]. As a mathematical exercise, our results in eqs. (3.14) are equivalent to a case discussed in [18] except for the last term in the last equation which comes from the tubular expansion of our potential.
The reasons why the above procedure correctly captures our general idea of Born-Oppenheimer type approximation and localization of wavefunction are as follows. The dependent rescaling in eq.

Analogue of linearized tachyon effective equation at leading order
Here we consider the transverse degrees of freedom to be frozen in the harmonic oscillator ground state and derive the effective Hamiltonian, as will be defined in eq.(3.22) below, for the longitudinal degree of freedom at the leading order. This will give us the linearized tachyon effective equation (see eq.(3.24)) at this order. We will explain this analogy later in §5.
The wavefunction under consideration is, such that, The expectation value of H up to first order in is gievn by, where we have used, Using the results, one finds for the effective Hamiltonian H ef f for T (x), , where, As the reason will be explained in §5, we identify the following equation as the analogue of linearized tachyon effective equation, this expansion in LM. More precisely, we will show that eqs.(2.3, 3.8) will still be valid, with the notations correctly interpreted, for the embedding M → LM with suitably constructed FNC and with additional equations, given in (4.49), that relate the relevant expansion coefficients to intrinsic geometric data of M.
The steps that we will follow are as follows. First we present an explicit construction of the tubular neighborhood in §4.1. A proof of existence was given through an explicit construction earlier by Stacey in [24]. Although the basic ideas are similar, the details of our construction are different and has been chosen to suit our purpose (in §4.2) better.

Explicit construction of the tubular neighborhood
Below we will first heuristically describe our construction and then specify it in more mathematical terms, in particular make connection with [24]. We will refer to geodesics and open neighborhoods in both LM and M in various places along the way. It should be clear from the context which space we are referring to.
The basic picture [12], true for any tubular neighborhood, that we will have in mind is given in fig.1. Given any point Q in the neighborhood, there exists a unique geodesic passing through Q that arrives at a unique point P on the submanifold orthogonally. This condition is not satisfied if two geodesics emerging orthogonally from the submanifold meet at a point outside.
Following the standard way, we will restrict ourselves to a region sufficiently close to the submanifold such that this does not happen. Recall that every point in the neighborhood in LM corresponds to a non-zero loop in M such that nearer the point resides to the submanifold of vanishing loops, smaller the loop it represents. It turns out that the above restriction corresponds to considering sufficiently small loops in M such that any given loop can be entirely encompassed within a single convex normal neighborhood [26] in M. This implies that a small loop should fit entirely into B p -the ball of largest RNC-radius with center at p ∈ M , for some p in the neighborhood.
Let us now consider the set of points lying on the geodesic QP in LM. This corresponds to a class of loops which progressively shrink to zero size as we approach P (see cartoon in fig.2). Therefore from the perspective of the interior of M this defines P to be some kind of an average value for all the loops in this class. Notice that such a definition of averaging is independent of the choice of coordinate system, simply because it only refers to geodesics. Given a loop-embedding Z in M, its average position, as defined above, can be found in the following way. In case M is flat, it is simply given by S 1 Z. This is basically because Minkowski space is also a vector space where one can define a radial vector. In a curved space one should make use of geodesics which look like radial vectors in RNC, the latter being related to general coordinates through the exponential map. Therefore, when M is curved, we first describe the loop in RNC centered at a suitable point, with coordinate say x. The choice of this base-point is not fixed, as the loop will in general fit into B x for a range of values of x. However, there is a unique value x within the allowed range for which the following condition is satisfied, Repeating this procedure for all the loops in C(U M ) one arrives at the following set [24], where π : U T M → U M is the projection map. Constancy of πŶ implies that the whole loop resides in the same fibre, unlike its configuration in M × M.
Since exponential map is a diffeomorphism, the above argument shows that the desired tubular neighborhood is diffeomorphic to the set in (4.26). The relevant diffeomorphism is a bundle map which is the collection of all the inverse exponential maps at all x ∈ U M , both being isomorphic to U M . This kind of a construction is called a local addition (see [27] for a precise definition), of which the exponential map is a standard example. In [24] construction of the local addition has been facilitated by embedding M in a higher dimensional Euclidean space. However, exponential map is more suitable for our purpose, as we will see in the next subsection where the aforementioned diffeomorphism will be explicitly constructed.

Construction of FNC
Here we would like to understand the loop space analogue of the metric expansion given in (2.3). As mentioned before, generically the tubular expansion coefficients are certain geometric quantities of the ambient manifold evaluated on the submanifold. Although these are not related to the intrinsic geometric properties of the submanifold in general, for loop space that is the case. Therefore, expressing the tubular expansion coefficients of LM in terms of the geometric data of M is the precise quantitative question that needs to be answered. The discussion in the previous subsection implicitly defines a procedure to answer this question which we pursue here.
There are two steps to be followed. Given a Riemannian structure on M, the space M×M acquires a natural direct product structure. The bundle map (diffeomorphism) in  The components of the vielbein are given by, , (4.29) Indices in square brackets refer to the non-coordinate basis in M × M. E (α) β are the vielbein components of M (with indices in round brackets referring to the non-coordinate basis) with metric components given by G αβ , In general the two copies of M can have different metrics which are diffeomorphic to each other. We have chosen x 1 and x 2 suitably so that these two metrics are same as given in eq.(4.30). We will denote the desired coordinate system FNC U T M byẑ = (x α ,ŷα) with α,α = 1, · · · , D 12 , which will be obtained below by following a series of coordinate transformations fromz.
The first step is to argue, as has been done in appendix D, that there exists a coordinate system, where x is a general coordinate system on ∆ U M×M = U M×M ∩∆ such that the transformed components of the vielbein (with an additional overall constant scaling of the metric, see appendix D) are given by the following expansion up to quadratic order in y .  12 Notice that the indices α 1 , α 2 , α andα all run over D = dim M dimensions. From the perspective of M they do not make any difference. However, from the perspective of M × M they do. 13 We use upper case symbols without aˇto denote tensors of M in general coordinates. RNC x refers to the RNC-system centered at x such that the vielbein components are given by E (α) β (x ) at the centre. q is an undetermined real number. We will argue toward the end of this subsection that the analogue of eq.(A.58) is satisfied up to quadratic order for arbitrary values of q.
Then the final step is to perform the following coordinate transformation: z a →ẑ a = (x α ,ŷα) such that, The transformed vielbein components are given by, which give the following results for the metric components, with,ω The tubular expansion in (4.34) is supposed to satisfy the analogue of the differential equation (A.58) in T M. This is given, in our notation adopted here, by, whered =ŷα ∂ ∂ŷα , a = α,α, b being 1 for b =β and −1, otherwise and, It is straightforward to to check that eq.(4.37) is satisfied by eqs. Tensors in LM are obtained from those in T M following a similar procedure [10] which we discuss in detail now. Let {êâ} = {ê α = ∂ ∂x α ,êα = ∂ ∂ŷα } and {dẑâ} = {dx α , dŷα} be the coordinate basis (in special coordinate system constructed in previous subsection) for the tangent and cotangent spaces Tẑ(T M) and T * z (T M) respectively atẑ in U T M . A rank (m, n) tensor is given by, The tubular expansion of the components take the following form, δŶα(σ) being a functional differential. The tensor corresponding to that in (4.41) is given by 15 , where t a 1 ···am b 1 ···bn (x, y) = tâ 1 ···âmb 1 ···bn (x,Ŷ (σ))e −i(a 1 +···+am)σ+i(b 1 +···+bn)σ . (4.46) 15 The definition (4.45, 4.46) is equivalent to the following alternative expression whereẑâ(σ) = (x α ,Ŷα(σ)).

Similar expression holds for the tubular expansion,
(4.49) In particular, the above shows that M → LM is a totally geodesic submanifold. See also comments below eq.(C.79). With this we end our discussion of the explicit construction of the tubular neighborhood of M → LM and the relevant FNC.

Analogy with finite dimensional model
In §3 we discussed a finite-dimensional analogue of LSQM. The primary goal of this analysis was to work out the relevant details (that one would eventually like to understand for LSQM) in a finite-dimensional set up which is free of divergences. Here we will spell out how precisely to interpret the analysis of §3 in the context of string theory. Our final goal will be to understand the features of the expected tachyon effective equation the analogue of which is described by eqs. (3.22, 3.23, 3.24).
• The ambient space L of the toy model is the configuration space and therefore is considered to be of Euclidean signature in §3. M, on the other hand, is the extended configuration space (which includes time) of NLSM. In the context of LSQM the analysis of §3 should be viewed as a worldline type theory. This has the following consequences.
-The theory is supplemented with the standard ghost sector of bosonic string theory. When M is taken to be pseudo-Riemannain, the potential V (to be discussed further below) of the model will be maximized, instead of being minimized, on the submanifold along the time-like directions. This gives rise to the standard problem of negative norm states which is cured by the presence of ghost sector. With this understanding we will simply ignore this problem now onwards and assume M to have Euclidean signature.
-Hamiltonian is a constraint. The effective form of this constraint on the submanifold obtained by integrating out the transverse (internal) degrees of freedom is supposed to give the linearized equation of motion for the string field components on M. This explains why eq.(3.24) has been interpreted to be analogue of the linearized tachyon effective equation.
• In the context of finite dimension in §2 we followed certain notation and convention for coordinate indices, FNC, tensors and their tubular expansion (see first few paragraphs of §2). We followed the same rules in the context of LM in §4.2.2.
The prescription for translating any finite dimensional expression involving tubular expansion is simply to interpret the transverse indices (i.e. capital Latin indices) according to rules of loop space as described below eq.(C.75) and evaluate the barred quantities involved in terms of the intrinsic data of M following eqs.(4.49).
• We now explain the potential of the toy model. Equation (3.8) is simply the second equation in (4.49). This implies that the submanifold is totally geodesic 16 . 16 In the context of LM, as pointed out below eq.(C.79), this is also related to the fact that the The relevance of eq.(3.9) may be understood as follows. The potential V LM of LSQM, given in (C.80), can be written in terms of FNC as follows, • The leading order rescaled Hamiltonian given by the first equation in (3.14) is analogous to the non-zero mode contribution to the Hamiltonian in flat space. The resemblance can be made more explicit through the following redefinition: where α andα are the usual flat-space-oscillators [28]. The indexĀ → (α, −a) (see discussion below eq.(C.75)) corresponds to a negative mode while A to a positive submanifold is the fixed-point set of the reparametrization Killing vector v(z) in (C.76). Such a feature, however, is not shared by the toy model. 17 Note that rescaling of LSQM is defined by the same eqs.(3.10, 3.11) with the identification (5.51).
mode. The leading order Hamiltonian takes the following familiar form in this new notation, The last term is the zero-point energy which is divergent. Noticing that at the leading order the transverse dynamics exactly matches with that of the non-zero modes in flat space, this term can be treated in the usual manner. The point to be emphasized here is that in flat space such a term (after collecting the ghost contribution) finally gets related to the tachyon mass. The same is true here as we see from the first equation in (3.23). Notice also that this mass has the right scaling with respect to = α . In fact, demanding that the leading order transverse Hamiltonian in (3.14) be precisely same as the non-zero modes contribution to the Hamiltonian in flat space fixes the rescaling of the model as described by eqs.(3.10, 3.11). Such a condition is required to get the right flat space limit (where the tubular expansion becomes trivial) of our analysis. in [10]. We postpone this analysis for a future work and assume this is true for the time being.
We will now argue that, With this we end our discussion of how the computations in the finite-dimensional model discussed in §3 should be interpreted in the context of LSQM.

Conclusion
This work investigates how to make sense of a semi-classical limit of LSQM as discussed in [10]. In this limit the wavefunction gets localized on the submanifold M of vanishing loops in LM where M is the target space of the corresponding NLSM. The study involves first defining the procedure in a finite dimensional toy model (content of §2 and §3) and then figuring out how the actual loop space model can be understood through an analogy with the toy model (content of §4 and §5). The study shows that the linearized effective equation for the tachyon fluctuation at leading order in α -expansion is reproduced correctly with all the divergent terms being proportional to the Ricci scalar of M.
The present approach makes the usual picture of particle quantum mechanics quite explicit and therefore it is conceptually appealing. Given this, it is perhaps a good idea to work out the details of how the standard questions, such as Ricci-flatness as leading order condition for conformal invariance, low-energy effective equations of motion and, most importantly, higher order α corrections, should be understood in the current approach.
We hope that the analysis of the present work will be helpful for further study along this direction and its supersymmetrization.
We will conclude with a few remarks regarding the mathematical framework of §4.1 and §4.2 where a certain Riemannian structure on T M was discussed. A speciality of this Riemannian structure is that it views T M 0 → T M as a submanifold admitting a tubular neighborhood. Recall that an all-order understanding of tubular expansion of vielbein in a generic case is available through [14] (reviewed in appendix A). This implies that finding the desired Riemannian structure on T M is equivalent to finding all the tubular expansion coefficients in terms of intrinsic geometric data of M. In this work this has been done in a limited sense which proved sufficient for the present level of analysis of LSQM. It is possible that a more complete understanding of this question will be required for computing α corrections. We hope to come back to these questions in future.
Finally, we note that the mathematical framework discussed in §4.1 should also be relevant for a multi-
(A.63) C n r are the binomial coefficients and, [(y.D) s ρ(x, 0; y)] (a) (b) = D A 1 · · · D As r (a) CD(b) (x, 0)y A 1 · · · y As y C y D , where D a is the covariant derivative in L with respect to the metric g ab . Notice that all such derivatives are evaluated at the submanifold. Finally 18 , being the vielbein of the induced metric on M , Using the results in (A.62) we find the metric expansion given in (2.3).

B Tubular expansion of Hamiltonian
Here we will present the detailed computations required to carry out various steps of performing semi-classical expansion as defined in §3.1. The rescaling of wavefunction that takes us to the submanifold based description is given by, This leads to the following expression for H sub as defined in eq.(3.6), where, The contribution at O(y n ) in the tubular expansion of H sub is given by, H sub n = − 2 (K n + K n + K ⊥ n ) + V n , (B.70) 18 Recall, according to our rule for notation,ω α where, and 19 , This shows how the contribution at a given order in is related to terms in tubular expansion of various geometric quantities at different orders.

C A note on loop space and LSQM
The loop space LM associated to a Riemannian manifold M is the space of all smooth maps from a parametrized loop to M.
Here we will briefly note down some general features of LM and LSQM that are relevant for our discussion in this article. The above definition implies that given any element . Therefore, 19 Given a geometric quantity X, the notation X n has been explained below eq.(3.7). 20 Given X, the notation X has been defined in eq.(3.13). Following [10], we will work with a Fourier space representation of these coordinate functions. In this representation the general coordinates of a point l as considered above in LM are given by, where σ parametrizes the loop, ≡ dσ 2π and the loop embedding Z α (σ) (α being a target space index) is obtained by following the above definition. We adopt the following convention for an infnite-dimensional coordinate index. It is given by a lower case Latin alphabet, which in turn is associated to a pair containing the corresponding Greek alphabet (i.e. a target space index) and an integer, denoted by the same small Latin alphabet in text format. For example, a → (α, a), b → (β, b). We will also adopt a similar association between such a pair and the corresponding upper case Latin alphabet when the integer is non-zero, i.e. A → (α, a), B → (β, b) etc. only when a, b = 0. We use this type of notation in all our discussion involving an explicit coordinate system in LM.
Reparametrization of the loop corresponds to an isometry of the loop space which exists irrespective of the isometries of M. The generator of this isometry is given by, which satisfies the Killing vector equation in LM [8].
where D a is the covariant derivative on LM. The metric and affine connection on LM are given by, g ab (z) = G αβ (Z(σ))e i(a+b)σ , γ a bd (z) = Γ α βδ (Z(σ))e i(−a+b+d)σ , (C.78) respectively, where G αβ and Γ α βδ are the metric and affine connection on M respectively. Notice from eqs.(C.76) that the submanifold of vanishing loops, whcih is given by, is where the Killing vector field vanishes. This situation is similar to the consideration of Kobayashi's theorem in [29] (in finite dimensions), which claims that the space of fixed points of an isometry is a totally geodesic submanifold of even co-dimension. We will see in §4 that the submanifold of interest is indeed totally geodesic. Although, this has infinite number of transverse directions, from the discussion of the infinite dimensional coordinate index done below eq.(C.75), it is clear that for every transverse index A → (α, a), there is a pairĀ → (α, −a).
We now briefly recall the structure of LSQM following [10]. The classical NLSM Lagrangian on a flat worldsheet takes the following form in terms of the general coordinates in LM, where a dot indicates derivative with respect to the worldsheet time. Notice that the potential is proportional to the norm-square of the Killing vector field discussed above.
LSQM [10] is a formal -deformation of this classical system obtained by following De-Witt's argument in [30]. Therefore, it has the same mathematical structure as that of the toy model discussed in §3 with the configuration space replaced by the infinite dimensional loop space 21 . In particular, the matrix element of the Hamiltonian between two scalar states is given by the same equation as in (3.5), with various quantities now interpreted in the context of LM instead of L. For example, D 2 denotes the Laplacian in LM.
D Existence of (x , y )-system In the discussion of §4.2.1 we assumed that starting from the direct product coordinate systemz = (x 1 , x 2 ) (see eqs.(4.28, 4.29)) on M × M one can arrive at another, namely z = (x , y ), such that the transformed vielbein components are given, up to a constant conformal transformation, by eqs.(4.32) up to quadratic order in y . Here we will explicitly construct z in a region whose overlap with the diagonal submanifold is sufficiently small.
(D.87) Therefore, we seem to have arrived at a coordinate system z a = (x α , y α ) where x α is a general coordinate system on ∆ and y α is orthogonal to it. However, it has been constructed using exponential map with a fixed base point. Therefore, it is guranteed to be the right one, i.e. the one relevant to FNC, only near the base point (x 0 , x 0 ). Now onwards we restrict to a region around this point whose overlap with ∆ is sufficiently small. More precisely, we consider u to be at higher order in smallness with respect to y , implying that we neglect terms of order uy and u 2 with respect to those of order y 2 .
With this approximation the transformed vielbein components in z -system are given by, The 1 √ 2 factors arise because of the standard constant rescaling of the measure when we go to a diagonal. Now onwards, we will absorb this by applying a constant conformal transformation of the metric.
There is a further coordinate transformation which keeps the form of the expansions in (D.88) invariant within the same region of validity, yet making it more general. This is given by z → z = (x α , y α ) such that, x α = x α + (q − 1 6 )Ř αγδ β (x )y γ y δ x β , y α = y α , (D.89) where q is a real constant. The transformed vielbein components are given by eqs.(4.32).