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usually when people want to quantize the string on flat background, they will try to find the the OPE of embeddings (by solving a green function in a 2D space) and use them to find the energy-momentum tensor OPE, central charge, Virasoro modes and their algebra and then, covariant quantization.

BUT on a curved background (as one can expect in advance), for finding a the OPE of embeddings, you will reach to a non-linear differential equation with a point source! (I mean delta function). Im not sophisticated in GR, however I think it is possible to encounter such problems in GR. Does any one have any idea ? is there any more powerful approach for quantization (probably there is, but I'm not sure) for solving this problem?


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When we're covariantly processing a perturbative string theory on any background, we usually want to start with solving the free string theory (tree-level string theory, neglecting the splitting and joining of strings i.e. higher-topology Riemann surfaces). What does it mean to solve it? It means to find the eigenstates of the world sheet Hamiltonian. By the state-operator correspondence, this is equivalent to finding all local operators on the world sheet that transform as tensor fields with a well-defined dimension. These operators also have OPEs that physically contain the information about 3-string tree-level scattering etc.

The operators contain $:\exp(ik\cdot X(z)):$ whenever there are some flat large dimensions but their dependence on other dimensions, e.g. curved compactified ones, may be more complicated. In the vertex operator, the dependence on $X(z)$ without derivatives reproduces the dependence of point-like fields in GR on the spacetime coordinates. String theory, however, also adds excited strings that have factors like $\partial_z X$ or $\partial_{\bar z}X$ or higher derivatives in the vertex operators.

And this form of vertex operators that contains $X$ or its world sheet derivatives is only appropriate if the geometry is really "large enough", with distance scales much larger than the string scale. For a string-length-sized geometry, it often makes no sense to organize the operators in this way because the world sheet theory is hugely interacting. These interactions have the dimensionless interaction constant $\alpha'/R^2$ where $R$ is the typical size or curvature radius of the spacetime dimensions. Note that the strength of these interactions are controlled by $\alpha'$; it's a different suppression than the suppression by the string (merging/splitting) coupling $g_s$.

When the world sheet theory is strongly coupled e.g. $\alpha' / R^2$ is of order one or larger, one has to find the spectrum of operators – and the corresponding states of one string – by some methods that aren't mechanical. In vacua with spacetime supersymmetry, the BPS states may usually be found analytically by considerations based on SUSY even though the SUSY spectrum can't be fully extracted. In those cases, at least when there is enough SUSY, one may even calculate the leading interactions i.e. OPEs.

Even when $X(z)$ are good enough variables, the procedure of "solving the free string theory" is more complicated than a problem in GR because the strings may also be excited. S it's like "string field theory on a curved background" – the string field contains all the massive string-scale-mass-like fields aside from the light and massless fields. Of course, if you want to neglect the whole massive tower of string states, then you're back to GR and the vertex operators are in one-to-one correspondence with the normal modes of the massless fields in a corresponding GR problem.

The form of the vertex operators, much like $\exp(ik\cdot X(z))$ discussed above, mimics the dependence of the fields on the spacetime and the OPEs of such operators are analogous. However, one shouldn't expect that the well-behaved vertex operators may be constructed from arbitrary coordinates on a curved manifold such as $\theta,\phi$ on a sphere. Such ad hoc coordinates almost never have well-defined dimensions - they're not normal modes of fields on the manifold – and they suffer from various unnatural inequalities bounding their ranges and unnatural identifications of various points.

When the curvature is really really small, strings in a region around a point in the spacetime may be described by the usual $X(z)$ and the curvature may be added as a perturbation to the world sheet action – to supplement the action for a flat spacetime. The main terms in the OPE and Green's functions etc. are then inherited from the flat spacetime. After all, this is the method how we derive Einstein's equation from Weyl invariance in string theory. For the Weyl invariance, the beta-functions for all the coupling constants have to vanish. In the setup of this paragraph, the world sheet coupling constants encode the second derivatives of the metric tensor around a spacetime point (whose vicinity is treated as a perturbation of the flat space) and the beta-function for the "coupling constant" $g_{\mu\nu}$ is the Ricci (or Einstein?) tensor $R_{\mu\nu}$ times a universal constant. Its vanishing, required for the consistency of the world sheet theory (Weyl invariance), implies Einstein's equations in the spacetime (and similar equations for any fields that are predicted by string theory).

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One approach -- not universal, but reasonably general -- is to study a linear sigma model, whose (flat) target space is equipped with a potential. This will not be a CFT in general, but the IR behavior of this theory will be described by a nonlinear sigma model whose target space is the set of minima of the potential. So IF you can find an appropriate potential, you can cook up your nonlinear sigma model as a low energy approximation to a (presumably more solvable) linear sigma model.

There's a variation on this theme, where one studies a gauged linear sigma model, and obtains in the low energy a non-linear sigma model to a symplectic quotient. See Witten, Phases of N=2 Theories in Two Dimensions.

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