Significance of massive states in string theory

Question

A free superstring has an infinite tower of states with increasing mass. The massless states correspond to the fields of the corresponding SUGRA. In "Quantum Fields and Strings: A Course for Mathematicians", vol. II p. 899 we find that the massive states do not contribute anything new to the possible string backgrounds. Terms in the string action corresponding to coupling to a massive background field are nonrenormalizable and therefore disappear when we RG-flow to the IR fixed point, which is the CFT we actually use in quantum string theory. Actually it is explained for the bosonic string but I don't think the difference is essential

What is the physical meaning of this result?

Does it mean massive string states are solitons of the massless fields? If so, do these solitons exist in classical SUGRA?

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Answer 1

Massive string modes have masses of order the string mass $M_s$, independent of the string coupling $g_{str}$, whereas solitons have masses of order $1/g_{str}$ or $1/g_{str}^2$, depending on whether they are open or closed string solitons. So that putative matching does not work (the exponential degeneracy of massive string states would be another obstacle).

I believe the statement you are referring to does not have the wide ranging implications you draw from it, it has to do specifically with mechanics of computing S-matrix elements via string perturbation theory. In such computations in the background of massless modes, the contribution of the massive strings is already accounted by the usual procedure of summing over Riemann surfaces. This is explained nicely by this classic paper by Dine and Seiberg, Microscopic Knowledge From Macroscopic Physics In String Theory.

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Comments

let us [continue this discussion in chat](http://chat.stackexchange.com/rooms/2391/discussion-between-squark-and-moshe)

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You can phrase this on the worldsheet: you have a sigma model whose beta function vanishes to one-loop, but not exactly. This may or may not correspond to an actual classical solution to all orders in $\alpha'$. By systematically correcting the sigma model in perturbation theory (which corresponds to adding higher derivative terms to the spacetime action) you might get an exact SCFT. In such cases there is one to one correspondence between leading order and exact solutions. This is a non-trivial statement and in many cases depends on spacetime supersymmetry.

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My problem is that it seems there are much less exact SCFTs than solutions of the classical equations of motion in the field theoretic POV, because the massive field don't contribute new SCFTs. However the SCFTs are supposed to be the true classical phase space of string theory. So it seems as though the massive fields don't add new degrees of freedom to string theory. I realize something must be flawed in this logic since perturbative excitations are string states and there are massive string states.

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I think the distinction you are making is between linearized perturbations and exact solutions. Even in ordinary nonlinear field theory, like GR, the phase space of the theory is spanned, to leading order, by the linear perturbation (say plane wave solutions of the linearized eom), but the exact phase space may be completely different (e.g. Smaller). In ST the linear perturbations are massive string modes, and exact CFTs are solutions to the full eom.

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OK but the space of linearized perturbations is the tangent space of the phase space so it must have the same dimension (of course dimension is infinite either way but heuristically the number of degrees of freedoms must be preserved). Of course this has to take gauge symmetry into account.

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They are all correct. 3 is a subset of 1: in case the classical solution can be described as a sigma model, in other words when there is a large spacetime, then 3 is the way to describe the (sub)space of CFTs. 2 is not a statement about the vacuum, it says that the CFT has infinite number of primary operators, most of which are irrelevant. In parametrizing the space of CFTs, the irrelevant operators play no role, but there are other questions in classical and quantum string theory which are sensitive to them.

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Think about a similar statement in GR. You can specify completely a certain family of solutions (black holes) by their charge, mass and angular momentum. It doesn't mean that all the information in GR boils down to those 3 numbers (even when discussing black hole physics only).

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Answer 2

The interactions of a massive particle fall off exponentially with distance (massless particles have long-range interactions), the exponent determined by the mass. Mathematically, this dependence is governed by the quadratic term of the field in the action.

Now let's lump all fields together into a multi-index field. Then the vacuum state(s) corresponds to the minimum energy configuration(s), and the nearby shape of the landscape around that minimum (those minima) is determined by the massive fields. Adding more of these fields won't change the space of minima or the long-range behavior of interactions.

Or is that too nontechnical and hand-wavy?

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