string theory FAQ



String theory


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What is string theory?




More conceptually, the premise is that





There are a host of educated guesses of what non-perturbative string theory might be, if anything, but it remains unknown. At some point the term M-theory had been established for whatever that non-perturbative theory is, but even though it already has a name, it still remains unknown. (Or rather: its full incarnation remains unknown. What is well defined is 11-dimensional supergravity with some M-brane effects and gauge enhancement included, and that is what is presently being studied under the name “M-theory”, see for instance at M-theory on G2-manifolds; and see also at F-theory.)


Then why not consider perturbative pp-brane scattering for any pp?

The above motivation of perturbative string theory as the evident result of replacing the definition of an S-matrix perturbation theory, via 1-dimensional Feynman diagrams encoding worldlines of particles, by 2-dimensional diagrams (Riemann surfaces) encoding worldsheets of strings raises the evident question: why stop at strings? Why not consider an S-matrix built as the sum of the correlators of a worldvolume field theory for each p+1p+1-dimensional manifold, encoding the propagation of a membrane for p=2p = 2, and generally of (what is called) a p-brane?

To answers this, it is again crucial to distinguish between the perturbation theory and the non-perturbative theory. On the one hand, study of the string perturbation theory shows that indeed strings interact with and gives rise to pp-branes for many different values of pp. But the dynamics of all these higher dimensional branes itself seems to be intrinsically non-perturbative. What does not seem to exist is a sensible perturbation series for pp-brane scattering with p>1p \gt 1.

The reason is that it is hard and seems impossible to make sense of this. There are two technical problems:

  1. for p>1p \gt 1 then the standard worldvolume action functionals (Nambu-Goto action), are not renormalizable;

  2. for p>1p \gt 1 the moduli spaces of p+1-manifolds are not controllable.

So for p>1p \gt 1 one a) does not know how to define the “Feynman amplitudes” and b) even if one did, one does not know how against what to integrate them.

Each of these two problems in itself makes a pp-brane perturbation theory for p>1p \gt 1 be hard to come by.

That is, incidentally, the very reason for the term “M-theory”. First there had been the observation that the super 1-brane in 10d target spacetimes is accompanied by a 2-brane in 11d target spacetime, now called the M2-brane (for the history see Mike Duff, The World in Eleven Dimensions). This suggested the evident idea that there ought to be perturbation theory for 2-branes – called membranes, hence that there ought to be “membrane theory” in direct analogy with “string theory”. But the above two problems make a direct such analogy unlikely. Nevertheless, since there might be a less obvious, more sophisticated kind of analogy, Edward Witten proposed to say “M-theory” as an abbreviation, not to commit himself to what exactly might be really going on, and leaving open for the future if “M” is for “membrane” or for something else:


M stands for magic, mystery, or membrane, according to taste (Witten 95)

What are the equations of string theory?

All local field theories in physics are prominently embodied by key equations, their equations of motion. For instance classical gravity (general relativity) is essentially the theory of Einstein's equations, quantum mechanics is governed by the Schrödinger equation, and so forth.

But perturbative string theory is not a local field theory. Instead it is an S-matrix theory (see What is string theory?). Therefore instead of being given by an equation that picks out the physical trajectories, it is given by a formula for how to compute scattering amplitudes. That formula is the string perturbation series: it says that the probability amplitude for n inn_{in} asymptotic states of strings coming in (into a particle collider experiment, say), scattering, and n outn_{out} other asymptotic string states emerging (and hitting a detector, say) is a sum over all Riemann surfaces with (n in,n out)(n_{in}, n_{out})-punctures of the n-point functions of the given 2d CFT that defines the string vacuum (see at What is a string vacuum?).

More in detail, a string background is equivalently a choice of 2d SCFT of central charge 15 (a “2-spectral triple”), and in terms of this the formula for the S-matrix element/scattering amplitude for a bunch of asymptotic string states ψ in 1,,ψ in n in\psi^1_{in}, \cdots, \psi^{n_{in}}_{in} coming in, and a bunch of states ψ out 1,,ψ out n out\psi^1_{out}, \cdots, \psi^{n_{out}}_{out} coming out is schematically of the form

S ψ in 1,,ψ in n in,ψ out 1,,ψ out n out=gλ gmodulispaceof(n in,n out)puncturedsuperRiemannsurfacesΣ g n in,n outofgenusg(SCFTCorrelatoroverΣofstatesψ in 1,,ψ in n in,ψ out 1,,ψ out n out) S_{\psi^1_{in}, \cdots, \psi^{n_{in}}_{in}, \psi^1_{out}, \cdots, \psi^{n_{out}}_{out}} \;=\; \underset{g \in \mathbb{N}}{\sum} \lambda^g \underset{ {moduli \; space \; of} \atop {{(n_{in},n_{out}) punctured} \atop {{super\; Riemann \; surfaces} \atop {{\Sigma^{n_{in}, n_{out}}_g} \atop {of\; genus\; g}}}} }{ \int } \left( SCFT \; Correlator \; over \; \Sigma \; of \; states\; {\psi^1_{in}, \cdots, \psi^{n_{in}}_{in}, \psi^1_{out}, \cdots, \psi^{n_{out}}_{out}} \right)

expressing the S-matrix element (scattering amplitude) shown on the left as a formal power series in the string coupling constant with coefficients the integrals over moduli space of super Riemann surfaces of the worldsheet correlators (nn-point functions) for the given incoming and outgoing string states.

With more technical details filled in, this formula reads as follows: for the bosonic string, as found in Polchinski 01, volume 1, equation (5.3.9)

and for the superstring, as found in Polchinski 01, volume 2, equation (12.5.24)

This is the equation that defines perturbative string theory.

And this is of just the same form as the Feynman perturbation series perturbative quantum field theory, the only difference being that the latter is more complicated: there one has to sum over Feynman diagrams with labeling for all intermediate particles (virtual particles) and with some choice of renormalization to make the integrals well defined, whereas here we simply sum over all super Riemann surfaces and that’s it. The different intermediate virtual particles as well as the renormalization counterterms are all taken care of by the higher string modes, encoded in the worldsheet CFT correlators.

There was a time in the 1960s, when quantum field theorists around Geoffrey Chew proposed that precisely such formulas for S-matrix elements should be exactly what defines a quantum field theory, this and nothing else. The idea was to do away with an explicit concept of spacetime and local interactions, and instead declare that all there is to be said about physics is what is seen by particles that probe the physics by scattering through it. This is an intrisically quantum approach, where there need not be any classical action functional defined in terms of spacetime geometry. Instead, all there is a formula for the outcome of scattering experiments.

Historically, this radical perspective fell out of fashion for a while with the success of QCD and the quark model in its formulation as as local field theory coming from an action functional: Yang-Mills theory.

But fashions come and go, and the original idea of Geoffrey Chew and the S-matrix approach continues to make sense in itself, and it is this form of a physical theory that perturbative string theory is an example of.

Ironically, more recently, the S-matrix-perspective also becomes fashionable again in Yang-Mills theory itself, with people noticing that scattering amplitudes at least in super Yang-Mills theory have good properties that are essentially invisible when expressing them as vast sums of Feynman diagram contributions as obtained from the action functional. For more on this see at amplituhedron.

On the other hand, there is also an analog of the second quantized field-theory-with-equations for string scattering: this is called string field theory, and this again is given by equations of motion. For instance the equations of motion of closed string field theory are of the form

Qψ+12ψψ+16ψψψ+=0, Q \psi + \tfrac{1}{2} \psi \star \psi + \tfrac{1}{6} \psi \star \psi \star \psi + \cdots = 0 \,,

where Ψ\Psi is the string field, QQ is the BRST operator and \star is the string field star product.

(For the bosonic string this is due to Zwiebach 92, equation (4.46), for the superstring this is in Sen 15 equation (2.22)).

The string field Ψ\Psi has infinitely many components, one for each excitation mode of the string. Its lowest excitations are the modes that correspond to massless fundamental particles, such as the graviton. Expanding the equations of motion of string field theory in mode expansions (“level expansion”) does reproduce the equations of motions of these fields as a perturbation series around a background solution and together with higher curvature corrections.

Why is string theory controversial?

As a theory of the observable universe that is supposed to checked by experiment and which predicts that all fundamental particles of the standard model of particle physics are secretly, if one probes at high enough energy, excitations of superstrings, string theory is an unproven hypothesis.

Current particle accelerator technology (notably the LHC) are about 15 orders of magnitude (hence far, far) away from the energy scale at which these strings would manifest themselves directly (at least for many models in string phenomenology).









Does string theory make predictions? How?









In string theory this happens with all the parameters. (Except for one single constant: the string coupling constant. From the perspective of “M-theory” even that disappears. See at string theory – scales.) There is no external choice of parameter, but there remains the choice of studying “solutions to the equations of motion” (which in string theory means: choices of 2d CFTs) which might model observed physics.

That is why in string theory instead of adjusting parameters one searches solutions. Since these are also called “string vacua” (see at What is a string vacuum?), one searches vacua. The infamous term “landscape of string theory vacua” refers to attempts to understand the space of possibilities here more globally. But very little is actually known to date.




Aside: How do physical theories generally make predictions, anyway?






  1. posit a theory;



  4. if experiment disagrees with the predictions then












Is string theory testable?









Is string theory causal, given that it is not local on the string scale?

In an S-matrix theory such as perturbative string theory (see above at What is string theory) the property of causality is embodied by the fact that the S-matrix shows certain analyticity features. (Therefore the S-matrix approach to quantum field theory is often referred to as “the analytic S-matrix”).

Since, as opposed to a fundamental particle, the string is extended, at the string scale string theory is not given by a local field theory. This superficially seems to suggest that at such scales also causality might be violated in string theory. However, computation shows that the string scattering S-matrix comes out suitably analytic and causal (e.g. Martinec 95).

A detailed analysis for how this comes about has been given in (Erler-Gross 04). They write:






Does string theory predict supersymmetry?

To understand this question and its answer, it is important to know that in general symmetries in physics (and in mathematics) come in a local and in a global flavor. For instance the theory of gravity is a theory which has as a local symmetry the Poincaré group, including the Lorentz group. But any model of the theory – a spacetime – may or may not have global Lorentz symmetry (“Killing vector fields”). In fact, the generic solution to the Einstein equations has no global Lorentz symmetry left. Indeed, global Lorentz symmetry of spacetime with all matter and force fields in it would mean that the world looks the same if we arbitrarily translate in some direction, or arbitrarily rotate in some plane. It would be rather bizarre to live in a spacetime with such a property!

(Mathematically the distinction is this: given a coset space G/HG/H, then the corresponding Klein geometry has global GG-symmetry, while the corresponding Cartan geometries have local GG-symmetry.)


(Mathematically this is now the situation of super Cartan geometry.)




(This result is known as the Goddard-Thorn no-ghost theorem with GSO projection.)

So: string theory implies that if there are fermions at all, then there is local supersymmetry, hence supergravity.

strings&fermionssupergravity strings \;\; \& \;\; fermions \;\;\; \Rightarrow \;\;\; supergravity


In fact the generic model has a priori no reason to preserve any low energy global supersymmetry below the Planck scale at all (see e.g. Giudice-Strumia 11, p. 1-2), just as the generic solution of Einstein's equations does not preserve any Lorentz group symmetry. The condition that a Kaluza-Klein compactification of 10-dimensional supergravity to 4d exhibits precisely one global supersymmetry is equivalent to the compactification space being a Calabi-Yau manifold. While this is a famous condition that has been extensively studied (see at supersymmetry and Calabi-Yau manifolds), nothing in the theory seemed to require KK-compactification on Calabi-Yau manifolds. These were originally considered because for phenomenological reasons it is (or was) expected that our observed world exhibits global low-energy supersymmetry.

However, recently arguments for a theoretical preference for N=1N=1 supersymmetric compactifications have been advanced after all (FSS19, Sec. 3.4, Acharya 19).




What is a string vacuum?

In perturbative quantum field theory a vacuum state is the information needed to turn a product of field observables such as Φ a(x)Φ b(y)\mathbf{\Phi}^a(x) \mathbf{\Phi}^b(y) into a function (or rather: generalized function/distribution) of the insertion points xx any yy, namely the n-point function (here 2-point function, also called the Hadamard propagator)

Φ a(x)Φ b(y) \langle \mathbf{\Phi}^a(x) \mathbf{\Phi}^b(y)\rangle

which may be regarded as the probability amplitude for a quantum in state bb at spacetime point yy to turn into a quantum in state aa at spacetime point xx, in the given state that the fields are in, which is defined thereby (see at state in AQFT).

In the worldline formalism of field theories these propagators arise from a 1-dimensional field theory on the “worldline” of (virtual) particles running from yy to xx.

Now by the very definition of perturbative string theory, these particles are replaced by strings whose dynamics is now encoded in a 2d field theory on the worldsheet of strings, specifically a 2d superconformal field theory (2d SCFT) of central charge 15. Hence now it is the 2d SCFT which defines the vacuum state that the perturbative string theory is in. This is then called a perturbative string theory vacuum.

In practice full 2d SCFTs are hard to construct, and often one considers them by perturbation theory of a “sigma-model” which is defined by a spacetime manifold equipped with extra fields (e.g. the B-field etc.). It turns out that to low order these background field configurations that define sigma-model 2d SCFTs are given by solutions to equations of motion of supergravity theories (e.g. type II supergravity for type II string theory, etc.)

Therefore often such supergravity solutions equipped with some extra data that makes them consistent CFT backgrounds at higher order are referred to as vacua for string theory. But this is in general a coarse approximation. The full vacua are the full 2d SCFTs that define the worldsheet theory of the string.

The collection of all string vacua, possibly subject to some assumptions, has come to be called the landscape of string theory vacua. See at What does it mean to say that string theory has a landscape of vacua?

How/why does string theory depend on “backgrounds”?







By design, all this applies also to perturbative string theory.




Did string theory provide any insight relevant in experimental particle physics?


Details on this are linked to at









What is the relationship between string theory and quantum field theory?

Recall from some of the previous answers:







Isn’t it fatal that the string perturbation series does not converge?


No. On the contrary, the Feynman perturbation series of every QFT of interest is supposed to have vanishing radius of convergence (Dyson 52) and be just an asymptotic series. Because the expansion parameter is the coupling constant and if the series had a finite radius of convergence, there would be also negative values of the coupling for which the correlators are convergent. See at non-perturbative effect for more.




How do strings model massive particles?





How is string theory related to the theory of gravity?


This means the following:








Do the extra dimensions lead to instability of 4 dimensional spacetime?

The parameters of size and shape of the compactified dimensions in string theory, and in fact in any Kaluza-Klein compactification, are called “moduli”. Since they are part of the higher dimensional metric, they are components of the higher dimensional field of gravity and hence are dynamical fields that evolve. The problem of their stability, hence the question whether there are dynamical mechanisms that make for instance the size of the compactified space remain stably at a given value, is famous as the problem of moduli stabilization in string theory.

This problem used to be open until around 2002. Then it was realized that vacuum expectation values (VEVs) of the higher form fields (“fluxes”) present in string theory generically induce effective potentials for moduli that may stabilize them, at least for fluctuations that preserve the given special holonomy (CY-compactifications of string theory or G2-holonomy compactifications for M-theory).

For type IIB string theory/F-theory this was argued in the influential article KKLT 03. An analogous moduli stabilization mechanism was also argued for M-theory on G2-manifolds by Acharya 02. It is the counting of all the many possible ways of stabilizing moduli via fluxes in type IIB that led to the now infamous discussion of the landscape of type II string theory vacua (see also below). In any case, there seems to be no lack of solutions of the stability problem.



In fact, more precisely, a perturbative string theory vacuum for string perturbation theory is a choice of super-2d CFT of central charge 15 (see What is a string vacuum?), and each of them induces such an effective background (by a mechanism indicated at 2-spectral triple). This imposes considerably more constraints than one has to solve to find a solution to just Einstein-Yang-Mills equations (for instance “modular invariance” of the 2d theory, etc.). As a result, it is considerably harder to find a backround string vacuum for string theory than for its non UV-complete non-renormalized effective Einstein-Yang-Mills-Dirac-Higgs theory.












write on page 9 of their accurate study of orientifold backgrounds:



Is string theory mathematically rigorous?

The core of perturbative string theory has a mathematically rigorous formulation. In fact much of mathematical physics and mathematical insight into quantum field theory as such has been gained from the study of the low-dimensional QFTs that constitute the worldvolume theories of the string and the various branes. For instance the axiomatization of QFT in the “FQFT” flavor (roughly dual to the AQFT picture) historically originates in insights gained in the study of (topological) string (namely the Moore-Seiberg axioms). On the other hand, the attempted implementations and applications of core string theory are vast and numerous, and when it finally comes to string phenomenology the usual level of rigor is just that common among practicing quantum field theorists. On the far end, deep aspects of string theory that are felt by many researchers to be of metaphysical relevance, such as the “landscape of string theory vacua” (see above) have led and are leading to speculations that are not anymore backed up by any disciplined reasoning.

More in detail:

The quantization of the string sigma-model may be obtained cleanly via the mathematical sound process of geometric quantization, see the references at string – Symplectic geometry and Geometric quantization. The famous Weyl anomaly of the string is formally understood in terms of anomalous action functionals, see for instance (Freed 86 2.). Various other obstructions to quantization (quantum anomalies) in the background fields for the string sigma-model such as notably the Freed-Witten-Kapustin anomaly, have been understood in fine detail in terms of obstructions in differential cohomology, see for instance (Distler-Freed-Moore 09).

Particularly well analyzed are the two special sectors of first quantized string theory, that of rational conformal field theory, which contains the example of strings propagating on Lie group manifolds – the Wess-Zumino-Witten model; as well as the example of topological strings. Rational conformal field theories indeed stand out as one non-trivial and rich class of QFTs which have been subject to complete mathematical classification (in the same sense in which mathematicians for instance do the classification of finite simple groups). For details on this classification see at FRS formalism.

For the topological string much more is true. The topological string has effectively become a subject in pure mathematics, with its rigorous axiomatization via the TCFT version of the cobordism hypothesis-theorem, its formulation as mathematical homological mirror symmetry, its relation to geometric Langlands duality etc. Here it is maybe noteworthy that all this mathematical insight into string physics rests on homotopy theory and higher category theory (the cobordism hypothesis, to wit, which governs the topological string, is a theorem in pure (infinity,n)-category theory). See also the question How does string theory involve homotopy theory, higher geometry and higher category theory?.

But the FQFT-axiomatics that serves to mathematically formalize the topological string is not restricted to the topological sector, it also applies to the physical string. For instance Huang’s theorem shows that the familiar description of physical string via vertex operator algebra is an instance of the FQFT-formalization. Indeed, in FRS formalism these two formalizations, vertex operator algebras (via their modular tensor categories of representations, and TQFT combined via the rigorous AdS3-CFT2 and CS-WZW correspondence give the classification of rational CFT). (In particular this says that in this low dimensionl holography and AdS-CFT duality is rigorous, of course this is far, far from true in higher dimensions.) This means that the rigorous quantization of the string in these approaches is “holographically” encoded in the quantization of Chern-Simons theory: the space of quantum states of 3d Chern-Simons theory is identified with the space of conformal blocks (correlators) of the string WZW-model on this surface. Here quantization of Chern-Simons theory is one of the best analyzed quantizations of a non-trivial rich QFT.

In summary this is a level of rigour with which the worldsheet 2d QFT of the string is understood which is well beyond of what one typically encounters for non-trivial interacting (non-free) QFT. And this is full non-perturbative quantum field theory (on the worldsheet!), not just the approximation in perturbation theory.

From here on, also string field theory (its action functional, that is), has a completely rigorous formulation in terms of operads and L-infinity algebras (Lie n-algebras for nn \to \infty).

Of course what does not have a rigorous formulation, not even a good non-rigorous formulation, is any conjectured non-perturbative (in spacetime!) completion of string theory (M-theory). But this is clear from the answer above at What is string theory.

A snapshot of the state of the art of rigorous foundations of string theory as of 2011 is in Mathematical Foundations of Quantum Field and Perturbative String Theory.









For the moment, see the following for more:


Textbooks on string theory and M-theory include the following (for more see at books about string theory):


Last revised on June 18, 2019 at 19:12:22. See the history of this page for a list of all contributions to it.