Types of quantum field thories
FQFT and cohomology
String theory is a theory in fundamental physics. Certainly it is, mathematically, a structure that contains in various limits a plethora of quantum field theories. Its interest for experimental high energy physics lies in the hypothesis that it provides a theory of everything in the sense of fundamental physics, but the jury on that is still out.
Below we indicate the basic idea and provide pointers to further details. See also the string theory FAQ.
Recall that in quantum field theory one approach to make sense of the path integral is the perturbation series expansion, which interprets the path integral for the scattering amplitudes (the S-matrix) as a certain sum over graphs of certain numbers assigned to each graph.
The graphs are called Feynman diagrams, the numbers assigned to them are called (renormalized) scattering amplitudes and the sum over graphs of (renormalized) amplitudes is the perturbation series or S-matrix.
Crucial for the motivation of the idea of string theory is the observation that this (renormalized) amplitude assigned to a graph is itself the correlator of a 1-dimensional quantum field theory on that graph: the “worldline quantum field theory” describing the (relativistic) quantum mechanics of these particles. This is usually a sigma-model with parameter space the given graph and target space the spacetime on which the fields live for which the perturbation series computes the path integral.
When made explicit this is called the worldline formalism for computing the quantum field perturbation series. (See there for more details.)
The premise of perturbative string theory is to replace the perturbation series over correlators of a 1-d QFT over graphs by a sum of correlators of a 2-dQFT over 2-dimensional surfaces – called worldsheets and hence produce an S-matrix this way. Again in simple cases this 2d QFT is a sigma-model whose target is the spacetime in which one computes interactions.
In analogy to the previous case, one thinks of the amplitude assigned this way to a surface as the amplitude for the boundary arcs – the strings – to interact in the way given by the surface.
Some of the motivations for considering this replacement of graphs by surfaces have been the following:
the 2-d correlators are better behaved in that they don’t have to be renormalized. The “counterterms” appearing in renormalization of ordinary QFT can be identified with contributions to the correlators that come from the linear extension of the strings (see the above reference for more on this);
there are fewer choices involved: a Feynman graph is really a decorated graph with the decoration from some more or less arbitrary index set, describing the nature of the particles associated with a given edge and the nature of the interaction associated with a given vertex. In the sum over surfaces there is no extra decoration (except on the boundary of the surfaces) and one finds that instead a single string diagram (a 2d QFT correlator for a given surface) encodes already a sum over (infinitely) many particle species decorations and all possible interaction decorations for them;
while there are fewer choices to be made by hand, it turns out that the effective particle content that does arise automatically from this prescription happens to be structurally of the kind one would hope for: the massless effective particles described by the string perturbation series happen to be gauge bosons, fermions charged under them, and, notably, gravitons. This is structurally exactly the Yang-Mills theory input of the standard model of particle physics combined with perturbative quantum gravity that one would hope to see.
These aspects have motivated the impression that the string perturbation series might be considerably closer to the true formalism of fundamental physics than ordinary perturbative quantum field theory. This impression is however offset by the following problems:
while the worldsheet 2d QFT whose correlators are summed over surfaces are themselves much easier to handle than the full target space quantum physics they are used to encode, a fully complete and rigorous theory of 2d QFT is available only in simple special cases.
In particular, even though there are fewer arbitrary choices involved in the string perturbation series as compared to the ordinary Feynman perturbation series, one crucial choice still present is that of this worldsheet 2d QFT. By the above, every choice of worldsheet QFT (called a choice of “vacuum”) corresponds to a choice of effective target space geometry (to be thought of as the one that the perturbation series computes the quantum perturbations about) and particle content (see 2-spectral triple for more on that). One would therefore like to understand the space of all worldsheet QFTs whose effective target space geometry and particle content is close to the one experimentally observed. After many years of rather naïve approaches to handle or not to handle this, it has more recently at least come to the general attention that there is something to be better understood here.
More fundamentally, already the role of the original perturbation series in quantum field theory is actually not fully understood. Its main success is the observation that truncating or resumming the perturbation series in a more or less ad hoc way, it does yield values that very well describe a plethora of real world measurements. One imagines that there is a non-perturbative definition of quantum field theory such that in certain well-defined circumstances the perturbation series does yield an approximation to it and is a posteriori justified. If so, there should be an analogous nonperturbative definition of string theory. There is a large ratio of speculations as to what that might be over solid results about it.
While therefore the premise of perturbative string theory is conceptually suggestive for various reasons, there is to date no connection to experimental phenomenology (apart from the fact that conceptual insights into string theory have helped analyze quantum field theoretic data, see at string theory results applied elsewhere). As a result much of the substantial outcome of string theory research is more in mathematical physics (if done well, at least), exploring the general theory space of quantum field theories and their UV-completions, than in realistic model building (though there is no lack of trying), where it remains very speculative. This has led to public or semi-public debates about the value of string theory for actual physics. See at criticism of string theory for pointers.
string coupling constant
Newton's constant (before compactification)
(see e.g. arXiv:0908.0333)
For both the bosonic string and the superstring the corresponding Polyakov action has a gauge anomaly for the conformal symmetry, depending on the dimension of target space, and on the strength of the dilaton background field. For vanishing dilaton field this anomaly vanishes exactly for for the bosonic model, and in for the superstring.
For target spaces of these dimensions one speaks of critical string theory. In as far as string theory is expected to have relevance for physics at all, it is usually expected to be in this critical dimension. But also noncritical string models can and have been considered.
In order for the action functional to be well-defined, the tensor product of these different anomaly line bundles over the bosonic configuration space must have trivial class (as bundles with connection, even). This gives rise to various further anomaly cancellation conditions:
For the heterotic string (necessarily closed) the anomaly cancellation condition is known as the Green-Schwarz mechanism : it says that the background fields of gravity and B-field must organize to a twisted differential string structure whose twist is given by the background Yang-Mills field.
For the open type II string the condition is known as the Freed-Witten anomaly cancellation condition: it says that the restriction of the B-field to any D-brane must consistute the twist of a twisted spin^c structure on the brane.
See also Diaconescu-Moore-Witten anomaly.
|brane||in supergravity||charged under gauge field||has worldvolume theory|
|black brane||supergravity||higher gauge field||SCFT|
|D-brane||type II||RR-field||super Yang-Mills theory|
|D0-brane||BFSS matrix model|
|D4-brane||D=5 super Yang-Mills theory with Khovanov homology observables|
|D1-brane||2d CFT with BH entropy|
|D3-brane||N=4 D=4 super Yang-Mills theory|
|(D25-brane)||(bosonic string theory)|
|NS-brane||type I, II, heterotic||circle n-connection|
|NS5-brane||B6-field||little string theory|
|D-brane for topological string|
|M-brane||11D SuGra/M-theory||circle n-connection|
|M2-brane||C3-field||ABJM theory, BLG model|
|M5-brane||C6-field||6d (2,0)-superconformal QFT|
|M9-brane/O9-plane||heterotic string theory|
|topological M2-brane||topological M-theory||C3-field on G2-manifold|
|topological M5-brane||C6-field on G2-manifold|
|solitons on M5-brane||6d (2,0)-superconformal QFT|
|self-dual string||self-dual B-field|
|3-brane in 6d|
Since the Witten genus in turn is the decategorification of the string orientation of tmf, this suggests that tmf-generalized (Eilenberg-Steenrod) cohomology classifies full string theories, in refinement of how the classification of D-brane charge (just the boundary conditions for open strings) is given by K-theory.
A non-trivial conistency check of this idea is announced in (Nikolaus 14).
Beyond the speculative hypothetized role of string theory as a theory behind observed particle physics, the theory has shed light on many aspects of quantum field theory, both on the conceptual structure of quantum field theory as such as well as on concrete theories and their concrete properties. Some of these string theory results enter crucially in computations that are used to interpret particle physics experiments such as the LHC.
For more see
A large body of references is organized at the
For another list of literature see the entry
A useful survey of the status of string theory as a theory of quantum gravity is in
Some reflections on the mathematical physics involved in string theory are in
An article summarizing information about cohomological models for aspects of string theory and listing plenty of useful further references is
A book trying to summarize the state of the art of capturing mathematical structures fundamental to the definition of perturbative string theory is
based on the insights of Green-Schwarz anomaly cancellation.
A nontrivial consistency check on the suggestion that this means that string backgrounds are classified in tmf is given in
Discussion of type II quantum anomalies is in
Richard Borcherds, 1998
Maxim Kontsevich, 1998
Edward Witten, 1990
Grigori Perelman, 2006
These tools used in the proof are all rather old, known for at least twenty years, and one may wonder why they have not before been put to use in connection with the Riemann moduli space. Maybe we lacked the inspiration that comes from the renewed interaction with physics, exemplified in conformal field theories.