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Types of quantum field thories
Perturbative string theory is something at least close to a categorification of the following description of perturbative quantum field theory in terms of sums over Feynman diagrams.
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 as a certain sum over graphs of certain numbers assigned to each graph.
The graphs are called Feynman graphs, the numbers assigned to them are called (renormalized) amplitudes and the sum over graphs of (renormalized) amplitudes is the perturbation series.
The amplitude assigned to a single graph with external edges is interpreted as the amplitude for “quanta” or “particles” of the fields in question to interact in the way indicated by the graph.
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. A useful discussion is for instance in
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. 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.
The action functional for the string-sigma model in general has a quantum anomaly of both kinds:
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.
Apart from the gauge anomaly, the action functional of the string-sigma-model also in general has an anomalous action functional , for two reasons:
The higher holonomy of the higher background gauge fields is in general not a function, but a section of a line bundle;
The fermionic path integral over the worldsheet-spinors of the superstring produces as section of a Pfaffian line bundle.
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.
A more detailed analysis of these type II anomalies is in (DFMI) and (DFMII).
Table of branes appearing in supergravity/string theory
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
We will also keep list of critics of string theory.
String theory leads a life somewhere in between the usual physics literature and the usual math literature. Large parts of it are still lacking a satisfactory mathematical formulation. But every now and then some aspect of string theory, some mathematical gadget or consequence found there is isolated and redefined independently and mathematically rigorously, retaining many features originally predicted. Famous examples are
quantum anomaly canellation Green-Schwarz mechanism
etc.
The question is what the formalism might be that unifies all this into one coherent theory of higher quantum field theory.
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
Discussion of type II anomalies is in