functorial quantum field theory
Reshetikhin?Turaev model? / Chern-Simons theory
FQFT and cohomology
An effect in non-perturbative quantum field theory that cannot be seen in perturbative quantum field theory is called a non-perturbative effect.
More in detail, theories with instanton field configurations (such as in Yang-Mills theory, hence in QCD and QED) or branes (such as in string theory), etc., are expected to have observables which as functions of the coupling constant $g$ are transseries of the form
where the first sum is the Feynman perturbation series itself and where the terms with a non-analytic dependence of the form $\exp(-A/g)$ or $\exp(-A/g^2)$ are the contributions of the instantons. Since all the derivatives of the functions $g \mapsto e^{-1/g}$ or $g \mapsto e^{-1/g^2}$ vanish at coupling constant $g = 0$, the Taylor series of this part of the observable does not appear in perturbative QFT, even though it is present. Therefore this is called a non-perturbative effect.
Related is resurgence theory. See also at perturbation theory – Divergence/convergence for more.
Notice that, while non-perturbative effects are generally the least understood, they are not exotic but ubiquitous: Essentially the entire parameter space of a generic “physical theory” is non-perturbative. In fact, the non-perturbative region of parameter space is the complement of an infinitesimally thickened point (labeled “perturbative” in the following graphics):
A central example of a non-perturbative effect is confinement (hence the “mass gap problem”) in Yang-Mills theory at low temperature. Perturbation theory is not suited to explain this (e.g Espiru 94, section 7).
At the other extreme of high temperature QCD, also the quark-gluon plasma, while now deconfined is thought to be strongly coupled.
Non-perturbative effects in hadron-physics affects the discussion of possible beyond-standard model physics as seen in
$\ell_P$, the Planck length in 11-dimensions;
$R_{10}$ the length (circumference) of the $S^1_{10}$ circle fiber for KK-compactification to 10 dimensions
and the string theory scales
$\ell_s$, the string length scale;
$g_{s}$, the string coupling constant of perturbative string theory.
Then under the duality between M-theory and type IIA string theory these scales are related as follows:
equivalently
equivalently
Hence a membrane instanton, which on a 3-cycle $C_3$ gives a contribution
becomes
if the cycle wraps, $C_3 = C_2 \cup S^1_{10}$, a worldsheet instanton
the cycle does not wrap, a spacetime instanton contribution, specifically a D2-brane instanton?
(This unification of the two different non-perturbative effects in perturbative string theory (worldsheet instantons and spacetime instantons), to a single type of effect (membrane instanton) in M-theory was maybe first made explicit in Becker-Becker-Strominger 95. Brief review includes Marino 15, sections 1.2 and 1.3).
Discussion of traditional algebraic quantum field theory as being about non-perturbative effects (and therefore lacking any interesting models in 3+1 dimensions, so far…):
Genral introduction and toy examples (e.g. phi^4-theory or anharmonic oscillator):
Mario Flory, Robert C. Helling, Constantin Sluka, Section 2 of: How I Learned to Stop Worrying and Love QFT (arXiv:1201.2714)
Discussion for phi^4 theory is in
Discussion for QCD includes
and in (super-)Yang-Mills theory and string theory is in
Marcos Mariño, Lectures on non-perturbative effects in large N gauge theories, matrix models and strings, Fortschritte der Physik 62.5‐6 (2014): 455-540 (arXiv:1206.6272)
Marcos Mariño, Non-perturbative effects in string theory and AdS/CFT, Spring School on Superstring Theory and Related Topics 2015 (pdf, pdf, recording)
In QCD (hadrodynamics) via operator product expansion:
Ulrich Nierste, Flavour Anomalies: Phenomenology and BSM Interpretations, 2018 (pdf)
Fred Jegerlehner, The Role of Mesons in Muon $g-2$ (arXiv:1809.07413)
In cosmology:
The form of the contribution of non-perturbative effects in string theory was originally observed in
The interpretation via D-branes of non-perturbative effects in the string coupling constant is due to
The identification of non-perturbative contributions in string theory with brane contributions is due to
Review includes
Reviews specifically in type II string theory include
Hugo Looyestijn, Non-perturbative effects in type IIA string theory, Master Thesis 2006 (pdf)
Angel Uranga, Non-perturbative effects and D-brane instanton resummation in string theory (pdf, pdf)
The relation of non-perturbative effects in string theory to M-theory goes back to
In Becker-Becker-Strominger 95 it was realized that the worldsheet instantons and D-brane instantons of string theory unify to membrane instantons (see Marino 15, section 1.3)
Last revised on September 22, 2023 at 14:11:50. See the history of this page for a list of all contributions to it.