nLab non-Lagrangian field theory

Idea

A field theory is called non-Lagrangian if it can not be described using a Lagrangian density. This is in contrast to Lagrangian field theories.

Examples

Example

A basic example of non-Lagrangian field theories used to be self-dual higher gauge theories: Superficially, these would be supposed to have higher Maxwell theory-type Lagrangian densities of the form $H_{2k+1} \wedge \star H_{2k+1}$, but if the self-duality constraint $\star H_{2k+1} = H_{2k+1}$ is imposed beforehand, then this expression vanishes identically instead of inducing the expected Euler-Lagrange equations of motion.

This issue concerns already the Green-Schwarz sigma model for the single (abelian) M5-brane and even more so the would-be theory of coincident M5-branes, involving a D=6 N=(2,0) SCFT with a nonabelian (higher) gauge field.

However, more complicated Lagrangians have been found (Pasti, Sorokin & Tonin 1997, Sen 2020) whose equations of motion do reproduce at least the abelian self-dual fields, and at least if one is willing to disregard some decoupled auxiliary fields, illustrating that the question of whether or not a “field theory” is or is not Lagrangian may require further specification to be well-defined.

Literature

• S.L. Lyakhovich, A.A. Sharapov, Quantizing non-Lagrangian gauge theories: an augmentation method, JHEP 0701:047 (2007) arXiv:hep-th/0612086

• P. C. Argyres, M. R. Plesser, N. Seiberg, E. Witten, New N=2 superconformal field theories in four-dimensions, Nucl. Phys. B461 (1996) 71–84 arXiv:hep-th/9511154

category: physics