nLab D=10 super Yang-Mills theory

Idea

super Yang-Mills theory in spacetime dimension 10.

Properties

Equations of motion from the Bianchi identity

The equations of motion of 10d super Yang-Mills happen to be equivalent in superspace to the Bianchi identity

$$D F = 0$$

subject to the constraint that the bispinorial part of the curvature 2-form vanishes

$$F_{\alpha \beta} = 0$$

(e.g. Witten 86, Atick-Dhar-Ratra 86, (4.14), Bonora-Bregola-Lechner-Pasti-Tonin 87, above (2.7)).

After embedding into heterotic supergravity this becomes parts of the torsion constraints of supergravity. See there.

In this superspace formulation the gaugino $\chi$ appears as the even-odd component of the super-curvature form

(1)$$F_{(1,1)} \;\propto\; \left(\overline{\psi} \Gamma_a \chi\right) \wedge e^a$$

(where $(e^a, \psi^\alpha)$ is the super vielbein). This is (Witten 86 (8), Bonora-Pasti-Tonin 87, below (11), Bonora-Bregola-Lechner-Pasti-Tonin 87, (2.27)).

Compactification to the point

The KK-compactification of $D=10$ super-Yang-Mills to the point is a theory whose fields are simply elements of the gauge Lie algebra, hence matrices if we have a matrix Lie algebra. The theory defined by this reduction is called the IKKT matrix model.

References

• Bengt Nilsson, Off-shell fields for the 10-dimensional supersymmetric Yang–Mills theory, Feb 1981 (Spire)

• Riccardo D'Auria, Pietro Fré, A. J. da Silva, Geometric structure of $N=1$, $D=10$ and $N=4$, $D=4$ super Yang-Mills theory, Nuclear Physics B 196 2 (1982) 205-239 [doi:10.1016/0550-3213(82)90036-0]

  (in D'Auria-Fré formulation)

• Joseph Atick, Avinash Dhar, and Bharat Ratra, Superspace formulation of ten-dimensional $N=1$ supergravity coupled to $N=1$ super Yang-Mills theory, Phys. Rev. D 33, 2824, 1986 (doi.org/10.1103/PhysRevD.33.2824)

• Edward Witten, Twistor-like transform in ten dimensions, Nuclear Physics B Volume 266, Issue 2, 17 March 1986 (spire:214192, doi:10.1016/0550-3213(86)90090-8)

• Loriano Bonora, Paolo Pasti, Mario Tonin, Superspace formulation of 10D SUGRA+SYM theory a la Green-Schwarz, Physics Letters B Volume 188, Issue 3, 16 April 1987, Pages 335–339 (doi:10.1016/0370-2693(87)91392-X)

• Loriano Bonora, M. Bregola, Kurt Lechner, Paolo Pasti, Mario Tonin, Anomaly-free supergravity and super-Yang-Mills theories in ten dimensions, Nuclear Physics B 296 4 (1988) [doi:10.1016/0550-3213(88)90402-6, inspire:247764]

• Martin Cederwall, Bengt Nilsson, Dimitrios Tsimpis, The structure of maximally supersymmetric Yang-Mills theory: constraining higher-order corrections, JHEP 0106:034,2001 (arXiv:hep-th/0102009)

• Martin Cederwall, Bengt Nilsson, Dimitrios Tsimpis, $D=10$ Super-Yang-Mills at $O(\alpha'^2)$, JHEP 0107:042,2001 (arXiv:hep-th/0104236)

• Mikhail Movshev, Albert Schwarz, On maximally supersymmetric Yang-Mills theories, Nucl. Phys. B681 (2004) 324-350 (arXiv:hep-th/0311132)

On scattering amplitudes in pure spinor-formalism:

• Maor Ben-Shahar, Max Guillen, 10D Super-Yang-Mills Scattering Amplitudes From Its Pure Spinor Action (arXiv:2108.11708)