nLab twistor string theory

Idea

It seems (Nair 1988, Witten 2004, Sec. 3) that the scattering amplitudes of $n$ gluons (n-point functions) in pure Yang-Mills theory on 4d Minkowski spacetime, regarded as distributions on the product space of $n$ copies of twistor space $\mathbb{C}P^3$, have support – after complexification – on (the diagonal image of) a complex curve in twistor space, hence on the image of a holomorphic function

(1)$$\phi \;\colon\; \Sigma \longrightarrow \mathbb{C}P^3$$

from a Riemann surface $\Sigma$ to complex projective 3-space. Here the genus $g$ of $\Sigma$ is bounded by the loop order $l$ of the scattering amplitude, $g \leq l$ and the degree of the map (1) depends on the number $q$ of negative helicity gluon states in the process, as $d \,=\, q - 1 + l$.

The idea of twistor string theory is to understand these complex curves in twistor space as worldsheet instantons (or rather D1-brane-instantons) of a string theory – specifically of the B-model topological string, whose target space is the twistor space $\mathbb{C}P^3$, or rather the supermanifold $\mathbb{C}^{3\vert 4}$ – and to use this to explain the structure of MHV amplitudes in N=4 D=4 super Yang-Mills theory.

Somehow. The original proposals (Witten 04, Berkovits 04) were only partially succesful as string theories (e.g. Skinner 09, Spradlin).

But the idea as such of scattering amplitudes localized on complex curves in twistor space remains fruitful in pure quantum field theory, for instance it seems to generalize to perturbative quantum gravity in the form of D=4 N=8 supergravity (Cachazo & Skinner 2012).

Related entries

• string theory results applied elsewhere

• homotopy of rational maps

References

Precursors:

• Edward Witten, An Interpretation of Classical Yang-Mills Theory, Phys. Lett. B 77 (1978) 394-398 (spire:129655, doi:10.1016/0370-2693(78)90585-3 pdf)

• Velayudhan Parameswaran Nair, A current algebra for some gauge theory amplitudes, Physics Letters B Volume 214, Issue 2, 17 November 1988, Pages 215-218 (doi:10.1016/0370-2693(88)91471-2, spire:24212)

Original articles:

• Edward Witten, Perturbative Gauge Theory As A String Theory In Twistor Space, Commun. Math. Phys. 252:189-258, 2004 (arXiv:hep-th/0312171)

• Nathan Berkovits, An Alternative String Theory in Twistor Space for $N = 4$ Super-Yang-Mills, Phys. Rev. Lett. 93 (2004) 011601 (arXiv:hep-th/0402045)

• Radu Roiban, Anastasia Volovich, All Googly Amplitudes from the B-model in Twistor Space, Phys. Rev. Lett. 93 (2004) 131602 (arXiv:hep-th/0402121)

• Radu Roiban, Marcus Spradlin, Anastasia Volovich, On the Tree-Level S-Matrix of Yang-Mills Theory, Phys. Rev. D70 : 026009, 2004 (arXiv:hep-th/0403190)

Survey:

• David Skinner, The geometry of scattering amplitudes, talk notes, November 2009 (pdf, pdf)

• Marcus Spradlin, Progress and Prospects in Twistor String Theory (pdf)

• Michael Atiyah, Maciej Dunajski, Lionel J. Mason, Section 7 of: Twistor theory at fifty: from contour integrals to twistor strings, Proc. R. Soc. A473: 20170530 (arXiv:1704.07464, doi:10.1098/rspa.2017.0530)

Further developments:

• Yvonne Geyer, Arthur E. Lipstein, Lionel Mason, Ambitwistor strings in 4-dimensions, Phys. Rev. Lett. 113, 081602 (2014) (arXiv:1404.6219)

Discussion for D=4 N=8 supergravity:

• Freddy Cachazo, David Skinner, Gravity from Rational Curves, Phys. Rev. Lett. 110, 161301, 2012 (arXiv:1207.0741, doi:10.1103/PhysRevLett.110.161301)

• Tim Adamo, Gravity with a cosmological constant from rational curves, JHEP 2015 098, 2015 (arXiv:1508.02554)