## Advances in Theoretical and Mathematical Physics

- Adv. Theor. Math. Phys.
- Volume 9, Number 6 (2005), 931-998.

### On supertwistors, the Penrose--Ward transform and $\CN=4$ super-Yang--Mills theory

Alexander D. Popov and Christian Sämann

#### Abstract

It was recently shown by Witten that B-type open topological string theory with the supertwistor space $\mathbb{C}P^{3|4}$ as a target space is equivalent to holomorphic Chern--Simons (hCS) theory on the same space. This hCS theory in turn is equivalent to self-dual $\mathcal{N}=4$ super-Yang–Mills (SYM) theory in four dimensions. We review the supertwistor description of self-dual and anti-self-dual $\mathcal{N}$-extended SYM theory as the integrability of SYM fields on complex $(2|\mathcal{N})$-dimensional superplanes and demonstrate the equivalence of this description to Witten's formulation. The equivalence of the field equations of hCS theory on an open subset of $\mathbb{C}P^{3|\mathcal{N}}$ to the field equations of self-dual $\mathcal{N}$-extended SYM theory in four dimensions is made explicit. Furthermore, we extend the picture to the full $\mathcal{N}=4$ SYM theory and, by using the known supertwistor description of this case, we show that the corresponding constraint equations are (gauge) equivalent to the field equations of hCS theory on a quadric in $\mathbb{C}P^{3|3}\times \mathbb{C}P^{3|3}$.

#### Article information

**Source**

Adv. Theor. Math. Phys., Volume 9, Number 6 (2005), 931-998.

**Dates**

First available in Project Euclid: 3 April 2006

**Permanent link to this document**

https://projecteuclid.org/euclid.atmp/1144070299

**Mathematical Reviews number (MathSciNet)**

MR2207571

**Zentralblatt MATH identifier**

1129.81082

#### Citation

Popov, Alexander D.; Sämann, Christian. On supertwistors, the Penrose--Ward transform and $\CN=4$ super-Yang--Mills theory. Adv. Theor. Math. Phys. 9 (2005), no. 6, 931--998. https://projecteuclid.org/euclid.atmp/1144070299