Anomalies

Abstract

Tree-level superamplitudes and the integrands of loop corrections are invariant under the Yangian of the superconformal symmetry ${\mathcal {Y}}({\mathfrak {psu}}(2,2|4)$, which is represented in momentum twistor space by the generators [1, 2].

Appendix: Component Expansions

The superconnection ${\mathbb {A}}$ on chiral superspace is determined by the superspace constraints and supergauge condition $\theta ^{\alpha a}{\mathcal {A}}_{\alpha a }=0$. Up to order fourth order in the fermions, the constraints are solved by the following expansion

$$\begin{aligned} {\mathcal {A}}&= A + \mathrm{i}|\theta ^a\rangle [\bar{\psi }_a| + \frac{\mathrm{i}}{2}|\theta ^a\rangle \langle \theta ^b|D\phi _{ab} -\frac{1}{3!}\varepsilon _{abcd}|\theta ^a\rangle \langle \theta ^b|\,D\langle \theta ^c\psi ^d\rangle \nonumber \\&\quad +\frac{\mathrm{i}}{4!}\varepsilon _{abcd}|\theta ^a\rangle \langle \theta ^b|\,D\langle \theta ^c|G|\theta ^d\rangle + \cdots \\ |{\mathcal {A}}_a\rangle&= \frac{\mathrm{i}}{2}\phi _{ab}|\theta ^b\rangle -\frac{1}{3}\varepsilon _{abcd}|\theta ^b\rangle \langle \theta ^c\psi ^d\rangle +\frac{\mathrm{i}}{8}\varepsilon _{abcd}|\theta ^b\rangle \langle \theta ^c|G|\theta ^d\rangle +\cdots \ .\nonumber \end{aligned}$$

(6.90)

The corresponding supercurvatures have component expansions up to second order in the fermions as follows

$$\begin{aligned} {\mathcal {F}}_{\dot{\alpha }\dot{\beta }}^+(x,\theta )&= F_{\dot{\alpha }\dot{\beta }}^{+} +\mathrm{i}\langle \theta ^a|D_{(\dot{\alpha }}\bar{\psi }_{\dot{\beta })a} +\frac{\mathrm{i}}{2}\langle \theta ^a|D_{(\dot{\alpha }}\langle \theta ^b|D_{\dot{\beta })}\phi _{ab} -\frac{\mathrm{i}}{2}\langle \theta ^a\theta ^b\rangle \left\{ \bar{\psi }_{a(\dot{\alpha }},\bar{\psi }_{\dot{\beta })b}\right\} +\cdots \nonumber \\ {\mathcal {F}}_{\alpha \beta }^-(x,\theta )&= F_{\alpha \beta }^{-} + \mathrm{i}\theta ^a_{(\alpha }D_{\beta )\dot{\beta }}\bar{\psi }_a^{\dot{\beta }} +\mathrm{i}\theta ^{\ a}_{(\alpha }\theta ^{\ b}_{\beta )} \left( \Box \phi _{ab} - \left\{ \bar{\psi }^{\dot{\alpha }}_{\ a},\bar{\psi }_{\dot{\alpha }b}\right\} \right) +\frac{1}{4}\theta ^{\gamma b}\theta ^a_{(\alpha }F^-_{\beta )\gamma }\phi _{ab}+\cdots \nonumber \\ {\mathcal {F}}_{\dot{\alpha }a}(x,\theta )&= \bar{\psi }_{\dot{\alpha }a}+\theta ^{\alpha b}D_{\alpha \dot{\alpha }}\phi _{ab} + \frac{\mathrm{i}\varepsilon _{abcd}}{3!}\theta ^{\alpha b}D_{\alpha \dot{\alpha }}\langle \theta ^c\psi ^d\rangle +\cdots \nonumber \\ {\mathcal {W}}_{ab}(x,\theta )&= \phi _{ab}+\mathrm{i}\varepsilon _{abcd}\langle \theta ^c\psi ^d\rangle +\frac{1}{2}\varepsilon _{abcd}\langle \theta ^c|G|\theta ^d\rangle + \frac{1}{4}\left[ \phi _{ac},\phi _{bd}\right] \langle \theta ^c\theta ^d\rangle + \cdots \, . \end{aligned}$$

(6.91)

See Refs. [32, 40] for further information.