Supersymmetry Ward Identities

• Supersymmetry Ward identities are mathematical relations deriving from superalgebra invariance that constrain the structure of superamplitudes and correlation functions in maximally supersymmetric theories.
• The method systematically eliminates redundant Grassmann variables and employs an algebraic basis expansion, simplifying the analysis of NMHV and higher MHV amplitudes using R-symmetry classifications.
• These identities not only underpin efficient superamplitude constructions but also serve as critical checks for the consistency of candidate ultraviolet counterterms in theories like N=4 SYM and N=8 supergravity.

Supersymmetry Ward identities are a set of relations, derived from the invariance of quantum field theories under supersymmetry and R-symmetry, that constrain the structure of correlation functions, $S$-matrix elements, and counterterms. In maximally supersymmetric theories such as $\mathcal{N}=4$ super Yang–Mills (SYM) and $\mathcal{N}=8$ supergravity, these constraints are so powerful that they allow the systematic reduction of all $(\text{Next-to})^K$-MHV (maximally helicity-violating) superamplitudes to a finite, algebraically complete set of ordinary component amplitudes, uniquely distinguished by their R-symmetry content. The Ward identities also play a key role in the analysis of potential ultraviolet counterterms and provide the formal underpinning of basis-expansion techniques in superamplitude constructions.

1. Structure of Supersymmetry Ward Identities in Superamplitudes

In maximally supersymmetric gauge and gravity theories, all $n$-point tree and loop-level superamplitudes are subject to Ward identities originating from both $Q$- and $\bar Q$-supersymmetry (the Poincaré part of the superalgebra) and $SU(\mathcal{N})_R$ R-symmetry. For $\mathcal{N}=4$ SYM, the external state space is efficiently organized using on-shell superfields where each leg is associated with Grassmann variables $\eta_{iA}$ ($A=1,\dots,4$). The $n$-point “superamplitude” in the NMHV sector has the canonical form: $$\mathcal{A}_n^{\text{NMHV}} = \delta^{(8)}\left(\sum_{i=1}^n |i\rangle \eta_{i}\right) P_4(\eta)$$ where $P_4(\eta)$ is a Grassmann polynomial of degree four. The full supersymmetry algebra imposes linear relations among the coefficients of this polynomial, corresponding to the requirement that the amplitude is annihilated by all supercharges.

The generic expansion of $P_4$ is: $$P_4 = \sum_{i, j, k, l=1}^n q_{ijkl} \, \eta_{i1} \eta_{j2} \eta_{k3} \eta_{l4}$$ with $q_{ijkl}$ completely symmetric in its indices due to $SU(4)_R$ invariance. The Ward identities stipulate, for each supersymmetry generator $Q^A$,

$$Q^A \, \mathcal{A}_n^{\text{NMHV}} = 0 \implies \sum_{i=1}^{n-2} [i] c_{ijkl} = 0 \quad \text{for every } (j,k,l)$$

where $[i]$ denotes a spinor contraction and $c_{ijkl}$ are redefined coefficients after eliminating two Grassmann variables using the delta functions.

2. Algorithmic Solution of NMHV Ward Identities

To solve these identities for an arbitrary number of particles:

1. Elimination of Redundant Grassmann Variables: The eight-dimensional Grassmann delta function can be “solved” for two of the $\eta$ variables by the expansion

$$\delta^{(8)}\left(\sum_{i} |i\rangle \eta_{i}\right) = \frac{1}{\langle n{-}1,n \rangle^4} \delta^{(4)}\left(\sum_{i} \langle n{-}1,i \rangle \eta_{i}\right) \delta^{(4)}\left(\sum_{i} \langle n,i \rangle \eta_{i}\right).$$

The variables $\eta_{n-1,a}$ and $\eta_{n,a}$ can thus be written as linear combinations of the remaining $(n-2)$ $\eta_{i,a}$.

1. Systematic Solution of Ward Identities: The requirement $Q^A P_4=0$ gives a linear system for the coefficients $c_{ijkl}$. Introducing two arbitrarily chosen “reference” lines $s$ and $t$ among $1,\ldots,n{-}2$, the solution exhibits that all coefficients with at least one index equal to $s$ or $t$ can be solved in terms of those with all indices distinct from $s$ and $t$, yielding a basis with $\binom{n-1}{4}$ independent elements.
2. Construction of Supersymmetry-Invariant Polynomials: The algebraic basis is naturally paired with manifestly SUSY- and R-invariant Grassmann polynomials $X_{ijkl}$ constructed by acting with the full set of eight $Q$-supercharges:

$$X_{ijkl} = \delta^{(8)}(Q_a) \left[ m_{i,s,t;1} m_{j,s,t;2} m_{k,s,t;3} m_{l,s,t;4} \right] \Biggl/ \left( [st]^4 \langle n{-}1,n \rangle^4 \right)$$

where $m_{i,s,t;a}$ are degree-one invariants under $Q$.

1. Projection to Component Amplitudes: The $c_{ijkl}$ coefficients—now free of redundancy—are shown to be the on-shell “basis” amplitudes, i.e., physical amplitudes with specified external state labels and $SU(4)$ indices, extracted by appropriate Grassmann differentiation.

The resulting final form for the NMHV superamplitude is: $$\mathcal{A}_n^{\text{NMHV}} = \sum_{1 \le i \le j \le k \le l \le n - 4} \mathcal{A}_n(\{i, j, k, l\} ++--)\, X_{(ijkl)}$$ where $X_{(ijkl)}$ is the symmetrization over all permutations.

3. Algebraic Basis and R-symmetry Classification

The set of independent basis amplitudes corresponds to the set of semi-standard tableaux of an $\mathcal{N} \times K$ rectangular Young diagram, with $\mathcal{N}=4$ and $K=1$ in the NMHV case. For general $(\text{Next-to-})^K$MHV amplitudes, one systematically classifies basis amplitudes by semi-standard tableaux associated to the R-symmetry group, leading to a complete basis expansion for all tree- and loop-level superamplitudes in maximally supersymmetric theories.

This algebraic organization drastically reduces the complexity of the space of amplitudes: for $n$ external particles in the NMHV sector, only $\binom{n-1}{4}$ basis functions are needed. All other components are linearly related through (super)symmetry constraints.

4. Implications for Ultraviolet Counterterms and Supergravity

The solution to the supersymmetry Ward identities provides direct constraints on the structure of possible UV counterterms in $\mathcal{N}=8$ supergravity. Any candidate counterterm (for instance, those of the $D^8 R^4$ type in maximal supergravity) must have all its $n$-point matrix elements compatible with the complete set of Ward identities. The basis-expansion formalism enables explicit checks of whether potential counterterms are actually supersymmetric at the on-shell matrix element level, despite being manifestly clique-invariant under $SU(8)_R$. This underpins the “matrix element approach” to counterterms, where potential ultraviolet divergences are analyzed not just at the action level but directly in terms of the allowed matrix elements.

5. Technical Details: Delta Function Manipulation and Symmetrization

The method relies crucially on the use of Schouten identities and spinor algebra to reduce the Grassmann delta function and eliminate redundant degrees of freedom. The systematic construction and symmetrization of polynomial invariants in the basis expansion is performed by careful combinatorial bookkeeping, ensuring full R-symmetry and supersymmetry invariance. The Appendix of (Elvang et al., 2010) provides explicit elimination formulae and symmetrization algorithms valid for arbitrary $n$.

Explicitly, the elimination of two $\eta$'s via the delta function introduces normalization factors like $\langle n{-}1\,n \rangle^{-4}$, and the symmetrized invariants $X_{(ijkl)}$ result from acting with all eight supercharges. This guarantees that the resulting basis amplitudes respect both the on-shell SUSY and R-symmetry constraints.

6. Extension to Loop Level and Supergravity

The approach is manifestly valid at both tree and loop level, as the Ward identities themselves are unaffected by loop corrections (modulo possible anomalies, which are absent in maximally supersymmetric settings at tree and low loop orders). The identical structure is found for both $\mathcal{N}=4$ SYM and $\mathcal{N}=8$ supergravity, up to the appropriate replacement of R-symmetry groups and the corresponding dimension of the Grassmann delta function. In practice, the procedure for “solving” the Ward identities and constructing the corresponding algebraic basis carries over without modification.

7. Summary and Applications

• The SUSY Ward identities fully determine the structure of NMHV and higher-$K$ MHV superamplitudes in maximally supersymmetric gauge and gravity theories.
• All superamplitudes can be written as a sum over an algebraic basis of component amplitudes, classified by semi-standard tableaux of Young diagrams and multiplied by invariant Grassmann polynomials.
• The procedure is applicable to all $n$-point amplitudes at both tree and loop level, and it provides a systematic tool for studying both the structure of on-shell amplitudes and the supersymmetry properties of UV counterterms.
• These results form the foundation for modern amplitude-based and on-shell methods in maximally supersymmetric field theories, as well as for the analysis of allowed counterterm structures in supergravity (Elvang et al., 2010).