Superconformal Ward Identities

Superconformal Ward identities correlate spacetime dependence, R-symmetry dependence, and the contributions of different conformal components in a supermultiplet. For selected BPS four-point functions they reduce many tensor functions to a protected part plus a smaller set of unrestricted functions. The reduction is specific to the spacetime dimension, superconformal algebra, external multiplet, and polarization convention; there is no universal equation obtained by replacing “conformal” with “superconformal.” Already in four-dimensional $\mathcal N=1$ theories, covariant two- and three-point structures and their Ward identities depend on the chosen superfields Osborn 1999, abstract.

Required background. Protected Data as Bootstrap Input supplies the multiplet and normalization record. Localized Transformations and Ward Identities supplies contact-term reasoning.

Kinematics with R-symmetry polarizations

For identical scalar superconformal primaries $\mathcal O(x,Y)$, encode a symmetric-traceless R-symmetry representation with a null polarization $Y$. A four-point function can be written schematically as

$$\langle\mathcal O_1\mathcal O_2\mathcal O_3\mathcal O_4\rangle =\mathcal K(x_i,Y_i)\, \mathcal G(z,\bar z;\alpha,\bar\alpha),$$

where

$$u=z\bar z, \qquad v=(1-z)(1-\bar z)$$

are spacetime cross ratios, while $\alpha,\bar\alpha$ or an equivalent pair $\sigma,\tau$ encode R-symmetry cross ratios. The prefactor $\mathcal K$ fixes weights and a tensor-structure basis.

Crossing permutes both pairs of variables and acts by a matrix on the R-symmetry structures. Choosing $u,v$ correctly while leaving the $Y$-tensor basis implicit does not define a crossing system. For four-dimensional $\mathcal N=2,4$ chiral primaries, the null-polarization construction reduces the R-symmetry dependence to two invariants and organizes the channels with harmonic polynomials Nirschl and Osborn 2005, abstract.

Solving the Ward constraints

Supersymmetry gives differential constraints

$$\mathscr W_A\mathcal G=\mathcal C_A,$$

where $\mathscr W_A$ differentiates spacetime and R-symmetry variables and $\mathcal C_A$ contains contact or protected terms fixed by the chosen multiplet. The operators $\mathscr W_A$ must be imported with the supercharge and polarization conventions that derived them.

For many half-BPS correlators, the solution can be organized as

$$\mathcal G =\mathcal G_{\rm protected} +\mathcal P(z,\bar z;\alpha,\bar\alpha) \,\mathcal H(z,\bar z),$$

or as a finite sum of such terms. The rational or single-variable protected piece is fixed by shortening data, while $\mathcal H$ contains long-multiplet dynamics. The polynomial $\mathcal P$ is not optional: it enforces the zeroes and tensor relations required by the Ward identities.

This form is a pattern, not a dimension-independent formula. Four-dimensional $\mathcal N=2$ and $\mathcal N=4$ half-BPS correlators provide classic realizations Dolan and Osborn 2002, §§4–6; different algebras and external representations yield different numbers of functions and protected pieces.

A reproducible reduction

Given an imported Ward system:

1. declare the external supermultiplet and which conformal component is inserted;
2. choose and normalize a complete R-symmetry tensor basis;
3. strip a stated kinematic prefactor;
4. write every differential Ward equation, including contact terms;
5. solve the equations before imposing crossing;
6. separate pieces fixed by shortening or cohomology from unrestricted functions;
7. substitute back into the full correlator and verify every Ward residual;
8. transform under generators of the permutation group and derive crossing for the reduced functions.

The protected-input flow appears below. The Ward step consumes an accepted multiplet convention; it does not repair missing normalization or recombination information.

Ward reduction acts only after the external multiplet, R-symmetry tensors, and protected source conventions are fixed. Its output is a reduced set of dynamical functions plus a separately identified protected contribution. The diagram is schematic.

An equivalent table is:

Input to the Ward step	Produced object	Independent check
External component and prefactor	dimensionless reduced correlator	weights at every point
R-symmetry representation	polynomial tensor basis	projector completeness
Supercharge convention	differential operators $\mathscr W_A$	algebra closure on the correlator
Protected multiplets	$\mathcal G_{\rm protected}$	OPE limits and imported coefficients
Ward solution	unrestricted functions $\mathcal H_r$	substitution residual vanishes
Permutation action	reduced crossing matrix	group relations and Bose/graded signs

Contact and shortening qualifications

Ward identities are distributional. At coincident points, variations can produce delta-function terms and mixing with descendants. A solution derived for separated points cannot be integrated through collisions without a regulator and local counterterms.

Short multiplets at an isolated point can recombine into a long multiplet when parameters change. The protected subtraction must therefore be compatible with the recombination identity used by the upstream multiplet classification. Otherwise a threshold contribution can appear once in $\mathcal G_{\rm protected}$ and again in the long sector.

Common pitfalls

Using a Ward solution from a different external representation. The R-symmetry polynomial degree and protected terms depend on the external multiplet. Similar notation does not make the systems interchangeable.

Subtracting the protected piece twice. If a superblock convention already includes a short contribution, removing it again from $\mathcal G$ corrupts crossing. State where the subtraction occurs.

Ignoring contacts when integrating the identity. Separated-point Ward equations do not fix coincident-source terms. Localization or integrated-correlator uses require their own subtraction map.

Exercises

Why must the crossing transformation act on both $(z,\bar z)$ and the R-symmetry variables?

Solution

A permutation moves both spacetime points and the R-symmetry polarizations attached to them. The prefactor and tensor structures therefore transform together. Acting only on spacetime variables generally mixes or misnormalizes the R-symmetry channels.

References

• Dolan, F. A., and Osborn, H. “Superconformal Symmetry, Correlation Functions and the Operator Product Expansion.” Nuclear Physics B 629 (2002): 3–73. arXiv. DOI.
• Nirschl, M., and Osborn, H. “Superconformal Ward Identities and Their Solution.” Nuclear Physics B 711 (2005): 409–479. arXiv. DOI.
• Osborn, H. “N=1 Superconformal Symmetry in Four-Dimensional Quantum Field Theory.” Annals of Physics 272 (1999): 243–294. arXiv. DOI.