Instanton Zero Modes and Selection Rules for Exact Terms

An instanton can contribute to a local superpotential only when its collective-coordinate measure reduces to precisely the two fermionic modes represented by $d^2\theta$. Additional unlifted zero modes force operator insertions, higher F-terms, or a vanishing contribution. The index gives the candidate modes; interactions, background expectation values, normalizability, and infrared control decide what survives.

Required background. R-symmetry, anomalies, and the holomorphic scale fixes the instanton factor and anomalous charges. Zero modes, collective coordinates, and measures explains how saddle-point zero modes become integration variables.

Helpful background. ’t Hooft anomaly matching gives an independent check on the global charges of candidate low-energy terms.

The two-zero-mode criterion

Work on Euclidean $\mathbb R^4$ with a four-dimensional $\mathcal N=1$ gauge theory and instanton number $k=1$. For a left-handed Weyl fermion in representation $R$, the index theorem gives

$$n_R=2T(R)$$

zero modes of the instanton chirality, with $\operatorname{tr}_R(T^aT^b)=T(R)\delta^{ab}$. The adjoint gaugino therefore has $2T(G)$ modes. For $SU(N_c)$,

$$n_\lambda=2N_c, \qquad n_{\psi_Q}=1$$

for each fundamental Weyl fermion because $T(\mathbf{N_c})=1/2$.

Two gaugino modes are universal supersymmetric modes: applying the broken supercharges to the bosonic instanton generates them. Their Grassmann integral is the superspace measure $d^2\theta$. A local superpotential contribution has the form

$$\Delta S=\int d^4x\,d^2\theta\;W_{\mathrm{inst}}+\text{h.c.},$$

so every other fermionic collective coordinate must be lifted by an interaction or saturated by an operator insertion. If more than two modes remain with neither mechanism, the superpotential amplitude vanishes.

This is a necessary criterion, not a sufficient proof. One must also verify that the bosonic size and orientation integrals are controlled, the instanton saddle is in the domain of the effective theory, the proposed term has the correct dimensions and global charges, and no cancellation removes it.

How zero modes are lifted or absorbed

Interactions insert fields into the collective-coordinate integral. Three common mechanisms are:

Gauge Yukawa lifting. The component interaction

$$\sqrt2g\,\phi_i^\dagger T^a\psi_i\lambda^a+\text{h.c.}$$

uses a scalar expectation value to pair one matter zero mode with one gaugino zero mode. On a Higgs branch this can lift modes and simultaneously regulate the large-instanton region.

Superpotential insertions. A mass or Yukawa term can pair matter zero modes. The resulting amplitude carries the corresponding holomorphic masses or background fields, whose charges must be included.

External chiral operators. If zero modes are saturated by external fermion or field-strength insertions, the instanton computes a correlation function or a higher chiral operator rather than a field-independent superpotential coefficient.

Lifting is an explicit term in the semiclassical action, not a subtraction from the index. The index counts zero modes of the quadratic operator at the saddle; interactions determine which Grassmann integrals can be saturated.

The collective-coordinate construction and its supersymmetric applications are worked through in Shifman 2022, §§ 10.19–10.20.

SQCD and the exceptional case Nf = Nc − 1

In $SU(N_c)$ SQCD with $N_f$ pairs $Q^i,\widetilde Q_i$, a unit instanton starts with

$$2N_c\ \text{gaugino modes} \quad\text{and}\quad 2N_f\ \text{matter modes}.$$

At a generic meson expectation value, each matter mode can be paired with a gaugino mode through the gauge Yukawa coupling. After all $2N_f$ matter modes are lifted, the number of remaining gaugino modes is

$$2N_c-2N_f=2(N_c-N_f).$$

Exactly two remain only when

$$N_f=N_c-1.$$

In this case a generic quark expectation value completely Higgses $SU(N_c)$. For sufficiently large expectation value, the gauge coupling at the instanton size is weak and the instanton-size integral is cut off. The one-instanton calculation is therefore semiclassically controlled and produces

$$W_{k=1}\propto \frac{\Lambda_h^{2N_c+1}}{\det M}, \qquad M^i{}_j=Q^i\widetilde Q_j.$$

The power $2N_c+1$ is $b_0=3N_c-(N_c-1)$. Dimensions and anomalous charges check the result, while the normalized collective-coordinate integral fixes the convention-dependent coefficient. This controlled case is the dynamical anchor of the Affleck–Dine–Seiberg result Affleck, Dine, and Seiberg 1984, pp. 493–534.

For $N_f<N_c-1$, lifting all matter modes leaves more than two gaugino modes, and a generic quark expectation value leaves an unbroken nonabelian subgroup. The exact superpotential still exists, but it is not a direct single four-dimensional instanton contribution in the original theory. Gaugino condensation in the unbroken subgroup and holomorphic decoupling provide the appropriate dynamics Intriligator and Seiberg 1996, §§ 3.1–3.2.

For $N_f\geq N_c$, the simple one-instanton measure cannot reduce to precisely the universal pair for a superpotential on the generic branch. Other chiral observables or quantum moduli-space constraints may occur, but their zero-mode saturation must be specified separately.

A decision procedure

For any proposed instanton-generated term:

1. State the saddle and boundary conditions. Give $G$, matter representations, topological charge, spacetime, masses, and background expectation values.
2. Count quadratic zero modes. Use $2|k|T(R)$ for each Weyl fermion, including the adjoint gaugino.
3. Identify the universal measure. A four-dimensional $\mathcal N=1$ superpotential needs the two supersymmetric modes and no other unabsorbed fermion modes.
4. List every lifting vertex or insertion. Check gauge indices, flavor indices, holomorphic parameters, and charges.
5. Classify the output. Two remaining modes suggest a superpotential; additional external insertions specify a correlator or higher F-term; unsaturated integrations give zero.
6. Check bosonic control. Inspect the instanton-size integral, unbroken gauge group, weak-coupling regime, and possible boundary contributions.
7. Cross-check the answer. Match dimension, R-charge, anomalous spurion charge, decoupling, and singularities.

The fourth step is where many incorrect arguments hide: saying that modes “are lifted” without displaying the interaction and required background is not a calculation.

Condensates are not automatically superpotentials

Pure $SU(N_c)$ super-Yang–Mills has $2N_c$ gaugino zero modes in a unit instanton. The instanton can saturate a correlator schematically of the form

$$\left\langle (\lambda\lambda)(x_1)\cdots(\lambda\lambda)(x_{N_c}) \right\rangle,$$

but it does not directly have the two-mode measure of a local superpotential. Extracting a one-point gaugino condensate requires additional dynamical reasoning—such as factorization in a gapped supersymmetric vacuum—and careful treatment of infrared contributions. A formal small-instanton calculation in the un-Higgsed theory is not made reliable merely by holomorphy.

Likewise, a term with four unlifted fermion modes may encode a multi-fermion F-term rather than zero. “Not a superpotential” and “no instanton information” are different conclusions.

Common pitfalls

Index count equals the final amplitude. The index gives the quadratic zero modes. Interactions and operator insertions determine how their Grassmann integrals are saturated.

Two modes are sufficient. The bosonic collective-coordinate integral can be infrared divergent or outside weak coupling. Charges, dimensions, and decoupling remain independent checks.

Every ADS superpotential is one-instanton generated. Direct four-dimensional one-instanton control holds at $N_f=N_c-1$. Results for smaller $N_f$ are extended by other controlled dynamics and holomorphy.

Exercises

For $SU(3)$ SQCD, classify the direct one-instanton contribution for $N_f=2$ and $N_f=1$ after gauge-Yukawa lifting at a generic quark expectation value.

Solution

There are six gaugino modes. For $N_f=2$, four matter modes pair with four gaugino modes, leaving two; the generic expectation value completely Higgses $SU(3)$, so a superpotential calculation can be controlled. For $N_f=1$, two matter modes lift two gaugino modes, leaving four. A direct superpotential contribution therefore fails the two-mode criterion; the exact $N_f=1$ result is reached by decoupling or by dynamics of the unbroken subgroup.

A mass insertion $m\widetilde QQ$ saturates two matter zero modes. Explain why the resulting instanton amplitude must contain $m$ rather than $m^*$.

Solution

The relevant insertion comes from the chiral superpotential and saturates same-chirality fermion modes. A chiral F-term depends holomorphically on the chiral mass source $m$; $m^*$ belongs to the antichiral sector and would violate both chirality and the spurionic charges.

References

• Ian Affleck, Michael Dine, and Nathan Seiberg, “Dynamical Supersymmetry Breaking in Supersymmetric QCD,” Nuclear Physics B 241 (1984), 493–534, DOI.
• Kenneth A. Intriligator and Nathan Seiberg, “Lectures on Supersymmetric Gauge Theories and Electric–Magnetic Duality,” Nuclear Physics B Proceedings Supplements 45BC (1996), 1–28, §§ 3.1–3.2, arXiv, DOI.
• Mikhail Shifman, Advanced Topics in Quantum Field Theory: A Lecture Course, 2nd ed., Cambridge University Press (2022), §§ 10.19–10.20, DOI.