Explicit two-instanton and instanton–anti-instanton derivation of the $O(\mathrm{e}^{-2S_0})$ effective action of the supersymmetric (2+1)-dimensional Georgi–Glashow model

In the supersymmetric extension of the (2+1)-dimensional Georgi–Glashow model (the pure three-dimensional $\mathcal{N}=2$ $SU(2)$ gauge theory and the parity-invariant $\mathcal{N}=1$ supersymmetric Higgs-model variant), monopole-instantons carry two fermion zero modes and generate a supersymmetric sine-Gordon effective theory whose bosonic potential — the term that gives the photon multiplet its mass and, in the $\mathcal{N}=2$ model, produces the runaway $V_{\rm eff}=m^2\mathrm{e}^{-8\pi\phi/e}$ — is of order $\mathrm{e}^{-2S_0}$. Affleck, Harvey and Witten (1982) obtained this potential only by supersymmetric completion of the one-instanton amplitude and explicitly acknowledged that they could not derive it from the two-instanton, two-anti-instanton or instanton–anti-instanton sectors, nor evaluate the determinant and zero-mode integrals fixing its coefficient. Later literature settled the qualitative questions: the instanton superpotential is exact, the Coulomb branch is lifted, and no finite-field supersymmetric vacuum exists. The explicit semiclassical multi-instanton computation itself, however, has not been performed for the three-dimensional supersymmetric model: the modern bion analyses handle $\mathbb{R}^3\times S^1$-compactified four-dimensional theories, and the multi-instanton resurgence analyses handle the non-supersymmetric Polyakov model. The problem is to derive explicitly the $O(\mathrm{e}^{-2S_0})$ terms of this model's effective action — including the topologically trivial instanton–anti-instanton sector, the one-instanton coefficients $A(\Phi)$ and $S_0(\Phi)$, and the non-holomorphic Kähler-potential corrections that the exact superpotential does not fix.

Background

The Georgi–Glashow model in $2+1$ dimensions is an $SU(2)$ gauge theory with a Higgs scalar in the adjoint representation; a vacuum expectation value $\phi$ of this scalar breaks $SU(2)$ to its $U(1)$ subgroup, so that perturbatively the theory contains a massless photon together with heavy charged $W$-bosons and a massive Higgs field. Polyakov showed in 1977 that this perturbative picture is incomplete: because the unbroken $U(1)$ is compact (its magnetic flux is quantized), the Euclidean version of the theory in three dimensions admits 't Hooft–Polyakov monopole solutions acting as instantons — localized Euclidean tunneling events rather than particles. A dilute gas of these monopole-instantons disorders the Higgs phase. Introducing the dual photon $\gamma$ — a compact scalar whose winding number is the magnetic flux of the photon — the gas sums to a sine-Gordon effective Lagrangian $\frac{1}{2}(\partial\gamma)^2+\zeta\cos(4\pi\gamma/e+\alpha)$, where $\zeta\sim\mathrm{e}^{-S_0}$ is the instanton fugacity (the weight of a single instanton in the gas), $S_0=4\pi\phi/e$ is the one-instanton action in the Bogomol'nyi–Prasad–Sommerfeld (BPS) limit in which the monopole saturates a lower bound on its energy, $e$ is the three-dimensional gauge coupling (with $e^2$ of mass dimension one), and $\alpha$ is the coefficient of a topological (total-derivative) term in the action. This potential gives the photon an exponentially small mass and produces linear confinement of electric charges. Affleck, Harvey and Witten (1982) asked what instantons do when the model is made supersymmetric, analyzing two supersymmetric variants. The first is the parity-invariant supersymmetric Higgs-model variant: the field content is enlarged by a gauge-singlet scalar $\Phi_{\rm S}$ with a tree-level superpotential $W=g\Phi_{\rm S}(\Phi^{2}-\Phi_{0}^{2})$, so that parity is a symmetry and the only perturbatively massless multiplet is the photon multiplet — the photon together with its fermionic superpartner, the photino $\lambda$. The second is the $\mathcal{N}=2$ model, obtained by adding the adjoint Majorana fermions required by $\mathcal{N}=2$ supersymmetry in three dimensions (four real supercharges; the theory is the dimensional reduction of four-dimensional $\mathcal{N}=1$ super-Yang–Mills, and with vanishing bare superpotential it is pure three-dimensional $\mathcal{N}=2$ $SU(2)$ gauge theory), in which the scalar expectation value $\phi$ is undetermined in perturbation theory and labels the Coulomb branch. In both variants the monopole-instanton background carries two fermion zero modes by the Jackiw–Rebbi/Callias index theorem — zero-energy fermionic modes localized on the monopole — so the single-instanton amplitude is a fermion bilinear, and completing the effective action to a supersymmetric one fixes the effective theory. In the parity-invariant variant the dual photon $\gamma$ and photino $\lambda$ combine into a chiral superfield $\Gamma=\gamma+\bar\theta\lambda+\frac{1}{2}\bar\theta\theta F$ (whose components are a scalar, a fermion and an auxiliary field), and the superpotential — a holomorphic function of the chiral superfields whose derivatives fix the scalar potential and Yukawa couplings — is $W=2A\,\mathrm{e}^{-S_0}\cos(4\pi\Gamma/e+\alpha)$, with $A$ the one-instanton one-loop determinant factor: the supersymmetric sine-Gordon theory. In the $\mathcal{N}=2$ model, whose low-energy fields are the dual photon, the photino and the scalar $\phi$ itself, completion yields the effective action $\frac{1}{2}(\partial\gamma)^{2}+\frac{1}{2}(\partial\phi)^{2}+\bar\lambda i\not\partial\lambda+m\,\mathrm{e}^{-4\pi\phi/e}\big(\lambda^{\mathrm{T}}\gamma_{0}\lambda\,\mathrm{e}^{i4\pi\gamma/e}+\bar\lambda\gamma_{0}\bar\lambda^{\mathrm{T}}\mathrm{e}^{-i4\pi\gamma/e}\big)$, in which the complex scalar $Z=\phi+i\gamma$ (the Coulomb-branch coordinate combining the scalar and the dual photon) makes the $\mathcal{N}=2$ structure manifest; here $\gamma_{0}$ is a Dirac matrix and $m$ is the mass scale fixed by the one-instanton amplitude (the determinant factor $A$ together with the zero-mode profile integrals). Supersymmetry is unbroken, but the bosonic potential is of order $\mathrm{e}^{-2S_0}$ — two-instanton order, parametrically smaller than in Polyakov's non-supersymmetric model: in the parity-invariant variant it is $\propto[\sin(4\pi\gamma/e+\alpha)]^2$ and gives the photon multiplet its mass, and in the $\mathcal{N}=2$ model it is the runaway $V_{\rm eff}(\phi)=m^2\mathrm{e}^{-8\pi\phi/e}$ on the Coulomb branch. The crucial point for the present problem is how this $O(\mathrm{e}^{-2S_0})$ potential was obtained. Affleck, Harvey and Witten derived it by supersymmetric completion of the one-instanton amplitude — the potential is required to close the superfield $F$-term (the superfield component that carries the superpotential) — and explicitly acknowledged that the two-instanton, two-anti-instanton and instanton–anti-instanton contributions that would generate it directly were never computed, writing that they were 'not able to show explicitly how this term arises'. The instanton–anti-instanton sector is topologically trivial and contains no classical solution of the field equations, so there is no BPS saddle to expand around; the coefficients $A(\Phi)$ (a function of the undetermined scalar expectation values $\Phi$) and the zero-mode profile integrals fixing the absolute normalization were likewise asserted rather than evaluated. Later literature settled the qualitative questions this left open. In 1997 Aharony, Hanany, Intriligator, Seiberg and Strassler, and independently de Boer, Hori and Oz, determined the instanton-generated superpotentials of general three-dimensional $\mathcal{N}=2$ gauge theories. For the pure $SU(2)$ theory — precisely Affleck–Harvey–Witten's $\mathcal{N}=2$ model — the instanton superpotential $W\sim 1/Y\sim\mathrm{e}^{-\Phi}$ is exact: here $Y$ is a monopole-operator coordinate (monopole operators are gauge-invariant disorder operators creating monopole-instanton events) and $\Phi$ is the complex field on the Coulomb branch (the moduli space parametrized by the scalar expectation value together with the dual photon) combining the scalar and the dual photon; exactness follows from holomorphy (analyticity of the superpotential, which strongly constrains its quantum corrections) and the global $U(1)_R$ symmetry. The Coulomb branch is completely lifted (a potential is generated on it, removing the classical degeneracy), there is no supersymmetric vacuum, and the semiclassical runaway is exact. Modern literature calls this instanton term, generalized to theories with adjoint matter as $W=\eta V_+V_-$ in monopole operators $V_\pm$, the AHW superpotential. Holomorphy, however, fixes only the superpotential: the non-holomorphic data of the effective theory — the Kähler potential, which determines the kinetic terms, and hence the absolute normalization of the fugacity — are not determined by it. In parallel, an explicit semiclassical technology for precisely such multi-instanton sectors has been developed in adjacent settings. In four-dimensional gauge theories with adjoint fermions compactified on $\mathbb{R}^3\times S^1$, monopole-instantons again carry two fermion zero modes; Ünsal showed that correlated monopole–anti-monopole pairs ('magnetic bions'), with total action $2S_0$, generate the $O(\mathrm{e}^{-2S_0})$ dual-photon potential and confinement, and Anber and Poppitz computed the microscopic structure of these bions explicitly. The resurgence program — the framework relating the large-order growth of perturbation theory to non-perturbative saddles — of Argyres–Ünsal and Poppitz–Ünsal, Pazarbaşı–Ünsal's cluster-expansion analysis of the multi-instanton sectors of the non-supersymmetric Polyakov model, and Nekrasov's study of instanton–anti-instanton saddles treat the topologically trivial multi-instanton sectors systematically. For the $\mathcal{N}=4$ cousin of the Georgi–Glashow model, Dorey, Khoze, Mattis, Tong and Vandoren computed the one-instanton measure on the Coulomb branch explicitly. The corresponding computation for the three-dimensional supersymmetric Georgi–Glashow model itself has not been performed.

Problem Statement

Derive explicitly, by direct semiclassical evaluation of the multi-instanton path integral, the terms of order $\mathrm{e}^{-2S_0}$ in the low-energy effective action of the supersymmetric (2+1)-dimensional Georgi–Glashow model (the $\mathcal{N}=2$ theory and the parity-invariant $\mathcal{N}=1$ supersymmetric Higgs-model variant) — the terms that Affleck, Harvey and Witten inserted by supersymmetric completion of the one-instanton amplitude and stated they could not derive. The computation determines (1) the two-instanton, two-anti-instanton and instanton–anti-instanton contributions to the effective action of the dual photon $\gamma$ and its superpartners, demonstrating explicitly how the bosonic potential $\propto[\sin(4\pi\gamma/e+\alpha)]^2$ — the term that gives the photon multiplet its mass in the parity-invariant variant and whose $\mathcal{N}=2$-model counterpart is the runaway $V_{\rm eff}=m^2\mathrm{e}^{-8\pi\phi/e}$ — arises from these sectors, with a controlled treatment of the Coulomb and scalar-exchange interaction between the constituents and of the relative-separation quasi-zero-mode integral (the nearly-flat direction describing the separation of the pair) in the topologically trivial instanton–anti-instanton valley; (2) the explicit one-instanton input coefficients: the one-loop determinant factor $A(\Phi)$ as a function of the scalar expectation values, the fermion zero-mode profile integrals, and the action $S_0(\Phi)$ beyond the leading Prasad–Sommerfeld estimate $S_0=4\pi\phi/e$, so that the absolute normalization of the instanton fugacity and of the photon-multiplet mass scale is fixed rather than parametrically estimated; (3) the non-holomorphic corrections of order $\mathrm{e}^{-2S_0}$ (the Kähler-potential/kinetic terms of the dual-photon effective theory), which are not determined by the exact holomorphic superpotential. The holomorphic part of (1) is already constrained: the instanton superpotential $W\sim\mathrm{e}^{-\Phi}$ (with $\Phi$ the complex Coulomb-branch field) is exact, and any derivation must reproduce the potential it implies; the genuinely open quantitative content is the explicit multi-instanton computation itself and the non-holomorphic data of (3). Questions already resolved by the exact-superpotential results — the existence of a finite-field supersymmetric vacuum (there is none; the runaway is exact) and the general model-building question of dynamical supersymmetry breaking in $2+1$ dimensions, which modern work addresses through monopole-operator dynamics at large $N$ rather than through the dilute instanton gas — are settled background for this problem.

References

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