Higher-Spin Dictionaries for Chern–Simons Matter and Parity Phases

Large-$N$ Chern–Simons theories coupled to one fundamental vector furnish parity-violating, weakly broken higher-spin CFTs. Their separated-point planar correlators are controlled by an effective rank and one continuous ’t Hooft coupling, which maps to a phase in the bulk higher-spin interactions. Integer level shifts, global gauge group, line operators, and background contact terms carry additional discrete information that the continuous planar map does not determine.

Required background. Higher-Spin and Vector-Model Dualities supplies the parity-even endpoints; Parity Anomalies and Background Contact Terms fixes quantized three-dimensional data.

Helpful background. Supersymmetric Parents of Mirror, Particle–Vortex, and Bosonization Webs provides the duality context; String and M-Theory Duality Webs and Parameter Maps illustrates why global data cannot be reconstructed from a local coupling alone.

Planar parameters and parity structures

Consider $U(N)_k$ Chern–Simons theory with one fundamental boson or fermion, in a specified regularization and global form. Its planar limit keeps

$$N,k\to\infty, \qquad \lambda=\frac{N}{k_{\mathrm{ren}}}$$

fixed. The relation between the bare and renormalized integer level depends on the regulator; for a Yang–Mills regulator a non-Abelian factor receives a sign-dependent rank shift. One must state which $k$ appears in $\lambda$.

Slightly broken higher-spin symmetry constrains a normalized three-point function to

$$\langle JJJ\rangle =\frac{\widetilde N}{1+\widetilde\lambda^2}\langle JJJ\rangle_{\mathrm{bos}} +\frac{\widetilde N\widetilde\lambda^2}{1+\widetilde\lambda^2}\langle JJJ\rangle_{\mathrm{fer}} +\frac{\widetilde N\widetilde\lambda}{1+\widetilde\lambda^2}\langle JJJ\rangle_{\mathrm{odd}},$$

up to operator-normalization conventions Maldacena and Zhiboedov 2013, §§ 4–5. In common quasi-boson conventions, $\widetilde\lambda=\tan(\pi\lambda/2)$ and the bulk parity phase is $\theta_0=\arctan\widetilde\lambda$ modulo discrete identifications. The precise formula is dictionary- and convention-dependent, so it must be fixed by one parity-odd correlator rather than assumed from notation.

First application: normalize one parity-odd coefficient

Choose the stress tensor and two scalar or current insertions, normalize all two-point functions, and project the three-point function onto the unique allowed parity-odd tensor structure. Planar Chern–Simons matter perturbation theory gives a coefficient proportional to

$$\frac{\widetilde N\widetilde\lambda}{1+\widetilde\lambda^2} =\frac{\widetilde N}{2}\sin(2\theta_0).$$

Matching it to the tree-level bulk answer fixes $g_{\mathrm{HS}}^2\sim1/\widetilde N$ and $\theta_0$ in the same normalization. The parity-even coefficients then provide genuine predictions. Exact planar correlators and their bosonization map give strong quantitative support for this structure Aharony, Gur-Ari, and Yacoby 2012, §§ 3–5.

This match concerns separated points. A background gauge field can carry a local Chern–Simons term

$$S_{\mathrm{ct}}=\frac{i\kappa}{4\pi}\int A\wedge dA,$$

whose integer-quantized shift changes contact terms but not separated-point tensors. The fractional part of an allowed contact coefficient can be physical once the global symmetry and spin structure are fixed Closset et al. 2012, §§ 2–4.

Global data beyond the planar phase

$U(N)$, $SU(N)$, and quotient gauge groups can share local planar correlators while differing in monopole sectors, line operators, one-form symmetries, and level-rank maps. Bosonic and fermionic Chern–Simons vector theories may be dual after ranks, levels, deformations, and contact terms are transformed. A bulk dictionary that records only $\theta_0$ has not specified those sectors. Nor has it supplied a Kaluza–Klein spectrum, string coupling, or nonperturbative higher-spin path integral.

Adversarial control: change regulator and global form

Translate the same calculation between a dimensional and Yang–Mills regulator, including the integer level shift, then compare quantized background contact terms. Next replace $U(N)$ by a theory with the same Lie algebra but different quotient and test a monopole or line operator. If the continuous $λ$ and separated-point three-point function agree while the discrete observable differs, the planar parity dictionary is valid but globally incomplete.

The controlled regime is the vector large-$N$ limit with fixed $\lambda$ and separated-point correlators at stated order in $1/N$. There is no large higher-spin gap, no Einstein truncation, and no inferred $\alpha'$ or Kaluza–Klein hierarchy. The evidence ceiling is a striking planar correlator and duality web; it is not a globally complete, nonperturbative bulk equivalence. Discrete anomaly and line data hand back to the three-dimensional gauge-theory treatment.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Aharony, O., Gur-Ari, G., and Yacoby, R. (2012). “d=3 Bosonic Vector Models Coupled to Chern–Simons Gauge Theories.” Journal of High Energy Physics 2012(3), 037. DOI.
• Closset, C., Dumitrescu, T. T., Festuccia, G., Komargodski, Z., and Seiberg, N. (2012). “Comments on Chern–Simons Contact Terms in Three Dimensions.” Journal of High Energy Physics 2012(9), 091. DOI.
• Maldacena, J., and Zhiboedov, A. (2013). “Constraining Conformal Field Theories with a Slightly Broken Higher Spin Symmetry.” Classical and Quantum Gravity 30, 104003. DOI.