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Comments on F-maximization and R-symmetry in 3D SCFTs

Vasilis Niarchos

TL;DR

This work tests the $F$-maximization principle as a nonperturbative method to determine the exact $U(1)$ R-symmetry in 3D ${\cal N}=2$ SCFTs by analyzing a large-$N$ Chern-Simons-Matter theory with a single adjoint. Using localization, the $S^3$ partition function is reduced to a matrix model and solved numerically in the saddle-point regime to extract $R(\lambda)$ as a function of the 't Hooft coupling $\lambda=N/k$, with $F=-\log|Z_{S^3}|$ maximized over trial R-charges. The results show $R(\lambda)$ monotonically decreases and remains near the nonperturbative bounds, matching perturbative expectations at small $\lambda$, but reveal tensions with a proposed Seiberg-like duality, indicating possible necessary refinements of the $F$-maximization framework or decoupling-field effects. The paper further conjectures a qualitative picture for the exact R-symmetry of the ${\bf \hat{A}}$ theory at strong coupling, including potential oscillatory behavior around a simple bounding curve and implications for deformations $W_{n+1}=\mathrm{Tr} X^{n+1}$ and for the $F$-theorem.

Abstract

We report preliminary results on the recently proposed F-maximization principle in 3D SCFTs. We compute numerically in the large-N limit the free energy on the three-sphere of an N=2 Chern-Simons-Matter theory with a single adjoint chiral superfield which is known to exhibit a pattern of accidental symmetries associated to chiral superfields that hit the unitarity bound and become free. We observe that the F-maximization principle produces a U(1) R-symmetry consistent with previously obtained bounds but inconsistent with a postulated Seiberg-like duality. Potential modifications of the principle associated to the decoupling fields do not appear to be sufficient to account for the observed violations.

Comments on F-maximization and R-symmetry in 3D SCFTs

TL;DR

This work tests the -maximization principle as a nonperturbative method to determine the exact R-symmetry in 3D SCFTs by analyzing a large- Chern-Simons-Matter theory with a single adjoint. Using localization, the partition function is reduced to a matrix model and solved numerically in the saddle-point regime to extract as a function of the 't Hooft coupling , with maximized over trial R-charges. The results show monotonically decreases and remains near the nonperturbative bounds, matching perturbative expectations at small , but reveal tensions with a proposed Seiberg-like duality, indicating possible necessary refinements of the -maximization framework or decoupling-field effects. The paper further conjectures a qualitative picture for the exact R-symmetry of the theory at strong coupling, including potential oscillatory behavior around a simple bounding curve and implications for deformations and for the -theorem.

Abstract

We report preliminary results on the recently proposed F-maximization principle in 3D SCFTs. We compute numerically in the large-N limit the free energy on the three-sphere of an N=2 Chern-Simons-Matter theory with a single adjoint chiral superfield which is known to exhibit a pattern of accidental symmetries associated to chiral superfields that hit the unitarity bound and become free. We observe that the F-maximization principle produces a U(1) R-symmetry consistent with previously obtained bounds but inconsistent with a postulated Seiberg-like duality. Potential modifications of the principle associated to the decoupling fields do not appear to be sufficient to account for the observed violations.

Paper Structure

This paper contains 10 sections, 33 equations, 3 figures, 1 table.

Table of Contents

  1. R-symmetry in three-dimensional ${\cal N}=2$ SCFTs
  2. ${\bf \hat{A}}$ theory
  3. Definition and known facts
  4. A brief note on moduli spaces
  5. The partition function on $S^3$ and $F$-maximization in the large-$N$ limit
  6. The R-charge at $\lambda\sim {\cal O}(1)$
  7. The R-charge at $\lambda\gg 1$
  8. Comments on a potential discrepancy
  9. Insights into the exact R-symmetry of the ${\bf \hat{A}}$ theory
  10. Open problems

Figures (3)

  • Figure 1: A typical distribution of the eigenvalues $t_i$$(i=1,\ldots, N)$ in the complex plane. This particular plot has been obtained for $N=100$, $\lambda=1$ and $R=0.28$.
  • Figure 2: The numerically computed R-charge curve in the regime of $\lambda\sim {\cal O}(1)$. The dashed curve represents the lower bounding function $\frac{1}{2(1+\lambda)}$.
  • Figure 3: Numerical results in the regime of very strong 't Hooft coupling. The black and red curves represent the lower bounding function $\frac{1}{2(1+\lambda)}$ and the upper bounding function $\frac{2}{2\lambda+1}$ respectively.