[Paper Review] Singlet Vector Models on Lens Spaces

Djordje Radicevic|arXiv (Cornell University)|Sep 30, 2012

Black Holes and Theoretical Physics4 citations

TL;DR

This paper computes the exact partition function and free energy of singlet vector models—large-N Chern-Simons-matter theories—at infinite level on lens spaces $L(p,1)$. Using $ extrm{Z}_p$ orbifolding of $S^3$, it identifies a $p^N$-like proliferation of light topological states at small volumes, revealing nontrivial topological configurations absent in standard Vasiliev higher-spin theories, with explicit free energy expressions derived via zeta function regularization and heat kernel methods.

ABSTRACT

We present exact computations of partition functions of singlet vector models (infinite level Chern-Simons-matter theories) on lens spaces L(p, 1). We identify light topological configurations and their spectra, and we comment on the relevance of our results in studying both the UV completions of Vasiliev's higher-spin theories and the dS/CFT correspondence in the large N limit.

Motivation & Objective

• To compute the exact partition function and free energy of infinite-level Chern-Simons-matter theories on lens spaces $L(p,1)$.
• To identify and characterize light, topologically induced configurations (states) in singlet vector models on nontrivial three-manifolds.
• To examine the implications of these topological states for the UV completion of Vasiliev's higher-spin gravity and the dS/CFT correspondence.
• To provide a systematic analytic framework for computing matter free energies on orbifolded manifolds using zeta function regularization and heat kernel techniques.

Proposed method

• The theory is formulated as a $U(N)$ Chern-Simons-matter theory at infinite level, decoupling gauge fields from matter and reducing to a sum over flat connections.
• The moduli space of vacua is parameterized by homomorphisms from $\pi_1(L(p,1)) = \mathbb{Z}_p$ to $U(N)$, classified by $p$-component integer vectors $\mathbf{N} = \{N_I\}$ with $\sum N_I = N$.
• For each vacuum sector labeled by $I$, the matter free energy is computed via zeta function regularization of the scalar Laplacian on $L(p,1)$, using spectral sums $\zeta_1(s;p,\alpha)$ and $\zeta_2(s;p,\alpha)$.
• The heat kernel method is employed to express the free energy as a Fourier expansion in terms of Wilson loop holonomies $e^{2\pi i I/p}$, with coefficients derived from spherical harmonic expansions.
• Analytic continuation is applied to assign finite values to divergent zeta sums at $s=0$, enabling explicit computation of $F_{\mathrm{M}}(p,I)$.
• Numerical results are tabulated for $p=1$ to $8$, and consistency checks are performed by summing over all $I$ to recover the known free energy on $S^3$.

Experimental results

Research questions

• RQ1 What is the structure of the partition function and free energy of a singlet vector model on lens spaces $L(p,1)$ at infinite Chern-Simons level?
• RQ2 How do topological configurations—specifically, light states induced by nontrivial holonomies—emerge in these theories, and what is their degeneracy?
• RQ3 To what extent do these topological states affect the duality between higher-spin gravity and the singlet vector model?
• RQ4 Can the free energy on lens spaces be consistently computed using zeta function regularization and heat kernel methods, and does it reproduce known results on $S^3$?
• RQ5 How does the presence of flat connections on $L(p,1)$ modify the spectrum and thermodynamics of the theory compared to $S^3$?

Key findings

• The matter free energy on $L(p,1)$ is computed exactly for all $p$ and $I$, with explicit expressions derived via zeta function regularization of spectral sums.
• For $p=2$, the free energies are $F_{\mathrm{M}}(2,0) = -\frac{C}{\pi} + \frac{1}{16}\left(\ln 4 - \frac{3\zeta(3)}{\pi^2}\right)$ and $F_{\mathrm{M}}(2,1) = \frac{C}{\pi} + \frac{1}{16}\left(\ln 4 - \frac{3\zeta(3)}{\pi^2}\right)$, where $C \approx 0.916$ is Catalan’s constant.
• The sum of $F_{\mathrm{M}}(p,I)$ over all $I = 0, \dots, p-1$ for any $p$ yields $\frac{1}{8}\left(\ln 4 - \frac{3\zeta(3)}{\pi^2}\right) \approx 0.128$, matching the known free energy of a complex scalar on $S^3$.
• A $p^N$-like abundance of light topological states emerges at small volume (large $p$), indicating a nontrivial topological sector not present in the original Vasiliev theory.
• The free energy exhibits $I$-dependence that is obscured by zeta regularization but becomes transparent in the heat kernel approach, where it appears as a Fourier series in $e^{2\pi i I/p}$.
• The results provide a consistent framework for studying the UV completion of Vasiliev theory and challenge a naïve extrapolation of the dS/CFT correspondence to nontrivial topologies.