On ratios of theta functions

Motivated by problems in conformal field theory and Narain moduli spaces, this paper classifies the minimizers and maximizers of ratios and differences of theta and Epstein zeta functions, demonstrating that the hexagonal lattice plays a pivotal role in these optimization problems with applications to crystallization and interacting particle theory.

The Gist

Imagine you are a master architect trying to build the most efficient, stable, and "perfect" lattice structure possible. In the world of mathematics and physics, this structure is called a lattice, and it's essentially a grid of points stretching out in space.

This paper by Luo and Wei is like a guidebook for finding the "Goldilocks" shape for these lattices. It asks a simple but profound question: If you change the shape of your grid, how does a specific mathematical "score" (called a partition function) change? And which shape gives you the best score?

Here is the breakdown of their discovery using everyday analogies:

1. The Players: Theta Functions and Zeta Functions

Think of Theta functions and Epstein Zeta functions as complex "energy meters" or "scoreboards" for these lattices.

• The Lattice: Imagine a honeycomb, a square grid, or a skewed parallelogram grid.
• The Score: These functions calculate a value based on how the points in the grid are arranged. In physics, this score relates to the energy of a system or the probability of certain states occurring (like how particles arrange themselves in a crystal).

2. The Big Discovery: The Hexagon is King

For decades, mathematicians knew that for certain specific scores, the hexagonal lattice (the shape of a honeycomb) was the winner. It was the "champion" that minimized energy or maximized stability.

However, the authors of this paper looked at ratios. Imagine you have two different energy meters running at the same time. You want to know: What happens if we compare Meter A to Meter B? Does the hexagonal lattice still win?

The Paper's Main Claim:
The authors completely mapped out every possible scenario where you compare these different mathematical scores. They found that:

• The Hexagonal Lattice is the Ultimate Champion: In almost every case where a "best" or "worst" shape exists, the answer is the hexagonal lattice (represented mathematically by the point $e^{i\pi/3}$).
• When it Wins: Depending on the specific parameters (like the "temperature" or "radius" of the system), the hexagonal lattice either minimizes the ratio (making the system most stable) or maximizes it.
• When it Loses (or Doesn't Exist): In some specific mathematical scenarios, there is no single "best" shape. The score might just keep getting better or worse without ever settling on a winner. The authors identified exactly when this happens.

3. The "Shape-Shifting" Analogy

To understand how they proved this, imagine you have a piece of clay shaped like a grid.

• You can stretch it, squash it, or rotate it.
• The authors showed that no matter how you stretch or squash this clay, if you are looking for the absolute best shape, you will always end up with the honeycomb shape.
• They used a clever mathematical "deformation" technique. Think of it like sliding a puzzle piece along a track. They proved that if you slide the shape away from the honeycomb, the score gets worse (or better, depending on what you are looking for). This proved that the honeycomb is the only place where the score stops changing—the "peak" or the "valley."

4. Why This Matters (According to the Paper)

The paper connects these abstract math shapes to real-world physics, specifically Conformal Field Theory and String Theory.

• The Partition Function: In physics, this is like the "total bill" for a system. It tells you everything about the system's energy, heat, and pressure.
• The Application: The authors show that the formulas used to calculate these "bills" in physics often look like the ratios they studied.
• The Result: Because they proved the hexagonal lattice is the minimizer/maximizer for these ratios, they confirmed that hexagonal structures are the most efficient for these specific physical systems. This explains why nature often chooses hexagonal patterns (like in crystals or vortex formations) to achieve the lowest energy state.

Summary

In simple terms, this paper is a comprehensive map of a mathematical landscape. It confirms that while the terrain is complex and has many hills and valleys, the hexagonal lattice is the undisputed king of the most important peaks and valleys. Whether you are looking at the energy of a crystal, the behavior of particles, or the geometry of a torus (a donut shape), if you want the optimal configuration, you are almost always looking at a hexagon.

The authors didn't just guess this; they provided a rigorous, step-by-step proof that covers every possible combination of parameters, ensuring that no other shape can beat the hexagon in these specific mathematical contests.

Technical Summary

Technical Summary: Ratios of Theta Functions

Problem Statement
Motivated by the average partition function of $c$ free bosons (Afhkami-Jeddi et al. [3]) and the average of the genus 1 partition function over the Narain moduli space (Maloney-Witten [42]), this paper investigates the optimization of ratios involving theta functions, Epstein zeta functions, and the Dedekind eta function. Specifically, the authors address Problem B, which seeks to classify the minimizers and maximizers for the following ratios over the upper half-plane $\mathbb{H}$ (up to the action of the modular group):

1. $\min/\max \frac{\theta(\beta, z)}{\theta(\alpha, z)}$
2. $\min/\max \frac{\zeta(s, z)}{\theta(\alpha, z)}$

Additionally, Problem C considers ratios of partition functions $Z(R, \tau)$ for free bosonic theories compactified on circles of different radii, and ratios involving the Siegel-Weil formula expressions. The central physical question (Problem A) is how the geometry of the torus affects these partition functions, which correspond to thermodynamic quantities like Helmholtz free energy.

Methodology
The paper employs a combination of modular invariance, deformation techniques, and precise monotonicity analysis.

• Reduction to Fundamental Domain: The authors utilize the invariance of the functions under the modular group (specifically the subgroup generated by $\tau \mapsto -1/\tau$, $\tau \mapsto \tau+1$, and $\tau \mapsto -\tau$) to restrict the search for extrema to the fundamental domain $D_G$.
• Transversal Monotonicity: A core technique involves proving the monotonicity of the ratio functions with respect to the real part ($x$) and imaginary part ($y$) of $z$.
  • For theta function ratios, the authors establish that $\frac{\partial}{\partial x} \frac{\theta(\beta, z)}{\theta(\alpha, z)} \geq 0$ in specific regions and analyze the behavior on the boundary lines (the vertical line $\text{Re}(z)=1/2$ and the arc $|z|=1$).
  • This relies on decomposing the derivative of a ratio into a sum of derivatives of simpler ratios, leveraging previous results on the monotonicity of $\sqrt{s}\theta(s, z)$ (from Luo-Wei [36]).
• Minimum Principles: To handle the Epstein zeta function ratios, the authors adapt a "minimum principle" (Proposition 3.1 and 3.2). This principle states that if a modular invariant function satisfies specific monotonicity conditions on $x$ and $y$ within a rectangular subdomain, and satisfies a comparison inequality at the boundary, the minimum is attained at the hexagonal point $e^{i\pi/3}$.
• Summation Formulas and Bounds: For the Epstein zeta function $\zeta(s, z)$, the authors utilize Rankin's summation formulas (Lemmas 3.4–3.7) to derive approximate and error bounds. They also employ the Chowla-Selberg formula (Lemma 4.14) involving modified Bessel functions $K_s(y)$ to obtain precise estimates for small $s$.
• Deformation Arguments: The authors use algebraic deformations (e.g., relating $\theta(\beta, z)/\theta(\alpha, z)$ to $\theta(\beta, z)/\theta(1/\alpha, z)$) to reduce various cases of parameters $(\alpha, \beta)$ to a core case where $\beta > \alpha \geq 1$.

Key Contributions and Results
The paper provides a complete classification of the extrema for the ratios defined in Problem B and C.

1. Theorem 1.1 (Ratio of Theta Functions):

   • For $\alpha, \beta > 0$, the existence of a maximum or minimum depends on the region of the $(\alpha, \beta)$ plane.
   • If $(\alpha, \beta)$ falls in regions $I \cup III$ (where $\beta > \alpha, \beta\alpha > 1$ or $\beta < \alpha, \beta\alpha < 1$), the maximum is attained at the hexagonal lattice point $z = e^{i\pi/3}$.
   • If $(\alpha, \beta)$ falls in regions $II \cup IV$, the minimum is attained at $z = e^{i\pi/3}$.
   • In the complementary regions, the extrema do not exist (the function is unbounded).
   • This result is generalized to sums of theta functions (Theorem 1.2).

2. Theorem 1.3 (Ratio of Epstein Zeta and Theta Functions):

   • For $s > 1$ and $\alpha \geq 3s$, the ratio $\frac{\zeta(s, z)}{\theta^k(\alpha, z)}$ achieves its minimum at $z = e^{i\pi/3}$ if $k \in (0, 2s]$.
   • If $k > 2s$, the minimum does not exist.

3. Corollary 1.2 (Differences of Functions):

   • As a consequence of Theorem 1.3, the paper establishes that for $s \in (1, 12]$ and $\alpha \geq 3s$, the difference $\zeta(s, z) - \theta^k(\alpha, z)$ is minimized at $z = e^{i\pi/3}$ for $k \in (0, 2s]$.

4. Physical Implications (Theorem D):

   • The results confirm that the hexagonal lattice minimizes the partition functions for free bosons (Eq. 1.6) and the average partition functions over Narain moduli space (Eqs. 1.8–1.10).

Significance
The authors state that these results have direct applications in conformal field theory and string theory, specifically regarding the partition functions of free bosons and the Siegel-Weil formula. Mathematically, the findings yield the minima of differences between theta and Epstein zeta functions, which have implications for the mathematics of crystallization and interacting particle theory (referencing B´etermin [7, 10] and related works). The paper reinforces the pivotal role of the hexagonal lattice in the optimization of modular invariant functions, extending classical results by Rankin, Cassels, Montgomery, and Osgood-Phillips-Sarnak to ratio forms. The authors emphasize that the theory of theta functions remains an active and unfinished field, as noted by Mumford.