Learning BPS Spectra and the Gap Conjecture

TL;DR

This work addresses the BPS spectra of 3d $\mathcal{N}=2$ theories associated to plumbed $3$-manifolds via the GPPV invariants $\hat{Z}_b(q;Y)$. The authors encode the $q$-series into vector representations built from the exponent sequence and the gaps between exponents, and apply principal component analysis (PCA) to a fixed H-shaped plumbing graph to extract latent structure. They find that the first principal component captures nearly all the variance for exponent-based representations, while for exponent-gap representations the first two gaps dominate the leading component, indicating that early terms of the $q$-series carry the most diagnostic information about the BPS spectrum. This demonstrates the utility of explainable ML in revealing hidden spectral structure and suggests a potentially universal role for initial exponent gaps in BPS counting across plumbed $3$-manifolds, with implications for understanding the quantum geometry/topology through low-order $q$-series data.

Abstract

We explore statistical properties of BPS q-series for 3d N=2 strongly coupled supersymmetric theories that correspond to a particular family of 3-manifolds Y. We discover that gaps between exponents in the q-series are statistically more significant at the beginning of the q-series compared to gaps that appear in higher powers of q. Our observations are obtained by calculating saliencies of q-series features used as input data for principal component analysis, which is a standard example of an explainable machine learning technique that allows for a direct calculation and a better analysis of feature saliencies.

Figures (4)

• Figure 1: 'H-shaped' plumbing graph with framing $\mathbf{m}=(a_1,a_2,a_3,a_4,a_5,a_6)$.
• Figure 2: Plumbing graphs related by Neumann moves. These plumbing graphs related by these moves result in homeomorphic $3$-manifolds.
• Figure 3: Summary of the exponent relevance analysis using PCA. We note that the first principal component $\mathbf{b}_1$ covers the vast majority of the proportional variance of the $q$-series vectors as shown in (a), (d), (g) and (j). This implies that for all vector representations, the vector spaces dimensionally reduce effectively to a 1-dimensional subspace. A closer look reveals that the components of $q$-exponent vector $\mathbf{e}$ and the corresponding normalized vector $\mathbf{e}^\prime$ do contribute to the first principal component when the exponent values becomes larger, as expected and shown in (b) and (e). For the $q$-exponent gap vector $\mathbf{r}$, we see however that only the first 2 components significantly contribute to the first principal exponent as shown in (h). When normalized, this contribution is evenly distributed to all the components of the normalized $q$-exponent gap vector $\mathbf{r}^\prime$ as shown in (k). Similar observations can be made in (c), (f), (i) and (l) for the second principal components, whose contribution is significantly less.
• Figure 4: The change in proportional variance $p(\lambda_i)$ for the first 2 principal components $\mathbf{b}_1$ and $\mathbf{b}_2$ when one takes only the first $D$ components $r_1, \dots , r_D$ of the $q$-exponent gap vectors $\mathbf{r}$ is shown in (a). Focusing on the first principal component $\mathbf{b}_1$, the relative contributions to $\mathbf{b}_1$ of the vector components $r_1$ and $r_2$ under changing the number of components of $\mathbf{r}$ are shown in (b) and (c), respectively.