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Characterizing 4-string contact interaction using machine learning

Harold Erbin, Atakan Hilmi Fırat

TL;DR

The paper tackles the challenge of constructing the geometry of closed string field theory contact interactions, which requires solving Strebel differential data on moduli spaces. It introduces neural networks to learn the accessory parameters of Strebel quadratics on the 4-punctured sphere and an indicator for the vertex region, enabling automatic derivation of local coordinates and mapping radii and facilitating off-shell amplitude calculations. The approach reproduces known results, notably the 4-tachyon contact term $v_4$, with high consistency to the literature and demonstrates symmetries and analytic structure of the accessory parameter $a(\xi,\xi^*)$. This ML-based framework is argued to be scalable to $n$-string contact interactions, offering a geometry-driven, flexible path to CSFT computations beyond four punctures.

Abstract

The geometry of 4-string contact interaction of closed string field theory is characterized using machine learning. We obtain Strebel quadratic differentials on 4-punctured spheres as a neural network by performing unsupervised learning with a custom-built loss function. This allows us to solve for local coordinates and compute their associated mapping radii numerically. We also train a neural network distinguishing vertex from Feynman region. As a check, 4-tachyon contact term in the tachyon potential is computed and a good agreement with the results in the literature is observed. We argue that our algorithm is manifestly independent of number of punctures and scaling it to characterize the geometry of $n$-string contact interaction is feasible.

Characterizing 4-string contact interaction using machine learning

TL;DR

The paper tackles the challenge of constructing the geometry of closed string field theory contact interactions, which requires solving Strebel differential data on moduli spaces. It introduces neural networks to learn the accessory parameters of Strebel quadratics on the 4-punctured sphere and an indicator for the vertex region, enabling automatic derivation of local coordinates and mapping radii and facilitating off-shell amplitude calculations. The approach reproduces known results, notably the 4-tachyon contact term , with high consistency to the literature and demonstrates symmetries and analytic structure of the accessory parameter . This ML-based framework is argued to be scalable to -string contact interactions, offering a geometry-driven, flexible path to CSFT computations beyond four punctures.

Abstract

The geometry of 4-string contact interaction of closed string field theory is characterized using machine learning. We obtain Strebel quadratic differentials on 4-punctured spheres as a neural network by performing unsupervised learning with a custom-built loss function. This allows us to solve for local coordinates and compute their associated mapping radii numerically. We also train a neural network distinguishing vertex from Feynman region. As a check, 4-tachyon contact term in the tachyon potential is computed and a good agreement with the results in the literature is observed. We argue that our algorithm is manifestly independent of number of punctures and scaling it to characterize the geometry of -string contact interaction is feasible.
Paper Structure (16 sections, 48 equations, 12 figures, 6 tables)

This paper contains 16 sections, 48 equations, 12 figures, 6 tables.

Table of Contents

  1. Introduction
  2. The geometry of string contact interaction
  3. Strebel quadratic differential
  4. Complex length and loss function
  5. Strebel differential on 4-punctured sphere
  6. Local coordinates, mapping radii, and vertex region
  7. Neural networks for accessory parameter and indicator function
  8. Accessory parameter neural network
  9. Indicator function neural network
  10. Off-shell 4-tachyon contact term
  11. Discussion and future directions
  12. Some details on numerical evaluations
  13. Continuous square root
  14. The integral for the complex length
  15. The integral for the mapping radii
  16. ...and 1 more sections

Figures (12)

  • Figure 1: The trajectory structure of the Strebel differential when punctures are at $P = \{0, 1, 0.8734-0.6242i, \infty\}$ (left). We marked the positions of punctures and zeros by crosses and plusses respectively. The inaccuracies around the zeros are due to evaluating the trajectories as an expansion after \ref{['eq:LocCord']}. The critical graph is a tetrahedron whose sketch on $\mathbb{CP}^1$ is shown on the right.
  • Figure 2: Part of the path of integration in \ref{['eq:InvLocCo']} can be deformed to the dotted path. The integration over the dotted path produces a real number, resulting in an irrelevant phase for the local coordinates \ref{['eq:InvLocCo']}. In extension, this part also doesn't contribute to the mapping radii \ref{['eq:map_rad']} below.
  • Figure 3: An example of an artificial neural network with 3 hidden layers containing $n_i$ nodes each. It inputs the position of unfixed punctures (moduli) and outputs the (independent) accessory parameters.
  • Figure 4: The summary of mathematical operations performed by artificial neural networks.
  • Figure 5: An example of training set $\mathcal{S}$ for 4-puncture spheres with $|\mathcal{S}| = 10^5$. Notice we have excluded small circles centered at $0,1,\infty$ where 4-punctured sphere is close to degeneration and only sampled points from the remaining triply-connected region (training region).
  • ...and 7 more figures