Modular Shadows

Worldsheet modular invariance underpins the UV finiteness of string theory and motivates the study of modular invariant regulators and their ``modular shadows’’ in the low-energy field theory (Smith, 2025). Complementary to the tropical approach, this project analyses modular integrals using spectral methods and Rankin–Selberg techniques, focusing initially on one-loop partition functions for string theory compactified on a three-torus. This provides a sufficient setting for the characterisation of modular shadows as missing modular data whose spectral content captures the UV regularising effects of the string in the field theory limit.

<div class="row justify-content-sm-center">
  <div class="col-sm-8 mt-3 mt-md-0">
    {% include figure.liquid path="assets/img/6.jpg" title="example image" class="img-fluid rounded z-depth-1" %}
  </div>
  <div class="col-sm-4 mt-3 mt-md-0">
    {% include figure.liquid path="assets/img/11.jpg" title="example image" class="img-fluid rounded z-depth-1" %}
  </div>
</div>