Remarks on Vafa-Witten theory, or how to sit on a wall
Abstract
We prove three results about refined SU(N) VW invariants on Hirzebruch surfaces. First, we crucially simplify the modular anomaly equation satisfied by the generating series of these invariants by expressing it in terms of complementary generalized error functions, which requires extending these error functions to their singular loci. Second, we evaluate the variation of the modular completions of the generating series upon change of polarization. Finally, a representation of the generating functions in terms of indefinite theta series allows us to compute VW invariants at the walls of marginal stability, and to express them in terms of the invariants on the two sides of the wall and those of lower ranks. All the results exhibit a universal structure suggesting their wider applicability.
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BibTeX
@misc{alexandrov2026remarks,
title = {{Remarks on Vafa-Witten theory, or how to sit on a wall}},
author = {Sergei Alexandrov and Aradhita Chattopadhyaya},
howpublished = {HAL},
year = {2026},
month = aug,
eprint = {2608.25007},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05742470v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Remarks on Vafa-Witten theory, or how to sit on a wall
AU - Sergei Alexandrov
AU - Aradhita Chattopadhyaya
PY - 2026
DA - 2026/08/25
PB - HAL
UR - https://hal.science/hal-05742470v1
AB - We prove three results about refined SU(N) VW invariants on Hirzebruch surfaces. First, we crucially simplify the modular anomaly equation satisfied by the generating series of these invariants by expressing it in terms of complementary generalized error functions, which requires extending these error functions to their singular loci. Second, we evaluate the variation of the modular completions of the generating series upon change of polarization. Finally, a representation of the generating functions in terms of indefinite theta series allows us to compute VW invariants at the walls of marginal stability, and to express them in terms of the invariants on the two sides of the wall and those of lower ranks. All the results exhibit a universal structure suggesting their wider applicability.
ER -