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PreprintCC BY 4.0HAL

Quantum index, Arnold-Rokhlin surfaces, and real enumerative geometry

  • Ilia Itenberg
  • Eugenii Shustin
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Abstract

The main goal of this note is to relate two different Welschinger-type rules of signs that were used for definition of real enumerative invariants of toric surfaces (the invariants considered being relative to the toric boundary). The relation between them intertwines Mikhalkin's quantum index and geometry of real and complex point sets of counted real curves. As a by-product, we suggest a new definition of quantum index, which can be used for refined invariant enumeration of real algebraic curves on surfaces in a non-toric setup.

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Cite

BibTeXRIS
BibTeX
@misc{itenberg2026quantum,
  title = {{Quantum index, Arnold-Rokhlin surfaces, and real enumerative geometry}},
  author = {Ilia Itenberg and Eugenii Shustin},
  howpublished = {HAL},
  year = {2026},
  month = jul,
  doi = {10.48550/arxiv.2607.21251},
  eprint = {2607.21251},
  archivePrefix = {arXiv},
  url = {https://hal.science/hal-05793102v1},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Quantum index, Arnold-Rokhlin surfaces, and real enumerative geometry
AU  - Ilia Itenberg
AU  - Eugenii Shustin
PY  - 2026
DA  - 2026/07/23
PB  - HAL
DO  - 10.48550/arxiv.2607.21251
UR  - https://hal.science/hal-05793102v1
AB  - The main goal of this note is to relate two different Welschinger-type rules of signs that were used for definition of real enumerative invariants of toric surfaces (the invariants considered being relative to the toric boundary). The relation between them intertwines Mikhalkin's quantum index and geometry of real and complex point sets of counted real curves. As a by-product, we suggest a new definition of quantum index, which can be used for refined invariant enumeration of real algebraic curves on surfaces in a non-toric setup.
ER  -