Virasoro conjecture
In algebraic geometry, the Virasoro conjecture states that a certain generating function encoding Gromov–Witten invariants of a smooth projective variety is fixed by an action of half of the Virasoro algebra. The Virasoro conjecture is named after theoretical physicist Miguel Ángel Virasoro. {{harvs|txt | last3=Xiong | first3=Chuan-Sheng | last2=Hori | first2=Kentaro | last1=Eguchi | first1=Tohru | title=Quantum cohomology and Virasoro algebra | url=https://dx.doi.org/10.1016/S0370-2693(97)00401-2 | doi=10.1016/S0370-2693(97)00401-2 |mr=1454328 | year=1997 | journal=Physics Letters B | issn=0370-2693 | volume=402 | issue=1 | pages=71–80}}
proposed the Virasoro conjecture as a generalization of Witten's conjecture. {{harvs|txt|last=Getzler|first=Ezra|authorlink=Ezra Getzler|year=1999}} gave a survey of the Virasoro conjecture.
The proof of the genus 0 Virasoro conjecture for all smooth projective varieties (or more generally, compact symplectic manifolds) was first given by Xiaobo Liu and Gang Tian (1998).{{Citation |last1=Liu |first1=Xiaobo |title=Virasoro Constraints For Quantum Cohomology |date=1998-10-20 |id=arXiv:math/9806028 |last2=Tian |first2=Gang|arxiv=math/9806028 |bibcode=1998math......6028L }}
References
{{Reflist}}
- {{Citation | last1=Getzler | first1=Ezra |author1-link=Ezra Getzler| editor1-last=Wiśniewski | editor1-first=Jarosław | editor2-last=Szurek | editor2-first=Michał | editor3-last=Pragacz | editor3-first=Piotr | title=Algebraic geometry: Hirzebruch 70 (Warsaw, 1998) | arxiv=math/9812026 | publisher=American Mathematical Society | location=Providence, R.I. | series=Contemporary Mathematics | isbn=978-0-8218-1149-8 |mr=1718143 | year=1999 | volume=241 | chapter=The Virasoro conjecture for Gromov-Witten invariants | pages=147–176| bibcode=1998math.....12026G | doi=10.1090/conm/241/03634 }}
- {{Citation | last3=Xiong | first3=Chuan-Sheng | last2=Hori | first2=Kentaro | last1=Eguchi | first1=Tohru | title=Quantum cohomology and Virasoro algebra | doi=10.1016/S0370-2693(97)00401-2 |mr=1454328 | year=1997 | journal=Physics Letters B | issn=0370-2693 | volume=402 | issue=1 | pages=71–80| arxiv=hep-th/9703086 | bibcode=1997PhLB..402...71E }}
- {{Citation |last1=Liu |first1=Xiaobo |title=Virasoro Constraints For Quantum Cohomology |date=1998-10-20 |arxiv=math/9806028 |bibcode=1998math......6028L |id=arXiv:math/9806028 |last2=Tian |first2=Gang}}
Category:Unsolved problems in geometry
{{algebraic-geometry-stub}}