Higher Spin Curves and Cohomological Field Theories

更高的自旋曲线和上同调场论

• 批准号: 0105788
• 负责人: Tyler Jarvis
• 金额: $ 7.09万
• 依托单位: Brigham Young University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-07-01 至 2005-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0105788&HistoricalAwards=false
• 关键词: Higher; Spin; Curves; Cohomological; Field

The Investigator studies intersection theory on and thecohomological field theory arising from the moduli of algebraicspin curves and stable spin maps. Especially important in thisstudy is the exploration of the plethora of similarities betweenGromov-Witten theory and cohomological field arising from higherspin curves and spin maps. Also important in this research is thestudy of the many relations between these cohomological fieldtheories and integrable hierarchies, including significant recentprogress toward proving some remarkable conjectures like theW-algebra conjecture and Generalized Witten Conjecture.The research in this project is concerned with connections betweenalgebraic geometry and physics, and it has implications for bothsubjects. Algebraic geometry, which is the study of solutions topolynomial equations, and especially the sub-discipline ofalgebraic curves, which is one of the main foci of this research,have many applications. The most notable of these applicationsare in secure electronic communications (cryptography and errorcorrecting codes). Recent research has shown that algebraicgeometry also has significant ties to high-energy physics andplays an important role in helping us understand the fundamentalnature of the universe. Conversely, physics has helped us betterunderstand some important aspects of algebraic geometry and itsapplications. This research is focused on studying and furtherdeveloping some of these important links.

研究者主要研究代数自旋曲线和稳定自旋映射的模的交理论和上同调场论。在这项研究中，特别重要的是探索Gromov-Witten理论和由更高自旋曲线和自旋映射产生的上同调场之间的大量相似之处。在这个研究中同样重要的是研究这些上同调场论和可积族之间的许多关系，包括最近在证明一些显著的猜想如W-代数猜想和广义维滕猜想方面的重大进展。 代数几何是研究多项式方程的解的学科，尤其是代数曲线的分支学科，是代数几何研究的主要焦点之一，有着广泛的应用。 这些应用中最引人注目的是安全电子通信（密码学和纠错码）。 最近的研究表明，代数几何也有显着的联系，高能物理和发挥重要作用，帮助我们了解宇宙的基本性质。 相反，物理学帮助我们更好地理解代数几何及其应用的一些重要方面。 本研究的重点是研究和进一步发展这些重要环节。