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<p>For any smooth projective variety with semisimple quantum cohomology, in 2014, Dubrovin, Liu, Yang and Zhang proposed an integrable hierarchy for its associated Hodge integrals, but the loop equation is yet to be established. In this paper, we explicitly derive Dubrovin-Zhang type loop equation for general Hodge integrals of any smooth projective variety (not necessarily with semisimple quantum cohomology) using Virasoro constraints. As applications, we study the loop equations and their solutions for Hodge integrals of a point and <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathbb{C}\mathbb{P}^1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">C</mi> <msup> <mrow> <mi mathvariant="double-struck">P</mi> </mrow> <mn>1</mn> </msup> </mrow> </math></EquationSource> </InlineEquation>.</p>

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Dubrovin-Zhang Type Loop Equations for Hodge Integrals with Target Varieties

  • 期刊论文

  • 发表日期：

    2026年2月5日

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  • Xin Wang

摘要

For any smooth projective variety with semisimple quantum cohomology, in 2014, Dubrovin, Liu, Yang and Zhang proposed an integrable hierarchy for its associated Hodge integrals, but the loop equation is yet to be established. In this paper, we explicitly derive Dubrovin-Zhang type loop equation for general Hodge integrals of any smooth projective variety (not necessarily with semisimple quantum cohomology) using Virasoro constraints. As applications, we study the loop equations and their solutions for Hodge integrals of a point and \(\mathbb{C}\mathbb{P}^1\) C P 1 .