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Yangian-Invariant Fishnet Integrals in Two Dimensions as Volumes of Calabi-Yau Varieties
Claude Duhr1, Albrecht Klemm1,2, Florian Loebbert1
1Bethe Center for Theoretical Physics, Universität Bonn, Bonn D-53115, Germany.
We found that Yangian-invariant fishnet integrals in 2D relate to Calabi-Yau manifolds. Their values are linked to quantum volumes, with instanton corrections appearing for loop orders above two.
Area of Science:
- Mathematical Physics
- String Theory
- Quantum Field Theory
Background:
- Yangian-invariant fishnet integrals are significant in theoretical physics.
- Calabi-Yau manifolds are crucial in string theory and algebraic geometry.
Purpose of the Study:
- To establish a connection between ℓ-loop Yangian-invariant fishnet integrals and Calabi-Yau manifolds.
- To compute these integrals using properties of Calabi-Yau manifolds and mirror symmetry.
Main Methods:
- Utilizing the periods of Calabi-Yau manifolds to compute integral values.
- Employing Yangian generators to derive the Picard-Fuchs differential ideal.
- Applying mirror symmetry to identify integral values with quantum volumes.
Main Results:
- A direct relationship between ℓ-loop fishnet integrals and Calabi-Yau ℓ-folds was established.
- Integral values correspond to the quantum volumes of mirror Calabi-Yau manifolds.
- Classical volumes are corrected by instanton contributions for loop orders ℓ ≥ 3.
Conclusions:
- The study provides a novel method for calculating Yangian-invariant fishnet integrals.
- This connection offers new insights into the interplay between quantum field theory and geometry.
- Results for 2- and 3-loop train track integrals are presented for arbitrary kinematics.
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