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Periodicity and integrability for the cube recurrence

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Abstract

Zamolodchikov periodicity is a property of T- and Y-systems, arising in the thermodynamic Bethe ansatz. Zamolodchikov integrability was recently considered as its affine analog in our joint work with P. Pylyavskyy. Here we prove periodicity and integrability for similar discrete dynamical systems based on the cube recurrence, also known as the discrete BKP equation. The periodicity part was conjectured by Henriques in 2007.

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Notes

  1. More precisely, we fix some big disc D and intersect this union of two rays with D.

  2. Indeed, take any (green) edge e of C and consider the lozenge L containing it. It has a red and a blue vertex, and one of them therefore necessarily lies inside of C because they lie on different sides of e.

  3. Except for the move (M1) which may change \([v_1]\) if \(v_2\) belongs to some other cycle in \({\mathbf {C}}\). However, the move (M1) does not change \([v_2]\) and \([v_3]\) so an analogous argument applies in this case.

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Acknowledgements

The author is indebted to Pavlo Pylyavskyy for his numerous contributions to this project, and to the anonymous referee for careful reading of the manuscript and several suggestions that led to improvements in the exposition.

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  1. Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, 02139, USA

    Pavel Galashin

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Correspondence to Pavel Galashin.

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Galashin, P. Periodicity and integrability for the cube recurrence. Math. Z. 299, 1533–1563 (2021). https://doi.org/10.1007/s00209-020-02667-6

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