| 1. |
- Dunin-Barkowski, Petr, et al.
(author)
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Combinatorial structure of colored HOMFLY-PT polynomials for torus knots
- 2019
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In: Communications in Number Theory and Physics. - : INT PRESS BOSTON, INC. - 1931-4523 .- 1931-4531. ; 13:4, s. 763-826
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Journal article (peer-reviewed)abstract
- We rewrite the (extended) Ooguri-Vafa partition function for colored HOMFLY-PT polynomials for torus knots in terms of the free-fermion (semi-infinite wedge) formalism, making it very similar to the generating function for double Hurwitz numbers. This allows us to conjecture the combinatorial meaning of full expansion of the correlation differentials obtained via the topological recursion on the Brini-Eynard-Marino spectral curve for the colored HOMFLY-PT polynomials of torus knots. This correspondence suggests a structural combinatorial result for the extended Ooguri-Vafa partition function. Namely, its coefficients should have a quasi-polynomial behavior, where non-polynomial factors are given by the Jacobi polynomials (treated as functions of their parameters in which they are indeed non-polynomial). We prove this quasi-polynomiality in a purely combinatorial way. In addition to that, we show that the (0,1)- and (0,2)-functions on the corresponding spectral curve are in agreement with the extension of the colored HOMFLY-PT polynomials data, and we prove the quantum spectral curve equation for a natural wave function obtained from the extended Ooguri-Vafa partition function.
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| 2. |
- Freiberg, Tristan, et al.
(author)
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Poisson distribution for gaps between sums of two squares and level spacings for toral point scatterers
- 2017
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In: Communications in Number Theory and Physics. - 1931-4523 .- 1931-4531. ; 11:4, s. 837-877
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Journal article (peer-reviewed)abstract
- We investigate the level spacing distribution for the quantum spectrum of the square billiard. Extending work of Connors-Keating, and Smilansky, we formulate an analog of the Hardy-Littlewood prime k-tuple conjecture for sums of two squares, and show that it implies that the spectral gaps, after removing degeneracies and rescaling, are Poisson distributed. Consequently, by work of Rud-nick and Ueberschar, the level spacings of arithmetic toral point scatterers, in the weak coupling limit, are also Poisson distributed. We also give numerical evidence for the conjecture and its implications.
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| 3. |
- Paquette, N. M., et al.
(author)
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Monstrous BPS-algebras and the superstring origin of moonshine
- 2016
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In: Communications in Number Theory and Physics. - 1931-4531 .- 1931-4523. ; 10:3, s. 433-526
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Journal article (peer-reviewed)abstract
- We provide a physics derivation of Monstrous moonshine. We show that the McKay-Thompson series T-g, g epsilon M, can be interpreted as supersymmetric indices counting spacetime BPS-states in certain heterotic string models. The invariance groups of these series arise naturally as spacetime T-duality groups and their genus zero property descends from the behaviour of these heterotic models in suitable decompactification limits. We also show that the space of BPS-states forms a module for the Monstrous Lie algebras m(g), constructed by Borcherds and Carnahan. We argue that m(g) arise in the heterotic models as algebras of spontaneously broken gauge symmetries, whose generators are in exact correspondence with BPS-states. This gives mg an interpretation as a kind of BPS-algebra.
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