12 research outputs found
Topological recursion for monotone orbifold Hurwitz numbers: a proof of the Do-Karev conjecture
We prove the conjecture of Do and Karev that the monotone orbifold Hurwitz
numbers satisfy the Chekhov-Eynard-Orantin topological recursion.Comment: 11 pages. V2: Updated grant acknowledgments of A.P. and mail address
of R.
Stimulated Raman adiabatic passage-like protocols for amplitude transfer generalize to many bipartite graphs
Adiabatic passage techniques, used to drive a system from one quantum state
into another, find widespread application in physics and chemistry. We focus on
techniques to spatially transport a quantum amplitude over a strongly coupled
system, such as STImulated Raman Adiabatic Passage (STIRAP) and Coherent
Tunnelling by Adiabatic Passage (CTAP). Previous results were shown to work on
certain graphs, such as linear chains, square and triangular lattices, and
branched chains. We prove that similar protocols work much more generally, in a
large class of (semi-)bipartite graphs. In particular, under random couplings,
adiabatic transfer is possible on graphs that admit a perfect matching both
when the sender is removed and when the receiver is removed. Many of the
favorable stability properties of STIRAP/CTAP are inherited, and our results
readily apply to transfer between multiple potential senders and receivers. We
numerically test transfer between the leaves of a tree, and find surprisingly
accurate transfer, especially when straddling is used. Our results may find
applications in short-distance communication between multiple quantum
computers, and open up a new question in graph theory about the spectral gap
around the value 0.Comment: 17 pages, 3 figures. v2 is made more mathematical and precise than v
Cut-and-join equation for monotone Hurwitz numbers revisited
We give a new proof of the cut-and-join equation for the monotone Hurwitz
numbers, derived first by Goulden, Guay-Paquet, and Novak. Our proof in
particular uses a combinatorial technique developed by Han.
The main interest in this particular equation is its close relation to the
quadratic loop equation in the theory of spectral curve topological recursion,
and we recall this motivation giving a new proof of the topological recursion
for monotone Hurwitz numbers, obtained first by Do, Dyer, and Mathews.Comment: 7 pages. v2: Added a second proof of lemma 2.3, using Jucys-Murphy
elements, and expanded motivatio
Half-Spin Tautological Relations and Faber's Proportionalities of Kappa Classes
We employ the -spin tautological relations to provide a particular
combinatorial identity. We show that this identity is a statement equivalent to
Faber's formula for proportionalities of kappa-classes on ,
. We then prove several cases of the combinatorial identity, providing
a new proof of Faber's formula for those cases
Loop equations and a proof of Zvonkine's -ELSV formula
We prove the 2006 Zvonkine conjecture that expresses Hurwitz numbers with
completed cycles in terms of intersection numbers with the Chiodo classes via
the so-called -ELSV formula, as well as its orbifold generalization, the
-ELSV formula, proposed recently in [KLPS17].Comment: 22 pages. Version 4: improved exposition of the proo
Taking limits in topological recursion
When does topological recursion applied to a family of spectral curves
commute with taking limits? This problem is subtle, especially when the
ramification structure of the spectral curve changes at the limit point. We
provide sufficient (straightforward-to-use) conditions for checking when the
commutation with limits holds, thereby closing a gap in the literature where
this compatibility has been used several times without justification. This
takes the form of a stronger result of analyticity of the topological recursion
along suitable families. To tackle this question, we formalise the notion of
global topological recursion and provide sufficient conditions for its
equivalence with local topological recursion. The global version facilitates
the study of analyticity and limits. For nondegenerate algebraic curves, we
reformulate these conditions purely in terms of the structure of its underlying
singularities. Finally, we apply this to study deformations of -spectral curves, spectral curves for weighted Hurwitz numbers, and provide
several other examples and non-examples (where the commutation with limits
fails).Comment: 83 pages, 12 figure
The tautological ring of Mg,n via Pandharipande-Pixton-Zvonkine r-spin relations
We use relations in the tautological ring of the moduli spaces Mg,n derived by Pandharipande, Pixton, and Zvonkine from the Givental formula for the r-spin Witten class in order to obtain some restrictions on the dimensions of the tautological rings of the open moduli spacesMg,n. In particular, we give a new proof for the result of Looijenga (for n = 1) and Buryak et al. (for n > 2) that dimRg-1(Mg,n) ≤ n. We also give a new proof of the result of Looijenga (for n = 1) and Ionel (for arbitrary n > 1) that Ri(Mg,n) = 0 for i > g and give some estimates for the dimension of Ri(Mg,n) for i ≤ g - 2