Flows on the $$\mathbf{PGL(V)}$$-Hitchin Component

Sun, Zhe; Wienhard, Anna; Zhang, Tengren

doi:10.1007/s00039-020-00534-4

Cited by 13 publications

(6 citation statements)

References 17 publications

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“…The Poisson bracket associated to a surface group has been heavily studied in [10], [25]; and in the context of Hitchin representations the link between the symplectic structure, coordinates and cluster algebras discovered by Fock-Goncharov in [8], has generated a lot of attention: for instance see [23], [20], [24], [7] and [22] for more results, and also relations with the swapping algebra [16].…”

Section: Introductionmentioning

confidence: 99%

Simple length rigidity for Hitchin representations

Bridgeman

Canary

Labourie

2020

Advances in Mathematics

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Nous montrons qu'une représentation de Hitchin est déterminée par les rayons spectraux des images de courbes simples et non séparantes. Comme application, nous caractérisons les isométries de la function d'intersection pour les composantes de Hitchin en dimension 3, ainsi que pour les composantes auto-duales en toutes dimensions. Un outil important de notre démonstration est un résultat de transversalité sur les quadruplets positifs de drapeaux.-------We show that a Hitchin representation is determined by the spectral radii of the images of simple, non-separating closed curves. As a consequence, we classify isometries of the intersection function on Hitchin components of dimension 3 and on the self-dual Hitchin components in all dimensions. As an important tool in the proof, we establish a transversality result for positive quadruples of flags.

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Section: Introductionmentioning

confidence: 99%

Simple length rigidity for Hitchin representations

Bridgeman

Canary

Labourie

2020

Advances in Mathematics

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“… Xie [Xie13] writes down the same local coordinates (up to a multiplicative factor of 3) for , , , , , as well as the one-honeycomb-webs and . The definition of these local coordinates can be checked experimentally by studying the highest terms of the Fock–Goncharov -trace polynomials; see the introduction as well as [Kim20, Proposition 5.80] (and [Kim21, Proposition 3.15]). Moreover, it appears that these coordinates fit into a broader geometric context [SWZ20, Theorem 8.22(2)]. The coordinates in the -setting are geometric intersection numbers; see the introduction. In contrast, the -coordinates depend on the choice of orientation of . …”

Section: Global Coordinates For Nonelliptic Websmentioning

confidence: 99%

Tropical Fock–Goncharov coordinates for -webs on surfaces I: construction

Douglas,

Sun

2024

Forum of Mathematics, Sigma

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For a finite-type surface $\mathfrak {S}$ , we study a preferred basis for the commutative algebra $\mathbb {C}[\mathscr {R}_{\mathrm {SL}_3(\mathbb {C})}(\mathfrak {S})]$ of regular functions on the $\mathrm {SL}_3(\mathbb {C})$ -character variety, introduced by Sikora–Westbury. These basis elements come from the trace functions associated to certain trivalent graphs embedded in the surface $\mathfrak {S}$ . We show that this basis can be naturally indexed by nonnegative integer coordinates, defined by Knutson–Tao rhombus inequalities and modulo 3 congruence conditions. These coordinates are related, by the geometric theory of Fock and Goncharov, to the tropical points at infinity of the dual version of the character variety.

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“…The merit of this point of view is that one can understand the various invariants geometrically. The enlarged set of coordinates (compared to the N " 2) can then be understood essentially as the statement that there are more projective invariants 22 , besides the cross-ratio, one can define, such as triple ratios [82] (see also [60,[86][87][88][89] and also [49] for an interpretation of the Fenchel-Nielsen length coordinates in terms of projective invariants). From the usual moduli space (4.2) point of view, these describe invariants of the representations ρ : π 1 pΣq Ñ PSLp3, Rq.…”

Section: Higher Genus and Convex Projective Structuresmentioning

confidence: 99%

Higher spin JT gravity and a matrix model dual

Kruthoff

2022

Preprint

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We propose a generalization of the Saad-Shenker-Stanford duality relating matrix models and JT gravity to the case in which the bulk includes higher spin fields. Using a PSLpN, Rq BF theory we compute the disk and generalization of the trumpet partition function in this theory. We then study higher genus corrections and show how this differs from the usual JT gravity calculations. In particular, the usual quotient by the mapping class group is not enough to ensure finite answers and so we propose to extend this group with additional elements that make the gluing integrals finite. These elements can be thought of as large higher spin diffeomorphisms. The cylinder contribution to the spectral form factor then behaves as T N ´1 at late times T , signaling a deviation from conventional random matrix theory. To account for this deviation, we propose that the bulk theory is dual to a matrix model consisting of N ´1 commuting matrices associated to the N ´1 conserved higher spin charges.We find further evidence for the existence of the additional mapping class group elements by interpreting the bulk gauge theory geometrically and employing the formalism developed by Gomis et al. in the nineties. This formalism introduces additional (auxiliary) boundary times so that each conserved charge generates translations in those new directions. This allows us to find an explicit description for the PSLp3, Rq Schwarzian theory for the disk and trumpet and view the additional mapping class group elements as ordinary Dehn twists, but in higher dimensions.

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Flows on the $$\mathbf{PGL(V)}$$-Hitchin Component

Cited by 13 publications

References 17 publications

Simple length rigidity for Hitchin representations

Simple length rigidity for Hitchin representations

Tropical Fock–Goncharov coordinates for -webs on surfaces I: construction

Higher spin JT gravity and a matrix model dual

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