Non-commutative matrix integrals and representation varieties of surface groups in a finite group

Mulase, Motohico; Yu, Josephine

doi:10.5802/aif.2157

Cited by 14 publications

(19 citation statements)

References 46 publications

(69 reference statements)

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“…Remark 4.12. There is a different proof of the graph independence theorem, using a topological idea of deforming graphs similar to the one used in [23].…”

Section: The Edge-contraction Axiomsmentioning

confidence: 99%

Edge contraction on dual ribbon graphs and 2D TQFT

Dumitrescu

Mulase

2018

Journal of Algebra

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We present a new set of axioms for 2D TQFT formulated on the category of cell graphs with edge-contraction operations as morphisms. We construct a functor from this category to the endofunctor category consisting of Frobenius algebras. Edge-contraction operations correspond to natural transformations of endofunctors, which are compatible with the Frobenius algebra structure. Given a Frobenius algebra A, every cell graph determines an element of the symmetric tensor algebra defined over the dual space A * . We show that the edge-contraction axioms make this assignment depending only on the topological type of the cell graph, but not on the graph itself. Thus the functor generates the TQFT corresponding to A.

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“…Remark 4.12. There is a different proof of the graph independence theorem, using a topological idea of deforming graphs similar to the one used in [23].…”

Section: The Edge-contraction Axiomsmentioning

confidence: 99%

Edge contraction on dual ribbon graphs and 2D TQFT

Dumitrescu

Mulase

2018

Journal of Algebra

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“…for each edge e with vertices v and v ′ (the case v = v ′ is allowed). This structure is equivalent to the (somewhat more cumbersome) concept of a Möbius graph defined in terms of "ribbon graphs with Z 2-grading on edges" as in [10,53,54].…”

Section: Iii24 Moduli Spaces Of Marked G Conf -Structured Surfacesmentioning

confidence: 99%

“…For example, for unoriented surfaces, the cyclic category Λ is replaced by the dihedral category Ξ, see [47]. Applying our formalism to Ξ, we get a concept known as a Möbius graph [10,53,54] but formulated in a somewhat more conceptual way.…”

Section: Introductionmentioning

confidence: 99%

Crossed simplicial groups and structured surfaces

Dyckerhoff

Kapranov

2015

Contemporary Mathematics

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We propose a generalization of the concept of a ribbon graph suitable to provide combinatorial models for marked surfaces equipped with a G-structure. Our main insight is that the necessary combinatorics is neatly captured in the concept of a crossed simplicial group as introduced, independently, by Krasauskas and Fiedorowicz-Loday. In this context, Connes' cyclic category leads to ribbon graphs while other crossed simplicial groups naturally yield different notions of structured graphs which model unoriented, N-spin, framed, etc, surfaces. Our main result is that structured graphs provide orbicell decompositions of the respective G-structured moduli spaces. As an application, we show how, building on our theory of 2-Segal spaces, the resulting theory can be used to construct categorified state sum invariants of G-structured surfaces.

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“…with β 2 gives rise to enumerations of graphs embedded in possibly non-orientable surfaces [73,119]. Then, the expansion (1.1.6) also contains half-integer g's.…”

Section: Enumerative Geometry and N-fold Integralsmentioning

confidence: 99%

Asymptotic Expansion of a Partition Function Related to the Sinh-model

Borot¹,

Guionnet²,

Kozłowski³

2016

Mathematical Physics Studies

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This paper develops a method to carry out the large-N asymptotic analysis of a class of Ndimensional integrals arising in the context of the so-called quantum separation of variables method. We push further ideas developed in the context of random matrices of size N, but in the present problem, two scales 1/N α and 1/N naturally occur. In our case, the equilibrium measure is N α -dependent and characterised by means of the solution to a 2×2 Riemann-Hilbert problem, whose large-N behaviour is analysed in detail. Combining these results with techniques of concentration of measures and an asymptotic analysis of the Schwinger-Dyson equations at the distributional level, we obtain the large-N behaviour of the free energy explicitly up to o(1). The use of distributional Schwinger-Dyson is a novelty that allows us treating sufficiently differentiable interactions and the mixing of scales 1/N α and 1/N, thus waiving the analyticity assumptions often used in random matrix theory.1 gborot@mpim-bonn.mpg.de 2 guionnet@math.mit.edu 3 karol.kozlowski@ens-lyon.fr 2 An opening discussionThe present work develops techniques enabling one to carry out the large-N asymptotic analysis of a class of multiple integrals that arise as representations for the correlation functions in quantum integrable models solvable by the quantum separation of variables. We shall refer to the general class of such integrals as the sinh model: [9,59,60,82,106,105,139,146]. The expressions obtained there are either directly of the form given above or are amenable to this form (with, possibly, a change of the integration contour from R N to C N , with C a curve in C) upon elementary manipulations. Furthermore, a degeneration of z N [W] arises as a multiple integral representation for the partition function of the six-vertex model subject to domain wall boundary conditions [93]. In the context of quantum integrable systems, the number N of integrals defining z N is related to the number of sites in a model (as, e.g. in the case of the compact or non-compact XXZ chains or the lattice regularisations of the Sinh or Sine-Gordon models) or to the number of particles (as, e.g. in the case of the quantum Toda chain). From the point of view of applications, one is mainly interested in the thermodynamic limit of the model, which is attained by sending N to +∞. For instance, in the case of an integrable lattice discretisation of some quantum field theory, one obtains in this way an exact and non-perturbative description of a quantum field theory in 1 + 1 dimensions and in finite volume. This limit, at the level of z N [W], translates itself in the need to extract the large N-asymptotic expansion of ln z N [W] up to o(1). It is, in fact, the constant term in the expansion of ln zwith W ′ some deformation of W that gives rise to the correlation functions of the underlying quantum field theory in finite volume. These applications to physics constitute the first motivation for our analysis. From the purely mathematical side, the motivation of our works stems ...

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Non-commutative matrix integrals and representation varieties of surface groups in a finite group

Cited by 14 publications

References 46 publications

Edge contraction on dual ribbon graphs and 2D TQFT

Edge contraction on dual ribbon graphs and 2D TQFT

Crossed simplicial groups and structured surfaces

Asymptotic Expansion of a Partition Function Related to the Sinh-model

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