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Journal articles on the topic 'Monodromy representation'

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Published: 2 June 2025

Last updated: 31 July 2025

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1

IMAYOSHI, YOICHI, MANABU ITO, and HIROSHI YAMAMOTO. "A REDUCIBILITY PROBLEM FOR MONODROMY OF SOME SURFACE BUNDLES." Journal of Knot Theory and Its Ramifications 13, no. 05 (2004): 597–616. http://dx.doi.org/10.1142/s0218216504003330.

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A surface bundle determines a monodromy representation recording the twisting of the fiber under transport around a closed path in the base space. The fascinating relation between these monodromy representations and the Thurston classification of surface automorphisms will be studied. In this note we deal with a simple and interesting case: the fibrations of Fadell and Neuwirth.

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2

GIORGADZE, G. "MONODROMY APPROACH TO QUANTUM COMPUTING." International Journal of Modern Physics B 16, no. 30 (2002): 4593–605. http://dx.doi.org/10.1142/s0217979202014607.

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In this work, a gauge approach to quantum computing is considered. It is assumed that there exists a classical procedure for placing certain classical system in a state described by a holomorphic vector bundle with connection with logarithmic singularities. This bundle and its connection are constructed with the aid of unitary operators realizing the given algorithm using methods of the monodromic Riemann–Hilbert problem. Universality is understood in the sense that for any collection of unitary matrices there exists a connection with logarithmic singularities whose monodromy representation in

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3

LAWRENCE, R. J. "A TOPOLOGICAL APPROACH TO REPRESENTATIONS OF THE IWAHORI-HECKE ALGEBRA." International Journal of Modern Physics A 05, no. 16 (1990): 3213–19. http://dx.doi.org/10.1142/s0217751x90002178.

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In this paper a topological construction of representations of the [Formula: see text]-series of Hecke algebras, associated with 2-row Young diagrams, will be announced. This construction gives the representations in terms of the monodromy representation obtained from a vector bundle over the configuration space of η points in the complex plane. The fibres are homology spaces of configuration spaces of points in a punctured complex plane, with a suitable twisted local coefficient system, and there is thus a natural flat connection on the vector bundle. It is also shown that there is a close co

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4

Tanabé, Susumu. "On monodromy representation of period integrals associated to an algebraic curve with bi-degree (2,2)." Analele Universitatii "Ovidius" Constanta - Seria Matematica 25, no. 1 (2017): 207–31. http://dx.doi.org/10.1515/auom-2017-0016.

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AbstractWe study a problem related to Kontsevich's homological mirror symmetry conjecture for the case of a generic curve Y with bi-degree (2,2) in a product of projective lines ℙ1× ℙ1. We calculate two differenent monodromy representations of period integrals for the affine variety X(2,2)obtained by the dual polyhedron mirror variety construction from Y. The first method that gives a full representation of the fundamental group of the complement to singular loci relies on the generalised Picard-Lefschetz theorem. The second method uses the analytic continuation of the Mellin-Barnes integrals

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5

V’yugin, I. V. "Irreducible Fuchsian system with reducible monodromy representation." Mathematical Notes 80, no. 3-4 (2006): 478–84. http://dx.doi.org/10.1007/s11006-006-0165-9.

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6

Goto, Yoshiaki, and Keiji Matsumoto. "The monodromy representation and twisted period relations for Appell’s hypergeometric function F 4." Nagoya Mathematical Journal 217 (March 2015): 61–94. http://dx.doi.org/10.1017/s0027763000026957.

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AbstractWe consider the systemF4(a, b, c)of differential equations annihilating Appell's hypergeometric seriesF4(a,b,c;x). We find the integral representations for four linearly independent solutions expressed by the hypergeometric seriesF4. By using the intersection forms of twisted (co)homology groups associated with them, we provide the monodromy representation ofF4(a, b, c)and the twisted period relations for the fundamental systems of solutions ofF4.

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7

Vyugin, Il'ya Vladimirovich, and Lada Andreevna Dudnikova. "Stable vector bundles and the Riemann-Hilbert problem on a Riemann surface." Sbornik: Mathematics 215, no. 2 (2024): 141–56. http://dx.doi.org/10.4213/sm9781e.

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The paper is devoted to holomorphic vector bundles with logarithmic connections on a compact Riemann surface and the applications of the results obtained to the question of solvability of the Riemann-Hilbert problem on a Riemann surface. We give an example of a representation of the fundamental group of a Riemann surface with four punctured points which cannot be realized as the monodromy representation of a logarithmic connection with four singular points on a semistable bundle. For an arbitrary pair of a bundle and a logarithmic connection on it we prove an estimate for the slopes of the ass

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8

Goto, Yoshiaki, and Keiji Matsumoto. "The monodromy representation and twisted period relations for Appell’s hypergeometric function F4." Nagoya Mathematical Journal 217 (March 2015): 61–94. http://dx.doi.org/10.1215/00277630-2873714.

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AbstractWe consider the system F4 (a, b, c) of differential equations annihilating Appell's hypergeometric series F4(a,b,c;x). We find the integral representations for four linearly independent solutions expressed by the hypergeometric series F4. By using the intersection forms of twisted (co)homology groups associated with them, we provide the monodromy representation of F4(a, b, c) and the twisted period relations for the fundamental systems of solutions of F4.

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9

GOTO, Yoshiaki, and Keiji MATSUMOTO. "Irreducibility of the monodromy representation of Lauricella's $F_C$." Hokkaido Mathematical Journal 48, no. 3 (2019): 489–512. http://dx.doi.org/10.14492/hokmj/1573722015.

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10

Shimizu, Koji. "A -adic monodromy theorem for de Rham local systems." Compositio Mathematica 158, no. 12 (2022): 2157–205. http://dx.doi.org/10.1112/s0010437x2200776x.

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We study horizontal semistable and horizontal de Rham representations of the absolute Galois group of a certain smooth affinoid over a $p$ -adic field. In particular, we prove that a horizontal de Rham representation becomes horizontal semistable after a finite extension of the base field. As an application, we show that every de Rham local system on a smooth rigid analytic variety becomes horizontal semistable étale locally around every classical point. We also discuss potentially crystalline loci of de Rham local systems and cohomologically potentially good reduction loci of smooth proper mo

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11

Kamgarpour, Masoud, GyeongHyeon Nam, and Anna Puskás. "Arithmetic geometry of character varieties with regular monodromy." Representation Theory 29, no. 11 (2025): 347–78. https://doi.org/10.1090/ert/693.

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We count points on a family of smooth character varieties with regular semisimple and regular unipotent monodromies. We show that these varieties are polynomial count and obtain an explicit expression for their E E -polynomials using complex representation theory of finite reductive groups. As an application, we give an example of a cohomologically rigid representation which is not physically rigid.

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12

Saito, Takeshi. "Hilbert modular forms and p-adic Hodge theory." Compositio Mathematica 145, no. 5 (2009): 1081–113. http://dx.doi.org/10.1112/s0010437x09004175.

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AbstractFor the p-adic Galois representation associated to a Hilbert modular form, Carayol has shown that, under a certain assumption, its restriction to the local Galois group at a finite place not dividing p is compatible with the local Langlands correspondence. Under the same assumption, we show that the same is true for the places dividing p, in the sense of p-adic Hodge theory, as is shown for an elliptic modular form. We also prove that the monodromy-weight conjecture holds for such representations.

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13

Bolibrukh, A. A. "Construction of a Fuchsian equation from a monodromy representation." Mathematical Notes of the Academy of Sciences of the USSR 48, no. 5 (1990): 1090–99. http://dx.doi.org/10.1007/bf01236293.

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14

Joshi, Nalini, and Yang Shi. "Exact solutions of a q -discrete second Painlevé equation from its iso-monodromy deformation problem. II. Hypergeometric solutions." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 468, no. 2146 (2012): 3247–64. http://dx.doi.org/10.1098/rspa.2012.0224.

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This is the second part of our study of the solutions of a q -discrete second Painlevé equation ( q -P II ) of type ( A 2 + A 1 ) (1) via its iso-monodromy deformation problem. In part I, we showed how to use the q -discrete linear problem associated with q -P II to find an infinite sequence of exact rational solutions. In this paper, we study the case giving rise to an infinite sequence of q -hypergeometric-type solutions. We find a new determinantal representation of all such solutions and solve the iso-monodromy deformation problem in closed form.

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15

LEE, H. C., M. L. GE, M. COUTURE, and Y. S. WU. "STRANGE STATISTICS, BRAID GROUP REPRESENTATIONS AND MULTIPOINT FUNCTIONS IN THE N-COMPONENT MODEL." International Journal of Modern Physics A 04, no. 09 (1989): 2333–70. http://dx.doi.org/10.1142/s0217751x89000947.

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The statistics of fields in low dimensions is studied from the point of view of the braid group Bn of n strings. Explicit representations MR for the N-component model, N=2 to 5, are derived by solving the Yang-Baxter-like braid group relations for the statistical matrix R, which describes the transformation of the bilinear product of two N-component fields under the transposition of coordinates. When R2≠1 the statistics is neither Bose-Einstein nor Fermi-Dirac; it is strange. It is shown that for each N, the N+1 parameter family of solutions obtained is the most general one under a given set o

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16

BRODA, B. "A THREE-DIMENSIONAL COVARIANT APPROACH TO MONODROMY (SKEIN RELATIONS) IN CHERN-SIMONS THEORY." Modern Physics Letters A 05, no. 32 (1990): 2747–51. http://dx.doi.org/10.1142/s0217732390003206.

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A genuinely three-dimensional covariant approach to the monodromy operator (skein relations) in the context of Chern-Simons theory is proposed. A holomorphic path-integral representation for the holonomy operator (Wilson loop) and for the non-abelian Stokes theorem is used.

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17

ALDROVANDI, E., L. BONORA, V. BONSERVIZI, and R. PAUNOV. "FREE FIELD REPRESENTATION OF TODA FIELD THEORIES." International Journal of Modern Physics A 09, no. 01 (1994): 57–86. http://dx.doi.org/10.1142/s0217751x94000042.

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We study the following problem: Can a classical sl n Toda field theory be represented by means of free bosonic oscillators through a Drinfeld-Sokolov construction? We answer affirmatively in the case of a cylindrical space-time and for real hyperbolic solutions of the Toda field equations. We establish in fact a one-to-one correspondence between such solutions and the space of free left and right bosonic oscillators with coincident zero modes. We discuss the same problem for real singular solutions with nonhyperbolic monodromy.

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18

Saber, Hicham, and Abdellah Sebbar. "Equivariant functions and vector-valued modular forms." International Journal of Number Theory 10, no. 04 (2014): 949–54. http://dx.doi.org/10.1142/s1793042114500092.

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For any discrete group Γ and any two-dimensional complex representation ρ of Γ, we introduce the notion of ρ-equivariant functions, and we show that they are parametrized by vector-valued modular forms. We also provide examples arising from the monodromy of differential equations.

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19

Corlette, Kevin, and Carlos Simpson. "On the classification of rank-two representations of quasiprojective fundamental groups." Compositio Mathematica 144, no. 5 (2008): 1271–331. http://dx.doi.org/10.1112/s0010437x08003618.

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AbstractSuppose that X is a smooth quasiprojective variety over ℂ and ρ:π1(X,x)→SL(2,ℂ) is a Zariski-dense representation with quasiunipotent monodromy at infinity. Then ρ factors through a map X→Y with Y either a Deligne–Mumford (DM) curve or a Shimura modular stack.

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20

PERRINE, SERGE. "MONODROMY ARISING FROM THE MARKOFF THEORY." International Journal of Modern Physics B 20, no. 11n13 (2006): 1819–32. http://dx.doi.org/10.1142/s0217979206034339.

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In a former work, recalling what the Markoff theory is, we summarized some existing links with the group GL(2, ℤ) of 2 × 2 matrices. We also quoted the relation with conformal punctured toruses. The monodromy representation of the Poincaré group of such a torus was considered. Here we explicit the corresponding solution of the associated Riemann-Hilbert problem, and the resulting Fuchs differential equation. We precisely describe how the calculus runs. The main result is the description of a complete family of Fuchs differential equations with, as the monodromy group, the free group with two g

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21

ARNAUDON, D., A. DOIKOU, L. FRAPPAT, É. RAGOUCY, and N. CRAMPÉ. "ANALYTICAL BETHE ANSATZ FOR OPEN SPIN CHAINS WITH SOLITON NONPRESERVING BOUNDARY CONDITIONS." International Journal of Modern Physics A 21, no. 07 (2006): 1537–54. http://dx.doi.org/10.1142/s0217751x06029077.

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We present an "algebraic treatment" of the analytical Bethe ansatz for open spin chains with soliton nonpreserving (SNP) boundary conditions. For this purpose, we introduce abstract monodromy and transfer matrices which provide an algebraic framework for the analytical Bethe ansatz. It allows us to deal with a generic [Formula: see text] open SNP spin chain possessing on each site an arbitrary representation. As a result, we obtain the Bethe equations in their full generality. The classification of finite dimensional irreducible representations for the twisted Yangians are directly linked to t

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22

HATAKENAKA, ERI. "Invariants of 3-manifolds derived from covering presentations." Mathematical Proceedings of the Cambridge Philosophical Society 149, no. 2 (2010): 263–95. http://dx.doi.org/10.1017/s0305004110000198.

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AbstractBy a covering presentation of a 3-manifold, we mean a labelled link (i.e., a link with a monodromy representation), which presents the 3-manifold as the simple 4-fold covering space of the 3-sphere branched along the link with the given monodromy. It is known that two labelled links present a homeomorphic 3-manifold if and only if they are related by a finite sequence of some local moves. This paper presents a method for constructing topological invariants of 3-manifolds based on their covering presentations. The proof of the topological invariance is shown by verifying the invariance

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23

Némethi, A., and I. Sigray. "Ont he monodromy representation of polynomial maps in n variables." Studia Scientiarum Mathematicarum Hungarica 39, no. 3-4 (2002): 361–67. http://dx.doi.org/10.1556/sscmath.39.2002.3-4.7.

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For a non-constant polynomial map f: Cn?Cn-1 we consider the monodromy representation on the cohomology group of its generic fiber. The main result of the paper determines its dimension and provides a natural basis for it. This generalizes the corresponding results of [2] or [10], where the case n=2 is solved. As applications, we verify the Jacobian conjecture for (f,g) when the generic fiber of f is either rational or elliptic. These are generalizations of the corresponding results of [5], [7], [8], [11] and [12], where the case n=2 is treated.

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24

KIMURA, Tosihusa, and Kazuhisa SHIMA. "ON THE MONODROMY REPRESENTATION OF AN IRREDUCIBLE HYPERGEOMETRIC DIFFERENTIAL EQUATION." Memoirs of the Faculty of Science, Kyushu University. Series A, Mathematics 46, no. 1 (1992): 137–67. http://dx.doi.org/10.2206/kyushumfs.46.137.

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25

SHIMA, Kazuhisa. "On the monodromy representation of a reducible hypergeometric differential equation." Japanese journal of mathematics. New series 18, no. 2 (1992): 403–38. http://dx.doi.org/10.4099/math1924.18.403.

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26

Costa, Antonio F., Milagros Izquierdo, and Gonzalo Riera. "One-Dimensional Hurwitz Spaces, Modular Curves, and Real Forms of Belyi Meromorphic Functions." International Journal of Mathematics and Mathematical Sciences 2008 (2008): 1–18. http://dx.doi.org/10.1155/2008/609425.

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Hurwitz spaces are spaces of pairs(S,f)whereSis a Riemann surface andf:S→ℂ^a meromorphic function. In this work, we study1-dimensional Hurwitz spacesℋDpof meromorphicp-fold functions with four branched points, three of them fixed; the corresponding monodromy representation over each branched point is a product of(p−1)/2transpositions and the monodromy group is the dihedral groupDp. We prove that the completionℋDp¯of the Hurwitz spaceℋDpis uniformized by a non-nomal indexp+1subgroup of a triangular group with signature(0;[p,p,p]). We also establish the relation of the meromorphic covers with el

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27

Kapovich, Michael, and John J. Millson. "Quantization of Bending Deformations of Polygons In , Hypergeometric Integrals and the Gassner Representation." Canadian Mathematical Bulletin 44, no. 1 (2001): 36–60. http://dx.doi.org/10.4153/cmb-2001-006-3.

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AbstractThe Hamiltonian potentials of the bending deformations of n-gons instudied in [KM] and [Kly] give rise to a Hamiltonian action of the Malcev Lie algebra𝓟nof the pure braid groupPnon the moduli spaceMrofn-gon linkages with the side-lengthsr = (r1, … , rn)in. Ife∈Mris a singular point wemay linearize the vector fields in𝓟nate. This linearization yields a flat connection ∇ on the spaceof n distinct points on. We show that the monodromy of ∇ is the dual of a quotient of a specialized reduced Gassner representation.

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28

Katz, Nicholas M., Antonio Rojas-León, and Pham Huu Tiep. "Rigid local systems with monodromy group the Conway group Co2." International Journal of Number Theory 16, no. 02 (2019): 341–60. http://dx.doi.org/10.1142/s1793042120500189.

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We first develop some basic facts about hypergeometric sheaves on the multiplicative group [Formula: see text] in characteristic [Formula: see text]. Certain of their Kummer pullbacks extend to irreducible local systems on the affine line in characteristic [Formula: see text]. One of these, of rank [Formula: see text] in characteristic [Formula: see text], turns out to have the Conway group [Formula: see text], in its irreducible orthogonal representation of degree [Formula: see text], as its arithmetic and geometric monodromy groups.

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DESENONGES, NICOLAS HUSSENOT. "Analytic continuation of holonomy germs of Riccati foliations along Brownian paths." Ergodic Theory and Dynamical Systems 37, no. 6 (2016): 1887–914. http://dx.doi.org/10.1017/etds.2015.132.

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Consider a Riccati foliation whose monodromy representation is non-elementary and parabolic and consider a non-invariant section of the fibration whose associated developing map is onto. We prove that any holonomy germ from any non-invariant fibre to the section can be analytically continued along a generic Brownian path. To prove this theorem, we prove a dual result about complex projective structures. Let $\unicode[STIX]{x1D6F4}$ be a hyperbolic Riemann surface of finite type endowed with a branched complex projective structure: such a structure gives rise to a non-constant holomorphic map $

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30

Mochizuki, Atsushi. "The Casson–Walker invariant of 3-manifolds with genus one open book decompositions." Journal of Knot Theory and Its Ramifications 28, no. 06 (2019): 1950018. http://dx.doi.org/10.1142/s0218216519500184.

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In this paper, we give two formulae of values of the Casson–Walker invariant of 3-manifolds with genus one open book decompositions; one is a formula written in terms of a framed link of a surgery presentation of such a 3-manifold, and the other is a formula written in terms of a representation of the mapping class group of a 1-holed torus. For the former case, we compute the invariant through the combinatorial calculation of the degree 1 part of the LMO invariant. For the latter case, we construct a representation of a central extension of the mapping class group through the action of the deg

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31

Bonsante, Francesco, Gabriele Mondello, and Jean-Marc Schlenker. "Minimizing immersions of a hyperbolic surface in a hyperbolic 3-manifold." American Journal of Mathematics 145, no. 4 (2023): 995–1049. http://dx.doi.org/10.1353/ajm.2023.a902953.

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abstract: Let $(S,h)$ be a closed hyperbolic surface and $M$ be a quasi-Fuchsian $3$-manifold. We consider incompressible maps from $S$ to $M$ that are critical points of an energy functional $F$ which is homogeneous of degree $1$. These ``minimizing'' maps are solutions of a non-linear elliptic equation, and reminiscent of harmonic maps---but when the target is Fuchsian, minimizing maps are minimal Lagrangian diffeomorphisms to the totally geodesic surface in $M$. We prove the uniqueness of smooth minimizing maps from $(S,h)$ to $M$ in a given homotopy class. When $(S,h)$ is fixed, smooth min

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32

BRODA, BOGUSLAW. "QUANTUM THEORY OF NON-ABELIAN DIFFERENTIAL FORMS AND LINK POLYNOMIALS." Modern Physics Letters A 09, no. 07 (1994): 609–21. http://dx.doi.org/10.1142/s0217732394003841.

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A topological quantum field theory of non-Abelian differential forms is investigated from the point of view of its possible applications to description of polynomial invariants of higher-dimensional two-component links. A path-integral representation of the partition function of the theory, which is a highly on-shell reducible system, is obtained in the framework of the antibracket-antifield formalism of Batalin and Vilkovisky. The quasi-monodromy matrix, giving rise to corresponding skein relations, is formally derived in a manifestly covariant non-perturbative manner.

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BASU-MALLICK, B. "COLORED EXTENSIONS OF GLq(2) QUANTUM GROUP AND RELATED NONCOMMUTATIVE PLANES." International Journal of Modern Physics A 10, no. 19 (1995): 2851–64. http://dx.doi.org/10.1142/s0217751x95001352.

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An infinite-dimensional quantum group, containing the standard GLq(2) and GLp,q(2) cases as different subalgebras, is constructed by using a colored braid group representation. It turns out that all algebraic relations occurring in this “colored” quantum group can be expressed in the Heisenberg-Weyl form, for a nontrivial choice of corresponding basis elements. Moreover a novel quadratic algebra, defined through Kac-Moody-like generators, is obtained by making some power series expansion of related monodromy matrix elements. The structure of invariant noncommutative planes associated with this

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34

Heller, Sebastian. "The asymptotic behavior of the monodromy representation of the associated family of a compact CMC surface." Bulletin of the London Mathematical Society 48, no. 5 (2016): 729–34. http://dx.doi.org/10.1112/blms/bdw036.

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35

Joyner, Sheldon T. "On an extension of the universal monodromy representation for $\mathbb{P}^1 \backslash \{ 0, 1,\infty \}$." Communications in Number Theory and Physics 8, no. 3 (2014): 369–402. http://dx.doi.org/10.4310/cntp.2014.v8.n3.a1.

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36

Lee, Min Ho. "Mixed Hilbert modular forms and families of abelian varieties." Glasgow Mathematical Journal 39, no. 2 (1997): 131–40. http://dx.doi.org/10.1017/s001708950003202x.

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In [18] Shioda proved that the space of holomorphic 2-forms on a certain type of elliptic surface is canonically isomorphic to the space of modular forms of weight three for the associated Fuchsian group. Later, Hunt and Meyer [6] made an observation that the holomorphic 2-forms on a more general elliptic surface should in fact be identified with mixed automorphic forms associated to an automorphy factor of the formfor z in the Poincaré upper half plane ℋ, g = and χ(g) = , where g is an element of the fundamental group Γ⊂PSL(2, R) of the base space of the elliptic fibration, χ-Γ→SL(2, R) the m

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37

Suzuki, Osamu, and Zhidong Zhang. "A Method of Riemann–Hilbert Problem for Zhang’s Conjecture 1 in a Ferromagnetic 3D Ising Model: Trivialization of Topological Structure." Mathematics 9, no. 7 (2021): 776. http://dx.doi.org/10.3390/math9070776.

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A method of the Riemann–Hilbert problem is applied for Zhang’s conjecture 1 proposed in Philo. Mag. 87 (2007) 5309 for a ferromagnetic three-dimensional (3D) Ising model in the zero external field and the solution to the Zhang’s conjecture 1 is constructed by use of the monoidal transform. At first, the knot structure of the ferromagnetic 3D Ising model in the zero external field is determined and the non-local behavior of the ferromagnetic 3D Ising model can be described by the non-trivial knot structure. A representation from the knot space to the Clifford algebra of exponential type is cons

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38

Kaneko, Jyoichi, Keiji Matsumoto, and Katsuyoshi Ohara. "The structure of a local system associated with a hypergeometric system of rank 9." International Journal of Mathematics 31, no. 03 (2020): 2050021. http://dx.doi.org/10.1142/s0129167x20500214.

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We study a local system associated with a system [Formula: see text] of hypergeometric differential equations in two variables of rank [Formula: see text] with seven parameters [Formula: see text] and [Formula: see text]. We modify the fundamental system of solutions to [Formula: see text] given in [A system of hypergeometric differential equations in two variables of rank 9, Internat. J. Math. 28 (2017), 1750015, 34 pp] so that it is valid even in cases where [Formula: see text] satisfy some integral conditions. By using this fundamental system, we show the irreducibility of the monodromy rep

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39

Sorensen, Claus M. "Level-raising for Saito–Kurokawa forms." Compositio Mathematica 145, no. 4 (2009): 915–53. http://dx.doi.org/10.1112/s0010437x09004084.

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AbstractThis paper provides congruences between unstable and stable automorphic forms for the symplectic similitude group GSp(4). More precisely, we raise the level of certain CAP representations Π arising from classical modular forms. We first transfer Π to π on a suitable inner form G; this is achieved by θ-lifting. For π, we prove a precise level-raising result that is inspired by the work of Bellaiche and Clozel and which relies on computations of Schmidt. We thus obtain a $\tilde {\pi }$ congruent to π, with a local component that is irreducibly induced from an unramified twist of the Ste

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40

Joshi, Nalini, and Yang Shi. "Exact solutions of a q -discrete second Painlevé equation from its iso-monodromy deformation problem: I. Rational solutions." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 467, no. 2136 (2011): 3443–68. http://dx.doi.org/10.1098/rspa.2011.0167.

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In this paper, we present a new method of deducing infinite sequences of exact solutions of q -discrete Painlevé equations by using their associated linear problems. The specific equation we consider in this paper is a q -discrete version of the second Painlevé equation ( q -P II ) with affine Weyl group symmetry of type ( A 2 + A 1 ) (1) . We show, for the first time, how to use the q -discrete linear problem associated with q -P II to find an infinite sequence of exact rational solutions and also show how to find their representation as determinants by using the linear problem. The method, w

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41

Qian, Zicheng. "Dilogarithm and higher ℒ-invariants for 𝒢ℒ₃(𝐐_{𝐩})". Representation Theory of the American Mathematical Society 25, № 12 (2021): 344–411. http://dx.doi.org/10.1090/ert/567.

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The primary purpose of this paper is to clarify the relation between previous results in [Ann. Sci. Éc. Norm. Supér. 44 (2011), pp. 43–145], [Amer. J. Math. 141 (2019), pp. 661–703], and [Camb. J. Math. 8 (2020), p. 775–951] via the construction of some interesting locally analytic representations. Let E E be a sufficiently large finite extension of Q p \mathbf {Q}_p and ρ p \rho _p be a p p -adic semi-stable representation G a l ( Q p ¯ / Q p ) → G L 3 ( E ) \mathrm {Gal}(\overline {\mathbf {Q}_p}/\mathbf {Q}_p)\rightarrow \mathrm {GL}_3(E) such that the associated Weil–Deligne representation

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42

Mondello, Gabriele. "Topology of representation spaces of surface groups in $\mathrm{PSL}_2 (\mathbb{R})$ with assigned boundary monodromy and nonzero Euler number." Pure and Applied Mathematics Quarterly 12, no. 3 (2016): 399–462. http://dx.doi.org/10.4310/pamq.2016.v12.n3.a3.

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Adrega de Moura, Adriano. "Elliptic dynamical $R$-matrices from the monodromy of the $q$-Knizhnik–Zamolodchikov equations for the standard representation of $U_q(\widetilde{\mathfrak{sl}}_{n+1})$." Asian Journal of Mathematics 7, no. 1 (2003): 91–114. http://dx.doi.org/10.4310/ajm.2003.v7.n1.a6.

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Cushman, Richard, and Jędrzej Śniatycki. "Classical and Quantum Spherical Pendulum." Symmetry 14, no. 3 (2022): 496. http://dx.doi.org/10.3390/sym14030496.

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The seminal paper by Niels Bohr followed by a paper by Arnold Sommerfeld led to a revolutionary Bohr–Sommerfeld theory of atomic spectra. We are interested in the information about the structure of quantum mechanics encoded in this theory. In particular, we want to extend Bohr–Sommerfeld theory to a full quantum theory of completely integrable Hamiltonian systems, which is compatible with geometric quantization. In the general case, we use geometric quantization to prove analogues of the Bohr–Sommerfeld quantization conditions for the prequantum operators Pf. If a prequantum operator Pf satisf

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Catanese, Fabrizio, and Michael Dettweiler. "Vector bundles on curves coming from variation of Hodge structures." International Journal of Mathematics 27, no. 07 (2016): 1640001. http://dx.doi.org/10.1142/s0129167x16400012.

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Fujita’s second theorem for Kähler fibre spaces over a curve asserts, that the direct image [Formula: see text] of the relative dualizing sheaf splits as the direct sum [Formula: see text], where [Formula: see text] is ample and [Formula: see text] is unitary flat. We focus on our negative answer [F. Catanese and M. Dettweiler, Answer to a question by Fujita on variation of Hodge structures, to appear in Adv. Stud. Pure Math.] to a question by Fujita: is [Formula: see text] semiample? We give here an infinite series of counterexamples using hypergeometric integrals and we give a simple argumen

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Hui, Chun Yin. "-independence for compatible systems of (mod ) representations." Compositio Mathematica 151, no. 7 (2015): 1215–41. http://dx.doi.org/10.1112/s0010437x14007969.

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Let$K$be a number field. For any system of semisimple mod $\ell$Galois representations$\{{\it\phi}_{\ell }:\text{Gal}(\bar{\mathbb{Q}}/K)\rightarrow \text{GL}_{N}(\mathbb{F}_{\ell })\}_{\ell }$arising from étale cohomology (Definition 1), there exists a finite normal extension$L$of$K$such that if we denote${\it\phi}_{\ell }(\text{Gal}(\bar{\mathbb{Q}}/K))$and${\it\phi}_{\ell }(\text{Gal}(\bar{\mathbb{Q}}/L))$by$\bar{{\rm\Gamma}}_{\ell }$and$\bar{{\it\gamma}}_{\ell }$, respectively, for all$\ell$and let$\bar{\mathbf{S}}_{\ell }$be the$\mathbb{F}_{\ell }$-semisimple subgroup of$\text{GL}_{N,\mat

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FELDER, G., V. TARASOV, and A. VARCHENKO. "MONODROMY OF SOLUTIONS OF THE ELLIPTIC QUANTUM KNIZHNIK–ZAMOLODCHIKOV–BERNARD DIFFERENCE EQUATIONS." International Journal of Mathematics 10, no. 08 (1999): 943–75. http://dx.doi.org/10.1142/s0129167x99000410.

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The elliptic quantum Knizhnik–Zamolodchikov–Bernard (qKZB) difference equations associated to the elliptic quantum group Eτ,η(sl2) is a system of difference equations with values in a tensor product of representations of the quantum group and defined in terms of the elliptic R-matrices associated with pairs of representations of the quantum group. In this paper we solve the qKZB equations in terms of elliptic hypergeometric functions and describe the monodromy properties of solutions. It turns out that the monodromy transformations of solutions are described in terms of elliptic R-matrices ass

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Kamada, Seiichi. "Graphic descriptions of monodromy representations." Topology and its Applications 154, no. 7 (2007): 1430–46. http://dx.doi.org/10.1016/j.topol.2006.04.024.

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Aaltonen, Martina. "Monodromy representations of completed coverings." Revista Matemática Iberoamericana 32, no. 2 (2016): 533–70. http://dx.doi.org/10.4171/rmi/894.

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Sergeev, Sergey. "Functional Bethe Ansatz for a sinh-Gordon Model with Real q." Symmetry 16, no. 8 (2024): 947. http://dx.doi.org/10.3390/sym16080947.

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Recently, Bazhanov and Sergeev have described an Ising-type integrable model which can be identified as a sinh-Gordon-type model with an infinite number of states but with a real parameter q. This model is the subject of Sklyanin’s Functional Bethe Ansatz. We develop in this paper the whole technique of the FBA which includes: (1) Construction of eigenstates of an off-diagonal element of a monodromy matrix. The most important ingredients of these eigenstates are the Clebsh-Gordan coefficients of the corresponding representation. (2) Separately, we discuss the Clebsh-Gordan coefficients, as wel

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