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Journal articles on the topic 'Monodromy operator'

Author: Grafiati

Published: 4 June 2021

Last updated: 16 February 2022

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1

Dimca, Alexandru. "Tate properties, polynomial-count varieties, and monodromy of hyperplane arrangements." Nagoya Mathematical Journal 206 (June 2012): 75–97. http://dx.doi.org/10.1017/s0027763000010540.

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AbstractThe order of the Milnor fiber monodromy operator of a central hyperplane arrangement is shown to be combinatorially determined. In particular, a necessary and sufficient condition for the triviality of this monodromy operator is given.It is known that the complement of a complex hyperplane arrangement is cohomologically Tate and, if the arrangement is defined over ℚ, has polynomial count. We show that these properties hold for the corresponding Milnor fibers if the monodromy is trivial.We construct a hyperplane arrangement defined over ℚ, whose Milnor fiber has a nontrivial monodromy operator, is cohomologically Tate, and has no polynomial count. Such examples are shown not to exist in low dimensions.

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2

Dimca, Alexandru. "Tate properties, polynomial-count varieties, and monodromy of hyperplane arrangements." Nagoya Mathematical Journal 206 (June 2012): 75–97. http://dx.doi.org/10.1215/00277630-1548502.

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AbstractThe order of the Milnor fiber monodromy operator of a central hyperplane arrangement is shown to be combinatorially determined. In particular, a necessary and sufficient condition for the triviality of this monodromy operator is given.It is known that the complement of a complex hyperplane arrangement is cohomologically Tate and, if the arrangement is defined over ℚ, has polynomial count. We show that these properties hold for the corresponding Milnor fibers if the monodromy is trivial.We construct a hyperplane arrangement defined over ℚ, whose Milnor fiber has a nontrivial monodromy operator, is cohomologically Tate, and has no polynomial count. Such examples are shown not to exist in low dimensions.

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3

Wang, Na, and Ke Wu. "Vertex operators, t-boson model and weighted plane partitions in finite boxes." Modern Physics Letters B 32, no. 05 (February 20, 2018): 1850061. http://dx.doi.org/10.1142/s0217984918500616.

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We consider two different subjects: the algebra of Hall–Littlewood functions and t-boson model. Tsilevich and Sułkowski, respectively, give that the creation operator [Formula: see text] in the monodromy matrix of t-boson model can be represented by [Formula: see text], where [Formula: see text] and [Formula: see text] are vertex operators closely related to the Hall–Littlewood functions. In this paper, we obtain that the annihilation operator [Formula: see text] in the monodromy matrix and other relations of t-boson model can also be realized in the algebra of Hall–Littlewood functions. Meanwhile, we get that the generating functions of weighted plane partitions in finite boxes can be obtained from the operators [Formula: see text].

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4

BHATTACHARYA, GAUTAM, and SASANKA GHOSH. "ALGEBRAIC BETHE ANSATZ SCHEME FOR RELATIVISTIC INTEGRABLE FIELD THEORIES IN CONTINUUM-: SINE-GORDON MODEL." International Journal of Modern Physics A 04, no. 03 (February 1989): 627–47. http://dx.doi.org/10.1142/s0217751x89000303.

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The linear problem associated with the Lax operator of the classical sine-Gordon theory can be recast into the monodromy matrix form that can be extended to quantum theory as well. Product of the quantum monodromy matrices will have contributions from the singularities arising out of the operator product expansions of sine-Gordon field. This enables one to find the star-triangle relations. This is a generalization of the method used by Thacker for the non-relativistic nonlinear Schrödinger field theory. In the infinite volume limit, it leads to an unambiguous description of the algebra involving the scattering data operators. Starting from a vacuum the module of physical states are constructed by the application of chains of the scattering operators and they turn out to have definite eigenvalues of energy and momentum. The mass-spectra and two-particle S matrix elements can be calculated straightforwardly and they coincide with the well-known results of Zamolodchikov and Zamolodchikov. It is also shown that the particle states can be created by a generalized Jordon-Wigner transformation of the free fermionic creation operators — thus establishing a link between the S matrix element and certain co-cycles.

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5

BRODA, B. "A THREE-DIMENSIONAL COVARIANT APPROACH TO MONODROMY (SKEIN RELATIONS) IN CHERN-SIMONS THEORY." Modern Physics Letters A 05, no. 32 (December 30, 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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6

Helm, David, and Eric Katz. "Monodromy Filtrations and the Topology of Tropical Varieties." Canadian Journal of Mathematics 64, no. 4 (August 1, 2012): 845–68. http://dx.doi.org/10.4153/cjm-2011-067-9.

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AbstractWe study the topology of tropical varieties that arise from a certain natural class of varieties. We use the theory of tropical degenerations to construct a natural, “multiplicity-free” parameterization of Trop(X) by a topological space ГXand give a geometric interpretation of the cohomology of ГXin terms of the action of a monodromy operator on the cohomology ofX. This gives bounds on the Betti numbers of ГXin terms of the Betti numbers ofXwhich constrain the topology of Trop(X). We also obtain a description of the top power of the monodromy operator acting on middle cohomology ofXin terms of the volume pairing on ГX.

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7

Yudovich, V. I. "Periodic differential equations with self-adjoint monodromy operator." Sbornik: Mathematics 192, no. 3 (April 30, 2001): 455–78. http://dx.doi.org/10.1070/sm2001v192n03abeh000554.

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8

Dat, Jean-François. "Opérateur de Lefschetz sur les tours de Drinfeld et Lubin–Tate." Compositio Mathematica 148, no. 2 (January 25, 2012): 507–30. http://dx.doi.org/10.1112/s0010437x11007214.

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AbstractWe define and study a Lefschetz operator on the equivariant cohomology complex of the Drinfeld and Lubin–Tate towers. For ℓ-adic coefficients we show how this operator induces a geometric realization of the Langlands correspondence composed with the Zelevinski involution for elliptic representations. Combined with our previous study of the monodromy operator, this suggests a possible extension of Arthur’s philosophy for unitary representations occurring in the intersection cohomology of Shimura varieties to the possibly non-unitary representations occurring in the cohomology of Rapoport–Zink spaces. However, our motivation for studying the Lefschetz operator comes from the hope that its geometric nature will enable us to realize the mod-ℓ Langlands correspondence due to Vignéras. We discuss this problem and propose a conjecture.

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9

OVCHINNIKOV, A. A. "CONSTRUCTION OF MONODROMY MATRIX IN THE F-BASIS AND SCALAR PRODUCTS IN SPIN CHAINS." International Journal of Modern Physics A 16, no. 12 (May 10, 2001): 2175–93. http://dx.doi.org/10.1142/s0217751x01003743.

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We present in a simple terms the theory of the factorizing operator introduced recently by Maillet and Sanches de Santos for the spin-1/2 chains. We obtain the explicit expressions for the matrix elements of the factorizing operator in terms of the elements of the monodromy matrix. We use this results to derive the expression for the general scalar product for the quantum spin chain. We comment on the previous determination of the scalar product of Bethe eigenstate with an arbitrary dual state. We also establish the direct correspondence between the calculations of scalar products in the F-basis and the usual basis.

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10

Korotyaev, E. L. "ON THE EIGENFUNCTIONS OF THE MONODROMY OPERATOR OF THE SCHRÖDINGER OPERATOR WITH A TIME-PERIODIC POTENTIAL." Mathematics of the USSR-Sbornik 52, no. 2 (February 28, 1985): 423–38. http://dx.doi.org/10.1070/sm1985v052n02abeh002898.

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11

MARKMAN, EYAL. "INTEGRAL CONSTRAINTS ON THE MONODROMY GROUP OF THE HYPERKÄHLER RESOLUTION OF A SYMMETRIC PRODUCT OF A K3 SURFACE." International Journal of Mathematics 21, no. 02 (February 2010): 169–223. http://dx.doi.org/10.1142/s0129167x10005957.

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Let S[n]be the Hilbert scheme of length n subschemes of a K3 surface S. H2(S[n],ℤ) is endowed with the Beauville–Bogomolov bilinear form. Denote by Mon the subgroup of GL [H*(S[n],ℤ)] generated by monodromy operators, and let Mon2be its image in OH2(S[n],ℤ). We prove that Mon2is the subgroup generated by reflections with respect to +2 and -2 classes (Theorem 1.2). Thus Mon2does not surject onto OH2(S[n],ℤ)/(±1), when n - 1 is not a prime power.As a consequence, we get counterexamples to a version of the weight 2 Torelli question for hyperKähler varieties X deformation equivalent to S[n]. The weight 2 Hodge structure on H2(X,ℤ) does not determine the bimeromorphic class of X, whenever n - 1 is not a prime power (the first case being n = 7). There are at least 2ρ(n - 1) - 1distinct bimeromorphic classes of X with a given generic weight 2 Hodge structure, where ρ(n - 1) is the Euler number of n - 1.The second main result states, that if a monodromy operator acts as the identity on H2(S[n],ℤ), then it acts as the identity on Hk(S[n],ℤ), 0 ≤ k ≤ n + 2 (Theorem 1.5). We conclude the injectivity of the restriction homomorphism Mon → Mon2, if n ≡ 0 or n ≡ 1 modulo 4 (Corollary 1.6).

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12

Deshmukh, Venkatesh. "Approximate Stability Analysis and Computation of Solutions of Nonlinear Delay Differential Algebraic Equations with Time Periodic Coefficients." Journal of Vibration and Control 16, no. 7-8 (June 2010): 1235–60. http://dx.doi.org/10.1177/1077546309341137.

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Approximate stability analysis of nonlinear delay differential algebraic equations (DDAEs) with periodic coefficients is proposed with a geometric interpretation of evolution of the linearized system. Firstly, a numerical algorithm based on direct integration by expansion in terms of Chebyshev polynomials is derived for linear analysis. The proposed algorithm is shown to have deeper connections with and be computationally less cumbersome than the solution of the underlying semi-explicit system via a similarity transformation. The stability of time periodic DDAE systems is characterized by the spectral radius of a “monodromy matrix”, which is a finite-dimensional approximation of a compact infinite-dimensional operator. The monodromy matrix is essentially a map of the Chebyshev coefficients (or collocation vector) of the state from the delay interval to the next adjacent interval of time. The computations are entirely performed with the original system to avoid cumbersome transformations associated with the semi-explicit form of the system. Next, two computational algorithms, the first based on perturbation series and the second based on Chebyshev spectral collocation, are detailed to obtain solutions of nonlinear DDAEs with periodic coefficients for consistent initial functions.

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13

LUZYANINA, TATYANA, and KOEN ENGELBORGHS. "COMPUTING FLOQUET MULTIPLIERS FOR FUNCTIONAL DIFFERENTIAL EQUATIONS." International Journal of Bifurcation and Chaos 12, no. 12 (December 2002): 2977–89. http://dx.doi.org/10.1142/s0218127402006291.

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Floquet multipliers determine the local asymptotic stability of a periodic solution and, in the context of parameter dependence, determine also its bifurcations. This paper deals with numerical aspects of the computation of the Floquet multipliers for three classes of functional differential equations: ordinary differential equations (ODEs), differential equations with constant delay (DDEs) and differential equations with state-dependent delay (sd-DDEs). Using a collocation approach for computing periodic solutions, we obtain an approximation of the (corresponding) monodromy operator, a monodromy matrix. The eigenvalues of this matrix form an approximation to the Floquet multipliers. The accuracy of the computed multipliers is an important issue in bifurcation analysis of a dynamical system. As far as we know, no prior work on the study of the convergence and accuracy of computed Floquet multipliers for DDEs and sd-DDEs exists. We analyze the dependency of the accuracy of the computed multipliers on the parameters and on the type of collocation approximation. In particular, we show that the accuracy of the computed trivial multiplier is not always comparable to the accuracy of the computed periodic solution and the accuracy of the other computed multipliers.

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14

Kim, Jung Hoon, Tomomichi Hagiwara, and Kentaro Hirata. "Spectrum of Monodromy Operator for a Time-Delay System With Application to Stability Analysis." IEEE Transactions on Automatic Control 60, no. 12 (December 2015): 3385–90. http://dx.doi.org/10.1109/tac.2015.2422479.

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15

DUCLOS, P., E. SOCCORSI, P. ŠŤOVÍČEK, and M. VITTOT. "ON THE STABILITY OF PERIODICALLY TIME-DEPENDENT QUANTUM SYSTEMS." Reviews in Mathematical Physics 20, no. 06 (July 2008): 725–64. http://dx.doi.org/10.1142/s0129055x08003390.

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The main motivation of this article is to derive sufficient conditions for dynamical stability of periodically driven quantum systems described by a Hamiltonian H(t), i.e. conditions under which it holds true sup t ∈ ℝ|〈ψt, H(t)ψt〉| < ∞ where ψt denotes a trajectory at time t of the quantum system under consideration. We start from an analysis of the domain of the quasi-energy operator. Next, we show, under certain assumptions, that if the spectrum of the monodromy (Floquet) operator U(T, 0) is pure point then there exists a dense subspace of initial conditions for which the mean value of the energy is uniformly bounded in the course of time. Further, we show that if the propagator admits a differentiable Floquet decomposition then ‖H(t)ψt‖ is bounded in time for any initial condition ψ0, and one employs the quantum KAM algorithm to prove the existence of this type of decomposition for a fairly large class of H(t). In addition, we derive bounds uniform in time on transition probabilities between different energy levels, and we also propose an extension of this approach to the case of a higher order of differentiability of the Floquet decomposition. The procedure is demonstrated on a solvable example of the periodically time-dependent harmonic oscillator.

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16

Vazquez, Eli A., and Joaquin Collado. "Finite Dimensional Approximation of the Monodromy Operator of a Periodic Delay Differential Equation with Piecewise Constant Orthonormal Functions." Applied Mathematics 09, no. 11 (2018): 1315–37. http://dx.doi.org/10.4236/am.2018.911086.

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17

Dolgii, Yu F., and E. V. Ul’yanov. "Singular numbers of the monodromy operator and sufficient conditions of the asymptotic stability of periodic system of differential equations with fixed delay." Proceedings of the Steklov Institute of Mathematics 259, S2 (December 2007): S95—S110. http://dx.doi.org/10.1134/s0081543807060065.

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18

GIORGADZE, G. "MONODROMY APPROACH TO QUANTUM COMPUTING." International Journal of Modern Physics B 16, no. 30 (November 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 involves these matrices.

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19

Wang, Xiang-Sheng, and Jianhong Wu. "Seasonal migration dynamics: periodicity, transition delay and finite-dimensional reduction." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 468, no. 2139 (October 5, 2011): 634–50. http://dx.doi.org/10.1098/rspa.2011.0236.

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We study a model of bird migration between the summer breeding ground and the winter refuge site. The model involves time lags for the transition time between the patches, and the model parameters are periodic in time as the biological activities related to the migration and reproduction are seasonal. It has been shown in previous studies that the model system exhibits threshold dynamics: either all solutions converge to the trivial solution, or the system has a positive and globally attractive periodic solution. Two issues remain and will be addressed in this paper: how to express the threshold condition in terms of model parameters explicitly (rather than the abstract spectral radius of a certain monodromy operator) and how to describe the temporal pattern of the positive periodic solution. We make an interesting and surprising observation that the delay differential system is completely characterized by a finite-dimensional ordinary differential system and then a finite-dimensional map in the sense that the bird population at the initial time of spring migration determines the future status of the system. As a consequence, we derive the threshold condition, explicitly in terms of the model parameters, for the extinction and persistence of the considered bird species.

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20

Ozoegwu, Chigbogu Godwin. "Estimating stability boundaries of distributed-delay oscillators via two-stage numerical integration." Journal of Vibration and Control 24, no. 9 (September 14, 2016): 1728–40. http://dx.doi.org/10.1177/1077546316668466.

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A procedure based on the concept of full-discretization and numerical integration is established in this work for the stability analysis of periodic distributed-delay oscillators governed by delay integro-differential equations (DIDEs). DIDEs can be found as models of mechanical systems suffering from distributed-delay feedback, such as regenerative machine tool vibrations modeled by distributed force and wheel shimmy. Unstable vibrations in such systems are systematically avoided/controlled if the boundaries between the stable and unstable subspaces are established. The presented method involves the two-stage application of numerical integration to the governing DIDE. While the first-stage application discretizes and converts the distributed delay to fine series of short discrete delays, the second-stage application results in discrete solutions paving the way for a new method of constructing a finite monodromy operator. The error and convergence of the method are studied. It is found that the presented method is of the same convergence as that of the well-accepted first-order semi-discretization method, but more computationally efficient in terms of time savings. A number of case study DIDEs that have already been studied in the literature using methods of semi-discretization and spectral finite elements are studied with the presented method. It is seen that the presented method is valid as it produces stability results that compare well with those of the earlier works.

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21

Chicherin, D., and R. Kirschner. "Monodromy operators and symmetric correlators." Journal of Physics: Conference Series 474 (November 29, 2013): 012014. http://dx.doi.org/10.1088/1742-6596/474/1/012014.

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22

Razumov, A. V. "Monodromy operators for higher rank." Journal of Physics A: Mathematical and Theoretical 46, no. 38 (September 4, 2013): 385201. http://dx.doi.org/10.1088/1751-8113/46/38/385201.

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23

Previato, Emma. "Monodromy of boussinesq elliptic operators." Acta Applicandae Mathematicae 36, no. 1-2 (August 1994): 49–55. http://dx.doi.org/10.1007/bf01001542.

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24

Beukers, F. "Unitary monodromy of Lamé differential operators." Regular and Chaotic Dynamics 12, no. 6 (December 2007): 630–41. http://dx.doi.org/10.1134/s1560354707060068.

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25

HOFMANN, JÖRG, and DUCO VAN STRATEN. "SOME MONODROMY GROUPS OF FINITE INDEX IN." Journal of the Australian Mathematical Society 99, no. 1 (March 30, 2015): 48–62. http://dx.doi.org/10.1017/s1446788715000014.

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We determine the index of five of the seven hypergeometric Calabi–Yau operators that have finite index in$\mathit{Sp}_{4}(\mathbb{Z})$and in two cases give a complete description of the monodromy group. Furthermore, we find six nonhypergeometric Calabi–Yau operators with finite index in$\mathit{Sp}_{4}(\mathbb{Z})$, most notably a case where the index is one.

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26

Phan, Quang Sang. "Spectral monodromy of non-self-adjoint operators." Journal of Mathematical Physics 55, no. 1 (January 2014): 013504. http://dx.doi.org/10.1063/1.4855475.

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27

Giorgadze, G. "Gates for quantum computing induced from monodromy operators." Physics of Particles and Nuclei Letters 4, no. 2 (March 2007): 173–75. http://dx.doi.org/10.1134/s1547477107020185.

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28

Oblomkov, A. A. "Monodromy-free Schrödinger operators with quadratically increasing potentials." Theoretical and Mathematical Physics 121, no. 3 (December 1999): 1574–84. http://dx.doi.org/10.1007/bf02557204.

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29

Liţcanu, Răzvan. "Lamé operators with finite monodromy—a combinatorial approach." Journal of Differential Equations 207, no. 1 (December 2004): 93–116. http://dx.doi.org/10.1016/j.jde.2004.08.012.

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30

PONSOT, B. "RECENT PROGRESS IN LIOUVILLE FIELD THEORY (REVIEW)." International Journal of Modern Physics A 19, supp02 (May 2004): 311–35. http://dx.doi.org/10.1142/s0217751x0402049x.

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An explicit construction for the monodromy of the Liouville conformal blocks in terms of Racah-Wigner coefficients of the quantum group [Formula: see text] is proposed. As a consequence, crossing-symmetry for four point functions is analytically proven, and the expression for the correlator of three boundary operators is obtained.

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31

ITOH, TAKASHI, and YASUHIKO YAMADA. "EXPLICIT EXCHANGE RELATIONS IN THE SL(2) WESS-ZUMINO-NOVIKOV-WITTEN MODEL." International Journal of Modern Physics A 06, no. 18 (July 30, 1991): 3283–91. http://dx.doi.org/10.1142/s0217751x91001593.

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We discuss, in the SL(2) Wess-Zumino-Novikov-Witten model, the quantum exchange relations for chiral vertex operators with arbitrary spin by using a free-field realization. We give a recursion formula for the monodromy matrices and the solution explicitly. They coincide with the q-6j symbol of the relevant quantum group Uq(sl(2)) .

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32

Dunkl, Charles. "Differential-difference operators and monodromy representations of Hecke algebras." Pacific Journal of Mathematics 159, no. 2 (June 1, 1993): 271–98. http://dx.doi.org/10.2140/pjm.1993.159.271.

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33

Millson, John J., and Valerio Toledano Laredo. "Casimir Operators and Monodromy Representations of Generalised Braid Groups." Transformation Groups 10, no. 2 (June 2005): 217–54. http://dx.doi.org/10.1007/s00031-005-1008-6.

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34

dos Santos, João Pedro Pinto. "Local solutions to positive characteristic non-Archimedean differential equations." Compositio Mathematica 143, no. 6 (November 2007): 1465–77. http://dx.doi.org/10.1112/s0010437x07003089.

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AbstractIn the complex domain, one can integrate (solve) holomorphic ordinary differential equations (ODEs) near a non-singular point. We study the existence of solutions in the case of a positive characteristic base field k which is complete with respect to a non-Archimedean absolute value. ODEs are substituted by modules over a ring of analytic functions endowed with an action of all differential operators. The monodromy groups associated to the corresponding category are computed.

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35

Coleman, Robert, and Adrian Iovita. "The Frobenius and monodromy operators for curves and abelian varieties." Duke Mathematical Journal 97, no. 1 (March 1999): 171–215. http://dx.doi.org/10.1215/s0012-7094-99-09708-9.

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36

Trgo, A. "Monodromy Matrix for Linear Difference Operators with Almost Constant Coefficients." Journal of Mathematical Analysis and Applications 194, no. 3 (September 1995): 697–719. http://dx.doi.org/10.1006/jmaa.1995.1324.

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37

GRILLO, S., and H. MONTANI. "RECTANGULAR YANG–BAXTER ALGEBRAS AND ALTERNATING A-TYPE INTEGRABLE VERTEX MODELS." International Journal of Geometric Methods in Modern Physics 02, no. 06 (December 2005): 1063–80. http://dx.doi.org/10.1142/s0219887805000946.

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Given a couple of Yang–Baxter operators 𝖱[k] and 𝖱[l] corresponding to integrable anisotropic vertex models of Ak-1 and Al-1 type, respectively, we construct and study a class of related lattice models whose monodromy matrices alternate between the mentioned operators. In order to do that, we use a natural generalization of the idea of coproduct in a bialgebra, that appears in the scenario of non-commutative algebraic geometry, related to the notion of internal homomorphisms of quantum spaces. We build up all eigenstates and eigenvalues of the transfer matrix by means of algebraic Bethe ansatz technics, where not only one vector, but a pseudovacuum subspace is needed for the process of diagonalization.

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38

Große-Klönne, Elmar. "Frobenius and monodromy operators in rigid analysis, and Drinfel'd's symmetric space." Journal of Algebraic Geometry 14, no. 3 (September 1, 2005): 391–437. http://dx.doi.org/10.1090/s1056-3911-05-00402-9.

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39

Bogner, Michael, and Stefan Reiter. "Some Fourth Order CY-type Operators with Non-symplectically Rigid Monodromy." Experimental Mathematics 26, no. 1 (July 8, 2016): 98–113. http://dx.doi.org/10.1080/10586458.2015.1127186.

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40

Bartlett, Bruce. "Fusion categories via string diagrams." Communications in Contemporary Mathematics 18, no. 05 (July 18, 2016): 1550080. http://dx.doi.org/10.1142/s0219199715500807.

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We use the string diagram calculus to give graphical proofs of the basic results of Etingof, Nikshych and Ostrik [On fusion categories, Ann. Math. 162 (2005) 581–642; arXiv:math/0203060, doi:10.4007/annals.2005.162.581] on fusion categories. These results include: the quadruple dual is canonically isomorphic to the identity, positivity of the paired dimensions, and Ocneanu rigidity. We introduce the pairing convention as a convenient graphical framework for working with fusion categories. We use this framework to express the pivotal operators as a product of the apex associator monodromy and the pivotal indicators. We also characterize pivotal structures as solutions of an explicit set of algebraic equations over the complex numbers, refining a formula of Wang.

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41

HIRATA, Kentaro, Tomomichi HAGIWARA, and Atsushi ITOKAZU. "Numerical Methods for Spectrum Computation of Monodromy Operators via Non-Causal Hold Discretization." SICE Journal of Control, Measurement, and System Integration 6, no. 1 (2013): 45–53. http://dx.doi.org/10.9746/jcmsi.6.45.

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42

Peters, Chris, and Morihiko Saito. "Lowest weights in cohomology of variations of Hodge structure." Nagoya Mathematical Journal 206 (June 2012): 1–24. http://dx.doi.org/10.1017/s0027763000010503.

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AbstractLetXbe an irreducible complex analytic space withj:U ↪ Xan immersion of a smooth Zariski-open subset, and let 𝕍 be a variation of Hodge structure of weightnoverU. Assume thatXis compact Kähler. Then, provided that the local monodromy operators at infinity are quasi-unipotent,IHk(X, 𝕍) is known to carry a pure Hodge structure of weightk+n, whileHk(U, 𝕍) carries a mixed Hodge structure of weight at leastk+n. In this note it is shown that the image of the natural mapIHk(X, 𝕍) →Hk(U, 𝕍) is the lowest-weight part of this mixed Hodge structure. In the algebraic case this easily follows from the formalism of mixed sheaves, but the analytic case is rather complicated, in particular when the complementX — Uis not a hypersurface.

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43

Peters, Chris, and Morihiko Saito. "Lowest weights in cohomology of variations of Hodge structure." Nagoya Mathematical Journal 206 (June 2012): 1–24. http://dx.doi.org/10.1215/00277630-1548466.

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AbstractLetXbe an irreducible complex analytic space withj:U ↪ Xan immersion of a smooth Zariski-open subset, and let 𝕍 be a variation of Hodge structure of weightnoverU. Assume thatXis compact Kähler. Then, provided that the local monodromy operators at infinity are quasi-unipotent,IHk(X, 𝕍) is known to carry a pure Hodge structure of weightk+n, whileHk(U, 𝕍) carries a mixed Hodge structure of weight at leastk+n. In this note it is shown that the image of the natural mapIHk(X, 𝕍) →Hk(U, 𝕍) is the lowest-weight part of this mixed Hodge structure. In the algebraic case this easily follows from the formalism of mixed sheaves, but the analytic case is rather complicated, in particular when the complementX — Uis not a hypersurface.

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44

ABDURRAHMAN, A., F. ANTON, M. A. NAMAZIE, and C. NÚÑEZ. "N=1 SUPERCONFORMAL MINIMAL MODEL CORRELATION FUNCTIONS ON THE TORUS." International Journal of Modern Physics A 10, no. 28 (November 10, 1995): 3985–4040. http://dx.doi.org/10.1142/s0217751x95001868.

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The Coulomb gas formalism is employed to construct contour integral representations of two-point correlation functions on the torus for the N=1 superconformal unitary discrete series, characterized by the single integer p. (For the particular case of the tricritical Ising model, these include the energy and vacancy density operators.) Modular and monodromy properties of the superconformal blocks are examined, and the generalization to superconformal theories of Verlinde’s results on modular transformations and the fusion algebra are discussed in some detail. For p odd the relevant modular matrix is (with respect to a particular basis) symmetric and unitary, as in ordinary rational conformal theory. However, for p even, there appears to be an obstruction due to the Ramond vacuum state.

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Masui, Yoichiro, Kentaro Hirata, and Tomomichi Hagiwara. "On Numerical Computation of the Spectrum of Monodromy Operators Based on Higher-Order Hold Approximation." Transactions of the Institute of Systems, Control and Information Engineers 29, no. 7 (2016): 324–35. http://dx.doi.org/10.5687/iscie.29.324.

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Olsen, Andrew, and Daleen Kruger. "Examining the musical ‘Sprachvermögen’ in Hendrik Hofmeyr’s operatic monodrama Saartjie." Journal of the Musical Arts in Africa 16, no. 1-2 (July 3, 2019): 45–75. http://dx.doi.org/10.2989/18121004.2019.1686230.

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Tsuchiya, Akihiro, and Yukihiro Kanie. "Vertex operators in the conformal field theory on P 1 and monodromy representations of the braid group." Letters in Mathematical Physics 13, no. 4 (May 1987): 303–12. http://dx.doi.org/10.1007/bf00401159.

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Karanth, D., D. Richmond, and J. R. Schmidt. "A first course in the Yang–Baxter equation." Canadian Journal of Physics 86, no. 10 (October 1, 2008): 1153–76. http://dx.doi.org/10.1139/p08-059.

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A crash course on the Yang–Baxter equation and its applications in the statistical mechanics of lattice vertex models is presented. Using the simple example of a one-dimensional lattice gas, the basic terminology and standard mathematical procedures of statistical mechanics are illustrated. The coproduct notation is introduced via a discussion of the two-dimensional dimer model, and the transfer matrix formulation is elaborated further. The algebraic Bethe Ansatz (ABA) is introduced in the context of the ice model, leading to the Yang–Baxter equations. The motivation for introducing the ABA is developed, emphasizing the similarity to the quantum oscillator problem, in features such as the generation of “excited states”, and the determination of the eigenvalue spectrum. Finally, these methods from the theory of lattice models are applied to the case of a quantum many-body problem, the Heisenberg chain. From the condition that the monodromy matrices satisfy the Yang–Baxter equation, a complete set of mutually commuting operators is derived, and the Heisenberg chain problem is completely solved.PACS Nos.: 02.20.Uw, 24.10.Cn, 87.10.Hk, 05.20.–y

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Balatsky, Alexander. "Spin Singlet Quantum Hall Effect and Nonabelian Landau-Ginzburg Theory." International Journal of Modern Physics B 06, no. 05n06 (March 1992): 765–88. http://dx.doi.org/10.1142/s0217979292000463.

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In this paper we present a theory of Singlet Quantum Hall Effect (SQHE). We show that the Halperin-Haldane SQHE wave function can be written in the form of a product of a wave function for charged semions in a magnetic field and a wave function for the Chiral Spin Liquid of neutral spin-½ semions. We introduce field-theoretic model in which the electron operators are factorized in terms of charged spinless semions (holons) and neutral spin-½ semions (spinons). Broken time reversal symmetry and short ranged spin correlations lead to SU(2)k=1 Chern-Simons term in Landau-Ginzburg action for SQHE phase. We construct appropriate coherent states for SQHE phase and show the existence of SU(2) valued gauge potential. This potential appears as a result of “spin rigidity” of the ground state against any displacements of nodes of wave function from positions of the particles and reflects the nontrivial monodromy in the presence of these displacements. We argue that topological structure of SU(2)k=1 Chern-Simons theory unambiguously dictates semion statistics of spinons.

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Sorensen, Claus M. "Level-raising for Saito–Kurokawa forms." Compositio Mathematica 145, no. 4 (July 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 Steinberg representation of the Klingen parabolic. To transfer $\tilde {\pi }$ back to GSp(4), we use Arthur’s stable trace formula. Since $\tilde {\pi }$ has a local component of the above type, all endoscopic error terms vanish. Indeed, by results due to Weissauer, we only need to show that such a component does not participate in the θ-correspondence with any GO(4); this is an exercise in using Kudla’s filtration of the Jacquet modules of the Weil representation. We therefore obtain a cuspidal automorphic representation $\tilde {\Pi }$ of GSp(4), congruent to Π, which is neither CAP nor endoscopic. It is crucial for our application that we can arrange for $\tilde {\Pi }$ to have vectors fixed by the non-special maximal compact subgroups at all primes dividing N. Since G is necessarily ramified at some prime r, we have to show a non-special analogue of the fundamental lemma at r. Finally, we give an application of our main result to the Bloch–Kato conjecture, assuming a conjecture of Skinner and Urban on the rank of the monodromy operators at the primes dividing N.

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