On link invariants and topological string amplitudes

We explicitly show that the new polynomial invariants for knots, upto nine crossings, agree with the Ooguri–Vafa conjecture relating Chern–Simons gauge theory to topological string theory on the resolution of the conifold.     

Some topological invariants for three-dimensional flows

We deal here with vector fields on three manifolds. For a system with a homoclinic orbit to a saddle-focus point, we show that the imaginary part of the complex eigenvalues is a conjugacy invariant. We show also that the ratio of the real part of the complex eigenvalue over the real one is invariant under topological equivalence. For a system with two saddle-focus points and an orbit connecting the one-dimensional invariant manifold of those points, we compute a conjugacy invariant related to the eigenvalues of the vector field at the singularities. (c) 2001 American Institute of Physics.     

From Seiberg-Witten invariants to topological Green-Schwarz string

In this note we describe the physics of equivalence of the Seiberg-Witten invariants of 4-manifolds and certain Gromov-Witten invariants defined by pseudo-holomorphic curves. We show that physics of the pseudo-holomorphic curves should be governed by the N=2 Green-Schwarz string.     

探究弦球链系统的拓扑不变量

周期性弦球链体系作为典型的声子晶体体系,其能带结构已得到了广泛而深入的研究.在实际观测体系能谱的过程中,会探测到一类特殊的能量本征态——边缘态(能量处在带隙间,且波函数局域在体系边缘的态),同时观察到其存在具有一定的鲁棒性.由于实验上观测到的弦球链边缘态的性质与电子体系中拓扑绝缘体的边缘态性质的相似性,可以利用同一套能带拓扑的语言研究弦球链体系.本文通过数值计算能带的拓扑不变量,揭示了弦球链体系非平庸的拓扑性质,从而证明了实验上探测到的边缘态是拓扑保护的边缘态.基于数值计算的结果,讨论了体系的拓扑相变与边缘态的输运性质     

Topological invariants for anomalous Floquet higher-order topological insulators

We review the recent development in constructing higher-order topological band insulators under strong periodic drivings. In particular, we focus on various approaches in formulating the anomalous Floquet topological invariants beyond (quasi-)static band topology, and compare their different physical consequences.     

Composite representation invariants and unoriented topological string amplitudes

Sinha and Vafa [1] had conjectured that the SO   Chern–Simons gauge theory on S³$S^3$ must be dual to the closed A  -model topological string on the orientifold of a resolved conifold. Though the Chern–Simons free energy could be rewritten in terms of the topological string amplitudes providing evidence for the conjecture, we needed a novel idea in the context of Wilson loop observables to extract cross-cap c=0,1,2$c=0,1,2$ topological amplitudes. Recent paper of Marino [2] based on the work of Morton and Ryder [3] has clearly shown that the composite representation placed on the knots and links plays a crucial role to rewrite the topological string cross-cap c=0$c=0$ amplitude. This enables extracting the unoriented cross-cap c=2$c=2$ topological amplitude. In this paper, we have explicitly worked out the composite invariants for some framed knots and links carrying composite representations in U(N)$U(N)$ Chern–Simons theory. We have verified generalised Rudolph's theorem, which relates composite invariants to the invariants in SO(N)$\mathit{SO}(N)$ Chern–Simons theory, and also verified Marino's conjectures on the integrality properties of the topological string amplitudes. For some framed knots and links, we have tabulated the BPS integer invariants for cross-cap c=0$c=0$ and c=2$c=2$ giving the open-string topological amplitude on the orientifold of the resolved conifold.     

New gauge anomalies and topological invariants in various dimensions

The very precise combined HERA data provides a testing ground in which the relevance of novel QCD regimes, other than the successful linear DGLAP evolution, in small-x inclusive DIS data can be ascertained. We present a study of the dependence of the AAMQS fits, based on the running coupling BK non-linear evolution equations (rcBK), on the fitted dataset. This allows for the identification of the kinematical region where rcBK accurately describes the data, and thus for the determination of its applicability boundary. We compare the rcBK results with NNLO DGLAP fits, obtained with the NNPDF methodology with analogous kinematical cuts. Further, we explore the impact on LHC phenomenology of applying stringent kinematical cuts to the low-x HERA data in a DGLAP fit.     

On Dijkgraaf-Witten-type invariants

We explicitly construct a series of lattice models based upon the gauge group Z _p which have the property of subdivision invariance, when the coupling parameter is quantized and the field configurations are restricted to satisfy a type of mod-p flatness condition. The simplest model of this type yields the Dijkgraaf-Witten invariant of a 3-manifold and is based upon a single link, or 1-simplex, field. Depending upon the manifold's dimension, other models may have more than one species of field variable, and these may be based on higher-dimensional simplices.Supported by Stichting voor Fundamenteel Onderzoek der Materie (FOM).     

Topological invariants for standard model: From semi-metal to topological insulator

We consider topological invariants describing semimetal (gapless) and insulating (gapped) states of the quantum vacuum of Standard Model and possible quantum phase transitions between these states.     

A combinatorial approach to topological quantum field theories and invariants of graphs

The combinatorial state sum of Turaev and Viro for a compact 3-manifold in terms of quantum 6j-symbols is generalized by introducing observables in the form of coloured graphs. They satisfy braiding relations and allow for surgeries and a discussion of cobordism theory. Application of these techniques give the dimension and an explicit basis for the vector space of the topological quantum field theory associated to any Riemann surface with arbitrary coloured punctures.Dedicated to H. Araki and E. Lieb on the occasion of their 60^th birthdaysSupported by DFG, SFB 288 Differentialgeometrie und Quantenphysik     

On universal Vassiliev invariants

Using properties of ordered exponentials and the definition of the Drinfeld associator as a monodromy operator for the Knizhnik-Zamolodchikov equations, we prove that the analytic and the combinatorial definitions of the universal Vassiliev invariants of links are equivalent.Supported by Fonds national suisse de la recherche scientifiqueURA 1436 du CNRS, associée à l'Ecole Normale Supérieure de Lyon et au laboratoire d'Annecyle-Vieux de Physique des Particules     

关于含时不变量的一些普遍性质

讨论了量子力学中含时不变量的一些普遍性质,其中有些是文献中未曾提及的．     

On the two-vector invariants of point groups

The paper is devoted to group-theoretical analysis of a function of two vectors, invariant under some point group (special attention is paid to the O_h group). In particular, the invariants in the direct product of spaces transforming according to the lth and l'th irreducible representations of the rotation group are studied. A compact formula determining the number of such invariants for the group O_h is found. It is shown that all possible invariants in the considered product space can be constructed from all possible scalar products of vector functions of both vectors transforming according to complex conjugate irreducible representations. The addition theorem for these functions is proved.     

A new way to construct topological invariants of non-Hermitian systems with the non-Hermitian skin effect

The non-Hermitian skin effect breaks the conventional bulk–boundary correspondence and leads to non-Bloch topological invariants. Inspired by the fact that the topological protected zero modes are immune to perturbations, we construct a partner of a non-Hermitian system by getting rid of the non-Hermitian skin effect. Through adjusting the imbalance hopping, we find that the existence of zero-energy boundary states still dictate the bulk topological invariants based on the band-theory framework. Two non-Hermitian Su–Schrieffer–Heeger(SSH) models are used to illuminate the ideas. Specially, we obtain the winding numbers in analytical form without the introduction of the generalized Brillouin zone. The work gives an alternative method to calculate the topological invariants of non-Hermitian systems.     

Quantum groups at odd roots of unity and topological invariants of 3-manifolds

Topological invariants of three-manifolds are constructed using quantum groups associated with theA, B, C andD series of Lie algebras at odd roots of unity. These invariants are also explicitly computed for the lens spaces.     

On topological excitations with isovector topological charges

It is shown that degenerate systems with order parameter ψ taking on values in compact homogeneous subspaces T _G or G/T _G (where G is a simple compact group and T _G is its maximum commutative subgroup) possess a rich collection of topological excitations (vortices or, correspondingly, instantons) with isovector topological charges. The corresponding homotopy groups are found for all G. The possibility of a topological interpretation of the quantum numbers of the groups and particles is discussed. Pis’ma Zh. éksp. Teor. Fiz. 63, No. 10, 758–762 (25 May 1996)     

On the invariants of the complex Ginzburg-Landau equation

Using Lie group theory, both the invariants and the similarity variables of the complex Ginzburg—Landau equation V_xx = a(V_t + bV) + cV|V|^2kwherea,b,c?Iandk?I are constructed.