Tree-level recursion relation and dual superconformal symmetry of the ABJM theory

We propose a recursion relation for tree-level scattering amplitudes in three-dimensional Chern-Simons-matter theories. The recursion relation involves a complex deformation of momenta which generalizes the BCFW-deformation used in higher dimensions. Using background field methods, we show that all tree-level superamplitudes of the ABJM theory vanish for large deformations, establishing the validity of the recursion formula. Furthermore, we use the recursion relation to compute six-point and eight-point component amplitudes and match them with independent computations based on Feynman diagrams or the Grassmannian integral formula. As an application of the recursion relation, we prove that all tree-level amplitudes of the ABJM theory have dual superconformal symmetry. Using generalized unitarity methods, we extend this symmetry to the cut-constructible parts of the loop amplitudes.     

Construction of AFLT States for W_NH Symmetry,Analytic Continuation and Integrability on AGT Relation

We propose a Hamiltonian to the construction of the AFLT states for WNsymmetry. We generalize the AGT relation to generic(extended) conformal field theory with 1 ≤ c ∞. We analyze the triangular structure hidden in the AGT relation with WNsymmetry in detail and the triangular structure implies the integrability.     

Kac determinant and singular vector of the level N representation of Ding–Iohara–Miki algebra

In this paper, we obtain the formula for the Kac determinant of the algebra arising from the level N representation of the Ding–Iohara–Miki algebra. It is also discovered that its singular vectors correspond to generalized Macdonald functions (the q-deformed version of the AFLT basis).

     

Construction of Recursion Formulas for Lax Pairs of (2+1)-Dimensional Equation:Modified Generalized Dispersive Long Wave Equation

In this article, we study the Lax pairs of (2+1)-dimensional equation: the modified generalized dispersive long wave (MGDLW) equation. Based on the well-known binary Darboux transformation, we dig out the recursion formulas of the first part of the Lax pairs. Then by further discussion and doing some revisional work, we make the recursion formulas fit for the second part of Lax pairs. At last, some solutions to the MGDLW equation are worked out by using the recursion formula.     

Computation of Open Gromov–Witten Invariants for Toric Calabi–Yau 3-Folds by Topological Recursion,a Proof of the BKMP Conjecture

The BKMP conjecture (2006–2008) proposed a new method to compute closed and open Gromov–Witten invariants for every toric Calabi–Yau 3-folds, through a topological recursion based on mirror symmetry. So far, this conjecture has been verified to low genus for several toric CY3folds, and proved to all genus only for \({\mathbb{C}^3}\). In this article we prove the general case. Our proof is based on the fact that both sides of the conjecture can be naturally written in terms of combinatorial sums of weighted graphs: on the A-model side this is the localization formula, and on the B-model side the graphs encode the recursive algorithm of the topological recursion.One can slightly reorganize the set of graphs obtained in the B-side, so that it coincides with the one obtained by localization in the A-model. Then it suffices to compare the weights of vertices and edges of graphs on each side, which is done in two steps: the weights coincide in the large radius limit, due to the fact that the toric graph is the tropical limit of the mirror curve. Then the derivatives with respect to Kähler radius coincide due to the special geometry property implied by the topological recursion.     

Does the quenched reduced U(∞) chiral model break spontaneously in weak coupling?

The U(N) chiral model, when quenched using Parisi's rule, has a [U(1) × U(1)]^N/U(1) global invariance. To determine whether this symmetry breaks spontaneously in weak coupling for N=∞, a one-loop calculation of the distribution of eigenvalues of the single U(N) matrix of the model is performed. This distribution is shown to be uniform on the unit circle and hence, no symmetry breaking occurs. Further, the order parameter | tr U|²/N², which should be zero at N=∞ in the absence of spontaneous symmetry breaking, is evaluated in the weak coupling phase for one, two and three dimensions for N varying from 2 to 50 by Monte Carlo simulation of the quenched model. The data indicate that this parameter indeed goes to zero as N→∞ implying that the symmetry does not break.     

耦合Burgers系统的Painlevé性质分析，对称和对称约化

本文给出了耦合Burgers系统的Painlevé性质，逆强对称算子，无穷多对称和李对称约化。通过把强对称和逆强对称算子重复多次作用到耦合Burgers模型的一些平庸对称，如恒等变换，空间平移变换和标度变换上，我们得到了三族无穷多对称。这些对称构成了无穷维李代数。用其中的有限维子代数——点李代数对模型进行对称约化，得到了模型的群不变解。     

Levinson-durbin''''s recursive solution |k_i| =1 for voice linear prediction coefficients

AsweknowLPCisoneofthemostpowerfulvoiceana1ysistechniques.Thebasicprinci-pleofLPCisthatavoicesamp1ecanbcpredictedasa1inearcombinationofthepastsamples.TheLPCsystemtransformfunctioncanbedenotedasThusthebasicquestionofLPCistodctermineasetofvoiccpredictorcoefficients{a,}.TheLpcoefficients{a,}canbedeterminedbyminimizingthesumofthesquareddifferences(overafiniteinterval)betweentherealvoicesampleandthelinearlypredictedones.ThisapproachgivesthcfollowingresultTheminimummean-squaredpredictionerrorca…     

Real space renormalization group approach to the two-dimensional antiferromagnetic Heisenberg model (I) – The singlet triplet gap

The low energy behaviour of the two-dimensional antiferromagnetic Heisenberg model is studied in the sector with total spins S = 0,1,2 by means of a renormalization group procedure, which generates a recursion formula for the interaction matrix Δ_S ⁽ⁿ⁺¹⁾ of 4 neighbouring “n clusters” of size 2ⁿ × 2ⁿ, n = 1,2,3,... from the corresponding quantities Δ_S ⁽ⁿ⁾. Conservation of total spin S is implemented explicitly and plays an important role. It is shown, how the ground state energies E_S ⁽ⁿ⁺¹⁾, S = 0,1,2 approach each other for increasing n, i.e. system size. The most relevant couplings in the interaction matrices are generated by the transitions 〈S’,m’;n+1|S_q ^*|S,m;n+1〉 between the ground states |S,m;n+1〉 (m = -S,...,S) on an (n+1)-cluster of size 2ⁿ⁺¹ × 2ⁿ⁺¹, mediated by the staggered spin operator S_q ^*.     

DYNAMICAL SYMMETRY BREAKING IN MODELS OF SPIMOR FIELDS WITH QUARTIC INTERACTIONS IN (1+1) DIMENSIONS

A nonperturbative method, namely, variarional method together with canonical transformations, is developtd to study dyndcal symmetry breaking. This method has been applied in the models of two dimensional massless fermion fields with quartic interactions. The results imply that the mechanism of dynamical symmetry breaking bears some analogy to the phenomenon of superconductivity. The new vacuum |õ> is just a relativistic BCS groundstate, In this vacuum |õ>, we can observe a quasi-particle with mass "MP" Furthermore, correlative vacuum |õ> exists and the mesons emerge with masses "O" and "2M_F". It is also shown that dynamical symmetry breaking always occurs in the models with infrared slavery and asymptotic freedom, while it is meaningless to dis-cuss dynamical symmetry breaking in infrared stable theory.     

Thermodynamic properties of the mixed spin-1/2 and spin-1 Ising chain with both longitudinal and transverse single-ion anisotropies

The mixed spin-1/2 and spin-1 Ising chain with both longitudinal and transverse single-ion anisotropies is solved exactly by means of a mapping to the spin-1/2 Ising chains with alternating transverse fields and the Jordan-Wigner transformation. Within this scheme, the thermodynamic quantities of this model are rigorously determined by a recursion formula derived for the partition function based on the reduced spin-1/2 transverse Ising model. The corresponding thermodynamic properties are calculated and discussed.     

Two new supersymmetric equations of Harry Dym type and their supersymmetric reciprocal transformations

A new N=1$N=1$ supersymmetric Harry Dym equation is constructed by applying supersymmetric reciprocal transformation to a trivial supersymmetric Harry Dym equation, and its recursion operator and Lax formulation are also obtained. Within the framework of symmetry approach, a class of 3rd order supersymmetric equations of Harry Dym type are considered. In addition to five known integrable equations, a new supersymmetric equation, admitting 5th order generalized symmetry, is shown to be linearizable through supersymmetric reciprocal transformation. Furthermore, its Lax representation and recursion operator are given so that the integrability of this new equation is confirmed.     

无简并微扰论公式的研究

本文介绍了Ｗｉｇｎｅｒ公式，Ｇｏｌｄｓｔｏｎｅ公式与Ｓｃｈｒｏｄｉｎｇｅｒ公式及其递推形式，比较表明，Ｓｃｈｒｏｄｉｎｇｅｒ公式的递推形式更容易利用计算机程序实现。文中还给出了逐级计算非连通项到任意级的方法。     

包含非中心电耦极矩的环状非谐振子势场赝自旋对称性的三对角化表示

提出了一个包含非中心电耦极矩分量的环状非谐振子势模型,在能够负载Dirac波动算子三对角化表示的完全平方可积L~2空间讨论了这一势场的赝自旋对称性.利用三对角化矩阵方案,使得求解Dirac方程转换为寻求波函数展开系数满足的三项递推关系式.角向波函数和径向波函数分别以Jacobi多项式和Laguerre多项式表示.由径向分量展开系数递推关系式的对角化条件得到束缚态的能量谱,显示出这一势模型具有严格的赝自旋对称性     

Solution of the Dirac equation in the tridiagonal representation with pseudospin symmetry for an anharmonic oscillator and electric dipole ring-shaped potential

An anharmonic oscillatory potential is proposed in which a noncentral electric dipole is included. The pseudospin symmetry for this potential is investigated by working in a complete square integrable basis that supports a tridiagonal matrix representation of the wave operator. The resulting three-term recursion relation for the expansion coefficients of the wavefunctions (both angular and radial) are presented. The angular/radial wavefunction is written in terms of Jacobi/Laguerre polynomials. The discrete spectrum of the bound states is obtained by the diagonalization of the radial recursion relation. The algebraic properties of the energy equation are also discussed, showing the exact pseudospin symmetry.     

Some properties of the moshinsky coefficients

Moshinsky has suggested a method of calculating Talmi transformation coefficients in which the coefficients with two radial quantum numbers equal to zero are calculated first and then recursion formulas over radial quantum numbers are applied. The coefficients with two radial quantum numbers equal to zero are shown to be expressed as factorials, in terms of a particular type of coefficient with all radial quantum numbers equal to zero, and a simple formula for these coefficients is given. Some interesting properties of the Moshinsky coefficient sums are found and a simple formula expressing the coefficients of the “three-body” hyperspherical function transformation in terms of generalized Moshinsky coefficients is obtained. Recursion formulas and symmetry relations for the Moshinsky coefficients are also discussed.     

Formulas to Solve the （2＋1）—Dimensional System：The （2＋1）—Extension of Classical Boussinesq System

We deal with the (2 1)-extension of classical Boussinesq system,which can reduce to several meaningful (1 1)-dimensional systems.By studying its lax paire,we put forward invariances of Lax pair at first,then a recursion formula depending on an arbitrary function is derived,At last,some solutions of the (2 1)-extension of classical Boussinesq system are digged out by using the formula.     

Deconfinement phase transition in a two-dimensional model of interacting 2 × 2 plaquettes

The low energy behaviour of the two-dimensional antiferromagnetic Heisenberg model is studied in the sector with total spins S = 0,1,2 by means of a renormalization group procedure, which generates a recursion formula for the interaction matrix Δ_S ⁽ⁿ⁺¹⁾ of 4 neighbouring “n clusters” of size 2ⁿ × 2ⁿ, n = 1,2,3,... from the corresponding quantities Δ_S ⁽ⁿ⁾. Conservation of total spin S is implemented explicitly and plays an important role. It is shown, how the ground state energies E_S ⁽ⁿ⁺¹⁾, S = 0,1,2 approach each other for increasing n, i.e. system size. The most relevant couplings in the interaction matrices are generated by the transitions 〈S’,m’;n+1|S_q ^*|S,m;n+1〉 between the ground states |S,m;n+1〉 (m = -S,...,S) on an (n+1)-cluster of size 2ⁿ⁺¹ × 2ⁿ⁺¹, mediated by the staggered spin operator S_q ^*.     

Super-Landau models: Unitarity and hidden symmetries

We review a recent progress in constructing and studying superextensions of the Landau problem of a quantum particle moving on the two-sphere S ². These are the superspherical and superflag Landau models describing a nonrelativistic particle on the supermanifolds SU(2|1)/U(1|1) and SU(2|1)/[U(1) x U(1)]. Though in both models, under the “naive” definition of the inner product, there are negative-norm states, all norms can be made positive-definite by introducing a nontrivial “metric” in the space of states. So both models are unitary. The superflag model is shown to be quantum-equivalent to the supersphere model for the special choice of the intrinsic parameters. We also address the planar limit of the superspherical model, in which there arises the hidden world-line N =2 supersymmetry. An analog of this phenomenon in the superflag model is a dynamical enhancement of its SU(2|1) symmetry to SU(2|2).     

Density of states in coupled chains with off-diagonal disorder

We compute the density of states rho(varepsilon) in N coupled chains with random hopping. At zero energy, rho(varepsilon) shows a singularity that strongly depends on the parity of N. For odd N, rho(varepsilon) approximately 1/|varepsilonln (3)varepsilon|, with and without time-reversal symmetry. For even N, rho(varepsilon) approximately |lnvarepsilon| in the presence of time-reversal symmetry, while there is a pseudogap, rho(varepsilon) approximately |varepsilonlnvarepsilon|, in the absence of time-reversal symmetry.