Non-Abelian global strings at chiral phase transition

We construct non-Abelian global string solutions in the UL_(N)×UR_(N)$U{(N)}_{\mathrm{L}}\times U{(N)}_{\mathrm{R}}$ linear sigma model. These strings are the most fundamental objects which are expected to form during the chiral phase transitions, because the Abelian η^′${\eta }^{\prime }$ string is marginally decomposed into N   of them. We point out Nambu–Goldstone modes of CPN−1$\mathbf{C}{P}^{N-1}$ for breaking of SUV_(N)$\mathit{SU}{(N)}_{\mathrm{V}}$ arise around a non-Abelian vortex.     

Quantum spins on star graphs and the Kondo model

We study the XX model for quantum spins on the star graph with three legs (i.e., on a Y  -junction). By performing a Jordan–Wigner transformation supplemented by the introduction of an auxiliary space we find a Kondo Hamiltonian of fermions, in the spin 1 representation of su(2)$\mathit{su}(2)$, locally coupled with a magnetic impurity. In the continuum limit our model is shown to be equivalent to the 4-channel Kondo model coupling spin-1/2 fermions with a spin-1/2 impurity and exhibiting a non-Fermi liquid behavior. We also show that it is possible to find an XY model such that – after the Jordan–Wigner transformation – one obtains a quadratic fermionic Hamiltonian directly diagonalizable.     

Topological recursion for chord diagrams,RNA complexes,and cells in moduli spaces

We introduce and study the Hermitian matrix model with potential _Vs,t(x)=x²/2−stx/(1−tx)$V_{s,t}(x)=x^2/2-stx/(1-tx)$, which enumerates the number of linear chord diagrams with no isolated vertices of fixed genus with specified numbers of backbones generated by s and chords generated by t. For the one-cut solution, the partition function, correlators and free energies are convergent for small t and all s   as a perturbation of the Gaussian potential, which arises for st=0$st=0$. This perturbation is computed using the formalism of the topological recursion. The corresponding enumeration of chord diagrams gives at once the number of RNA complexes of a given topology as well as the number of cells in Riemann?s moduli spaces for bordered surfaces. The free energies are computed here in principle for all genera and explicitly in genus less than four.     

Three ways to solve the Poisson equation on a sphere with Gaussian forcing

Motivated by the needs of vortex methods, we describe three different exact or approximate solutions to the Poisson equation on the surface of a sphere when the forcing is a Gaussian of the three-dimensional distance, ∇²ψ=exp(-2?²(1-cos(θ))-C^Gauss(?)${\nabla }^2\psi =\exp (-2{?}^2(1-\cos (\theta ))-C^{\mathit{Gauss}}(?)$. (More precisely, the forcing is a Gaussian minus the “Gauss constraint constant”, C^Gauss$C^{\mathit{Gauss}}$; this subtraction is necessary because ψ$\psi$ is bounded, for any type of forcing, only if the integral of the forcing over the sphere is zero [Y. Kimura, H. Okamoto, Vortex on a sphere, J. Phys. Soc. Jpn. 56 (1987) 4203–4206; D.G. Dritschel, Contour dynamics/surgery on the sphere, J. Comput. Phys. 79 (1988) 477–483]. The Legendre polynomial series is simple and yields the exact value of the Gauss constraint constant, but converges slowly for large ?$?$. The analytic solution involves nothing more exotic than the exponential integral, but all four terms are singular at one or the other pole, cancelling in pairs so that ψ$\psi$ is everywhere nice. The method of matched asymptotic expansions yields simpler, uniformly valid approximations as series of inverse even powers of ?$?$ that converge very rapidly for the large values of ?  (?>40)$(?>40)$ appropriate for geophysical vortex computations. The series converges to a nonzero O(exp(-4?²))$O(\exp (-4{?}^2))$ error everywhere except at the south pole where it diverges linearly with order instead of the usual factorial order.     

Mechanical analogues of spin Hamiltonians and dynamics

Bloch et al. mapped the precession of the spin-half in a magnetic field of variable magnitude and direction to the rotations of a rigid sphere rolling on a curved surface utilizing SU(2)–SO(3)$\mathit{SU}(2)\text{–}\mathit{SO}(3)$ isomorphism. This formalism is extended to study the behaviour of spin–orbit interactions and the mechanical analogy for Rashba–Dresselhauss spin–orbit interaction in two dimensions is presented by making its spin states isomorphic to the rotations of a rigid sphere rolling on a ring. The change in phase of spin is represented by the angle of rotation of sphere after a complete revolution. In order to develop the mechanical analogy for the spin filter, we find that perfect spin filtration of down spin makes the sphere to rotate at some unique angles and the perfect spin filtration of up spin causes the rotations with certain discrete frequencies.     

Zero modes for the boundary giant magnons

We study the fermion zero-mode dynamics for open strings ending on the giant graviton branes. For the open string ending on the Z=0$Z=0$ brane, the quantization of the fermion zero-modes of boundary giant magnons reproduces the 256 states of the boundary degrees with the precise realization of the SU(2|2)×SU(2|2)$\mathit{SU}(2|2)\times \mathit{SU}(2|2)$ symmetry algebra. Also for the open string ending on the Y=0$Y=0$ brane, we reproduce the unique vacuum state from the fermion zero-modes.     

A solution to the non-Abelian duality problem

Dualities uniquely excel at resolving non-perturbative aspects of complex phase diagrams of interacting, Landau or topologically ordered, systems. However, traditional duality transformations fail for systems like the Heisenberg model and non-Abelian gauge theories. The bond-algebraic theory of quantum and classical dualities provides a solution to this long-standing conundrum, the so-called non-Abelian duality problem, by embedding traditional dualities into a more general transformation scheme that always preserves locality in any number of dimensions. Remarkably, it turns out to be unimportant whether a model?s group of symmetries is Abelian or non-Abelian. The capability of the bond-algebraic approach to handle finite and infinite systems with arbitrary boundary conditions has recently led to the discovery of holographic symmetries  , relating topological order, edge states, and generalized order parameters. We discuss the interplay between these distinguished boundary symmetries and our solution to the non-Abelian duality problem. To illustrate our technique we present, among others, novel dualities for the SU(2)$\mathit{SU}(2)$ principal chiral field and both U(1)$U(1)$ and SU(2)$\mathit{SU}(2)$ generalizations of the planar quantum compass model of orbital ordering.     

Interactions of non-Abelian global strings

Non-Abelian global strings are expected to form during the chiral phase transition. They have orientational zero modes in the internal space, associated with the vector-like symmetry SU_(N)L+R$\mathit{SU}{(N)}_{\mathrm{L}+\mathrm{R}}$ broken in the presence of strings. The interaction among two parallel non-Abelian global strings is derived for general relative orientational zero modes, giving a non-Abelian generalization of the Magnus force. It is shown that when the orientations of the strings are the same, the repulsive force reaches the maximum, whereas when the relative orientation becomes the maximum, no force exists between the strings. For the Abelian case we find a finite volume correction to the known result. The marginal instability of the previously known Abelian η^′${\eta }^{\prime }$ strings is discussed.     

Ternary Virasoro–Witt algebra

A 3-bracket variant of the Virasoro–Witt algebra is constructed through the use of su(1,1)$\mathit{su}(1,1)$ enveloping algebra techniques. The Leibniz rules for 3-brackets acting on other 3-brackets in the algebra are discussed and verified in various situations.     

A generalized Beraha conjecture for non-planar graphs

We study the partition function _ZG(nk,k)(Q,v)$Z_{G(nk,k)}(Q,v)$ of the Q  -state Potts model on the family of (non-planar) generalized Petersen graphs G(nk,k)$G(nk,k)$. We study its zeros in the plane (Q,v)$(Q,v)$ for 1?k?7$1?k?7$. We also consider two specializations of _ZG(nk,k)$Z_{G(nk,k)}$, namely the chromatic polynomial _PG(nk,k)(Q)$P_{G(nk,k)}(Q)$ (corresponding to v=−1$v=-1$), and the flow polynomial _ΦG(nk,k)(Q)${\Phi }_{G(nk,k)}(Q)$ (corresponding to v=−Q$v=-Q$). In these two cases, we study their zeros in the complex Q  -plane for 1?k?7$1?k?7$. We pay special attention to the accumulation loci of the corresponding zeros when n→∞$n\to \infty$. We observe that the Berker–Kadanoff phase that is present in two-dimensional Potts models, also exists for non-planar recursive graphs. Their qualitative features are the same; but the main difference is that the role played by the Beraha numbers for planar graphs is now played by the non-negative integers for non-planar graphs. At these integer values of Q, there are massive eigenvalue cancellations, in the same way as the eigenvalue cancellations that happen at the Beraha numbers for planar graphs.