Cylindric Versions of Specialised Macdonald Functions and a Deformed Verlinde Algebra

We define cylindric versions of skew Macdonald functions P _λ/μ (q, t) for the special cases q = 0 or t = 0. Fixing two integers n > 2 and k > 0 we shift the skew diagram λ/μ, viewed as a subset of the two-dimensional integer lattice, by the period vector (n, ?k). Imposing a periodicity condition one defines cylindric skew tableaux and associated weight functions. The resulting weighted sums over these cylindric tableaux are symmetric functions. They appear in the coproduct of a commutative Frobenius algebra which is a particular quotient of the spherical Hecke algebra. We realise this Frobenius algebra as a commutative subalgebra in the endomorphisms over a ${U_{q}\widehat{\mathfrak{sl}}(n)}$ Kirillov-Reshetikhin module. Acting with special elements of this subalgebra, which are noncommutative analogues of Macdonald polynomials, on a highest weight vector, one obtains Lusztig’s canonical basis. In the limit q = t = 0, this Frobenius algebra is isomorphic to the ${\widehat{\mathfrak{sl}}(n)}$ Verlinde algebra at level k, i.e. the structure constants become the ${\widehat{\mathfrak{sl}}(n)_{k}}$ Wess-Zumino-Novikov-Witten fusion coefficients. Further motivation comes from exactly solvable lattice models in statistical mechanics: the cylindric Macdonald functions discussed here arise as partition functions of so-called vertex models obtained from solutions to the Yang-Baxter equation. We show this by stating explicit bijections between cylindric tableaux and lattice configurations of non-intersecting paths. Using the algebraic Bethe ansatz the idempotents of the Frobenius algebra are computed.     

Analyticity of the partition function for finite quantum systems

The partition functionZ(β,λ)=Tre ^-β(T+λV) for a finite quantized system is investigated. If the interactionV is a relatively bounded operator with respect to the kinetic energyT withT-boundb<1,Z(β,λ) is shown to be a holomorphic function of β and λ for $$\left| {\arg \beta } \right|< arctg\frac{{\sqrt {1 - b^2 \left| \lambda \right|^2 } }}{{b\left| \lambda \right|}}and\left| \lambda \right|< b^{ - 1} .$$ Forb=0Z(β,λ) is an entire function of λ and holomorphic in β for Re β>0.     

Holomorphic Anomaly in Gauge Theory on ALE space

We consider four-dimensional Ω-deformed ${\mathcal{N} = 2}$ supersymmetric SU(2) gauge theory on A ₁ space and its lift to five dimensions. We find that the partition functions can be reproduced via special geometry and the holomorphic anomaly equation. Schwinger-type integral expressions for the boundary conditions at the monopole/dyon point in moduli space are inferred. The interpretation of the five-dimensional partition function as the partition function of a refined topological string on A ₁ × (local ${\mathbb{P}^{1} \times \mathbb{P}^1}$ ) is suggested.     

The Refined BPS Index from Stable Pair Invariants

A refinement of the stable pair invariants of Pandharipande and Thomas for non-compact Calabi–Yau spaces is introduced based on a virtual Bialynicki-Birula decomposition with respect to a ${\mathbb{C}^{*}}$ action on the stable pair moduli space, or alternatively the equivariant index of Nekrasov and Okounkov. This effectively calculates the refined index for M-theory reduced on these Calabi–Yau geometries. Based on physical expectations we propose a product formula for the refined invariants extending the motivic product formula of Morrison, Mozgovoy, Nagao, and Szendroi for local ${\mathbb{P}^1}$ . We explicitly compute refined invariants in low degree for local ${\mathbb{P}^2}$ and local ${\mathbb{P}^1\,\times\,\mathbb{P}^1}$ and check that they agree with the predictions of the direct integration of the generalized holomorphic anomaly and with the product formula. The modularity of the expressions obtained in the direct integration approach allows us to relate the generating function of refined PT invariants on appropriate geometries to Nekrasov’s partition function and a refinement of Chern–Simons theory on a lens space. We also relate our product formula to wall crossing.     

Gauge Theories on ALE Space and Super Liouville Correlation Functions

We present a relation between ${\mathcal{N}=2}$ quiver gauge theories on the ALE space ${\mathcal{O}_{\mathbb{P}^1}(-2)}$ and correlators of ${\mathcal{N}=1}$ super Liouville conformal field theory, providing checks in the case of punctured spheres and tori. We derive a blow-up formula for the full Nekrasov partition function and show that, up to a U(1) factor, the ${\mathcal{N}=2^*}$ instanton partition function is given by the product of the character of ${\widehat{SU}(2)_2}$ times the super Virasoro conformal block on the torus with one puncture. Moreover, we match the perturbative gauge theory contribution with super Liouville three-point functions.     

Three Attractive Osculating Walkers and a Polymer Collapse Transition

Consider n interacting lock-step walkers in one dimension which start at the points {0,2,4,...,2(n?1)} and at each tick of a clock move unit distance to the left or right with the constraint that if two walkers land on the same site their next steps must be in the opposite direction so that crossing is avoided. When two walkers visit and then leave the same site an osculation is said to take place. The space-time paths of these walkers may be taken to represent the configurations of n fully directed polymer chains of length t embedded on a directed square lattice. If a weight λ is associated with each of the i osculations the partition function is $Z_t^{(n)} (\lambda ) = \sum\nolimits_{i = 0}^{\left\lfloor {\tfrac{{(n - 1)t}}{2}} \right\rfloor } {z_{t,i}^{(n)} } \lambda ^i $ where z ⁽ⁿ⁾ _t,i is the number of t-step configurations having i osculations. When λ=0 the partition function is asymptotically equal to the number of vicious walker star configurations for which an explicit formula is known. The asymptotics of such configurations was discussed by Fisher in his Boltzmann medal lecture. Also for n=2 the partition function for arbitrary λ is easily obtained by Fisher's necklace method. For n>2 and λ≠0 the only exact result so far is that of Guttmann and Vöge who obtained the generating function $G^{(n)} (\lambda ,u) \equiv \sum\nolimits_{t = 0}^\infty {Z_t^{(n)} (\lambda )u^t } $ for λ=1 and n=3. The main result of this paper is to extend their result to arbitrary λ. By fitting computer generated data it is conjectured that Z ⁽³⁾ _t (λ) satisfies a third order inhomogeneous difference equation with constant coefficients which is used to obtain $$G^{(n)} (\lambda ,u) = \frac{{(\lambda - 3)(\lambda + 2) - \lambda (12 - 5\lambda + \lambda ^2 )u - 2\lambda ^3 u^2 + 2(\lambda - 4)(\lambda ^2 u^2 - 1){\text{ }}c(2u)}}{{(\lambda - 2 - \lambda ^2 u)(\lambda - 1 - 4\lambda u - 4\lambda ^2 u^2 )}}$$ where $c(u) = \tfrac{{1 - \sqrt {1 - 4u} }}{{2u}}$ , the generating function for Catalan numbers. The nature of the collapse transition which occurs at λ=4 is discussed and extensions to higher values of n are considered. It is argued that the position of the collapse transition is independent of n.     

Combinatorics of Generalized Bethe Equations

A generalization of the Bethe ansatz equations is studied, where a scalar two-particle S-matrix has several zeroes and poles in the complex plane, as opposed to the ordinary single pole/zero case. For the repulsive case (no complex roots), the main result is the enumeration of all distinct solutions to the Bethe equations in terms of the Fuss-Catalan numbers. Two new combinatorial interpretations of the Fuss-Catalan and related numbers are obtained. On the one hand, they count regular orbits of the permutation group in certain factor modules over ${\mathbb{Z}^M}$ Z M , and on the other hand, they count integer points in certain M-dimensional polytopes.     

Z c (4025) as the hadronic molecule with hidden charm

We have studied the loosely bound $D^{*}\bar{D}^{*}$ system. Our results indicate that the recently observed charged charmonium-like structure Z _c (4025) can be an ideal $D^{*}\bar{D}^{*}$ molecular state. We have also investigated its pionic, dipionic, and radiative decays. We stress that both the scalar isovector molecular partner Z _c0 and three isoscalar partners ${\tilde{Z}}_{c0,c1,c2}$ should also exist if Z _c (4025) is a $D^{*}\bar{D}^{*}$ molecular state in the framework of the one-pion-exchange model. Z _c0 can be searched for in the channel e ⁺ e ^?→Y→Z _c0(4025)(ππ)_P-wave where Y can be Y(4260) or any other excited 1^?? charmonium or charmonium-like states such as Y(4360), Y(4660), etc. The isoscalar $D^{*}\bar{D}^{*}$ molecular states ${\tilde{Z}}_{c0,c2}$ with 0⁺(0⁺⁺) and 0⁺(2⁺⁺) can be searched for in the three pion decay channel $e^{+}e^{-}\to Y \to {\tilde{Z}}_{c0,c2} (3\pi)^{I=0}_{\text{P-wave}}$ . The isoscalar molecular state ${\tilde{Z}}_{c1}$ with 0^?(1^+?) can be searched for in the channel ${\tilde{Z}}_{c1}\eta$ . Experimental discovery of these partner states will firmly establish the molecular picture.     

Linear collider test of a neutrinoless double beta decay mechanism in left–right symmetric theories

There are various diagrams leading to neutrinoless double beta decay in left?Cright symmetric theories based on the gauge group SU(2) _L ×SU(2) _R . All can in principle be tested at a linear collider running in electron?Celectron mode. We argue that the so-called ??-diagram is the most promising one. Taking the current limit on this diagram from double beta decay experiments, we evaluate the relevant cross section $e^{-} e^{-} \to W^{-}_{L} W^{-}_{R}$ , where $W^{-}_{L}$ is the Standard Model W-boson and $W^{-}_{R}$ the one from SU(2) _R . It is observable if the life-time of double beta decay and the mass of the W _R are close to current limits. Beam polarization effects and the high-energy behaviour of the cross section are also analyzed.     

Rules of thumb for theZ line shape

In this paper the theoretical parameters of theZ line shape, such asM _Z andΓ _Z, and the one photon exchange diagram are related to a set of parameters characterizing the experimental line shape. The latter are the peak height σ_max, peak position \(\sqrt {s_{\max } } \) and half peak positions \(\sqrt {s_ \pm } \) . The rules of thumb are accurate within 10 MeV. As a result we obtain approximate formulae which expressM _Z and Γ_Z in the measured \(\sqrt {s_{\max } } \) and \(\sqrt {s_ + } - \sqrt {s_ - } \) .     

M s −M u quark mass difference and the scalar form factor ofK μ3 0 reaction

We study through QCD sum rules the connection between the invariant quark mass difference \(\hat m_s - \hat m_u\) and the scalar form factor of the reactionK ⁰→π ^? μ ⁺ v _μ in the physical region. We use both theoretical information, (the value off ₊(0) and the Callan-Treiman relation, includingm _π ²/m _k ² corrections) and experimental one (the value ofλ ₀ from a linear fit) to give a lower bound for \(\hat m_s - \hat m_u\) . Taking the world most recent fitted value forλ ₀,λ ₀ = 0.025, which may be reasonably identified with the slope att=0, andf ₊ (0) ≈ 0.98, we obtain \(\hat m_s - \hat m_u\) ≥ 250 MeV for \(\Lambda _{\overline {MS} }\) = 150 MeV. The relevant hypotheses and experimental trends are discussed.     

Dynamics of tachyon and phantom field beyond the inverse square potentials

We investigate the cosmological evolution of the tachyon and phantom-tachyon scalar field by considering the potential parameter $\Gamma(=\frac{VV''}{V'^{2}}$ ) as a function of another potential parameter $\lambda(=\frac{V'}{\kappa V^{3/2}}$ ), which correspondingly extends the analysis of the evolution of our universe from a two-dimensional autonomous dynamical system to the three-dimensional case. It allows us to investigate the more general situation where the potential is not restricted to an inverse square potential. One particular result is that, apart from the inverse square potential, there are a large number of potentials which can give the scaling and dominant solution when the function Γ(λ) equals 3/2 for one or more values of λ _*, as well as that the parameter λ _* satisfies certain conditions. We also find that for a class of different potentials the possibilities for the dynamical evolution of the universe are actually the same and therefore undistinguishable.     

New Integrable Models from the Gauge/YBE Correspondence

We introduce a class of new integrable lattice models labeled by a pair of positive integers N and r. The integrable model is obtained from the Gauge/YBE correspondence, which states the equivalence of the 4d $\mathcal {N} =1$ $S^{1}\times S^{3}/ \mathbb {Z} _{r}$ index of a large class of SU(N) quiver gauge theories with the partition function of 2d classical integrable spin models. The integrability of the model (star-star relation) is equivalent with the invariance of the index under the Seiberg duality. Our solution to the Yang-Baxter equation is one of the most general known in the literature, and reproduces a number of known integrable models. Our analysis identifies the Yang-Baxter equation with a particular duality (called the Yang-Baxter duality) between two 4d $\mathcal {N} =1$ supersymmetric quiver gauge theories. This suggests that the integrability goes beyond 4d lens indices and can be extended to the full physical equivalence among the IR fixed points.     

Self-Dual Noncommutative {\phi^4} -Theory in Four Dimensions is a Non-Perturbatively Solvable and Non-Trivial Quantum Field Theory

We study quartic matrix models with partition function \({\mathcal{Z}[E, J] = \int dM}\) exp(trace \({(JM - EM^{2} - \frac{\lambda}{4} M^4)}\) ). The integral is over the space of Hermitean \({\mathcal{N} \times \mathcal{N}}\) -matrices, the external matrix E encodes the dynamics, \({\lambda > 0}\) is a scalar coupling constant and the matrix J is used to generate correlation functions. For E not a multiple of the identity matrix, we prove a universal algebraic recursion formula which gives all higher correlation functions in terms of the 2-point function and the distinct eigenvalues of E. The 2-point function itself satisfies a closed non-linear equation which must be solved case by case for given E. These results imply that if the 2-point function of a quartic matrix model is renormalisable by mass and wavefunction renormalisation, then the entire model is renormalisable and has vanishing β-function. As the main application we prove that Euclidean \({\phi^4}\) -quantum field theory on four-dimensional Moyal space with harmonic propagation, taken at its self-duality point and in the infinite volume limit, is exactly solvable and non-trivial. This model is a quartic matrix model, where E has for \({\mathcal{N} \to \infty}\) the same spectrum as the Laplace operator in four dimensions. Using the theory of singular integral equations of Carleman type we compute (for \({\mathcal{N} \to \infty}\) and after renormalisation of \({E, \lambda}\) ) the free energy density (1/volume) log \({(\mathcal{Z}[E, J]/\mathcal{Z}[E, 0])}\) exactly in terms of the solution of a non-linear integral equation. Existence of a solution is proved via the Schauder fixed point theorem. The derivation of the non-linear integral equation relies on an assumption which in subsequent work is verified for coupling constants \({\lambda \leq 0}\) .     

Diffraction of electrons in the CubicTi5O5 superstructure of titanium monoxide

Annealed titanium monoxide TiO_1.087 has been studied by the electron diffraction method. A cubic model of the Ti₅O₅ superstructure (Ti₅O₅ (Ti₉₀?₁₈O₉₀??₁₈)) of nonstoichiometric titanium monoxide Ti _x O _z has been proposed on the basis of experimental data and representations about the disorder-order transition channel. It has been shown that reflections observed on the electron diffraction pattern are identified in the space group $Pm\bar 3m$ . The period of the unit cell of the cubic Ti₅O₅ superstructure is larger than that for the B1 basic disordered structure of Ti _x O _z monoxide by a factor of 3. The disorder-order transition channel Ti _x O _z (space group $Fm\bar 3m$ )-Ti₅O₅ (space group $Pm\bar 3m$ ) includes 75 superstructure vectors of seven stars {k ₁₀}, {k ₇}, {k ₆₍₁₎}, {k ₆₍₂₎}, {k ₄₍₁₎}, {k ₄₍₂₎}, and {k ₁}. The distribution functions of Ti and O atoms over the sites of the cubic Ti₅O₅ (space group $Pm\bar 3m$ ) superstructure have been calculated.     

Hole Probabilities and Overcrowding Estimates for Products of Complex Gaussian Matrices

We consider eigenvalues of a product of n non-Hermitian, independent random matrices. Each matrix in this product is of size N×N with independent standard complex Gaussian variables. The eigenvalues of such a product form a determinantal point process on the complex plane (Akemann and Burda in J. Phys. A, Math. Theor. 45:465201, 2011), which can be understood as a generalization of the finite Ginibre ensemble. As N→∞, a generalized infinite Ginibre ensemble arises. We show that the set of absolute values of the points of this determinantal process has the same distribution as $\{R_{1}^{(n)},R_{2}^{(n)},\ldots\}$ , where $R_{k}^{(n)}$ are independent, and $(R_{k}^{(n)} )^{2}$ is distributed as the product of n independent Gamma variables Gamma(k,1). This enables us to find the asymptotics for the hole probabilities, i.e. for the probabilities of the events that there are no points of the process in a disc of radius r with its center at 0, as r→∞. In addition, we solve the relevant overcrowding problem: we derive an asymptotic formula for the probability that there are more than m points of the process in a fixed disk of radius r with its center at 0, as m→∞.     

Intermediate symmetries in theSO(10) model with16 \oplus \overline {16} \oplus 45 Higgses

Patterns of hierarchical symmetry breaking in theSO(10) model with Higgses in \(16 \oplus \overline {16} \oplus 45\) representations are studied. UsualSU(5) or flippedSU(5)?U(1) are shown to emerge as intermediate symmetries from the minimization of the scalar potential. Some low-energy implications are briefly discussed.     

The Lie–Rinehart Universal Poisson Algebra of Classical and Quantum Mechanics

The Lie–Rinehart algebra of a (connected) manifold ${\mathcal {M}}$ , defined by the Lie structure of the vector fields, their action and their module structure over ${C^\infty({\mathcal {M}})}$ , is a common, diffeomorphism invariant, algebra for both classical and quantum mechanics. Its (noncommutative) Poisson universal enveloping algebra ${\Lambda_{R}({\mathcal {M}})}$ , with the Lie–Rinehart product identified with the symmetric product, contains a central variable (a central sequence for non-compact ${{\mathcal {M}}}$ ) ${Z}$ which relates the commutators to the Lie products. Classical and quantum mechanics are its only factorial realizations, corresponding to Z  =  i z, z  =  0 and ${z = \hbar}$ , respectively; canonical quantization uniquely follows from such a general geometrical structure. For ${z =\hbar \neq 0}$ , the regular factorial Hilbert space representations of ${\Lambda_{R}({\mathcal{M}})}$ describe quantum mechanics on ${{\mathcal {M}}}$ . For z  =  0, if Diff( ${{\mathcal {M}}}$ ) is unitarily implemented, they are unitarily equivalent, up to multiplicity, to the representation defined by classical mechanics on ${{\mathcal {M}}}$ .     

Positivity and monotonicity properties ofC 0-semigroups. I

If exp {?tH}, exp {?tK}, are self-adjoint, positivity preserving, contraction semigroups on a Hilbert space ?=L ²(X;dμ) we write (*) $$e^{ - tH} \succcurlyeq e^{ - tK} \succcurlyeq 0$$ whenever exp {?tH}-exp {?tK} is positivity preserving for allt≧0 and then we characterize the class of positive functions for which (*) always implies $$e^{ - tf(H)} \succcurlyeq e^{ - tf(K)} \succcurlyeq 0.$$ This class consists of thef∈C ^∞(0, ∞) with $$( - 1)^n f^{(n + 1)} (x) \geqq 0,x \in (0,\infty ),n = 0,1,2, \ldots .$$ In particular it contains the class of monotone operator functions. Furthermore if exp {?tH} isL ^p (X;dμ) contractive for allp∈[1, ∞] and allt>0 (or, equivalently, forp=∞ andt>0) then exp {?tf(H)} has the same property. Various applications to monotonicity properties of Green's functions are given.     

The Dual Stream Function Representation of an Ideal Steady Fluid Flow and its Local Geometric Structure

Using the methodology of Lie groups and Lie algebras we determine new symmetry and equivalence classes of the stationary three-dimensional Euler equations by introducing potential functions that are based on the so-called dual stream function representation of the steady state velocity field u(x, y, z) = ?λ(x, y, z) × ?μ(x, y, z), which itself can only be defined locally. In particular an infinite dimensional Lie algebra for Beltrami fields is gained. We show that this Lie algebra generates canonical transformations of a Hamiltonian flow for the dual pair of variables \(\lambda \) and \(\mu \) . It enables us to make the classification of a two-dimensional Riemannian manifold \(M^{2}\) wherein \((\lambda ,\mu )\) presents the local coordinates of \(M^{2}\) . Furthermore the local geometry of this manifold is explored in detail. As a result an infinite set of locally conserved currents and charges in the context of a conformal field theory is finally observed.