Quantum ’t Hooft Loops of SYM $${{\mathcal N}=4}$$ as Instantons of YM2 in Dual Groups SU(N) and SU(N)/ZN

A relation between circular 1/2 BPS ’t Hooft operators in 4d N=4{{\mathcal N}=4} SYM and instantonic solutions in 2D Yang-Mills theory (YM₂) has recently been conjectured. Localization indeed predicts that those ’t Hooft operators in a theory with gauge group G are captured by instanton contributions to the partition function of YM₂, belonging to representations of the dual group ^L G. This conjecture has been tested in the case G = U(N) =  ^L G and for fundamental representations. In this paper, we examine this conjecture for the case of the groups G = SU(N) and ^L G = SU(N)/Z _N and loops in different representations. Peculiarities when groups are not self-dual and representations not “minimal” are pointed out.     

Fredholm Determinants and Pole-free Solutions to the Noncommutative Painlevé II Equation

We extend the formalism of integrable operators à la Its-Izergin-Korepin-Slavnov to matrix-valued convolution operators on a semi–infinite interval and to matrix integral operators with a kernel of the form \fracE₁^T(l) E₂(m)l+m{\frac{E_1^T(\lambda) E_2(\mu)}{\lambda+\mu}}, thus proving that their resolvent operators can be expressed in terms of solutions of some specific Riemann-Hilbert problems. We also describe some applications, mainly to a noncommutative version of Painlevé II (recently introduced by Retakh and Rubtsov) and a related noncommutative equation of Painlevé type. We construct a particular family of solutions of the noncommutative Painlevé II that are pole-free (for real values of the variables) and hence analogous to the Hastings-McLeod solution of (commutative) Painlevé II. Such a solution plays the same role as its commutative counterpart relative to the Tracy–Widom theorem, but for the computation of the Fredholm determinant of a matrix version of the Airy kernel.     

Bi-Hamiltonian structure of the extended noncommutative Toda hierarchy

The noncommutative Toda hierarchy is studied with the help of Moyal deformation by a reduction on the non-commutative two dimensional Toda hierarchy. Further we generalize the noncommutative Toda hierarchy to the extended noncommutative Toda hierarchy. To survey on its integrability, we construct the bi-Hamiltonian structure and noncommutative conserved densities of the extended noncommutative Toda hierarchy by means of the R-matrix formalism. This extended noncommutative Toda hierarchy can be reduced to the extended multicomponent Toda hierarchy, extended Z_N?-Toda hierarchy, extended Toda hierarchy respectively by reductions on Lie algebras.     

Relations between nonlinear kernels of the collision integral

Previously, to solve the Boltzmann equation by the moments method with expansion of the distribution function by spherical Hermit polynomials, a new computational method was suggested which allowed to construct nonlinear matrix elements of the collision integral with very large indices. This made it possible to substantially advance in construction of the distribution function. Limitations to convergence of the distribution function that appear in moment method are eliminated if we come to expansion by spherical harmonics from expansion by spherical Hermit polynomials. In this case, a complex five-fold collision integral is replaced by a set of comparatively simple integral operators, and kernels G _l1,l2 ^l (c, c ₁, c ₂) of these operators become the analog of matrix elements. We found the relations between expansions of the distribution function in the reference frames with various velocities of motion along marked axis. Starting from the invariance condition of the collision integral with respect to selection of such reference frames, we derived recurrent relations between the kernels with various indices. These relations allow us to construct any nonlinear kernel G _{l1, l2} ^l (c, c ₁, c ₂), if the kernel G _0,0⁰(c, c ₁, c ₂) is known.     

Noncommutative Manifolds, the Instanton Algebra¶and Isospectral Deformations

We give new examples of noncommutative manifolds that are less standard than the NC-torus or Moyal deformations of ℝ ⁿ . They arise naturally from basic considerations of noncommutative differential topology and have non-trivial global features. The new examples include the instanton algebra and the NC-4-spheres S ⁴ _θ. We construct the noncommutative algebras ?=C ^∞ (S ⁴ _θ) of functions on NC-spheres as solutions to the vanishing, ch _j (e) = 0, j < 2, of the Chern character in the cyclic homology of ? of an idempotent e∈M ₄ (?), e ²=e, e=e ^*. We describe the universal noncommutative space obtained from this equation as a noncommutative Grassmannian as well as the corresponding notion of admissible morphisms. This space Gr contains the suspension of a NC-3-sphere S ³ _θ distinct from quantum group deformations SU _q (2) of SU (2). We then construct the noncommutative geometry of S _θ ⁴ as given by a spectral triple ?, ℋ, D) and check all axioms of noncommutative manifolds. In a previous paper it was shown that for any Riemannian metric g _μν on S ⁴ whose volume form is the same as the one for the round metric, the corresponding Dirac operator gives a solution to the following quartic equation,

How do black holes predict the sign of the Fourier coefficients of siegel modular forms?

Single centered supersymmetric black holes in four dimensions have spherically symmetric horizon and hence carry zero angular momentum. This leads to a specific sign of the helicity trace index associated with these black holes. Since the latter are given by the Fourier expansion coefficients of appropriate meromorphic modular forms of Sp(2,\mathbbZ){Sp(2,{\mathbb{Z}})} or its subgroup, we are led to a specific prediction for the signs of a subset of these Fourier coefficients which represent contributions from single centered black holes only. We explicitly test these predictions for the modular forms which compute the index of quarter BPS black holes in heterotic string theory on T ⁶, as well as in \mathbbZ_N{{\mathbb{Z}}_N} CHL models for N = 2, 3, 5, 7.     

A Module Structure on the Symplectic Floer Cohomology

The aim of this paper is to offer an affirmative answer to the Floer conjectures in [2, p. 589] which states that there is a module structure on the Z _{2 N} -graded symplectic Floer cohomology for monotone symplectic manifolds. By constructing a Z-graded symplectic Floer cohomology as an integral lift of the Z _{2 N} -graded symplectic Floer cohomology, we gain control of the holomorphic bubbling spheres. This makes a module structure on the Z-graded Floer cohomology. There is a spectral sequence with E ¹ _*,* given by the Z-graded symplectic Floer cohomology. Such a spectral sequence converges to the Z _{2 N} -graded symplectic Floer cohomology. Hence we induce a module structure for the Z _{2 N} -graded symplectic Floer cohomology by the spectral sequence and algebraic topology methods. Received: 2 August 1999 / Accepted: 25 October 1999     

Mass Measurements of Exotic Nuclei around N=Z=40 with CSS2

Mass measurements of the N=Z nuclei ⁸⁰Zr, ⁷⁶Sr, ⁶⁸Se were performed for the first time and a new measurement was obtained for ⁸⁰Y, using the second cyclotron CSS2 of GANIL as a high-resolution spectrometer. Ions around N=Z were produced by fusion-evaporation in the inverse ⁵⁸Ni (4.32MeVA) + ²⁴Mg and ¹²C reactions. New masses were measured by a time-of-flight method, with a precision of 2⋅10⁻⁶, by using well-known masses as references. Study of the double binding energy difference δV _np is then performed leading to a strong N=Z Wigner effect around N=Z=40. Knowledge of new masses in this region also plays a crucial role in the modelling of the astrophysical rp process. This revised version was published online in July 2006 with corrections to the Cover Date.     

Riemannian Geometry of Quantum Groups¶and Finite Groups with Nonuniversal Differentials

We construct noncommutative “Riemannian manifold” structures on dual quasitriangular Hopf algebras such as ℂ _q [SU ₂] with its standard bicovariant differential calculus, using the quantum frame bundle approach introduced previously. The metric is provided by the braided-Killing form on the braided-Lie algebra on the tangent space and the n-bein by the Maurer–Cartan form. We also apply the theory to finite sets and in particular to finite group function algebras ℂ[G] with differential calculi and Killing forms determined by a conjugacy class. The case of the permutation group ℂ[S ₃] is worked out in full detail and a unique torsion free and cotorsion free or “Levi–Civita” connection is obtained with noncommutative Ricci curvature essentially proportional to the metric (an Einstein space). We also construct Dirac operators in the metric background, including on finite groups such as S ₃. In the process we clarify the construction of connections from gauge fields with nonuniversal calculi on quantum principal bundles of tensor product form. Received: 22 June 2000 / Accepted: 26 August 2001     

Dirac Operators on Quantum Projective Spaces

We construct a family of self-adjoint operators D _N , ${N\in{\mathbb Z}}We construct a family of self-adjoint operators D _N , N ? \mathbb Z{N\in{\mathbb Z}} , which have compact resolvent and bounded commutators with the coordinate algebra of the quantum projective space \mathbb CP^l_q{{\mathbb C}{\rm P}^{\ell}_q} , for any ℓ ≥ 2 and 0 < q < 1. They provide 0⁺-dimensional equivariant even spectral triples. If ℓ is odd and N=\frac12(l+1){N=\frac{1}{2}(\ell+1)} , the spectral triple is real with KO-dimension 2ℓ mod 8.     

Principal Fibrations from Noncommutative Spheres

We construct noncommutative principal fibrations S_θ⁷→S_θ⁴ which are deformations of the classical SU(2) Hopf fibration over the four sphere. We realize the noncommutative vector bundles associated to the irreducible representations of SU(2) as modules of coequivariant maps and construct corresponding projections. The index of Dirac operators with coefficients in the associated bundles is computed with the Connes-Moscovici local index formula. “The algebra inclusion is an example of a not-trivial quantum principal bundle.”     

A purely algebraic construction of a gauge and renormalization group invariant scalar glueball operator

This paper presents a complete algebraic proof of the renormalizability of the gauge invariant d=4 operator F _{μ ν} ²(x) to all orders of perturbation theory in pure Yang–Mills gauge theory, whereby working in the Landau gauge. This renormalization is far from being trivial as mixing occurs with other d=4 gauge variant operators, which we identify explicitly. We determine the mixing matrix Z to all orders in perturbation theory by using only algebraic arguments and consequently we can uncover a renormalization group invariant by using the anomalous dimension matrix Γ derived from Z. We also present a future plan for calculating the mass of the lightest scalar glueball with the help of the framework we have set up.     

Contributions of Spin–Lattice Relaxation to the Echo Decay of Planar Cu in High-Temperature Superconductors

Measurement ofT_2G, the Gaussian component of the spin-echo envelope of planar Cu nuclei in high-temperature superconductors, gives important information about the real part of the Cu electron spin susceptibility. In the traditional picture of the planar Cu echo decay, the internuclear coupling is assumed to remain static with respect to spin–lattice relaxation and mutual exchange fluctuations. In some circumstances, however, this assumption breaks down. We calculate the internuclear corrections arising from spin–lattice relaxation to the conventional theory ofT_2Gand show thatT_2Gcan be easily corrected for these effects. We argue that mutual exchanges due to the perpendicular indirect couplings are suppressed in these materials. For YBa₂Cu₄O₈, we find a correction on the order of 10% inT_2Gand using the corrected values we find that the isotope ratio⁶³T_2G/⁶⁵T_2Gagrees with theory.     

Cu nuclear spin-spin relaxation in YBa2Cu3O6.98 and YBa2Cu4O8

We have measured the temperature (T) dependence of the transverse relaxation rate (T _G ^–1 ) of the Cu(1) nuclear spin in YBa₂Cu₃O_6.98 (T _c=92 K) and YBa₂Cu₄O₈ (T _c=82 K). From the scaling ratio ofT _G ^–2 (Cu1) toT _G ^–2 (Cu2), we have estimated the strength of a covalent bonding between the CuO₂ plane and the CuO chain to be B0.38×A _zz. The experimentalT _G ^–1 (Cu1) in YBa₂Cu₄O₈ was of the same order of magnitude as the estimated one fromT _G ^–1 (Cu2). These results appear to indicate that the electrons in the CuO₂ plane fairly spread out of the plane in both compounds.     

Asymptotic Statistics of Zeroes for the Lamé Ensemble

The Lamé polynomials naturally arise when separating variables in Laplace's equation in elliptic coordinates. The products of these polynomials form a class of spherical harmonics, which are joint eigenfunctions of a quantum completely integrable (QCI) system of commuting, second-order differential operators P ₀=Δ, P ₁,…,P ^{N −1} acting on C ^∞(? ^N ). These operators naturally depend on parameters and thus constitute an ensemble. In this paper, we compute the limiting level-spacings distributions for the zeroes of the Lamé polynomials in various thermodynamic, asymptotic regimes. We give results both in the mean and pointwise, for an asymptotically full set of values of the parameters. Received: 17 January 2001 / Accepted: 14 May 2001     

Nuclear orientation study of the decay of204BiFe

The decay of²⁰⁴Bi nuclei (I =6⁺, T_1/2=11·22 h) oriented in an iron host was investigated on the JINR low-temperature nuclear orientation facility SPIN. The orientation parameterB ₂=1·17 (6) was obtained from the analysis of six prominent E1 gamma-transitions. From the measured normalized intensities of the gamma-rays observed some 70 values of multipole mixing ratios for the gamma-transitions in²⁰⁴Pb nucleus were determined for the first time. The spins 6, 6, 5 and 4 could be uniquely assigned to the²⁰⁴Pb negative parity levels at 3891·5 keV, 3768·4 keV, 3301·5 keV and 2338·2 keV, respectively. The spin-parity assignments of the levels at 4183·8 keV, 4094·2 keV, 3782·0 keV, 2506·9 keV and 2065·1 keV were confirmed as 6^–, 6^–, 5^–, 5^– and 5⁺, respectively. For the level at 3105·1 keV spin-parity 5^– was suggested and spinparity 7^– of the level at 2696·4 keV was called in question. The possible placements of the gammatransitions 3 1351·7 keV and 1353·4 keV in the decay scheme is discussed. The reorientation parameters for the long-living levels at 2264·2 keV (T _1/2=0·45 s) and 1273·9 keV (T _1/2= =265 ns) were determined asG ₂=0·41 (14) andG ₂=0·60 (17), respectively. For the isomeric level at 2185·7 keV (T _1/2=67·2 min) the value ofG ₂=0·88 (49) was proposed.The authors would like to express their thanks to T. I. Kracíková and M. Trhlík for the valuable discussions in the course of the evaluation of the experimental data.     

Deconstruction of gauge symmetry breaking by discrete symmetry and G unification

We deconstruct the non-supersymmetric SU(5) breaking by discrete symmetry on the space-time and in the Higgs mechanism deconstruction scenario. Also we explain the subtle point of how to exactly match the continuum results with the latticized results on the quotient space S ¹ /Z ₂ and . We also propose an effective deconstruction scenario and discuss the gauge symmetry breaking by the discrete symmetry on the theory space in this approach. As an application, we suggest the G^N unification where G^N is broken down to by the bifundamental link fields and the doublet-triplet splitting can be achieved. Received: 10 October 2002 / Revised version: 23 March 2003 / Published online: 13 May 2003 RID="a" ID="a" Current address: School of Natural Sciences, Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540, USA e-mail: tli@sns.ias.edu RID="b" ID="b" e-mail: liutao@sas.upenn.edu     

Correlation Functions for MN/SN Orbifolds

We develop a method for computing correlation functions of twist operators in the bosonic 2-d CFT arising from orbifolds M ^N /S _N , where M is an arbitrary manifold. The path integral with twist operators is replaced by a path integral on a covering space with no operator insertions. Thus, even though the CFT is defined on the sphere, the correlators are expressed in terms of partition functions on Riemann surfaces with a finite range of genus g. For large N, this genus expansion coincides with a 1/N expansion. The contribution from the covering space of genus zero is “universal” in the sense that it depends only on the central charge of the CFT. For 3-point functions we give an explicit form for the contribution from the sphere, and for the 4-point function we do an example which has genus zero and genus one contributions. The condition for the genus zero contribution to the 3-point functions to be non-vanishing is similar to the fusion rules for an SU(2) WZW model. We observe that the 3-point coupling becomes small compared to its large N limit when the orders of the twist operators become comparable to the square root of N – this is a manifestation of the stringy exclusion principle. Received: 20 July 2000 / Accepted: 17 December 2000     

Spin-Hall Effect with Quantum Group Symmetries

We construct a model of spin-Hall effect on a noncommutative four sphere S ⁴ _Θ with isospin degrees of freedom, coming from a noncommutative instanton, and invariance under a quantum group SO _θ. The corresponding representation theory allows to explicitly diagonalize the Hamiltonian and construct the ground state; there are both integer and fractional excitations. Similar models exist on higher dimensional spheres S _Θ ^N and projective spaces . Dedicated to Rafael Sorkin with friendship and respect.