Elliptic Hypergeometry of Supersymmetric Dualities II. Orthogonal Groups, Knots, and Vortices

We consider Seiberg electric-magnetic dualities for 4d ${\mathcal{N} = 1}$ SYM theories with SO(N) gauge group. For all such known theories we construct superconformal indices (SCIs) in terms of elliptic hypergeometric integrals. Equalities of these indices for dual theories lead both to proven earlier special function identities and new conjectural relations for integrals. In particular, we describe a number of new elliptic beta integrals associated with the s-confining theories with the spinor matter fields. Reductions of some dualities from SP(2N) to SO(2N) or SO(2N + 1) gauge groups are described. Interrelation of SCIs and the Witten anomaly is briefly discussed. Possible applications of the elliptic hypergeometric integrals to a two-parameter deformation of 2d conformal field theory and related matrix models are indicated. Connections of the reduced SCIs with the state integrals of knot theory, generalized AGT duality for (3 + 3)d theories, and a 2d vortex partition function are described.     

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.     

The phase and pole structure of the N*(1535) in \piN \rightarrow \piN and \gammaN \rightarrow \piN

The nature of some baryonic resonances is still an unresolved issue. The case of the N ^*(1535) is particularly interesting in this respect due to the nearby $ \eta$ N threshold and interference with the N ^*(1650) . The N ^*(1535) has been described as a threshold effect, as a genuine 3-quark resonance, or as dynamically generated from the interaction of the octet of baryons with the octet of mesons. In the scheme of dynamical generation, predictions for the interaction of the N ^*(1535) with the photon can be made. In this study, we simultaneously analyze the role of the N ^*(1535) in the $ \pi$ N $ \rightarrow$ $ \pi$ N and $ \gamma$ N $ \rightarrow$ $ \pi$ N reactions and compare to the respective amplitudes from partial-wave analyses. This test is very sensitive to the meson-baryon components of the N ^*(1535) .     

The decay of theT=1 isospin triplet inA=12 systems: IV. The energy dependence of the asymmetry coefficients\alpha _{\beta ^ \mp } of the beta-ray angular distribution in aligned12B and12N and the induced pseudotensor interaction

The asymmetry parameters \(\alpha _{\beta ^ \mp } \) of the beta-ray emitted from aligned¹²B and¹²N are evaluated as a function of the energy. The agreement with experimental differential data is excellent for both \(\alpha _{\beta ^ - } \) (W) and \(\alpha _{\beta ^ + } \) (W). This work confirms, using available nuclear model information, that no induced pseudotensor (IPT) interaction is required for a correct theoretical interpretation of the data. An upper limit for the IPT coupling constantf _T is determined from a simultaneous fit of \(\alpha _{\beta ^ - } \) (W) and \(\alpha _{\beta ^ + } \) (W).     

Elliptic Genera of Two-Dimensional {\mathcal{N} = 2} Gauge Theories with Rank-One Gauge Groups

We compute the elliptic genera of two-dimensional ${\mathcal{N} = (2, 2)}$ and ${\mathcal{N} = (0, 2)}$ -gauged linear sigma models via supersymmetric localization, for rank-one gauge groups. The elliptic genus is expressed as a sum over residues of a meromorphic function whose argument is the holonomy of the gauge field along both the spatial and the temporal directions of the torus. We illustrate our formulas by a few examples including the quintic Calabi–Yau, ${\mathcal{N} = (2, 2)}$ SU(2) and O(2) gauge theories coupled to N fundamental chiral multiplets, and a geometric ${\mathcal{N} = (0, 2)}$ model.     

Tau-Function Theory of Chaotic Quantum Transport with β = 1, 2, 4

We study the cumulants and their generating functions of the probability distributions of the conductance, shot noise and Wigner delay time in ballistic quantum dots. Our approach is based on the integrable theory of certain matrix integrals and applies to all the symmetry classes ${\beta \in \{1, 2, 4\}}$ of Random Matrix Theory. We compute the weak localization corrections to the mixed cumulants of the conductance and shot noise for β = 1, 4, thus proving a number of conjectures of Khoruzhenko et al. (in Phys Rev B 80:(12)125301, 2009). We derive differential equations that characterize the cumulant generating functions for all ${\beta \in \{1, 2, 4 \} }$ . Furthermore, when β = 2 we show that the cumulant generating function of the Wigner delay time can be expressed in terms of the Painlevé III′ transcendant. This allows us to study properties of the cumulants of the Wigner delay time in the asymptotic limit ${n \to \infty}$ . Finally, for all the symmetry classes and for any number of open channels, we derive a set of recurrence relations that are very efficient for computing cumulants at all orders.     

On the {{\mathcal{N}}=2} Superconformal Index and Eigenfunctions of the Elliptic RS Model

We define an infinite sequence of superconformal indices, ${{\mathcal{I}}_n}$ , generalizing the Schur index for ${{\mathcal{N}}=2}$ theories. For theories of class ${{\mathcal{S}}}$ we then suggest a recursive technique to completely determine ${{\mathcal{I}}_n}$ . The information encoded in the sequence of indices is equivalent to the ${{\mathcal{N}}=2}$ superconformal index depending on a maximal set of fugacities. Mathematically, the procedure suggested in this note provides a perturbative algorithm for computing a set of eigenfunctions of the elliptic Ruijsenaars–Schneider model.     

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.     

Entropic dimension for completely positive maps

We extend the concept of quantum dynamical entropyh _φ (γ) to cover the case of a completely positive map γ. Forh _φ (γ) = 0 we examine the limit $$h_\phi (N,\gamma ,\beta ) = \mathop {\lim }\limits_n (1/n^\beta )H_\phi (N,\gamma {\rm N},...,\gamma ^{n -- 1} N)$$ calling the turning point β₀ between zero and infiniteh _φ (N, γ, β) the “entropic dimension”D _N (γ). The application of this theory to a solvable irreversible quantum dynamical semigroup on a one-dimensional fermion lattice provides any value ofD _N (γ) between 0 and 1.     

On a third-order phase transition

The asymptotic behaviour of random variables of the general form $$\ln \sum\limits_{i = 1}^{\kappa ^N } {\exp (N^{1/p} \beta \zeta _i )} $$ with independent identically distributed random variables ζ _i is studied. This generalizes the random energy model of Derrida. In the limitN→∞, there occurs a particular kind of phase transition, which does not incorporate a bifurcation phenomenon or symmetry breaking. The hypergeometric character of the problem (see definitions of Sect. 4), its Φ-function, and its entropy function are discussed.     

Hadronic Few-Body Systems in Chiral Dynamics

Hadronic composite states are introduced as few-body systems in hadron physics. The Λ(1405) resonance is a good example of the hadronic few-body systems. It has turned out that Λ(1405) can be described by hadronic dynamics in a modern technology, which incorporates coupled channel unitarity framework and chiral dynamics. The idea of the hadronic ${\bar KN}$ composite state of Λ(1405) is extended to kaonic few-body states. It is concluded that, due to the fact that K and N have similar interaction nature in s-wave ${\bar K}$ couplings, there are few-body quasibound states with kaons systematically just below the break-up thresholds, like ${\bar KNN, \,\bar KKN}$ and ${\bar KKK}$ , as well as Λ(1405) as a ${\bar KN}$ quasibound state and f ₀(980) and a ₀(980) as ${\bar KK}$ .     

The Interpolating Airy Kernels for the \beta =1 and \beta =4 Elliptic Ginibre Ensembles

We consider two families of non-Hermitian Gaussian random matrices, namely the elliptic Ginibre ensembles of asymmetric $N$ -by- $N$ matrices with Dyson index $\beta =1$ (real elements) and with $\beta =4$ (quaternion-real elements). Both ensembles have already been solved for finite $N$ using the method of skew-orthogonal polynomials, given for these particular ensembles in terms of Hermite polynomials in the complex plane. In this paper we investigate the microscopic weakly non-Hermitian large- $N$ limit of each ensemble in the vicinity of the largest or smallest real eigenvalue. Specifically, we derive the limiting matrix-kernels for each case, from which all the eigenvalue correlation functions can be determined. We call these new kernels the “interpolating” Airy kernels, since we can recover—as opposing limiting cases—not only the well-known Airy kernels for the Hermitian ensembles, but also the complementary error function and Poisson kernels for the maximally non-Hermitian ensembles at the edge of the spectrum. Together with the known interpolating Airy kernel for $\beta =2$ , which we rederive here as well, this completes the analysis of all three elliptic Ginibre ensembles in the microscopic scaling limit at the spectral edge.     

Matrix De Rham Complex and Quantum A-infinity algebras

I establish the relation of the non-commutative BV-formalism with super-invariant matrix integration. In particular, the non-commutative BV-equation, defining the quantum A _∞-algebras, introduced in Barannikov (Modular operads and non-commutative Batalin–Vilkovisky geometry. IMRN, vol. 2007, rnm075. Max Planck Institute for Mathematics 2006–48, 2007), is represented via de Rham differential acting on the supermatrix spaces related with Bernstein–Leites simple associative algebras with odd trace q(N), and gl(N|N). I also show that the matrix Lagrangians from Barannikov (Noncommutative Batalin–Vilkovisky geometry and matrix integrals. Isaac Newton Institute for Mathematical Sciences, Cambridge University, 2006) are represented by equivariantly closed differential forms.     

Jack Superpolynomials with Negative Fractional Parameter: Clustering Properties and Super-Virasoro Ideals

The Jack polynomials ${P_\lambda^{(\alpha)}}$ at ???= ?(k?+?1)/(r ? 1) indexed by certain (k, r, N)-admissible partitions are known to span an ideal ${I_{N}^{(k,r)}}$ of the space of symmetric functions in N variables. The ideal ${I_{N}^{(k,r)}}$ is invariant under the action of certain differential operators which include half the Virasoro algebra. Moreover, the Jack polynomials in ${I_{N}^{(k,r)}}$ admit clusters of size at most k: they vanish when k?+?1 of their variables are identified, and they do not vanish when only k of them are identified. We generalize most of these properties to superspace using orthogonal eigenfunctions of the supersymmetric extension of the trigonometric Calogero-Moser-Sutherland model known as Jack superpolynomials. In particular, we show that the Jack superpolynomials ${P_\lambda^{(\alpha)}}$ at ???= ?(k?+?1)/(r ? 1) indexed by certain (k, r, N)-admissible superpartitions span an ideal ${\mathcal{I}_{N}^{(k,r)}}$ of the space of symmetric polynomials in N commuting variables and N anticommuting variables. We prove that the ideal ${\mathcal{I}_{N}^{(k,r)}}$ is stable with respect to the action of the negative-half of the super-Virasoro algebra. In addition, we show that the Jack superpolynomials in ${\mathcal {I}_{N}^{(k,r)}}$ vanish when k?+?1 of their commuting variables are equal, and conjecture that they do not vanish when only k of them are identified. This allows us to conclude that the standard Jack polynomials with prescribed symmetry should satisfy similar clustering properties. Finally, we conjecture that the elements of ${\mathcal{I}_{N}^{(k,2)}}$ provide a basis for the subspace of symmetric superpolynomials in N variables that vanish when k?+?1 commuting variables are set equal to each other.     

EPR studies on the di-nitrogen centers with nonequivalent atoms in a reddish-brown plastically deformed diamond

Two new di-nitrogen centers, which were labeled M2 and M3, were found together with known W7 and N4 centers in an unusual reddish-brown natural diamond. The following magnetic hyperfine interaction parameters (expressed in MHz) were determined for the two nitrogen atoms:A ₁ ⁽¹⁾ =117.95(5),A ₁ ⁽²⁾ =A ₁ ⁽³⁾ =84.48(5),A ₂ ⁽¹⁾ =7.1(1),A ₂ ⁽²⁾ =A ₂ ⁽³⁾ =6.6(1) for M2 andA ₁ ⁽¹⁾ =121.55(5),A ₁ ⁽²⁾ =A ₁ ⁽³⁾ =85.90(5),A ₂ ⁽¹⁾ =6.0(1),A ₂ ⁽²⁾ =5.4(1),A ₂ ⁽³⁾ =5.1(1) for M3. Hyperfine interaction tensors for the nitrogen atom N₁ with a larger interaction have axial symmetry about the <111> direction, but those for the other nitrogen atom, N₂, appear to be small, almost isotropic. Probable models of the M2 and M3 centers are suggested and discussed.     

Addendum to: A new numerical method for obtaining gluon distribution functions G(x,Q 2)=xg(x,Q 2), from the proton structure function F_{2}^{\gamma p}(x,Q^{2})

Since publication of M.M. Block in Eur. Phys. J. C 65, 1 (2010), we have discovered that the algorithm of Block (2010) does not work if g(s)→0 less rapidly than 1/s, as s→∞. Although we require that g(s)→0 as s→∞, it can approach 0 as ${1\over s^{\beta}}$ , with 0<β<1, and still be a proper Laplace transform. In this note, we derive a new numerical algorithm for just such cases, and test it for $g(s)={\sqrt{\pi}\over \sqrt{s}}$ , the Laplace transform of ${1\over\sqrt{v}}$ .     

Symplectic three-algebra and \mathcal{N}=6, Sp(2N)×U(1) superconformal Chern–Simons-matter theory

We introduce an antisymmetric metric into a 3-algebra and call it a symplectic 3-algebra. The $\mathcal{N}=6$ , Sp(2N)×U(1) superconformal Chern–Simons-matter theory with SU(4) R-symmetry in three dimensions is constructed by specifying the 3-brackets in a symplectic 3-algebra. We also demonstrate that the $\mathcal{N}=6$ , U(M)×U(N) theory can be recast into this symplectic 3-algebraic framework.     

Monotonicity of Quantum Ground State Energies: Bosonic Atoms and Stars

The N-dependence of the non-relativistic bosonic ground state energy ? ^B (N) is studied for quantum N-body systems with either Coulomb or Newton interactions. The Coulomb systems are “bosonic atoms,” with their nucleus fixed, and it is shown that $\mathcal {E}_{{C}}^{{B}}(N)/\mathcal {P}_{{C}}(N)$ grows monotonically in N>1, where ? _C (N)=N ²(N?1). The Newton systems are “bosonic stars,” and it is shown that when the Bosons are centrally attracted to a fixed gravitational “grain” of mass M>0, and N>2, then $\mathcal {E}_{{N}}^{{B}}(N;M)/\mathcal {P}_{\!{N}}(N)$ grows monotonically in N, where ? _N (N)=N(N?1)(N?2); in the translation-invariant problem (M=0), it is shown that when N>1 then $\mathcal {E}_{{N}}^{{B}}(N;0)/\mathcal {P}_{{C}}(N)$ grows monotonically in N, with ? _C (N) from the Coulomb problem. Some applications of the new monotonicity results are discussed.