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The open N = 2 string theory is defined on the 4D space-time with the split signature (+, +, −, −). The string field theory action of the open N = 2 string theory is described by the 4D Wess–Zumino–Witten (WZW4) model. The equation of motion of the WZW4 model is the Yang equation, which is equivalent to the anti-self-dual Yang–Mills equation. In this paper, we study soliton-type classical solutions of the WZW4 model in the split signature by calculating the action density of the WZW4 model. We find that the action density of the one-soliton solutions is localized on a 3D hyperplane. This shows that there would be codimension-one-solitonic objects, or equivalently some kind of three-branes in the open N = 2 string theory. We also prove that, in the asymptotic region of the space-time, the action density of the n -soliton solutions is a "non-linear superposition" of n one-solitons. This suggests the existence of n intersecting three-branes in the N = 2 strings. Finally we make a reduction to a (1 + 2)D real space-time to calculate the energy densities of the soliton solutions. We can successfully evaluate the energy distribution for the two-soliton solutions and find that there is no singularity in the interacting region. This implies the existence of smooth intersecting codimension-one branes in the whole region. Soliton solutions in the Euclidean signature are also discussed. [ABSTRACT FROM AUTHOR]

We investigate the quasi-Hopf twist of the quantum toroidal algebra of gl 1 as an elliptic deformation. Under the quasi-Hopf twist the underlying algebra remains the same, but the coproduct is deformed, where the twist parameter p is identified as the elliptic modulus. Computing the quasi-Hopf twist of the R matrix, we uncover the relation to the elliptic lift of the Nekrasov factor for instanton counting of the quiver gauge theories on ℝ4× T2. The same R matrix also appears in the commutation relation of the intertwiners, which implies an elliptic quantum KZ equation for the trace of intertwiners. We also show that it allows a solution which is factorized into the elliptic Nekrasov factors and the triple elliptic gamma function. [ABSTRACT FROM AUTHOR]

We derive a generalized Knizhnik-Zamolodchikov equation for the correlation function of the intertwiners of the vector and the MacMahon representations of Ding-Iohara-Miki algebra. These intertwiners are cousins of the refined topological vertex which is regarded as the intertwining operator of the Fock representation. The shift of the spectral parameter of the intertwiners is generated by the operator which is constructed from the universal R matrix. The solutions to the generalized KZ equation are factorized into the ratio of two point functions which are identified with generalizations of the Nekrasov factor for supersymmetric quiver gauge theories. [ABSTRACT FROM AUTHOR]

We review the problem of Bogomol'nyi–Prasad–Sommerfield (BPS) state counting described by the generalized quiver matrix model of Atiyah–Drinfield–Hitchin–Manin type. In four dimensions the generating function of the counting gives the Nekrasov partition function, and we obtain a generalization in higher dimensions. By the localization theorem, the partition function is given by the sum of contributions from the fixed points of the torus action, which are labeled by partitions, plane partitions and solid partitions. The measure or the Boltzmann weight of the path integral can take the form of the plethystic exponential. Remarkably, after integration the partition function or the vacuum expectation value is again expressed in plethystic form. We regard it as a characteristic property of the BPS state counting problem, which is closely related to the integrability. [ABSTRACT FROM AUTHOR]

We suggest a further generalization of the hypergeometric-like series due to M. Noumi and J. Shiraishi by substituting the Pochhammer symbol with a nearly arbitrary function. Moreover, this generalization is valid for the entire Shiraishi series, not only for its Noumi–Shiraishi part. The theta function needed in the recently suggested description of the double-elliptic systems [Awata et al. JHEP 2020:150, arXiv:2005.10563, (2020)], 6d N = 2* SYM instanton calculus and the doubly-compactified network models, is a very particular member of this huge family. The series depends on two kinds of variables, x → and y → , and on a set of parameters, which becomes infinitely large now. Still, one of the parameters, p is distinguished by its role in the series grading. When y → are restricted to a discrete subset labeled by Young diagrams, the series multiplied by a monomial factor reduces to a polynomial at any given order in p. All this makes the map from functions to the hypergeometric-like series very promising, and we call it Shiraishi functor despite it remains to be seen, what are exactly the morphisms that it preserves. Generalized Noumi–Shiraishi (GNS) symmetric polynomials inspired by the Shiraishi functor in the leading order in p can be obtained by a triangular transform from the Schur polynomials and possess an interesting grading. They provide a family of deformations of Macdonald polynomials, as rich as the family of Kerov functions, still very different from them, and, in fact, much closer to the Macdonald polynomials. In particular, unlike the Kerov case, these polynomials do not depend on the ordering of Young diagrams in the triangular expansion. [ABSTRACT FROM AUTHOR]

The mother functions for the eigenfunctions of the Koroteev-Shakirov version of quantum double-elliptic (Dell) Hamiltonians can be presented as infinite series in Miwa variables, very similar to the recent conjecture due to J. Shiraishi. Further studies should clear numerous remaining obstacles and thus solve the long-standing problem of explicitly constructing a Dell system, the top member of the Calogero-Moser-Ruijsenaars system, with the P Q-duality fully explicit at the elliptic level. [ABSTRACT FROM AUTHOR]

Purpose. To investigate the clinical manifestations, prognosis, and HLA-type of tubulointerstitial nephritis and uveitis syndrome (TINU) with long-term follow-up. Methods. Clinical data of five patients with TINU were retrospectively reviewed. Results. The mean age was 15.8 years. The mean follow-up periods were 54.0 months. The initial subjective symptoms were bulbar injection (100%), ocular pain (80%), and blurred vision (60%). The medical department that the patients visited first was ophthalmology in 4 (80%) cases. Urinalysis showed the characteristic increase of the β2 microglobulin in all (100%) patients. Uveitis and nephritis were diagnosed within 1 week from each other. Although two showed recurrences, the topical and systemic steroid treatment with mean duration of 14.1 months brought the resolution of nephritis and uveitis in all patients. Recurrence-free periods ranged from 12 to 71 months. The final visual outcome was 20/20 or better in all cases. HLA-DR4 or the allele of DRB1∗04 was present in all (100%) patients. Conclusions. TINU should be considered in the differential diagnosis in young patients with uveitis of unknown origin and renal dysfunction. Urinary β2 microglobulin level and HLA typing may help in the diagnosis of TINU. The prognosis for patients with TINU is generally good with steroid treatment. [ABSTRACT FROM AUTHOR]

Skin morphology formation on injection-molded isotactic polypropylene (PP) was investigated using micro-beam synchrotron wide-angle X-ray diffraction and numerical simulation. The 1-20 μm depth range was characterized with an X-ray beam of 0.273 μm χ 0.389 μm in size. From an evaluation of doping nucleating agents (NA) in PP, the NAs did not work at a depth of 1 μm. α-specified NA affected crystallization within a 5-μm depth. β-specified PP showed α-form crystallinity at the 5-20 μm depth. The mesomorphic crystal near the surface showed extremely high orientation. From viscoelastic flow simulation, PP molecules near the surface were oriented in the flow direction by extensional flow in the flow front, but freezing occurred faster than flow-induced crystallization. It was estimated that the delay of crystallization occurred during the transient temperature. The deformation rate did not cause a difference in crystal morphology near the surface, but the cooling rate did. POLYM. ENG. SCI., 2011. © 2010 Society of Plastics Engineers. [ABSTRACT FROM AUTHOR]

It has been argued that Nekrasov's partition function gives the generating function of refined BPS state counting in the compactification of M theory on local Calabi–Yau spaces. We show that a refined version of the topological vertex we previously proposed (arXiv:hep-th/0502061) is a building block of Nekrasov's partition function with two equivariant parameters. Compared with another refined topological vertex by Iqbal, Kozcaz and Vafa (arXiv:hep-th/0701156), our refined vertex is expressed entirely in terms of the specialization of the Macdonald symmetric functions which is related to the equivariant character of the Hilbert scheme of points on ℂ2. We provide diagrammatic rules for computing the partition function from the web diagrams appearing in geometric engineering of Yang–Mills theory with eight supercharges. Our refined vertex has a simple transformation law under the flop operation of the diagram, which suggests that homological invariants of the Hopf link are related to the Macdonald functions. [ABSTRACT FROM AUTHOR]

We argue that the Nekrasov's partition function with two equivariant parameters is reproduced by the method of topological vertex, if we introduce a refined version of the vertex. Our refined topological vertex is expressed in terms the specialization of the Macdonald functions which is related to the equivariant character of the Hilbert scheme of ℂ2. We provide diagrammatic rules for computing the partition function. [ABSTRACT FROM AUTHOR]

We build nearly topological quantum field theories in various dimensions. We give special attention to the case of eight dimensions for which we first consider theories depending only on Yang–Mills fields. Two classes of gauge functions exist which correspond to the choices of two different holonomy groups in SO(8), namely SU(4) and Spin(7). The choice of SU(4) gives a quantum field theory for a Calabi–Yau fourfold. The expectation values for the observables are formally holomorphic Donaldson invariants. The choice of Spin(7) defines another eight dimensional theory for a Joyce manifold which could be of relevance in M- and F-theories. Relations to the eight dimensional supersymmetric Yang–Mills theory are presented. Then, by dimensional reduction, we obtain other theories, in particular a four dimensional one whose gauge conditions are identical to the non-abelian Seiberg–Witten equations. The latter are thus related to pure Yang–Mills self-duality equations in 8 dimensions as well as to the N=1, D=10 super Yang–Mills theory. We also exhibit a theory that couples 3-form gauge fields to the second Chern class in eight dimensions, and interesting theories in other dimensions. [ABSTRACT FROM AUTHOR]

Local gauge invariance is discussed purely at the operator level in the manifestly covariant canonical formalism of the non-abelian gauge theory. In spite of the presence of gauge fixing, it is shown that local gauge transformations are faithfully represented in a certain four-dimensional commutation relation. [ABSTRACT FROM AUTHOR]

We propose a super Lax type equation based on a certain class of Lie superalgebra as a supersymmetric extension of generalized (modified) KdV hierarchy. We are able to construct an infinite set of conservation laws and the consistent time evolution generators for generalized modified super KdV equations. Thefirst few of the conserved currents, the (modified) super KdV equation and the super Miura transformation are worked out explicitly in the case of twisted affine Lie superalgebra OSp(2/2). [ABSTRACT FROM AUTHOR]

The origin of the nontriviality of observables in topological Yang-Mills theory is discussed from the viewpoint that BRST transformation is the exterior derivative. We show that the appearance of singularities due to the Gribov problem makes the Donaldson polynomials cohomologically nontrivial. [ABSTRACT FROM AUTHOR]

As a development of [1], we note that the ordinary Shiraishi functions have an insufficient number of parameters to describe generic eigenfunctions of double elliptic system (Dell). The lacking parameter can be provided by substituting elliptic instead of the ordinary Gamma functions in the coefficients of the series. These new functions (ELS-functions) are conjectured to be functions governed by compactified DIM networks which can simultaneously play the three roles: solutions to non-stationary Dell equations, Dell conformal blocks with the degenerate field (surface operator) insertion, and the corresponding instanton sums in 6d SUSY gauge theories with adjoint matter. We describe the basics of the corresponding construction and make further conjectures about the various limits and dualities which need to be checked to make a precise statement about the Dell description by double-periodic network models with DIM symmetry. We also demonstrate that the ELS-functions provide symmetric polynomials, which are an elliptic generalization of Macdonald ones, and compute the generation function of the elliptic genera of the affine Laumon spaces. In the particular U(1) case, we find an explicit plethystic formula for the 6d partition function, which is a non-trivial elliptic generalization of the (q, t) Nekrasov-Okounkov formula from 5d. [ABSTRACT FROM AUTHOR]

We argue the connection of Nekrasov's partition function in the Ω background and the moduli space of D-branes, suggested by the idea of geometric engineering and Gopakumar-Vafa invariants. In the instanton expansion of 풩 = 2 SU(2) Yang-Mills theory the Nakrasov's partition function with equivariant parameters ϵ1,ϵ2 of toric action on ℂ2 factorizes correctly as the character of SU(2)L × SU(2)R spin representation. We show that up to two instantons the spin contents are consistent with the Lefschetz action on the moduli space of D2-branes on (local) F0. We also present an attempt at constructing a refined topological vertex in terms of the Macdonald function. The refined topological vertex with two parameters of T2 action allows us to obtain the generating functions of equivariant χy and elliptic genera of the Hilbert scheme of n points on ℂ2 by the method of topological vertex. [ABSTRACT FROM AUTHOR]