SHc^c Realization of Minimal Model CFT: Triality, Poset and Burge Condition

Recently an orthogonal basis of $\mathcal{W}_N$-algebra (AFLT basis) labeled by $N$-tuple Young diagrams was found in the context of 4D/2D duality. Recursion relations among the basis are summarized in the form of an algebra SH$^c$ which is universal for any $N$. We show that it has an $\mathfrak{S}_3$ automorphism which is referred to as triality. We study the level-rank duality between minimal models, which is a special example of the automorphism. It is shown that the nonvanishing states in both systems are described by $N$ or $M$ Young diagrams with the rows of boxes appropriately shuffled. The reshuffling of rows implies there exists partial ordering of the set which labels them. For the simplest example, one can compute the partition functions for the partially ordered set (poset) explicitly, which reproduces the Rogers-Ramanujan identities. We also study the description of minimal models by SH$^c$. Simple analysis reproduces some known properties of minimal models, the structure of singular vectors and the $N$-Burge condition in the Hilbert space.Comment: 1+38 pages and 12 figures. v2: typos corrected + comments adde

Reflection states in Ding-Iohara-Miki algebra and brane-web for D-type quiver

Reflection states are introduced in the vertical and horizontal modules of the Ding-Iohara-Miki (DIM) algebra (quantum toroidal $\mathfrak{gl}_1$). Webs of DIM representations are in correspondence with $(p,q)$-web diagrams of type IIB string theory, under the identification of the algebraic intertwiner of Awata, Feigin and Shiraishi with the refined topological vertex. Extending the correspondence to the vertical reflection states, it is possible to engineer the $\mathcal{N}=1$ quiver gauge theory of D-type (with unitary gauge groups). In this way, the Nekrasov instanton partition function is reproduced from the evaluation of expectation values of intertwiners. This computation leads to the identification of the vertical reflection state with the orientifold plane of string theory. We also provide a translation of this construction in the Iqbal-Kozcaz-Vafa refined topological vertex formalism.Comment: 27 pages, 11 figures. Details of translation in terms of IKV refined topological vertex added in the second versio

(p,q)-webs of DIM representations, 5d N=1 instanton partition functions and qq-characters

Instanton partition functions of $\mathcal{N}=1$ 5d Super Yang-Mills reduced on $S^1$ can be engineered in type IIB string theory from the $(p,q)$-branes web diagram. To this diagram is superimposed a web of representations of the Ding-Iohara-Miki (DIM) algebra that acts on the partition function. In this correspondence, each segment is associated to a representation, and the (topological string) vertex is identified with the intertwiner operator constructed by Awata, Feigin and Shiraishi. We define a new intertwiner acting on the representation spaces of levels $(1,n)\otimes(0,m)\to(1,n+m)$, thereby generalizing to higher rank $m$ the original construction. It allows us to use a folded version of the usual $(p,q)$-web diagram, bringing great simplifications to actual computations. As a result, the characterization of Gaiotto states and vertical intertwiners, previously obtained by some of the authors, is uplifted to operator relations acting in the Fock space of horizontal representations. We further develop a method to build qq-characters of linear quivers based on the horizontal action of DIM elements. While fundamental qq-characters can be built using the coproduct, higher ones require the introduction of a (quantum) Weyl reflection acting on tensor products of DIM generators.Comment: 42 page

Sedimentary environments of mangrove swamp in the Funaura Bay, Iriomote Island, Okinawa Prefecture, Southwest Japan

The distribution of conch shell contained in clastic sediments in the mangrove swamps in the Funaura Bay, Iriomote Island, Okinawa Prefecture was studied. The sediments in the mangrove swamp are mainly composed of up to 90% sands. The sand clasts are inferred to be derived from the sandstone of Miocene Yaeyama Group. The conch shells are richer in the muddy fraction than the sandy fraction. Many Terebralia palustris inhabit the mangrove swamp. However few dead shells were also observed in the sediments. Effect of selective transportation hermit crabs is considered to be the cause of this distribution

<ORIGINAL ARTICLE>ESR spin trapping study on ascorbic acid as a scavenger of hydroxyl and hydrogen radicals

The present study was carried out to confirm the usefulness of the ESR spin trapping technique for detecting and measuring short-lived radicals and to clarify the ability of ascorbic acid as a radical scavenger against radicals generated from irradiated human serum. One hundred μl of human serum was irradiated with 1 Gy of X-ray radiation coexisting with 35 μl of DETAPAC, 50 μl of various concentrations (2×10^M～60×10^M) of ascorbic acid, and 15 μl of DMPO. After irradiation at each concentration, the ESR signals were recorded using an ESR spectrometer. These signals were identified, and then normalized as the relative signal intensities against the standard signal intensity of manganese oxide marker. The relative signal intensities were plotted, and the results showed that the signals were from DMPO-OH and DMPO-H, showing the generation of ・OH and ・H, and that the relative signal intensities obtained at each concentration decreased as the concentration of ascorbic acid increased