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[en] The structure of topological theory coupled to topological gravity is studied. We show that in this theory Q-exact terms do not decouple. This not decoupling in the action of the theory is connected with the existence of boundaries of the moduli space and leads to problems in defining the topological gravity for massive topological theories. Not decoupling of Q-exact observables leads to filtration of the gravitational descendants constructed from matter fields. Two a priori different preferred splittings of this filtration are constructed (one connected with the massive deformation of the theory and the other connected with the flatness of the connection on the space of theories). It is conjectured that they coincide
[en] An elementary introduction to the 2d/4d correspondences is given. After quickly reviewing the 2d q-deformed Yang–Mills theory and the Liouville theory, we will introduce 4d theories obtained by coupling trifundamentals to SU(2) gauge fields. We will then cleraly see that the supersymmetric partition function of these theories on and on S 4 is given, respectively, by the q-deformed Yang–Mills theory and the Liouville theory. After giving a short discussion on how this correspondence may be understood from the viewpoint of the 6d theory, we conclude the review by enumerating future directions. Most of the technical points will be referred to more detailed review articles. (topical review)
[en] In this article we study certain degenerations of the mirror curves associated with the Calabi-Yau threefolds X, and the effect of these degenerations on the refined topological string partition function of X. We show that when the mirror curve degenerates and become the union of the lower genus curves the corresponding partition function factorizes into pieces corresponding to the components of the degenerate mirror curve. Moreover we show that using degeneration of a generalised mirror curve it is possible to obtain the partition function corresponding to X from X.
[en] Two level parabosonic systems are the simplest systems in which the partition function exhibits intrinsic parastatistical behavior. The authors obtain the thermodynamic partition function for such systems and discuss its important features
[en] Starting with the Fourier representation of the two body potential and by partial integration over certain auxiliary variables, a simple proof is given for the linked cluster expansion of the partition function. The same starting point is also used to derive the functional integration form in both space and momentum variables, leading to the Wilson Hamiltonian
[en] We propose and demonstrate a limiting procedure in which, starting from the q-lifted version (or K-theoretic five-dimensional version) of the (W)AGT conjecture to be assumed in this paper, the Virasoro/W block is generated in the r-th root of unity limit in q in the 2d side, while the same limit automatically generates the projection of the five-dimensional instanton partition function onto that on the ALE space R4/Zr. This circumvents case-by-case conjectures to be made in a wealth of examples found so far. In the 2d side, we successfully generate the super-Virasoro algebra and the proper screening charge in the q→−1, t→−1 limit, from the defining relation of the q-Virasoro algebra and the q-deformed Heisenberg algebra. The central charge obtained coincides with that of the minimal series carrying odd integers of the N=1 superconformal algebra. In the r-th root of unity limit in q in the 2d side, we give some evidence of the appearance of the parafermion-like currents. Exploiting the q-analysis literatures, q-deformed su(n) block is readily generated both at generic q,t and the r-th root of unity limit. In the 4d side, we derive the proper normalization function for general (n,r) that accomplishes the automatic projection through the limit
[en] We propose a limiting procedure in which, starting from the q-lifted version (or K-theoretic five dimensional version) of the (W)AGT conjecture to be assumed, the Virasoro/W block is generated in the r-th root of unity limit in q in the 2d side, while the same limit automatically generates the projection of the five dimensional instanton partition function onto that on the ALE space R4/Zr. This proceeding is based on arXiv:1308.2068
[en] We consider the problem of reconstructing 3D objects via meshfree interpolation methods. In this framework, we usually deal with large data sets and thus develop an efficient local scheme via the well-known partition of unity method. The main contribution in this paper consists in constructing the local interpolants for the implicit interpolation by means of rational radial basis functions. Numerical experiments, devoted to test the accuracy of the scheme, confirm that the proposed method is particularly performing when 3D objects, or more in general implicit functions defined by scattered points, need to be approximated.
[en] We study the Andrews–Gordon–Bressoud (AGB) generalisations of the Rogers–Ramanujan q-series identities in the context of cylindric partitions. We recall the definition of r-cylindric partitions, and provide a simple proof of Borodin’s product expression for their generating functions, that can be regarded as a limiting case of an unpublished proof by Krattenthaler. We also recall the relationships between the r-cylindric partition generating functions, the principal characters of algebras, the minimal model characters of algebras, and the r-string abaci generating functions, providing simple proofs for each. We then set r = 2, and use two-cylindric partitions to re-derive the AGB identities as follows. Firstly, we use Borodin’s product expression for the generating functions of the two-cylindric partitions with infinitely long parts, to obtain the product sides of the AGB identities, times a factor , which is the generating function of ordinary partitions. Next, we obtain a bijection from the two-cylindric partitions, via two-string abaci, into decorated versions of Bressoud’s restricted lattice paths. Extending Bressoud’s method of transforming between restricted paths that obey different restrictions, we obtain sum expressions with manifestly non-negative coefficients for the generating functions of the two-cylindric partitions which contains a factor . Equating the product and sum expressions of the same two-cylindric partitions, and canceling a factor of on each side, we obtain the AGB identities. (paper)