Let @math be a conformal block, with @math consecutive channels @math , @math , in the conformal field theory @math , where @math is a @math minimal model, generated by chiral fields of spin @math , and labeled by two co-prime integers @math and @math , @math , while @math is a free boson conformal field theory. @math is the expectation value of vertex operators between an initial and a final state. Each vertex operator is labelled by a charge vector that lives in the weight lattice of the Lie algebra @math , spanned by weight vectors @math . We restrict our attention to conformal blocks with vertex operators whose charge vectors point along @math . The charge vectors that label the initial and final states can point in any direction. Following the @math AGT correspondence, and using Nekrasov's instanton partition functions without modification, to compute @math , leads to ill-defined expressions. We show that restricting the states that flow in the channels @math , @math , to states labeled by @math partitions that satisfy conditions that we call @math -Burge partitions, leads to well-defined expressions that we identify with @math . We check our identification by showing that a specific non-trivial conformal block that we compute, using the @math -Burge conditions satisfies the expected differential equation.
5. In @cite_34 , Fukuda, Nakamura, Matsup and Zhu studied the representation theory of @math , the central extension of the degenerate double affine Hecke algebra @cite_25 @cite_3 in the context of the minimal @math models. They found, among other results, that the states are labelled by @math -partitions that satisfy the @math -Burge conditions discussed in this work.
@cite_34 @cite_25 @cite_3
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Recently an orthogonal basis ofWN -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 S3 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 shued. The reshuing 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. We introduce a fermionic formula associated with any quantum affine algebra U q (X N (r) . Guided by the interplay between corner transfer matrix and the Bethe ansatz in solvable lattice models, we study several aspects related to representation theory, most crucially, the crystal basis theory. They include one-dimensional sums over both finite and semi-infinite paths, spinon character formulae, Lepowsky—Primc type conjectural formula for vacuum string functions, dilogarithm identities, Q-systems and their solution by characters of various classical subalgebras and so forth. The results expand [HKOTY1] including the twisted cases and more details on inhomogeneous paths consisting of non-perfect crystals. As a most intriguing example, certain inhomogeneous one-dimensional sums conjecturally give rise to branching functions of an integrable G 2 (1) -module related to the embedding G 2 (1) ↪ B 3 (1) ↪ D 4 1 . We prove that for any partition ( 1;:::; d2) of size ‘d there exists k 1 such that the tensor square of the irreducible representation of the symmetric group Sk‘d with respect to the rectangular partition (k‘;:::;k‘) contains the irreducible representation corresponding to the stretched partition (k 1;:::;k d 2). We also prove a related approximate version of this statement in which the stretching factor k is eectively bounded in terms of d. We further discuss the consequences for geometric complexity theory which provided the motivation for this work. Author(s): Johnson, William Andrew | Advisor(s): Scanlon, Thomas | Abstract: This dissertation is a collection of results in model theory, related in one way or another to fields, NIP theories, and elimination of imaginaries. The most important result is a classification of dp-minimal fields, presented in Chapter 9. We construct in a canonical fashion a non-trivial Hausdorff definable field topology on any unstable dp-minimal field. Usingthis we classify the dp-minimal pure fields and valued fields up to elementary equivalence. Furthermore we prove that every VC-minimal field is real closed or algebraically closed.In Chapter 11, we analyze the theories of existentially closed fields with several valuations and orderings, as studied by van den Dries. We show that these model complete theories are NTP2, and analyze forking, dividing, and burden in these theories. The theory of algebraically closed fields with n independent valuation rings turns out to be an example of such a theory. This provides a new and natural example of an NTP2 theory which is neithersimple nor NIP, nor even a conceptual hybrid of something simple and something NIP.In Chapter 8, we exhibit a bad failure of elimination of imaginaries in a dense o-minimal structure. We produce an exotic interpretable set which cannot be put in definable bijection with a definable set, after naming any amount of parameters. However, we show that these exotic interpretable sets are still amenable to some of the tools of tame topology: they mustadmit nice definable topologies locally homeomorphic to definable sets.Chapter 12 proves the existence of Z nZ-valued definable strong Euler characteristics on pseudofinite fields, which measure the non-standard “size” of definable sets, mod n. The non-trivial result is that these “sizes” are definable in families of definable sets. This could probably be proven e cohomology, but we give a more elementary proof relying heavily on the theory of abelian varieties.We also present simplified and new proofs of several model-theoretic facts, including the definability of irreducibility and Zariski closure in ACF (Chapter 10), and elimination of imaginaries in ACVF (Chapter 6). This latter fact was originally proven by Haskell, Hrushovski, and Macpherson. We give a proof that is drastically simpler, inspired by Poizat’s proofs of elimination of imaginaries in ACF and DCF.