Journal
JOURNAL OF HIGH ENERGY PHYSICS
Volume -, Issue 6, Pages -Publisher
SPRINGER
DOI: 10.1007/JHEP06(2016)137
Keywords
Conformal and W Symmetry; Conformal Field Models in String Theory; Integrable Field Theories; Supersymmetric gauge theory
Categories
Funding
- Institut Henri Poincare, Paris
- Russian Science Foundation [14-12-01383]
- Australian Research Council
- Russian Science Foundation [14-12-01383] Funding Source: Russian Science Foundation
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Current studies of W-N Toda field theory focus on correlation functions such that the W-N highest-weight representations in the fusion channels are multiplicity-free. In this work, we study W-3 Toda 4-point functions with multiplicity in the fusion channel. The conformal blocks of these 4-point functions involve matrix elements of a fully-degenerate primary field with a highest-weight in the adjoint representation of sl(3), and a fully-degenerate primary field with a highest-weight in the fundamental representation of sl(3). We show that, when the fusion rules do not involve multiplicities, the matrix elements of the fully-degenerate adjoint field, between two arbitrary descendant states, can be computed explicitly, on equal footing with the matrix elements of the semi-degenerate fundamental field. Using null-state conditions, we obtain a fourth-order Fuchsian differential equation for the conformal blocks. Using Okubo theory, we show that, due to the presence of multiplicities, this differential equation belongs to a class of Fuchsian equations that is different from those that have appeared so far in W-N theories. We solve this equation, compute its monodromy group, and construct the monodromy-invariant correlation functions. This computation shows in detail how the ambiguities that are caused by the presence of multiplicities are fixed by requiring monodromy-invariance.
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