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Novel construction and the monodromy relation for threepoint fuctions at weak coupling
"... In this article, we shall develop and formulate two novel viewpoints and properties concerning the threepoint functions at weak coupling in the SU(2) sector of the N = 4 super YangMills theory. One is a double spinchain formulation of the spinchain and the associated new interpretation of the o ..."
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In this article, we shall develop and formulate two novel viewpoints and properties concerning the threepoint functions at weak coupling in the SU(2) sector of the N = 4 super YangMills theory. One is a double spinchain formulation of the spinchain and the associated new interpretation of the operation of Wick contraction. It will be regarded as a skew symmetric pairing which acts as a projection onto a singlet in the entire SO(4) sector, instead of an inner product in the spinchain Hilbert space. This formalism allows us to study a class of threepoint functions of operators built upon more general spinchain vacua than the special configuration discussed so far in the literature. Furthermore, this new viewpoint has the significant advantage over the conventional method: In the usual “tailoring ” operation, the Wick contraction produces inner products between offshell Bethe states, which cannot be in general converted into simple expressions. In contrast, our procedure directly produces the socalled partial domain wall partition functions, which can be expressed as determinants. Using this property, we derive simple determinantal representation for a broader class of threepoint functions. The second new property uncovered in this work is the nontrivial identity satisfied by the threepoint functions with monodromy operators inserted. Generically this relation connects threepoint functions of different operators and can be regarded as a kind of SchwingerDyson equation. In particular, this identity reduces in the semiclassical limit to the triviality of the product of local monodromies Ω1Ω2Ω3 = 1 around the vertex operators, which played a crucial role in providing all important global information on the threepoint function in the strong coupling regime [arXiv:1312.3727]. This structure may provide a key to the understanding of the notion of “integrability ” beyond the spectral level.
3 MULTIPLE INTEGRAL FORMULAE FOR THE SCALAR PRODUCT OF ONSHELL AND OFFSHELL BETHE VECTORS IN SU(3)INVARIANT MODELS
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APCTP Pre2013017 Notes on Mayer Expansions and Matrix Models JeanEmile BOURGINE∗
"... Mayer cluster expansion is an important tool in statistical physics to evaluate grand canonical partition functions. It has recently been applied to the Nekrasov instanton partition function of N = 2 4d gauge theories. The associated canonical model involves coupled integrations that take the form ..."
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Mayer cluster expansion is an important tool in statistical physics to evaluate grand canonical partition functions. It has recently been applied to the Nekrasov instanton partition function of N = 2 4d gauge theories. The associated canonical model involves coupled integrations that take the form of a generalized matrix model. It can be studied with the standard techniques of matrix models, in particular collective field theory and loop equations. In the first part of these notes, we explain how the results of collective field theory can be derived from the cluster expansion. The equalities between free energies at first orders is explained by the discrete Laplace transform relating canonical and grand canonical models. In a second part, we study the canonical loop equations and associate them to similar relations on the grand canonical side. It leads to relate the multipoint densities, fundamental objects of the matrix model, to the generating functions of multirooted clusters. Finally, a method is proposed to derive loop equations directly on the grand canonical model. ∗email address: