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Form Factors and Action of on ∞-Cycles

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Let be a sequence of skew-symmetric polynomials in X 1,⋯,X l satisfying deg X j P n,l n−1, whose coefficients are symmetric Laurent polynomials in z 1 ,⋯,z n . We call p an ∞-cycle if holds for all n,l. These objects arise in integral representations for form factors of massive integrable field theory, i.e., the SU(2)-invariant Thirring model and the sine-Gordon model. The variables α a =−logX a are the integration variables and β j =logz j are the rapidity variables. To each ∞-cycle there corresponds a form factor of the above models. Conjecturally all form-factors are obtained from the ∞-cycles. In this paper, we define an action of on the space of ∞-cycles. There are two sectors of ∞-cycles depending on whether n is even or odd. Using this action, we show that the character of the space of even (resp. odd) ∞-cycles which are polynomials in z 1 ,⋯,z n is equal to the level (−1) irreducible character of with lowest weight −Λ0 (resp. −Λ1). We also suggest a possible tensor product structure of the full space of ∞-cycles.

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Correspondence to T. Miwa.

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Communicated by L. Takhtajan

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Jimbo, M., Miwa, T., Mukhin, E. et al. Form Factors and Action of on ∞-Cycles. Commun. Math. Phys. 245, 551–576 (2004). https://doi.org/10.1007/s00220-003-1024-0

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