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Form factors and action of U_{\sqrt{-1}}(sl_2~) on infinite-cycles

Abstract

Let p={Pn,l}n,l∈Zβ‰₯0nβˆ’2l=m{\bf p}=\{P_{n,l}\}_{n,l\in\Z_{\ge 0}\atop n-2l=m} be a sequence of skew-symmetric polynomials in X1,...,XlX_1,...,X_l satisfying deg⁑XjPn,l≀nβˆ’1\deg_{X_j}P_{n,l}\le n-1, whose coefficients are symmetric Laurent polynomials in z1,...,znz_1,...,z_n. We call p{\bf p} an ∞\infty-cycle if Pn+2,l+1∣Xl+1=zβˆ’1,znβˆ’1=z,zn=βˆ’z=zβˆ’nβˆ’1∏a=1l(1βˆ’Xa2z2)β‹…Pn,lP_{n+2,l+1}\bigl|_{X_{l+1}=z^{-1},z_{n-1}=z,z_n=-z} =z^{-n-1}\prod_{a=1}^l(1-X_a^2z^2)\cdot P_{n,l} holds for all n,ln,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 Ξ±a=βˆ’log⁑Xa\alpha_a=-\log X_a are the integration variables and Ξ²j=log⁑zj\beta_j=\log z_j are the rapidity variables. To each ∞\infty-cycle there corresponds a form factor of the above models. Conjecturally all form-factors are obtained from the ∞\infty-cycles. In this paper, we define an action of Uβˆ’1(sl~2)U_{\sqrt{-1}}(\widetilde{\mathfrak{sl}}_2) on the space of ∞\infty-cycles. There are two sectors of ∞\infty-cycles depending on whether nn is even or odd. Using this action, we show that the character of the space of even (resp. odd) ∞\infty-cycles which are polynomials in z1,...,znz_1,...,z_n is equal to the level (βˆ’1)(-1) irreducible character of sl^2\hat{\mathfrak{sl}}_2 with lowest weight βˆ’Ξ›0-\Lambda_0 (resp. βˆ’Ξ›1-\Lambda_1). We also suggest a possible tensor product structure of the full space of ∞\infty-cycles.Comment: 27 pages, abstract and section 3.1 revise

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    Last time updated on 03/01/2020