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

Abstract

Let $\mathbf{p}=\{P_{n,l}{\}}_{\genfrac{}{}{0ex}{}{n,l\in {\mathbb{Z}}_{\ge 0}}{n-2l=m}}$ be a sequence of skew-symmetric polynomials in $X_1,...,X_l$ satisfying ${°}_{X_j}{P}_{n,l}\le n-1$, whose coefficients are symmetric Laurent polynomials in $z_1,...,z_n$. We call $\mathbf{p}$ an $\infty$-cycle if $P_{n+2,l+1}{|}_{X_{l+1}=z^{-1},z_{n-1}=z,z_n=-z}=z^{-n-1}{\prod }_{a=1}^l(1-X_a^2{z}^2)\cdot P_{n,l}$ 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 ${\alpha }_a=-\log X_a$ are the integration variables and ${\beta }_j=\log z_j$ 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 $U_{\sqrt{-1}}({\widetilde{\mathfrak{sl}}}_2)$ on the space of $\infty$-cycles. There are two sectors of $\infty$-cycles depending on whether $n$ 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 $z_1,...,z_n$ is equal to the level $(-1)$ irreducible character of ${\widehat{\mathfrak{sl}}}_2$ with lowest weight $-{\Lambda }_0$ (resp. $-{\Lambda }_1$). We also suggest a possible tensor product structure of the full space of $\infty$-cycles.

27 pages, abstract and section 3.1 revised