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

Let $\mathbf{p}=\{P_{n,l}{\}}_{\genfrac{}{}{0px}{}{n,l\in {\mathbb{Z}}_{\ge 0}}{n-2l=m}}$ be a sequence of skew-symmetric polynomials in $X_1,\mathrm{.}\mathrm{.}\mathrm{.},X_l$ satisfying $\deg_{X_j}P_{n,l}\le n-1$,whosecoefficientsaresymmetricLaurentpolynomialsin$z_1,…,z_n$. We call $\mathbf{p}$ an $\mathrm{\infty }$-cycle if $P_{n+2,l+1}\bigl|{X{l+1}=z^{-1},z_{n-1}=z,z_n=-z} =z^{-n-1}\prod_{a=1}^l(1-X_a^2z^2)\cdot P_{n,l}$holdsforall$n,l$. 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 $\mathrm{\infty }$-cycle there corresponds a form factor of the above models. Conjecturally all form-factors are obtained from the $\mathrm{\infty }$-cycles. In this paper, we define an action of $U_{\sqrt{-1}}({\widetilde{\mathfrak{s}\mathfrak{l}}}_2)$ on the space of $\mathrm{\infty }$-cycles. There are two sectors of $\mathrm{\infty }$-cycles depending on whether $n$ is even or odd. Using this action, we show that the character of the space of even (resp. odd) $\mathrm{\infty }$-cycles which are polynomials in $z_1,\mathrm{.}\mathrm{.}\mathrm{.},z_n$ is equal to the level $(-1)$ irreducible character of ${\widehat{\mathfrak{s}\mathfrak{l}}}_2$ with lowest weight $-{\mathrm{\Lambda }}_0$ (resp. $-{\mathrm{\Lambda }}_1$). We also suggest a possible tensor product structure of the full space of $\mathrm{\infty }$-cycles.