  Let ${\bf p}=\{P_{n,l}\}_{n,l\in\Z_{\ge 0}\atop n-2l=m}$ be a sequence of
skew-symmetric polynomials in $X_1,...,X_l$ satisfying $\deg_{X_j}P_{n,l}\le
n-1$, whose coefficients are symmetric Laurent polynomials in $z_1,...,z_n$. We
call ${\bf p}$ an $\infty$-cycle if
$P_{n+2,l+1}\bigl|_{X_{l+1}=z^{-1},z_{n-1}=z,z_n=-z}
=z^{-n-1}\prod_{a=1}^l(1-X_a^2z^2)\cdot P_{n,l}$ 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 $\hat{\mathfrak{sl}}_2$ with lowest
weight $-\Lambda_0$ (resp. $-\Lambda_1$). We also suggest a possible tensor
product structure of the full space of $\infty$-cycles.
