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.