Let p={Pn,lβ}nβ2l=mn,lβZβ₯0βββ be a sequence of
skew-symmetric polynomials in X1β,...,Xlβ satisfying degXjββPn,lββ€nβ1, whose coefficients are symmetric Laurent polynomials in z1β,...,znβ. We
call p an β-cycle if
Pn+2,l+1ββXl+1β=zβ1,znβ1β=z,znβ=βzβ=zβnβ1βa=1lβ(1βXa2βz2)β Pn,lβ 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 Ξ±aβ=βlogXaβ are the integration
variables and Ξ²jβ=logzjβ are the rapidity variables. To each
β-cycle there corresponds a form factor of the above models.
Conjecturally all form-factors are obtained from the β-cycles.
In this paper, we define an action of
Uβ1ββ(sl2β) on the space of β-cycles.
There are two sectors of β-cycles depending on whether n is even or
odd. Using this action, we show that the character of the space of even (resp.
odd) β-cycles which are polynomials in z1β,...,znβ is equal to the
level (β1) irreducible character of sl^2β with lowest
weight βΞ0β (resp. βΞ1β). We also suggest a possible tensor
product structure of the full space of β-cycles.Comment: 27 pages, abstract and section 3.1 revise