Relations among tautological classes on
M
¯
g
,
n
\overline {\mathcal {M}}_{g,n}
are obtained via the study of Witten’s
r
r
-spin theory for higher
r
r
. In order to calculate the quantum product, a new formula relating the
r
r
-spin correlators in genus 0 to the representation theory of
s
l
2
(
C
)
{\mathsf {sl}}_2(\mathbb {C})
is proven. The Givental-Teleman classification of CohFT (cohomological field theory) is used at two special semisimple points of the associated Frobenius manifold. At the first semisimple point, the
R
R
-matrix is exactly solved in terms of hypergeometric series. As a result, an explicit formula for Witten’s
r
r
-spin class is obtained (along with tautological relations in higher degrees). As an application, the
r
=
4
r=4
relations are used to bound the Betti numbers of
R
∗
(
M
g
)
R^*(\mathcal {M}_g)
. At the second semisimple point, the form of the
R
R
-matrix implies a polynomiality property in
r
r
of Witten’s
r
r
-spin class.
In Appendix A (with F. Janda), a conjecture relating the
r
=
0
r=0
limit of Witten’s
r
r
-spin class to the class of the moduli space of holomorphic differentials is presented.