Sums of Recursion Operators

Abstract

Let $(M,\omega,\tau)_A$ be a $2n$-dimensional smooth manifold with a pair of symplectic forms $\omega$ and $\tau$ intertwined by a recursion operator $A \in \operatorname{End}(TM)$. We consider a codimension two submanifolds $Q \subset M$ with those restricted symplectic forms $(\omega|_Q,\tau|_Q)$. Assume that $TQ$ is $A$-invariant. We call the tuple $(M,\omega,\tau,Q)_A$ symplectic-recursion data. In this paper, we consider the problem of fibre connected sum of such two symplectic-recursion data $(M_0,\omega_0,\tau_0,Q_0)_{A_0}$ and $(M_1,\omega_1,\tau_1,Q_1)_{A_1}$. It is interesting to consider potential applications of this result to integrable systems and mathematical string theory.