2026/07/20 by Mohamad Alameddine, Alexander Hock
#math-ph #hep-th #math.AG #math.CV #math.MP
We investigate generalized topological recursion on compact spectral curves admitting exponential, and more generally essential, singularities as ramification points. Exploiting the global formulation of generalized topological recursion, we establish a contour deformation of the recursive residue formula that replaces contributions from these essential singularities by residues at meromorphic points. This provides a natural recursive framework for exponentially ramified spectral curves while remaining entirely within the generalized topological recursion formalism. Our formalism can also be viewed as a limiting case of the Bouchard-Eynard higher-order topological recursion, obtained when the order of a ramification point tends to infinity in a convergent manner, as occurs, for example, for exponential singularities while dy remains regular and non-vanishing. We further illustrate the resulting formalism through several examples, including transcendental functions and the x-y dual of the Mirzakhani curve.