An ind-scheme is an ind-object in the category of algebraic schemes. Various variants of ‘the’ category of formal schemes is a proper subcategory of the category of ind-schemes. While many analytic varieties have a structure of algebraic varieties, and loops in a manifold or in an analytic variety can be treated via infinite-dimensional manifolds, their algebraic analogues are examples of ind-schemes.
Mikhail Kapranov, Éric Vasserot, Vertex algebras and the formal loop space, Publications Mathématiques de l’IHÉS, 100 (2004), p. 209-269 numdam, djvu, pdf
M. Kapranov, É. Vasserot, Formal loops II : a local Riemann–Roch theorem for determinantal gerbes, Annales scientifiques de l’École Normale Supérieure, Sér. 4, 40 no. 1 (2007), p. 113-133, numdam, subscription req, math.AG/0509646
A. Beilinson, Vladimir Drinfel'd, Quantization of Hitchin’s integrable system and Hecke eigensheaves on Hitchin system, preliminary version (pdf)