We develop a variant of calculus of functors, and use it to relate the gauge group G(P) of a principal bundle P over M to the Thom ring spectrum (P-Ad)(-TM). If P has contractible total space, then the resulting Thom ring spectrum is LM-TM, which plays a central role in string topology. Cohen and Jones have recently observed that, in a certain sense, (P-Ad)(-TM) is the linear approximation of G(P). We prove an extension of that relationship by demonstrating the existence of higher-order approximations and calculating them explicitly. This also generalizes calculations done by Arone.
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