Abstract
We show that the nonperturbative dynamics of N=2 super-Yang-Mills theories in a self-dual Ω background and with arbitrary simple gauge group is fully determined by studying renormalization group equations of vacuum expectation values of surface operators generating one-form symmetries. The corresponding system of equations is a nonautonomous Toda chain, the time being the renormalization group scale. We obtain new recurrence relations which provide a systematic algorithm computing multi-instanton corrections from the tree-level one-loop prepotential as the asymptotic boundary condition of the renormalization group equations. We exemplify by computing the E_{6} and G_{2} cases up to two instantons.
Highlights
In an ideal world the nonperturbative structure of gauge theories should be computed by quantum equations of motion determined by a symmetry principle
In this Letter, we consider a class of theories where the nonperturbative effects are computed from the properties of surface operators charged under the center of the gauge group
We show that the renormalization group equation obeyed by the vacuum expectation value of such surface operators provides a recursion relation which fully determines, from the perturbative one-loop prepotential, all instanton contributions on the self-dual Ω background or, equivalently, the all-genus topological string amplitudes on the relevant geometric background [6]
Summary
Recursion relations (and the results) coming from blowup equations more involved and difficult to handle. The gauge theory interpretation of these τ functions is the vacuum expectation value of surface operators associated with the corresponding decomposition of the Lie algebra representation under which these are charged We expect these equations and their generalizations to describe chiral ring relations in the presence of a surface operator, which deserve further investigation. The τ functions corresponding to the affine nodes, that is the ones which can be removed from the Dynkin diagram leaving behind that of an irreducible simple Lie algebra, play a special role These are related to simple surface operators associated with elements of the center. Such surface operators are the generators of the one-form symmetry of the corresponding gauge theory [5] Since their magnetic charge is defined modulo the magnetic root lattice, a natural ansatz for their expectation value is. Up to Weyl reflections, the only solution to the above mentioned constraints is given by n1 1⁄4 ep − eq, n2 1⁄4 0, and m1 1⁄4 ep − e1, m2 1⁄4 −eq þ e1, leading to 1⁄21 þ ðep − eqÞ · σ2B0ðσ þ ep − eqÞB0ðσÞ
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