Instantons to the people: the power of one-form symmetries

We show that the non-perturbative dynamics of $\mathcal{N}=2$ super Yang-Mills theories in a self-dual $\Omega$-background and with an arbitrary simple gauge group is fully determined by studying renormalization group equations of vevs of surface operators generating one-form symmetries. The corresponding system of equations is a {\it non-autonomous} Toda chain, the time being the RG 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 RGE. We exemplify by computing the $E_6$ and $G_2$ cases up to two-instantons.

In an ideal world the non-perturbative structure of gauge theories should be computed by quantum equations of motion determined by a symmetry principle.The presence of extended operators generating higher form symmetries in quantum field theory is a powerful tool to concretely realise such a programme.A perturbative analysis in a weakly coupled regime, if any, would supply appropriate asymptotic conditions.In this letter we present a class of theories where the full non-perturbative result is fixed in such a framework.These are N = 2 super Yang-Mills theories in four dimensional self-dual Ω-background, which enjoy a one-form symmetry generated by surface operators [1].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.Actually, partition functions with surface operators display a very clear resurgent structure led by the summation over the magnetic fluxes [2].
The system of equations we study is a non-autonomous twisted affine Toda chain of type ( Ĝ) ∨ , where ( Ĝ) ∨ is the Langlands dual of the untwisted affine Kac-Moody algebra Ĝ.Each node of the corresponding affine Dynkin diagram defines a surface operator, the associated τfunction being its vacuum expectation value.The time flow corresponds in the gauge theory to the renormalization group.The resulting recurrence relations constitute a new effective algorithm to determine instanton contributions for all classical groups G. Let us remark that the τ -functions we obtain provide the general solution at the canonical rays for the Jimbo-Miwa-Ueno isomonodromic deformation problem [3,4] on the sphere with two-irregular punctures for all classical groups, which to the best of our knowledge was not known in the previous literature.The recursion relations we obtain are different from the blow-up equations of [5] further elaborated in [6].Indeed the latter necessarily involve the knowledge of the partition function in different Ω-backgrounds.This makes the recursion relations (and the results) coming from blow-up equations more involved and difficult to handle.However, we expect a relation between the two approaches to follow from blow-up relations in presence of surface defects.Indeed, the isomonodromic τ -function for the sphere with four regular punctures was obtained in a similar way from SU (2) gauge theory with N f = 4 in [7].In this letter we summarise our results and refer to a subsequent longer paper for a fully detailed discussion.
The τ -functions are labeled by the simple roots of the affinization of the Lie algebra of the gauge group α ∈ ∆, namely {τ α } α∈ ∆, and satisfy the equations where t := (Λ/ǫ) and the logarithmic Hirota derivative is given by D Given a simple root α, its coroot is as usual given by α ∨ = 2α/(α, α), where (•, •) is the scalar product defined by the affine Cartan matrix.Eq. ( 1) is the de-autonomization of the τ -form of the standard Toda integrable system [8,9] governing the classical Seiberg-Witten (SW) theory [10].The de-autonomization is induced by coupling the theory to a self-dual Ω-background (ǫ 1 , ǫ 2 ) = (ǫ, −ǫ) [11].In the autonomous limit ǫ → 0, τ -functions reduce to θ-functions on the classical SW curve [12], which were used to provide recursion relations on the coefficients of the SW prepotential in [13].The gauge theory interpretation of these τ -functions is the v.e.v. of surface operators associated to 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 presence of a surface operator, which deserve further investigation.Higher chiral observables should generate the flows of the full non-autonomous Toda hierarchy.The actual form of equations (1) depends on the Dynkin diagram.For the classical groups A, B and D these reduce to bilinear equations which we solve via general recursion relations.For C, E, F and G the resulting equations are of higher order and we study them case by case.The symmetries of the equations are given by the center of the group G, namely Moreover, the center is isomorphic to the coset of the affine coweight lattice by the affine coroot lattice, and coincides with the automorphism group of the affine Dynkin diagram.By a remark in [14], the coweights, and by extension the lattice cosets, corresponding to these nodes are the miniscule coweights, a representation of g being miniscule if all its weights form a single Weyl-orbit.This remark will be crucial while solving the τ -system.
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 rôle.Indeed, these are related to simple surface operators associated to elements of the center Z(G), and are bounded by fractional 't Hooft lines.Such surface operators are the generators of the one-form symmetry of the corresponding gauge theory, [1].Since their magnetic charge is defined modulo the magnetic root lattice, a natural Ansatz for their expectation value is the co-root lattice and (λ ∨ aff , α) = δ α aff ,α for any simple root α.The constant κ g = (−n g ) rg,s , where n g is the ratio of the squares of long vs. short roots and r g,s is the number of short simple roots.For simply laced, all roots are long and κ g = 1.
We will now show how the term t 2) is the full Nekrasov partition function in the self-dual Ωbackground upon the identification σ = a/ǫ, where a is the Cartan parameter.In the A n case, (2) is known as the Kiev Ansatz.In the A 1 case, it was used to give the general solution of Painlevé III 3 equation in [15] and further analysed in [16].
Let us remark that the τ -function (2) displays a clear resurgent structure, with "instantons" given by the magnetic fluxes in the lattice summed with "resurgent" coefficients B(σ|t) and trans-series parameter e 2π √ −1η , see [17] for a similar analysis in the Painlevé III 3 case.
The Ansatz (2) is consistent with equations (1).Indeed, after eliminating the τ -functions associated to the non-affine nodes, the resulting equation is bilinear and therefore the Ansatz (2) reduces to a set of recursion relations for the coefficients Z i (σ).The variables η and σ are the integration constants of the second order differential equations (1) and correspond to the initial position and velocity of the de-autonomized Toda particle.
Let us set more precisely the boundary conditions which we impose to the solutions of equations (1).We consider the asymptotic behaviour of the solutions at t → 0 and σ → ∞ as up to quadratic and log-terms [18].We will show that the solution of (1) which satisfies the above asymptotic condition is such that where G(z) is the Barnes' G-function and R is the adjoint representation of the group G.The expansion of the above function matches the one-loop gauge theory result upon the appropriate identification of the logbranch.This reads, in the gauge theory variables, as ln √ −1r • a/Λ ∈ R and matches the canonical Stokes rays obtained in [19].
Let us first focus on the A n case whose affine Dynkin diagram is The root lattice is , and all the fundamental weights are miniscule, namely where (1 p , 0 n+1−p ) stands for a vector whose first p entries are 1 and the remaining entries vanish.We label the τ -functions as τ α j ≡ τ j .The τ -system is given by the closed chain of differential equations with τ j = τ n+1+j .Since all the nodes in this case are affine we can use the Kiev Ansatz (2).Then, all the τfunctions are determined by τ 0 as τ j (σ|t) = τ 0 (σ + λ j |t).
It is therefore enough to solve the single equation Here and in the following we use the notation f (y ± x) ≡ f (y + x)f (y − x).The Ansatz (2) for τ 0 reads and by inserting it into (6) one gets after some simplifi- Now we simply equate the exponents.To fix B 0 (σ), we look at the lowest order in t.This produces a quadratic constraint and n + 1 linear constraints on the root lattice variables (n 1 , n 2 ) and (m 1 , m 2 ).Let us fix p, q ∈ {0, ...n+1}, p = q.Up to Weyl reflections, the only solution to the above mentioned constraints is given by n 1 = e p − e q , n 2 = 0 and m 1 = e p − e 1 , m 2 = −e q + e 1 , leading to This is solved by ( 4) up to a function periodic on the root lattice, which is set to one by the asymptotic condition (3).The higher order terms in (7) provide the recursion relations where B 0 (σ) is given by (4).For k = 1 we easily obtain The above coincide with one and two instanton contributions to the SU (n + 1) Nekrasov partition function as computed from supersymmetric localization [20,21].
Let us remark that the use of the τ -system (5) provides a completely independent tool to compute all instanton corrections just starting from the asymptotic behaviour (3).This procedure extends to all classical groups.
D n is a simply laced root system, with the checkerboard lattice 2 ).These correspond to the "legs" of the affine diagram.Whichever rank we consider, we always have the consistency conditions which are also equal if n = 4, due to the enhanced symmetry of D 4 .
, and the two miniscule weights are λ ∨ 0 = (0 n ) and λ ∨ 1 = (1, 0 n−1 ), corresponding to the "antennae" of the diagram.The τ -system coincides with that of D n+1 , with the modification that (i) there is no τ n+1 and (ii) that For n ≥ 3, the analysis proceeds as for D n except we may only use the left antennae and consider the first equation in (9).Therefore, we have a unified approach for both D n and B n .Explicitly, inserting (2) and τ 1 (σ|t) = τ 0 (σ + λ 1 |t) into the first of ( 9) we get after some simplification a formula analogous to (7) leading to quadratic and linear constraints on the lattice labels.By repeating the analysis similarly to the previous case, the equation, analogous to (8), fixing B 0 is The two cases are distinguished by the corresponding different asymptotic conditions (3).Indeed, we have Also the recursion relations are the same, upon using the appropriate root systems R: This result is in line with the contour integral formulae for the relevant Nekrasov partition functions.Indeed the poles in the D n and B n cases are the same, with different residues.From the above recursion relation we can compute the 1-instanton terms and the 2-instantons and so on.These are easily compared to [22].We now turn to the analysis of the other classical groups, which is more involved.Indeed, the τ -system reduces to higher order equations which produce more complicated recurrence relations to be solved by a case by case analysis.We performed explicit checks for C 3 , C 4 and C 5 up to two-instantons again in agreement with [22].
For the exceptional group E 6 we obtain the system where we used the notation D 2n := D 2 • D 2n−2 .The equations which specify B 0 be written as follows.
Choose the miniscule weight to be λ = (0 5 , (− 2 3 ) 3 ).Let p 1 , ...p 5 be a permutation of {1, ..., 5} and let δ := (( 12 ) 8 ).Then one gets from the lowest order in ( 11) The solution satisfying the asymptotic behaviour (3) is B We also solved the recurrence relation arising from (11) up to two-instantons.For one-instanton, our results agree with the ones of [23], while the two instantons result is a too huge formula to be reported here.We remark that (11) represents a completely novel way of obtaining equivariant volumes of instanton moduli spaces for exceptional groups.
Unimodular algebras G 2 , F 4 , E 8 have no outer automorphisms and consequently all the τ -functions associated to different nodes are independent.Therefore, the equations on the τ -function associated to the affine node turn out to be more difficult to solve.Let us display them for the G 2 case.