Instantons to the people: the power of one-form symmetries
aa r X i v : . [ h e p - t h ] F e b Instantons to the people: the power of one-form symmetries
Giulio Bonelli, Fran Globlek, Alessandro Tanzini
International School of Advanced Studies (SISSA), via Bonomea 265,34136 Trieste, Italy and INFN, Sezione di Trieste andInstitute for Geometry and Physics, IGAP, via Beirut 2, 34136 Trieste, Italy (Dated: February 3, 2021)Abstract: We show that the non-perturbative dynamics of N = 2 super Yang-Mills theories in aself-dual Ω -background and with an arbitrary simple gauge group is fully determined by studyingrenormalization group equations of vevs of surface operators generating one-form symmetries. Thecorresponding system of equations is a non-autonomous Toda chain, the time being the RG scale.We obtain new recurrence relations which provide a systematic algorithm computing multi-instantoncorrections from the tree-level one-loop prepotential as the asymptotic boundary condition of theRGE. We exemplify by computing the E and G cases up to two-instantons. In an ideal world the non-perturbative structure ofgauge theories should be computed by quantum equa-tions of motion determined by a symmetry principle. Thepresence of extended operators generating higher formsymmetries in quantum field theory is a powerful toolto concretely realise such a programme. A perturbativeanalysis in a weakly coupled regime, if any, would sup-ply appropriate asymptotic conditions. In this letter wepresent a class of theories where the full non-perturbativeresult 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 gener-ated by surface operators [1]. We show that the renor-malization group equation obeyed by the vacuum expec-tation value of such surface operators provides a recur-sion relation which fully determines, from the perturba-tive one-loop prepotential, all instanton contributions onthe self-dual Ω -background or, equivalently, the all-genustopological string amplitudes on the relevant geometricbackground. Actually, partition functions with surfaceoperators display a very clear resurgent structure led bythe summation over the magnetic fluxes [2].The system of equations we study is a non-autonomous twisted affine Toda chain of type ( ˆ G ) ∨ , where ( ˆ G ) ∨ is theLanglands dual of the untwisted affine Kac-Moody alge-bra ˆ G . Each node of the corresponding affine Dynkindiagram defines a surface operator, the associated τ -function being its vacuum expectation value. The timeflow corresponds in the gauge theory to the renormaliza-tion group. The resulting recurrence relations constitutea new effective algorithm to determine instanton contri-butions for all classical groups G . Let us remark thatthe τ -functions we obtain provide the general solutionat the canonical rays for the Jimbo-Miwa-Ueno isomon-odromic deformation problem [3, 4] on the sphere withtwo-irregular punctures for all classical groups, which tothe best of our knowledge was not known in the previousliterature. The recursion relations we obtain are differentfrom the blow-up equations of [5] further elaborated in[6]. Indeed the latter necessarily involve the knowledge ofthe partition function in different Ω -backgrounds. This makes the recursion relations (and the results) comingfrom blow-up equations more involved and difficult tohandle. However, we expect a relation between the twoapproaches to follow from blow-up relations in presenceof surface defects. Indeed, the isomonodromic τ -functionfor the sphere with four regular punctures was obtainedin a similar way from SU (2) gauge theory with N f = 4 in [7]. In this letter we summarise our results and refer toa subsequent longer paper for a fully detailed discussion.The τ -functions are labeled by the simple roots of theaffinization of the Lie algebra of the gauge group α ∈ ˆ∆ ,namely { τ α } α ∈ ˆ∆ , and satisfy the equations D ( τ β ) = − β ∨ · β ∨ t /h ∨ Y β ∈ ˆ∆ , β = α [ τ α ] − α · β ∨ (1)where t := (Λ /ǫ ) h ∨ and the logarithmic Hirota deriva-tive is given by D ( f ) = f ∂ t f − ( ∂ log t f ) . Given a sim-ple root α , its coroot is as usual given by α ∨ = 2 α/ ( α, α ) ,where ( · , · ) is the scalar product defined by the affineCartan matrix. Eq. (1) is the de-autonomization ofthe 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 the-ory to a self-dual Ω -background ( ǫ , ǫ ) = ( ǫ, − ǫ ) [11].In the autonomous limit ǫ → , τ -functions reduce to θ -functions on the classical SW curve [12], which wereused to provide recursion relations on the coefficients ofthe SW prepotential in [13]. The gauge theory inter-pretation of these τ -functions is the v.e.v. of surfaceoperators associated to the corresponding decompositionof the Lie algebra representation under which these arecharged. We expect these equations and their general-izations to describe chiral ring relations in presence ofa surface operator, which deserve further investigation.Higher chiral observables should generate the flows of thefull non-autonomous Toda hierarchy. The actual formof equations (1) depends on the Dynkin diagram. Forthe classical groups A , B and D these reduce to bilinearequations which we solve via general recursion relations.For C , E , F and G the resulting equations are of higherorder and we study them case by case. The symmetriesof the equations are given by the center of the group G ,namely g A n B n C n D n D n +1 E n F G Z ( G ) Z n +1 Z Z Z × Z Z Z − n Moreover, the center is isomorphic to the coset of theaffine coweight lattice by the affine coroot lattice, and co-incides with the automorphism group of the affine Dynkindiagram. By a remark in [14], the coweights, and by ex-tension the lattice cosets, corresponding to these nodesare the miniscule coweights, a representation of g beingminiscule if all its weights form a single Weyl-orbit. Thisremark will be crucial while solving the τ -system.The τ -functions corresponding to the affine nodes, thatis the ones which can be removed from the Dynkin di-agram leaving behind that of an irreducible simple Liealgebra, play a special rôle. Indeed, these are relatedto simple surface operators associated to elements of thecenter Z ( G ) , and are bounded by fractional ’t Hooft lines.Such surface operators are the generators of the one-formsymmetry of the corresponding gauge theory, [1]. Sincetheir magnetic charge is defined modulo the magneticroot lattice, a natural Ansatz for their expectation valueis τ α aff ( σ , η | κ g t ) = X n ∈ Q ∨ aff e π √− η · n t ( σ + n ) B ( σ + n | t ) (2)where B ( σ | t ) = B ( σ ) P i ≥ t i Z i ( σ ) with Z ( σ ) ≡ and Q ∨ aff = λ ∨ aff + Q ∨ , Q ∨ being the co-root lattice and ( λ ∨ aff , α ) = δ α aff ,α for any simple root α . The constant κ g = ( − n g ) r g ,s , where n g is the ratio of the squares oflong vs. short roots and r g ,s is the number of short simpleroots. For simply laced, all roots are long and κ g = 1 .We will now show how the term t σ B ( σ | t ) in (2) isthe full Nekrasov partition function in the self-dual Ω -background upon the identification σ = a /ǫ , where a isthe Cartan parameter. In the A n case, (2) is known asthe Kiev Ansatz. In the A case, it was used to givethe general solution of Painlevé III equation in [15] andfurther analysed in [16].Let us remark that the τ -function (2) displays a clearresurgent structure, with “instantons” given by the mag-netic fluxes in the lattice summed with “resurgent” coef-ficients B ( σ | t ) and trans-series parameter e π √− η , see[17] for a similar analysis in the Painlevé III case.The Ansatz (2) is consistent with equations (1). In-deed, after eliminating the τ -functions associated to thenon-affine nodes, the resulting equation is bilinear andtherefore the Ansatz (2) reduces to a set of recursion re-lations for the coefficients Z i ( σ ) . The variables η and σ are the integration constants of the second order differ-ential equations (1) and correspond to the initial positionand velocity of the de-autonomized Toda particle. Let us set more precisely the boundary conditionswhich we impose to the solutions of equations (1). Weconsider the asymptotic behaviour of the solutions at t → and σ → ∞ as log( B ) ∼ − X r ∈ R ( r · σ ) log ( r · σ ) (3)up to quadratic and log -terms [18]. We will show thatthe solution of (1) which satisfies the above asymptoticcondition is such that B ( σ ) = Z − loop ( σ ) ≡ Y r ∈ R G (1 + r · σ ) (4)where G ( z ) is the Barnes’ G-function and R is the ad-joint representation of the group G . The expansion ofthe above function matches the one-loop gauge theoryresult upon the appropriate identification of the log-branch. This reads, in the gauge theory variables, as ln (cid:2) √− r · a / Λ (cid:3) ∈ R and matches the canonical Stokesrays obtained in [19].Let us first focus on the A n case whose affine Dynkindiagram is τ τ τ j − τ j τ j +1 τ n The root lattice is Q = { n +1 P i =1 c i e i | n +1 P i =1 c i = 0 } , and all thefundamental weights are miniscule, namely λ i = 1 n + 1 (1 i , n +1 − i ) − in + 1 (1 n +1 ) , where (1 p , n +1 − p ) stands for a vector whose first p en-tries are and the remaining entries vanish. We labelthe τ -functions as τ α j ≡ τ j . The τ -system is given bythe closed chain of differential equations D ( τ j ) = − t n +1 τ j − τ j +1 , (5)with τ j = τ n +1+ j . Since all the nodes in this case areaffine we can use the Kiev Ansatz (2). Then, all the τ -functions are determined by τ as τ j ( σ | t ) = τ ( σ + λ j | t ) .It is therefore enough to solve the single equation D ( τ ( σ )) = − τ ( σ ± e ) . (6)Here and in the following we use the notation f ( y ± x ) ≡ f ( y + x ) f ( y − x ) . The Ansatz (2) for τ reads τ ( σ , η | t ) = P n ∈ Q, i ≥ e π √− n · η t ( σ + n ) + i B ( σ + n ) Z i ( σ + n ) and by inserting it into (6) one gets after some simplifi-cations X n , n ∈ Qi ,i ≥ e π √− n + n ) · η t n + n + i + i + σ · ( n + n ) × (cid:18) n − n + i − i + σ · ( n − n ) (cid:19) × B ( σ + n ) B ( σ + n ) Z i ( σ + n ) Z i ( σ + n )= − X m , m ∈ Qj ,j ≥ t m + m + e · ( m − m )+ j + j + σ · ( m + m ) × e π √− m + m ) · η B ( σ + m + e ) B ( σ + m − e ) × Z j ( σ + m + e ) Z j ( σ + m − e ) (7)Now we simply equate the exponents. To fix B ( σ ) ,we look at the lowest order in t . This produces aquadratic constraint and n + 1 linear constraints on theroot lattice variables ( n , n ) and ( m , m ) . Let us fix p, q ∈ { , ...n +1 } , p = q . Up to Weyl reflections, the onlysolution to the above mentioned constraints is given by n = e p − e q , n = 0 and m = e p − e , m = − e q + e ,leading to (1 + ( e p − e q ) · σ ) B ( σ + e p − e q ) B ( σ ) = − B ( σ + e p ) B ( σ − e q ) . (8)This is solved by (4) up to a function periodic on the rootlattice, which is set to one by the asymptotic condition(3). The higher order terms in (7) provide the recursionrelations k Z k ( σ ) = − X n + j + j = k n ∈ e + Q, j , 12 8 P i =1 ε i σ i ) We also solved the recurrence relation arising from (11)up to two-instantons. For one-instanton, our resultsagree with the ones of [23], while the two instantons resultis a too huge formula to be reported here. We remarkthat (11) represents a completely novel way of obtain-ing equivariant volumes of instanton moduli spaces forexceptional groups.Unimodular algebras G , F , E have no outer auto-morphisms and consequently all the τ -functions associ-ated to different nodes are independent. Therefore, theequations on the τ -function associated to the affine nodeturn out to be more difficult to solve. Let us display themfor the G case. τ τ τ In the normalization where its longest root has length 2,the G coroot lattice is the span Q ∨ = Z √ ( − , , ⊕ Z √ , − , . We introduce σ = ( σ , σ , σ ) but all ex-pressions should be restricted to σ + σ + σ = 0 . Byeliminating τ and τ , the τ -system reduces to the singleequation D ( τ − D ( τ )) = 3 t ( D ( τ )) . (12)The operator on the l.h.s. of (12) turns out to factorizeas D ( τ − D ( τ )) = ˜ D ( τ ) · D ( τ ) , where ˜ D ( τ ) is afourth order operator in τ and its derivatives. The trivialsolution of D ( τ ) = 0 is τ = at b which we discard beingincompatible with (2). In the remainder we insert τ ( σ , η | t ) = P n ∈ Q ∨ e π √− η · n (cid:0) − t (cid:1) ( σ + n ) B (cid:0) σ + n | − t (cid:1) and obtain, after a rescaling t 7→ − t , X { n k }∈ Q ∨ { i k }∈ N Y k =1 e π √− η · n k t ( σ + n k ) + i k B ( σ + n k ) Z i k ( σ + n k ) Y k We would like to thank M.Mariño and T. Nosaka for fruitful discussions. Thisresearch is partially supported by the INFN ResearchProjects GAST and ST & FI, by PRIN "Geometria dellevarietà algebriche" and by PRIN "Non-perturbative As-pects Of Gauge Theories And Strings". [1] Gaiotto, D., Kapustin, A., Seiberg, N., and Willett, B.,JHEP , 172 (2015), arXiv:1412.5148 [hep-th].[2] For an introduction to resurgence in QFT, see for ex-ample M. Marino, Instantons and Large N , (2015) Cam-bridge University Press. [3] Jimbo, M., Miwa, T., and Ueno, a. K., Physica D2 , 306(1981).[4] Jimbo, M., Miwa, T., and Ueno, a. K., Physica D2 , 407(1982).[5] Nakajima, H. and Yoshioka,K., Invent. 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