JHEP02(2015)160 = 2 and Springer N February 6, 2015 January 26, 2015 February 25, 2015 : : : 10.1007/JHEP02(2015)160 Received Accepted Published doi: Published for SISSA by b [email protected] , . 3 1501.05671 The Authors. and Gerald V. Dunne expansion of the ABJM matrix model, in which non-perturbative effects are c Nonperturbative Effects, Supersymmetric gauge theory

a The Nekrasov-Shatashvili limit for the low-energy behavior of , N [email protected] supersymmetric SU(2) gauge theories is encoded in the spectrum of the Mathieu ∗ ¨ Unsal, showing that for these spectral systems all non-perturbative effects are subtly = 2 Department of Physics, UniversityStorrs of CT Connecticut, 06269, U.S.A. E-mail: Maryland Center for FundamentalCollege Physics, Park, University MD, of 20742, Maryland, U.S.A. b a Open Access Article funded by SCOAP quantized elliptic curve. Keywords: ArXiv ePrint: related to complex space-time instantons.regimes arise In from this an paper exact WKBThis we analysis, approach study and also how join leads these smoothly to through very aand the simple different magnetic proof region. of a resurgenceencoded relation in found perturbation recently by theory, Dunne and identifies this with the Picard-Fuchs equation for the they are more properly describedperturbative by effects resurgent are trans-series deeply inexpansions which entwined. are perturbative convergent, In and but the non- neverthelessin gauge there are theory the non-perturbative electric effects expansion region dueprovides the coefficients, to a poles concrete spectral and analog of which ain phenomenon we the found large recently associate by Drukker, with Mari˜noand Putrov worldline instantons. This N and Lam´eequations, respectively.orders This Bohr-Sommerfeld correspondence relation, but iswhich this usually neglects is expressed non-perturbative very effects, via differentdyonic the in an region nature the the all- of spectral electric, expansions magnetic are divergent, and and dyonic indeed are regions. not Borel-summable, In so the gauge theory Abstract: G¨ok¸ceBa¸sar Resurgence and the Nekrasov-Shatashviliconnecting limit: weak and strongand coupling Lam´esystems in the Mathieu JHEP02(2015)160 31 17 22 10 23 28 34 6 33 6 40 19 26 = 2 supersymmetric SU(2) Yang-Mills N 33 SUSY gauge theory – 1 – ∗ = 2 SUSY gauge theory ¨ Unsal relation and its geometric interpre- N = 2 N = 2 supersymmetric SU(2) Yang-Mills theory possesses a rich 42 N 14 1 39 ]. The 15 – 1 [ barrier top 1 Here, perturbative and non-perturbative refers to the expansion in the Nekrasov deformation parameter 6.1 Resurgent analysis6.2 in the dyonic WKB, region Gelfand-Dikii expansion, and KdV, in the electric region tation actions 3.1 Dyonic region:3.2 resurgence from Electric all-orders region: WKB 3.3 convergent expansions and Magnetic the region:3.4 Nekrasov duality instanton and expansion Connecting analytic strong continuation across and the weak barrier coupling regimes 2.1 Mathieu equation:2.2 notation and Dyonic basic region: spectral2.3 properties resurgent trans-series expansions, Electric deep region: inside the convergent wells continued-fraction expansions, high above the 1 theories using thephysics resurgence formalism thatvacuum unifies structure. perturbative The space and of non-perturbative gauge inequivalent vacua, the moduliof space, the is gauge a theory, not manifold the gauge coupling. 1 Introduction In this paper we revisit the vacuum structure of A Simple analog of uniform asymptoticB behavior WKB coefficients 6 Lam´eequation and SU(2) 7 Conclusions 4 Worldline instantons and multi-photon vacuum5 pair production A simple proof of the Dunne- 3 All-orders WKB analysis of the Mathieu equation: actions and dual Contents 1 Introduction 2 Mathieu equation and SU(2) JHEP02(2015)160 ] , and 50 , (1.1) ~ (or the dyonic 49 c 0 limit, electric , for the N ], and the u → 24 ~ ]. Remarkably, ]. 28 – 12 , 25 ], in the 11 2 85 , analysis valid throughout 47 , variables, the coupling is directly identified with the . It is also possible to introduce yet region is a cross-over region in 46 , ], but this is not done in the current N u 32 two 12 uniform , , 31 11 . In their seminal work, Seiberg and Wit- magnetic i ]. A further conceptual and computational 2 ~ ). For a 23 ~ N – Tr Φ – 2 – h ≡ 16 N, ].) The ( λ = u 52 u , = 51 u ) Toda system, as in [ c : N ]. The moduli parameter N 45 – . In this paper we are discussing SU(2) gauge theory. The role of 30 c N , of the supersymmetric (SUSY) gauge theory. The interplay of per- ] that systematically unify perturbative and non-perturbative physics; , from the SU( c is not 48 N – ¨ Unsal in the spectra of certain quantum systems, which shows that all N expansion of the ABJM matrix model, in which non-perturbative effects are 46 , magnetic region has convergent perturbative expansions but there are nevertheless non- N 12 , and 11 An elementary but significant observation is that the energy eigenvalue, In this paper we extend this approach to show that there are additional non- electric Note that our 2 of the matrix model) is played in this context by the level number usual analyses of divergent perturbativeand expansions their [ associated resurgent trans-series representations, areN implicitly restricted toanother parameter, a paper. also the eigenvalue levelthe label entire spectrum, it is natural to define a “’t Hooft parameter”, and consider different limits of the two parameters, including a double-scaling limit. The spectral problems also providesby a Dunne simple and proofnon-perturbative of effects are a subtly surprising encoded resurgence in relation perturbation found theory [ Schr¨odingersystems should be viewed as a function of in the large related to complex space-time instantons,the and ’t which Hooft were expansion subsequentlywhich related coefficients non-perturbative to [ effects poles are large, in andwidth. in fact the This spectral correspondence bands and between gaps are the of Nekrasov-Shatashvili equal limit and monodromies of region is characterized by divergentseries perturbative [ expansions described bythe resurgent trans- perturbative effects associated withconcrete poles analog of of a the phenomenon expansion found coefficients. recently by (This Drukker, provides Mari˜noand a Putrov [ perturbative aspects of this relationgaps that reflect and the bands physics in of thegions, the spectra. non-perturbatively Schr¨odinger and small These spectra these havedyonic can three distinct be physical explicitly re- turbation associated theory with and the non-perturbative three physics physical is regimes, different in each region. The the prepotential in the Nekrasov-Shatashvili limitin is certain encoded simple in Schr¨odingersystems the through spectra theof of monodromy certain differential and states equations exact WKB [ eigenvalues properties of these Schr¨odingersystems, and thevalues scalar are (and dual identified scalar) with field actions expectation and dual actions in an all-orders WKB analysis. ten showed that this manifold(i.e. is tori) precisely and the the dynamics moduli of spaceric the low language of energy in genus-one effective the Riemann theory terms can surfaces of bebreakthrough elliptic formulated came in curves [ a with geomet- the introductionsubsequent direct of relation the to Nekrasov integrable partition models function and [ the Bethe ansatz [ parametrized by the scalar condensate JHEP02(2015)160 , Z 1 ~ )). (1.3) (1.4) (1.5) limit  2.30 N ] of the N 22 . – (see ( | ) 16 1), and dyons λ u , ( ]. D 0 (0 a is small, (1.2) ± 54 ) m – q λ N ( 51 n for large . Around these points ˜ φ dφ 2 F ) + λ 2 ), correspond to the two u Λ ( − ln u cos − 0 ( n 2 is the ’t Hooft coupling], and 2 a N ], where in the large D 0 2 e ]. The information about the ~ Λ = a λ q − 54 [ | 18 e – =0 − u φ dφ ∞ 2 λ X n correspond to lowest order WKB are integration cycles discussed in u √ 49 √ 2 cos D 0 p N/ ) and = and 2 ) = a ) , γ − u 2 λ Λ ( 2 1 e m ( Λ 0 γ ,q n − a e u/ q F ( u 1 n = Λ ) and 2 − p u − u ( 2 π cos 0 – 3 – , 0 0 a N Z Z , but of the form ,M λ 2 2 ) =0 ∞ π π X n u √ √ → ∞ ( energy eigenvalue has the form D 0 u ≡ ≡ th a ) = µ µ m N 2 1 q γ γ for small N, λ bosons and an infinite tower of dyonic tower with charges I I ( ) + u Z u N/λ regime the perturbative expansions are convergent, but the per- ( ) = ) = − 0 λ e u u ) = a ( ( ~ e 0 q D 0 a a N, ( ]. We show that there are in fact non-perturbative corrections to this ) = u are integers that denote the electric and magnetic charge of the state. 6 u ( m m q ,q e q Z and e 1). In the other, it consists of magnetic monopoles with charge q , There are three singular points in the moduli space where one or both of the cycles We first review some basic facts from the SUSY gauge theory side [ e q ( the rest of theelectric, paper magnetic we will and refersectors dyonic to in regions, the the local respectively. moduliof neighborhoods space the around In with these theory general, different points± consists there particle as of are spectra. the two In separate one sector, the spectrum Here acquire branch points.on They the are modulibosons, space almost the massless magnetic low monopoles energy and almost theory massless is dyons, respectively. described For by weakly coupled massive cycles for the MathieuBPS system, states hence of the the subscriptsmass “0” theory of [ is a BPS also state contained are in given these as cycles, where the central charge and where Λ is thedetail dynamically below, generated scale, in and section 3. In this form gauge theory quantities and the correspondingvalues spectral quantities. of The the vacuum expectation scalarindependent cycles field on and the its torus dual given as partner, This can bematrix compared models with associated recent with resultsthere the are concerning ABJM extra non-perturbative free non-perturbative terms energy contributionsthese of [ extra to the form terms are related to poles of thecorrespondence and expansion set coefficients our [ notation, in order to make the precise identification between the which is analogous to thetheory genus system expansion [ of theexpression, free of the energy form for aMoreover, in matrix this model large orturbative gauge coefficients have poles that are responsible for the non-perturbative splittings. corresponding to states withperturbative expression energy for well the below the energy barrier. More generally, the particular non-uniform limit in which the eigenvalue level number JHEP02(2015)160 ), 0 ) of a 0 ]. In ( (1.8) (1.9) (1.6) (1.7) a 0 27 F – , 24 ), is intro- 2 ] = 0, called 0 The first few ,  1 /a  3 | D 0 = 0 and contains a a ... k ( ], a particular one [ 4 u prepotential F + m  27 = 0. We denote the 0 – I . Λ 14 0 16 a 2 ) is ) 25   0 a πia 1.7 ,k ( π 0 31 ) 4 c 1469Λ 2 0 2 YM a , and =1 ( g ∞ + . ~ k X . i ) 10 0 2 0 2 + inst ≡ 0 πi a 2 12 Λ, the prepotential has the following ) F ,  1 0 πia 1  . -instanton corrections.  a + 3Λ |  ( k π k ) + a  2 4 256 0 ( 0 2 a 2 0 a  YM F ( a + Λ ) in the limit 0 θ . 0 Λ a a 6 0  ( → = – 4 – 0 2  pert 8 0 , 0 F πia ,k log F 1 2 0  0 5Λ F 2 0 c π 2 ∂a a 256 2 ∂ ) + i =1 0 ) = lim ∞ k X + a a + ( ( . 2 0 0 2 0 2 0 πi 4 a a , F 2 , where the BPS states with higher charges can decay into 0 Λ πia D 0 class τ 0 8 a 1 2 F = = = = 0 ) = 0 ) = 0 F 0 a are the theta parameter and the coupling constant (at the scale ∂a a ∂ ( . ( . 0 2 F inst YM Λ 0 g F ± = 1). These two regions are separated by a closed curve, and u ± , YM (1 θ ± An alternative way to calculate the instanton sum is through localization [ Another important object for the low energy effective theory is the curve of marginal stability A comment here concerning terminology: throughout the paper, we use the word “instanton” to describe = 2 SUSY SU(2) gauge theory, the instanton expansion in ( 3 parameter deformation of the gauge theory with two different objects:mechanical instantons (i) that the appearThe BPST in distinction instantons the should quantum that bewill mechanical appear apparent mostly description within refer in of the to the the context (ii). deformed 4d of gauge the gauge theory. discussion, theory; and (ii) for the the bulk quantum of the paper it obtains the Seiberg-Witten prepotential In this paper we will focus on the Nekrasov-Shatashvili limit [ However beyond the two-instanton level,computationally the very direct difficult. quantum field theory methods become this approach, aduced. two-parameter generalization The of parametersThe the characterize prepotential prepotential, of certain the SUSY deformed preserving theory space-time is deformations. calculable via localization technique and one The first two termsthe in last this term expansionterms is of are the the the sum instanton classical expansionthe over calculated and extraction non-perturbative via of one-loop standard the contributions, fieldN coefficients theory and from methods the agree Seiberg- with Witten solution. For example, for the where the gauge theory. Insemiclassical the expansion: electric region where the monopoles and dyons. Thisthe curve points is approximately an ellipse around which is a holomorphic function whose derivatives are with JHEP02(2015)160 0, = 2 → (1.13) (1.10) (1.11) (1.14) (1.12) 2 N k ... , is zero, the + m   u ψ . ... ) = 2 u ... 2 n ( 12 theory reduces to the 14 ) 2 a Λ ψ + a ~ D n ∗  a  8 a ~ 8 a, log τ n a ( . = 2 2 ;  3072 0 1495Λ , is encoded in the energy 2 ~ n πi 21Λ u a 256 2 K N inst 2 K + d =0 ∞ F iπ X n + − 8 10 =1 ∞ ) 4 a + X n 4 a a ) + ) = 2 2 Λ x ~ ~ ~ ) generalize to the Bohr-Sommerfeld 16 219Λ  a, 2048  + u, ( 1.4 ( . P ) 2 + 2 2 D πi 2  ~ a ), ( 6 2 pert u ψ 2 2Λ ,  4 a 4 ~ ~ F Λ = 1.3 . 64 ) + = 2 SUSY gauge theory, and Lam´etheory for log ) − = ψ – 5 – a  , a a ) + 2 ( 1 ) ( . N 2  πi ~ 4 n x | m πi ~ ~ a F a, 2 48 inst  0 ( ]. They can be expressed formally as all orders ( n ], and also with 0+1 dimensional (i.e. quantum me- . ) 2 1 8 − F + F = 2 SUSY SU(2) gauge theory, the expansions for the u ~ 0 28 ( 32 + cos( + , – n class → N =0 ∞ 2 a ) = (quadratic poly. in ) = limit, combined with vanishing elliptic parameter 2 2 ψ ψ X F n  31 25 ~ ~ , the periods ( n 2 2 2 ~ dx dx d d = = ) of a, a, ~ = 4 theory, while at large mass it reduces back to the ( ( 2 2 ) = lim . . 2 2 ~ ~ → ∞ ~ =0 1.8 ∞ N X n 2 inst − − pert a, ( F m F ]. In the limit where the mass of the hypermultiplet, ) = NS ~ 44 F : ) + – u, ~ ( is finite. 30 SUSY gauge theory, a model with a massive hypermultiplet in the adjoint a a, ∗ 2 ( ∗ . k ] to have a smooth decoupling limit in which the 2 = 2 : = 2 m = 2 class 6.2 is the Weierstrass elliptic function. Some minor rescaling is required [as discussed theory becomes N N F N ∗ P Furthermore, at nonzero In the Nekrasov-Shatashvili limit, there is a direct correspondence between the gauge = 2 = 2 theory, taking integrals of theWKB QM expansions: system [ where in section N such that Schr¨odingerequation SU(2) representation [ N theory. In particular,eigenvalue the of gauge the theory quantum moduli mechanical (QM) parameter, system described by the time independent theory and integrable modelschanical) [ Sine-Gordon theory for SU(2) Continuing the example ( perturbative and instanton contributions are prepotential in this limit as JHEP02(2015)160 – ]. 57 78 = 0, , (2.1) ¨ Unsal ]. The ~ 76 ]. Here 12 56 , , 11 of bands and 55 ], from the per- u ψ = 44 , ψ widths http://dlmf.nist. ) of the bands or gaps. 43 x cos( 2 locations + Λ 2 ψ 2 dx d 2 2 ~ ). However, this is only part of the story, . Using the direct correspondence, we can λ = 2 SUSY gauge theory ]. ) give only the 1.14 ]. In section 2 we review basic properties of the – 6 – −→ − 58 68 N 1.14 – 59 = 0 ] applied to exact quantization conditions for complex ψ 45 )) z -ensembles and the associated quantum geometry [ cos(2 β across the entire spectrum. This relation has also been formulated Q 2 − A + ( ]. In section 5 we discuss a Picard-Fuchs interpretation that explains the uniformly 00 ψ 75 – ): 69 In this paper we build a unified WKB analysis that spans all regions of the spectrum, The relation between the Nekrasov-Shatashvili limit and exact quantization conditions ], translated into notation that makes explicit the relation to the Nekrasov partition region, based on the Gelfand-Dikii expansion of the resolvent. We end with a summary 80 function. The standardgov/28 textbook form of the Mathieu equation is ( 2 Mathieu equation and SU(2) 2.1 Mathieu equation:In notation this and section we basic review spectral relevant facts properties about the spectrum of the Mathieu equation [ relation that connects the fluctuations aboutabout the perturbative the vacuum one-instanton with the sector, fluctuations Lam´esystem is and discussed with in all section 6, higherλ by multi-instanton means sectors of [ aand complementary approach comments to about the future large work. we describe a physical analogof of the the transition transition between betweeneffect different tunneling spectral [ regions, and in multi-photonsignificance terms pair of resurgence, production and also inof show the that the this Nekrasov-Shatashvili Schwinger SUSY limit gauge yields theory a perspective simple in proof terms of the recently found Dunne- for connecting strong and weak coupling [ using all-orders exact WKB analysisMathieu [ spectrum, and itsShatashvili relation limit. to Section the 3 Nekrasov contains partition the function uniform in all-orders the WKB Nekrasov- analysis. In section 4 ods, treated in terms of Whithamquantum dynamics mechanical [ systems. Another recenttum connection spectral between problems monodromies arises and in quan- theIndeed, study even of in the simple statistical zero-dimensional mechanics of models, Coulomb monodromy gases and [ resurgence are important different regions of theassociate ’t this with Hooft different parameter physical behavior in the correspondingof SUSY quantum gauge mechanical theory. systems hasspective also of been holomorphic discussed recentlywe in follow [ a complementary approach, building our analysis on elementary WKB meth- the leading order WKBas approximation these to Bohr-Sommerfeld ( expressions ( There is also important spectralgaps. information This information in is the connectedin non-perturbative to different the parts perturbative of information the in spectrum, quite corresponding different ways to different semiclassical behavior for the One recovers the Seiberg-Witten solution in the continuum limit obtained from JHEP02(2015)160 ] ). 47 = 1, 2.8 , (2.3) (2.2) u 46 ℏ ψ as a function of 1) u − , and exponentially ~ ZJJ 1.5 g E . follows the quadratic behavior 3.3 = (16 ψ follows the linear behavior in ( 2 u  1, are shown as dotted lines. Notice ) ~ 8 location ± x = = ψ location u 1.0 + cos( ZJJ 2 E 2 ,A d dx will be identified with the scalar condensate moduli = – 7 – 2 u ) ψ g 2  2 ) 2 (16 ~ 4Λ g x − , rather than the conventional lower-case ones, as the √ =  Q . This transition occurs at the top of the potential, where (2 Q ~ 2 0.5 and sin −→ g 1 A ). This eigenvalue 8 + 2.1 2 2 d dx have special meaning in the gauge theory discussion). We thus make the ]), who used a different scaling: 1 2 q , as in ( − ~ 12 ,  ) and ℏ the gaps are exponentially narrow, and the gap 11 ( a . The band spectrum of the Mathieu equation, expressing the eigenvalue u ~ ). The top and bottom of the potential, at 0.5 1.5 1.0 2.5 2.0 0.5 1.0 - - We also make explicit comparison with the work of Zinn-Justin and Jentschura [ 2.23 , the bands are exponentially narrow, and the band (see also [ We will mostly setsimple the dimensional scale scaling Λ arguments. = 1 in what follows, re-introducing it where necessary by (We use capital letters symbols identifications: in ( the smooth transition betweennarrow exponentially gaps narrow (unshaded) bands at (shaded) large shown at small as a straightequal line. width, and Note are that not in exponentially narrow, the as vicinity discussed of section the barrier top, the bands and gaps are of Figure 1 the parameter parameter in the SUSYband gauge shown theory. as The a bands~ solid are (blue) shaded line, in and grey,At the with large top the lower edge edges of of each each band as a dashed (red) line. At small JHEP02(2015)160 , u (2.4) (2.5) is to be ℏ = 0 as: E z 6 ). 2.4 , using the re- Q ), ( ], can be expressed 2.3 79 5 , ZJJ , or E g ), ( 78 ~ , ), normalized at 2.2 ~ with the SUSY gauge theory 76 . 1 + u u, ; # − z ) occurs at the top of the potential, ( ~ = 2 4 1 0 0 1 ψ " (2.6) . The band/gap edges correspond to ) θ ) by a simple half-period shift) ~ = ZJJ x ), can then be written in compact form , in terms of # u, z ) ) ( ; g E A ) and ~ ~ ]. The transition region between exponentially π ~ ψ ( 1 u, u, iθ u, 47 , e ψ ; 3 , or z – 8 – (0; (0; 1 + 16 ( 0 46 2 2 E 1 , ) = − ) = ψ ψ ψ θ π ) ) 12 = , ~ ~ + 11 z u, u, cos ( ( ), and with the scale Λ = 1. The rescaled eigenvalue ψ 2 (0; (0; 0 1 1 2.4 ψ ψ made using the above re-scalings ( " E [and therefore g , u )], for each Bloch parameter , as in ( ~ or E 2.4 ) evaluated one half period away from the normalization point: ) and exponentially narrow gaps (large A ~ ~ 1 = 16 ), ( u, ~ ; 2.3 π ) is the “exact quantization condition”, implicitly expressing the eigen- ( , as shown as a dotted line. Note that in the vicinity of the barrier top, the bands 1 ~ ), ( / ψ 2.6 ) 2.2 in terms of . Another view of the band spectrum of the Mathieu equation, expressed here in terms of = 2 ℏ 0 ( u 2 6 4 8 E E The exact Bloch spectral condition, or Floquet analysis [ Equation ( value scalings ( The Bloch boundary condition, in terms of Because of the direct identification ofscalar the rescaled condensate, eigenvalue we willwith describe conversions to the Mathieu spectrum in termsin of terms this of eigenvalue two independent solutions, and gaps are of equal width, and are not exponentiallyThus, small. we identify (note: we flipped the sign of cos( Figure 2 the rescaled eigenvalue compared with the normalizationsnarrow in bands [ (small where JHEP02(2015)160 . ~ ]. 81 ) is of lim- 2.6 we see a transition 1; (iii) deep inside the 1 , to narrow gaps, with ∼ : ~ ~ 3 1. In fact, this transition N and ∼ u N 1, exponentially narrow bands 1, and illustrates the same behavior, in , which emphasizes the transition ∼ 2 ~  / u 1, bands and gaps of equal width 1, exponentially narrow gaps ~ ]. Different expansions are suitable for ∼ N  , we see the transition from exponentially + 1) . In this transition region the bands and ~ 80 ~ 2 1, – u N . Figure N ~ 3.3 78 , to particle-on-a-circle behavior at large and ∼ − = ( 1, ~ – 9 – +1, 1 u E and  ∼ u u N . However the exact quantization condition ( . In figures π 1. In gauge theory language, these are the “electric”, “magnetic” 2  ~ and 1 N 1; (ii) barrier top region, where , as discussed in section  8 π ~ 1, and ∼ N ~ ∼ − N u 1, and . Three different spectral regimes for the Mathieu equation: (i) far above the barrier,  u dyonic: deep inside potential wells: magnetic: near the barrier top: electric: high above barrier top: There are clearly three interesting spectral regimes, which we identify with three in- The spectrum of the Mathieu equation consists of an infinite sequence of bands and • • • . Concrete approximations for the eigenvalues can be obtained from the exact quantiza- being an integer multiple of 1 teresting physical regions of the gauge theory, as shown in figure from narrow bands, with locationslocations approximately approximately linear quadratic in in terms of the rescaledfrom energy harmonic eigenvalue oscillator behavior at small gaps, as shown in figures narrow bands low inThe the cross-over spectrum, occurs to near exponentiallyoccurs the narrow when top gaps of higher in thegaps the potential, are of spectrum. where equal width, and neither is exponentially narrow. In figure such as trigonometric ordifferent Hermite regions functions in [ the spectrum,inside the as wells, is we familiar usein from the the elementary tight-binding wells, solid approximation while state in far physics. terms above of Deep the ‘atomic’ barrier states we bound use the ‘neary-free-electron model’ [ θ ited practical use unless we haveψ an explicit expression for thetion normalized Mathieu condition function by making expansions of the Mathieu functions in terms of other functions, Figure 3 where wells, where and “dyonic” regions, respectively, as shown in the figure. JHEP02(2015)160 , ], # 12 47 , (2.8) (2.9) ,  1 2 11 0, with 46 + → ) of [ N ~ .  are factorially 2.3 ], as explained 1 3 4 N + 1) ... ~ +1 2 47 ), translated into + ,  u N !) 3 − , plus perturbative ~  ≡ N # 46 , ( 1 2 + 2 ...  E n 1 2 + + 32 ; but we still have three , Γ(  + N 1 2 31  , +1 N fixed and + , we can consider the energy " → ∞ region gives information about N  ]: ) can be derived from the inver- ~ 2 N , therefore permits access to all 12 3 λ N N , 2 +2 λ ~ N  N 47 16 n 2 2.8 64 , # 405 11 ~ 16 − 9 (2.7) ≡ 32 + ) 46 π , # 3 ). λ 1 + region can be interpreted as + 1 4  : 2 12 λ N, λ , 1 2 ), ∼ − + λ (  3.28 ) 2 , the expansions for 1. (In fact, we show later that the cross-over u 11 ∼ − 1 2 + [ 1.1 ~  N )–( ) / ( ~ + and 1 2  N 1 ~ – 10 – n ) = fixed, requires  λ ~ + N N 3.21  N, http://dlmf.nist.gov/28.8.E1 λ  ( 8 N , c N, u 205 ( N 4  , or 17 1, or u ) can be derived in several ways. Straightforward " ~ + analysis, valid for all ∼ + 2 5 2.8 n ~ 16 4 λ  ~ 0, with and  , with 1 2 ) − : see eqs. ( 1, 1 2 N N  N + → ( 1 2 + , the situation is reversed.  . On the other hand, the large n 3.1 uniform ~ g 1 ~ N c λ + N  =0  ∞  N X n 4 5 2 33  ). A N " " ~ 8 π ) = 3 4 4 5 ~ ~ ∼ 16 16 1 + N, g λ − − 1 regime, which is weak-coupling in the QM sense but strongly-coupled in the ( E ∼ −  ) λ ~ For fixed level number There is an interesting inversion of the meaning of strong and weak coupling. In gauge In terms of the “ ’t Hooft coupling” ( N, ( fixed, which therefore gives information about the weak-coupling expansion for low-lying u These perturbative expansions areon therefore their own, non-Borel-summable, and and should so be extended are to incomplete real, unambiguous trans-series expansions [ corrections. Alternatively, this perturbative expansionsion ( of an all-orders-WKBin Bohr-Sommerfeld detail relation below [ in section divergent and non-alternating, as series in This perturbative expressionRayleigh-Schr¨odingerperturbation theory about ( the bottom ofacteristic each harmonic well leads oscillator to the form, char- In this gauge theory sense, theformal perturbative perturbative expansions expansion for reads theour energy notation: ( levels are divergent. The theory language, the electriccoupled. region is On the weakly other coupled, hand,in in while quantum terms the mechanical language of dyonic in the region the coupling scaling is ( strongly 2.2 Dyonic region: resurgent trans-series expansions, deep inside the wells region is at regions of the spectrum. ForN example, the small energy levels with the strong-coupling expansion for low-lying modes, with eigenvalue as a function of The semiclassical limit, different regions, where JHEP02(2015)160 (2.11) (2.12) (2.10) (2.13) ) ) 2 ~ ( O ! l + ...  # 1 − 3 4 ~ − 4 ) diverges at a rate as- +    = 8. 2.8  ~ ln 16 ... 1 2  dx  − k + · ) are related to one another in #
1) 8 N N 13 2 8 ~ ) + 1 ( /  − x 1 1 − × − nkl − n 2  ( c N + 4 8 ~ n cos(  ~ 2 · exp − ~ p  16 "  8 π 1 2 ], in contrast to an asymptotic perturbative 13  n π − · ]. Note that these log terms first appear at ~ 2 R + , − exp ) 5 2 2 1 84 2 – 11 – 1 n N N – / effect, and is real and unambiguous. The factor 8 √ − ( 1 ·  82 − 1 3 5 2 nkl N

" c  − ~ ]. This leading cancellation, at the two-instanton level, 1 / ~ 2 ~ − 1 32 =0 l k X 16 47   , − − ! =0 ∞ n 1 k X 46 π e ! +1) , ( ∼ N =0 N ∞ ∼ × X n 4( 12 , ) 2 single-instanton ], for the lowest band, the large-order behavior of the perturbative (0) ~ n ]. This ambiguous imaginary non-perturbative term is in fact can- ). 2 π 11 , c ) = ~ 12 ~ , / (0 r 2.8 ]. Mathematically speaking, the advantage of the trans-series is that it E 11 16 N, ∼ ( − 47 , u Im 46 band N information about the function being computed, and is rigorously equivalent to , exp [ u i ∆ 12 all , ]: 11 ∼ ± For example [ However, the non-Borel-summable perturbative series in ( The resurgent trans-series incorporates non-perturbative contributions at all multi- 47 , while the fluctuations about the instanton/anti-instanton saddle are given by coefficients of the resurgent trans-series expansion. coefficients is: term celled by an identicalinstanton term interaction coming [ from anis instanton just the gas tip analysiseral of of the Borel the iceberg: summation instanton/anti- of thesectors, perturbation cancellations occur theory, between and at imaginary those terms all coming produced orders, from by and lat- the these multi-instanton cancellations are encoded in relations between the This band splitting is a in the exponent is the instanton action: sociated with the two-instantonthereof], sector and [more lateral precisely, Borel the summation produces instanton/anti-instanton an part ambiguous imaginary non-perturbative the function wherever the functionexpansion exists such [ as ( instanton orders, the lowest of which is the exponentially small band width: instanton/anti-instanton interactions [ the 2-instanton level.intricate The ways, trans-series such coefficients thatergy all [ imaginary parts cancel,encodes leaving a real and unambiguous en- The trans-series isfluctuations, an and also expansion by in powers terms of of logarithms due instanton to factors, quasi-zero each modes multiplied coming by from 46 JHEP02(2015)160 , ) ~ 11  E, 9 32 (2.17) (2.16) (2.14) (2.15) ( ]. ), which + ZJJ 12 , 2 B B 11 E, g 4 ( 17 θ π  + ZJJ 2 3 A 4 ... √ E 2 2 cos B ...  + 5 3 = +   E + 11
3 5   E ] (we rewrite the following 5 E + 3  4 ZJJ 4 ~ E 4 23 ZJJ 16 4 B 47 E 4 A E ,    5 4 1 2 1021 + 4 245 2 46 ]: −  − 2 1 E 199 +  e 2 2 + 3 ...  12 ~  25 3 + , 16 Γ E B − 4 ~ E 2  16 + 3 11 8 ZJJ E  2  + 8 B + 525 B E 4 − 3 +  + 326 341 2 64 +  35 B ] have argued the remarkable result that the  405 E E + 2 ~  ): E 32 + g 2 E 47 + 64 – 12 – , 64 − + 3  64 3 17 32 721 215 4487 +  46 3 4 ~ B 16 1 4     ), we obtain [ 8 +  ~ 3 4 3 4  
205     ~ 16 − E, ~ 16 ~ ~ ~ ~ ( ) describes the fluctuations about the nonperturbative + 16 16 16 16 ZJJ instead of  + 5 B + ZJJ     1 4 g ZJJ A ~ B 2.13 16 4 − + E + + + B 1 2 + 16 e 1 2 33 2 ≡ ), which describes the perturbative series, and  B ) = ) = Γ ~ 4 ~ ~   E, g ~ ZJJ ( E, E, 16 ~ 16 ( ( B − −  ZJJ ZJJ ZJJ  ) is associated with the fluctuations about the perturbative vacuum sad- B − B A B ) are given by: ~ 32 ~  2.12 ) = E, ~ ( is the Bloch angle, and for the Mathieu potential, the two functions θ B, ZJJ ( A Zinn-Justin and Jentschura (ZJJ) [ ZJJ ]. The ZJJ exact quantization condition is written as [ E Inverting the expression for and where effectively describes the fluctuationstrans-series structure around follows the naturally from single-instanton.expression a of uniform In WKB the analysis, fact, analytic which12 the continuation shows properties it resurgent is of an theexpressions parabolic in cylinder terms functions of [ language, resurgence means thatthe the multi-instanton fluctuations expansion are about directly variouspersist related different to to saddle one all points another. orders in Such of resurgent the relations non-perturbative and quasi-zero-modeentire trans-series expansions can [ be generated fromtwo an functions: exact quantization condition, together with just Equation ( dle point, whileinstanton/anti-instanton saddle Equation point, ( the smallestuum action quantum saddle-point numbers. with the Thisthat same exist vac- is within an the trans-series, explicit and example a of direct manifestation the of relations resurgence. between In path coefficients integral Note the precise correspondence of the coefficients in these two very different expansions. JHEP02(2015)160 ]. In ]. For (2.20) (2.21) (2.18) (2.19) 12 , 12 , ) and the 11 11 2.4   ...... ), can be deduced B 4 ~ − + 17 ). Thus, only one of
 E, ~ ( 4 3 + B 3 ], it was shown that this B, + ( ZJJ B  2 5 A 12 ZJJ B , + 327  64 : 3 ... 135 2 E 3 ), the non-perturbative function  11  B  − B + ~ ZJJ ~ ~  32 2 16 2.17 ∂ 3 4 B  ), with the definitions ( ∂A − 8 ). In [ + ~ 1 + + 1120 2.8 205 1 2 5 B +  2.11 +   3 4 B + B ) is actually needed to generate the entire 4 + 4 2 ~ + : ... N 2  B 336 4 2 ), and hence also – 13 – N E, B − ~ = 55 B ( ~ 3 16 9 3 64 B   B B, −  ZJJ 8 3 4 ( ~ ~ A   32 = ~ ). 16 − ZJJ ~ ~ 1 16 16 − A + ZJJ    2.21 ). This surprising result is in fact consistent with the am- ∂B ~ ~ ) is equivalent to conventional Rayleigh-Schr¨odingerpertur- 16 + + ) and ∼ = 1 ∂E ~ ~ B, ( ): E, ZJJ E, ) = ( ~ ( A ) encodes the fluctuations about the single-instanton [ ~ 1 2 ZJJ ~ B, ZJJ − E ZJJ ( B, e B ( B, B with the level number ) can be re-expressed as a function of ( ZJJ ZJJ ZJJ B E ZJJ ∂B 2.16 A ), which is itself a statement of the Bloch boundary condition [ A ∂E 2.6 ) and ) in ( ~ ~ B, E, ( ( ZJJ ZJJ principle, information about the expansionsdifferent around other saddles saddles, are provided one connected.condition knows how ( This connectionsection is 5 provided we by discussproof the this of further, this exact surprising and quantization result use ( the gauge theory perspective to give a simple the two functions trans-series. Therefore, thenon-perturbative fluctuations saddles, about are precisely the encodedvacuum. single-instanton in saddle, In the and fluctuations other all about words,around the other the the perturbative vacuum, full trans-seriesbitious is goal encoded of in resurgence, the which claims perturbative that fluctuations the expansion about one saddle contains, in This implies thatimmediately the from function knowledge of the perturbative energy in agreement with thecorrespondence fluctuation is factor directly inA connected ( with a simple relation between the two functions example, the single-instanton fluctuation factor is given by The function bation theory, about the perturbative vacuum.A Using ( identification of Thus, the function which agrees with the perturbative expansion ( JHEP02(2015)160 , ]. ~ / 78 1 –   ): (2.22) 76  ...... N , and large + + N 16 16 ~ ~ 83 83   540 540 ...... − − + + 14 14 ]. These expansions are ~ ~ 14 14  55 55  77 ~ ~ [ 576 576 ... 2 ... 609 609 ~ − + + / 6400 6400 + 12 12   + − 16 16 ~ ~ ~ ~ 49 49  12 12 ...... ~ ~ 144 144 http://dlmf.nist.gov/28.6.i + + ) is treated as a perturbation. Standard ... x + + 1. Then for large level number 21391 1961 1961 12 12 + 5760 5760 ~ ~ . 6998400 10 10 6998400 1669068401  cos( 14 ~ ~ − − 2 11 11 ~ ~ 2 + − 2 36 36 ~ N 10 10 gap, are ( – 14 – 80392 45608 12 12 ~ ~ 5 5 68687 − + and ~ ~ 2304 th 5315625 5315625 8 8 16 16 1 ~ ~ 1 1 N + 289 6 6 + − + − , with fixed, the high spectral region can also be probed with 1002401 8 8 8 8 10 19440 19440 − − ~ ~ ~ ~ ~ 58 N 6 6 9 13 13 − + 1 1 80 80 ~ ~ → ∞ 433 317 8 8 − 3375 ~ ~ 3375 + + + − 5 N 6 763 6 6 4 4 54 54 ~ 7 1 and + − 1 1 2 2 ~ ~ 4 ~ ~ 4 4 + −  ~ ~ − + + − − 8 8 4 4 , leads to the above expansions in 1 , where the potential 0 and ~
4 4 2 2 2 15 ~ ~ 15 4
20 1 1 1 4 4 3 3 ~ ~ ~ ~ ~ π 2 → 2 Nx iθx , with a radius of convergence that increases quadratically with the level − − − ~ 16 + 4 4 + 9 + 9 + 16 + 0 1 1 +          exp 2 2 2 2 2 2 2 2 2 8 8 8 8 8 8 8 8 8 1 regime, which is strongly-coupled in the QM sense but weakly-coupled in the ~ ~ ~ ~ ~ ~ ~ ~ ~ ∼ and sin  ======. They are conventionally expressed as continued-fraction representations of the
by taking convergent ) ) ) ) λ 0 N barrier top 2 − − − − λ u denoting the top/bottom of the ( 4 (+) 4 (+) 2 ( 3 (+) 3 (+) 1 ( 2 ( 1 Nx Instead of taking u u u u u u u u ) ± N ( , the strong-coupling expansions for low-lying modes, converted to our notation and with associated with the exponentially smallspectrum, splittings as of are the clearly spectral seen gaps in in figures this region oflarge the the continued-fraction expressions for the energy eigenvalues give approximate expressions cos in fact index eigenvalues, and these continued-fractionNevertheless, expressions despite these are convergence themselves properties, there convergent are [ also non-perturbative effects, These expressions follow fromthe a straightforward “fully strong-coupling quantum” expansion regimeare which where governs derived the by kinetic perturbingfunction term is around dominates the over (degenerate) thedegenerate potential. free perturbation particle They theory on for a the circle gap whose wave edge states with even/odd wave-functions, In the gauge theory sense, the behavior~ is completely different. Foru small level label 2.3 Electric region: convergent continued-fraction expansions, high above the JHEP02(2015)160 , N . In ...... N (2.25) (2.27) (2.28) (2.23) (2.24) (2.29) (2.26) + + !   2 ) 2 ~ ... ~ ... 4 N ( + 8 +  we use Matone’s 2  − 12 ) 12 1 2 2 ~ ~ 1  u : for any finite a ~ ... N  7 2 ~  ∞ 3 5( ): +  55  4) 128 ). = 2  ) 9) − 2 ~ + 1 + ~  N 2 N  − 2.22 ... 8 . Because of the poles, the ( 1 a 2 N n ... + ( ). We now compare this with − N + 7 3 ) is identified with the multi- 8 8 + 21 1 2 128 a 1) 1.1  + 29 6 4)( . N 6 2 + − 2.23 2 5 8 21Λ is proportional to − 256 ) as: 2 ~  48 4 πia N 1 2 n a ~ 5Λ + 4 N − 1 2 − N c − N 4 256 ( 2.23 16 2 k n 4 5 a + 58 32( ~   b NS Λ Λ + 2 4 1) 2 ~ ) in the Nekrasov-Shatashvili limit of the 16 ) F ∂ + 5 2 2 ~ 2 64 N ∂  − 4 N 4 k 9 + Λ 2  + Λ 1.11 2 πia – 15 – − πi ≡  2 ~ 2 ~ 8 N iπ 2  2 a 2  ) + ... N + 2 ~ 64( ( ~ 2 2 1) ) = 2 N + 2 + ( a 1 Λ ~ n =1 a Λ − Y k http://dlmf.nist.gov/28.6.E14 10 1 1 − a, 2 a ( ]. This leads to a set of continued fraction relations that ). In order to relate the prepotential to log 1 log u N  78 9 2 πi 2( – 128 2 2 πi a 1.11 2 ~ . Note that each coefficient has poles at integer values of  76 + 48 + gap as ( − ~ 2 2 6  − 1 3.4 th N a N =  N

5 64 1 4 2 8 NS ~ + + F ) should be understood as an expansion about 2 2 ∼ a ], 1  ) 4 98 ~ ~ 2.23 2 – denotes the greatest integer less than or equal to + N N, c 96 2 (  2 a n u b 2 1  Now we demonstrate how the energy spectrum ( ∼ ∼ = 2 SU(2) SUSY gauge theory. First, rewrite ( u Plugging the expansion the instanton expansion in ( relation [ The second term on the right hand side is due to the perturbative part of the prepotential. where we have defined the “action” which is half the “’t Hooft coupling” defined previously ( one can just use the strong-coupling expansion expressions ininstanton ( expansion ofN the prepotential ( where expression ( This expression is obtained by substitutingand a Fourier equating mode coefficients ansatz for [ thegenerate Mathieu the function, expansion, andfurther the below connection in to section ourparticular, all-orders the WKB denominator approach of is the discussed coefficient of for the energy of the JHEP02(2015)160 ] , , ) ) ) ). N ± N N ( N 1 N 4 50 ( ~ u , k 2.10 2.23 (2.31) β 49 location , and we increases, ) does not N , completely 3.3 ) ~ 2.23 k N / ) as 4 ( 1 ~ k ). So, at finite β 2.22   =0 4 ∞ 2.30 k X N  ) clearly show that N 2 ~ 2, due to the poles in the 2 /  2.22 1) 2 ~   ) for  O − 2 expansion ( 2 . As the level index N ~ (2.30) ]. As mentioned already in the N 1 + of the gap. In fact, for a given 1 2 1)!)  and ~ = ( ]. We show in section 4 that in this 81 /  , N − 1 k 2 52 1 77  , N width ( ) matches precisely with the expansion in 2 ~ ]. It is clear from the continued-fraction 1 51 ~  ,N − 87 , as shown explicitly in section dependence of the expansion ( a, 2 N – ( N ~ 1 2 u (2 85 ~ – 16 – 2 1)!) ) is not the whole story. First, this gives the same 8 N ~ 2 −  ± 1 2.26 N ~ ) ): ( , whereas the expressions ( e 1 ) ) is only approximate, giving the approximate k N . Physically this indicates the appearance of degenerate N ), ( 4 − ± ( ( N ~ k  N N u 2 α 2.23 2.23 (2 ) also has a natural non-perturbative interpretation in terms ~ π ) are those given in the continued fraction expansion ( 2 2 =0 4 ∞ N ~ ) are convergent, the origin of this non-perturbative splitting is k X N 2.30 ( 2 k expansion of the ABJM matrix model, in which non-perturbative ∼ ∼ ). N α 8 2.22 2 N ~ gap N 2.26 ). Physically, this is simply the statement that the perturbation theory u ∆ ) = ) that the splitting is directly associated with the poles of the expansion 2.22 ~ ) that the pole terms arise at the two-instanton level, proportional to ( ) ± 2.23 ( N 2.23 u ) one sees that the expansion for 2.28 To make a sharper analogy with the trans-series (multi-instanton) expansion ( However, it is clear that ( http://dlmf.nist.gov/28.6.E15 terms, arises beyond thisthe order, gap and splitting has terms leadingis are behavior a missed given by one-instanton in the effect, ( see Bohr-Sommerfeld proportional from expansion. to ( The gapas expected splitting from the resurgent structure. where the coefficients However this identification is validcoefficients only up of to the order continued fraction expansion. The gap splitting, encoded in the regime the gap splitting ( of semiclassical configurations. near the bottom of the well, we can reorganize the large perturbation theory forIntroduction, both this edges is a ofconcerning concrete the analog the of large gap the [ effects results are Drukker, Mari˜noand related Putrov to [ complexto space-time poles instantons, in and the which ’t were Hooft subsequently related expansion coefficients [ Since the expansions ( quite different from theterms familiar with weak-coupling divergent analysis perturbativeexpansion that series ( associates [ non-perturbative coefficients at integer values of Bohr-Sommerfeld expansion ( of a narrowneglecting gap the high non-perturbative in splittingthe for the splitting the spectrum of the (see energythe levels figures splitting occurs drifts at to the higher order orders( 1 in perturbation theory and becomes exponentially small the second line of ( expression for both gapare edges, different from one another.match that Second, of the ( into ( JHEP02(2015)160 (3.1) (3.2) is used b C  g ~ C ... − 0 dx  2 / 000 7 ) V 0 V we will further identify this is used to determine the width V − 4 g dx 16 ˜ u 2 C g ( / C 5 2 ) − ) ] 0 V (band) 2 (gap) / V 60 − 4 ( , π 11 ) 0 ) u -2 ( 59 V V ~ C  − I 49( is used to determine the width of the band. The 1 2 u 2 6 b ( 2 ~ + ˜ – 17 – C  ~ − N C I  πN π (band) 4 b 13 (gap) 2 2 V dx ~ 2 C ~        − ~ − u =  1 2 √ 0 P C + C I I ~ N  2 N  π 2 √        . Geometrically, for the Mathieu problem there are two independent 4 = = P b I C is used to determine the location of the gap, while π 1 2 g ]: . The integration contours for the all-orders WKB expressions. The first figure shows is the local momentum and the contours go around the appropriate turning points, C denoting the energy eigenvalue, we have [ 68 P π – u -2 In the next section we show how we can actually obtain these non-perturbative gap actions 59 With proper analytic continuation, thisand quantization dualities condition permits between smooth the transitions and various spectral also regions, the connecting bottom weak and and top strong of coupling, the wells. The distinction between the various regions where as shown in figure cycles and they correspond towith the generators of the two cycles of the torus. More explicitly, 3 All-orders WKB analysis of the MathieuThe equation: all-orders actions WKBas and expression [ dual for the location of bands and gaps can be expressed splittings from an exact all-orderseffect WKB with analysis. worldline In section instantons,ionization in and monochromatic make time-dependent a laser pulses. direct physical analogy with multi-photon the paths for theto bands, determine when the location thesecond of energy figure the is shows band, well the while contour below paths the for barrier the top.of gaps, the The gap. when contour the Note that energy in is this well case above the relevant the turning barrier points top. are in The the complex plane. Figure 4 JHEP02(2015)160 4 of a (3.3) (3.4) (3.5) ... . Leading width ... 4 + + ]. At leading 2 ], and in more ~  94 2 8 94 – 1 2 – N + 91 90 , ∼
N ) 2 of the gap is determined ) ~ 89 of the band is determined  , u ~ N, 88 shown in figure ( width u g 1 + (1 + width ˜  C ) can be found immediately as P O or ˜ ∼ − ⇒ C b ) I ˜ 2.23 ) + ~ C u  Im N, 2 1 ( ~ / 1 u 3 − (1 + u  ) and ( ⇒  ∼ gap band 2.8 O L L exp dy – 18 – + 2 y y ∂u u ∂N 2 − − 2 π √ u 1 2 ∼ of the bands or gaps, we integrate the WKB integrands along r ∼ 1 N u − u dy Z ∆ 2 2 y y location . π √ − − 2 , shown here. In the deep band case, the u 1 3.3 ≈ gap r ~ L ]. For energies inside the wells there are real turning points. As the 1 1  − 67 1 2 Z – and 2 + 65 π √ 2 N band L  ≈ . To determine the ~ . N Perhaps less well-known is that there are corresponding expressions for the For example, the leading behavior in ( 5 where now the contours areorder around results the for dual bands contours anddetail gaps below in in the section barrier region are discussed in [ band deep in theorder, spectrum the and width of of a aand band gap an or high exponential: a in gap the is spectrum expressed as [ the product of a density-of-states factor The higher terms are discussed below, in the next subsections. For the location of the center of a deep band: energy approaches the barriermove apart top, again the along the turningand imaginary points axis come for energy together above and the coalesce, barrier top. and follows. See For figures the location of the center of a high gap: by integrating through thecoalesce barrier. and move As into the theby complex energy integrating plane: between goes in these above the complex the high turning gap barrier points. case, top, the the turningis points encoded in theStokes location lines of [ the turning points in the complex plane, and the associated Figure 5 the paths, JHEP02(2015)160 rd ! (3.9) (3.6) (3.7) (3.10) (3.11) (3.12) ... . − is the 3  3 dx σ u 2 / 2 5  − 2 ) ) 1 0  V 2 + 1  V − ( u ... , where K u 3  ( − σ has support from under ) 2 a K ) u + 1) u ( D dx ) ( 1 a 2 u 1 u − / ( − ) = 5 − has support from both above 1 2 2 ) i a cos ) cos π ( is defined between the turning 2 0 V a − (1 − cl . V Z 1 2 D (3.8) = − (  ) 2 6 a  ) 1, the leading WKB expression for u u 2 − ~ u 1 2 ( ( u ( 2  − ≤ D π − 0 π + du a − 1 u Z da , the associated action is real/pure imaginary. In N 2 + 1  1, the choice of the contours for the two ) 2 + 6 ≤  ) + V dx u u E 2 ~ R ( u ≤ 1 2  ( ~  − ∈ correspond to the gauge invariant observable D 0 ) implies that − a with the center of the band is implemented i u − a u E ~ , fixing the normalization to π 4 – 19 – u  3.7 u √ − ≤ − )] = ) ... 4 π ) )] = ) V dx ~ u 1 u ( , and = u u + ( 1 ( ( − − 1 = u, ∞ − 1 a u ( − D 0 u du [ u 2 1 a e u [ cos − − √ cos da √ u < e − 2 R − ) − given in ( π y π y Z 2 u R ≤ = − y ( y D

1 Z 0 a r a a − 2  r π dy 2 2 √ π 1 dy 2 √ and u − 1 u Z − . The real action in this spectral region can therefore be expressed as In the interval a = = Z 2 D π 4 a P P 2 √ 1 2 π − √ γ γ − I I can take complex values, and the analysis follows by analytic continuation. ] = ~ π π 1 1 a 4 4 ) = [ )] = u u ( and m ( I D 0 0 u . We set our normalization such that classically Φ i a a ) := ) := [ 2 u u e As discussed in the introduction, the WKB periods associated with the Mathieu and ( ( Here, and for the rest of the paper, the notion of “real”/“pure imaginary” action refers to the particular a 4 R D a Tr Φ choice of cycles such thatgeneral, when These expressions are the originalical Seiberg-Witten language solution, is which in identified thethe with quantum Picard-Fuchs the mechan- relation: leading order WKB expansion. Note that they satisfy The leading order terms of the actions are expressed in terms of elliptic integrals The exact quantizationby that requiring identifies independent actions and below the barrier andthe is therefore barrier complex and valued, whereas ispoints, pure imaginary. Furthermorea since linear combination: 3.1 Dyonic region:For resurgence energies from below all-orders the WKB the barrier band top, location is where givenreal by part the of usual Bohr-Sommerfeld the expressionis action, that pure while involves only the imaginary. the dual action is associated with under-the-barrier, and so Lam´eequations correspond to the vacuumdual expectation partner, value and of the theh energy scalar field eigenvalue ΦPauli and matrix, its and therefore JHEP02(2015)160 (3.18) (3.13) (3.20) (3.16) (3.17) (3.19) ]. For ]. This 98 36   – – u u 95 ) are 2 2 − 34 , 1 + 1   3.15 32 , K K (3.15) ) 31 [ u  ( ), which are given by  u 0 u ), using the conventions u a + 75) + 75) 2 3.7 2 2 − k   u u  1 u u 2 1 + − up to this order are given in 2 60 ) transformations applied to . Therefore by differentiating d 2 2  )  − du Z (1 − + 60 n V E , ( k 1 1 + . E 2 2 K κ u ) u   u u 2 u . + 75 k ( E E 0 − 3 + 2

+ 2 π 3 a n (3.14) + 93 + 93 d k  )  75 75 du = 3 3 + u du u denotes u u 0 u u n ( − − 0 4 2 0 d 2 − 2 2 a K k )(4 − u u 1 KK ) and the equation will still hold. 1 + u u  )( )  + 120  − n u − ( k d 153 153 ), the numerators in ( – 20 – du 4 K 3.12 K κ 4 K x 0 ) ) − − (1 d . The coefficients du + u u E  n 4 4 =0 (1 + 2 k X 2 10 u u 3  2 − u + ] in ( ) ~ 4 4 ) as well. The general form of the differential operators d 0 0 3 2 du u ) 28 a (1 (1 + u = cos( [ ( ) = 2 u ) is invariant under SL(2    2 u EK u D +2 +2 − ( V ) ) Re a  n 2 2 − 45 1 1 (1 3.12 a 9 u u 1 by i 48 2 π (1 − − 0 taking appropriate combinations one can generate the integrands π i 1 a (1 (1 u ) = ) = π π ) to the order u u 46080 is ( ( 1 2 46080 48 48 − , can be re-expressed as polynomials of 0 a a a 3.16 V ) which corresponds to changing the basis for the two cycles [ ) = ) = to )] = )] = D 0 u u u u n ( ( ( ( a , a D 2 D 1 2 1 . The first two next-to-leading order actions calculated from ( 0 a a a a [ [ a B e e with respect to ) up to total derivatives which vanish after integrating over turning points. For R R V ]]: Remarkably, the higher-order WKB actions can be obtained by acting on these leading 3.7 − 76 u We have verified ( appendix that relates and the same relations hold for the derivatives of √ in ( example, at the next two orders: the pair ( example, we can replace WKB actions with differential operatorsfollows with from respect to the the fact energy that for of [ The Picard-Fuchs equation ( which follows from the Legendre relation [here JHEP02(2015)160 th  N , the 1, as: (3.29) (3.21) (3.22) (3.25) (3.30) (3.23) (3.24) (3.26) ... E + 77 + ∼ −  ......  u )] as expan- + ... . Recall that u + + 1 32 1 2 ( 3 ... + 3 4 + n u E a + + 5  [  ) as a function of , respectively, for e 3 8 ~ N + 1) 525 E R + 1 32 + 1) u 4 ≡ N, u 1, as: + 1) ( u  30 log B u (3.27)  u 33554432 524288 4 ~  16 )] 8085( = E ∼ − π 245(  E + 1) and u u 2 ~ + 2 log 25 + + u + 1) ~ 2147483648 (3.28) ( 3 3  2 4 u π 141757( )] 53 2  ( 12288 1 + E ~ ), or in the notation of ZJJ, the ~ − + 4 π 16 + 1) − ( 35 + 1) E, 2 2.8 n + 1) u  3 64  ~ u a 16384 [ u ...  ... ) for the location of the energy bands 491520 + ( 1048576 e + 1) 25( 3 ~ + 5 − + 525( 16 1 + ]: − 3.9 R − u E  −  2  + n E 3 + (  2 47 3 , 2 a 1 ~ 2 + [ 64 + 1  32 721 e 46 134217728 + 1) − + 94 E – 21 – =0 ). Thus we have the following identification with ∞ u 4 ... ~ 10941( X n R 5 4 16 + 1) u ~  + 1)  ( 512 1   ) as an expansion in 2 2 2 + ~ ~ u  π + u ~ E, ~ 32768  5 16 ( 3( + 1 + + 1 32 ≡ 32 E ~ 35(  B, 2 16 3 log u u ) = + ( + 1) ZJJ E ~ 4   +  10240 +  2 E 245 , we can re-write these expressions for u B 3  π + 4 E, − E  1 log ( ~ , and next-to-next-to-highest powers of  16 ~  32 3 8388608 + 1)  ~ 384 + 1) 721( 16 E : ~  ~ u ≡ 16 16 u (35 log ZJJ ( 2048  − E + +  u + 1)   B 5( π + + 1) ), we recognize these expansions as the highest powers of 2 2 1 4 17 32 u E + 7 π ( + u and ) is non-Borel-summable. ( + 1) π ∼ ∼ ∼ 17 2 + 1 2.15 ~ 1 u 1 + 1) −    ( 128 262144 u ) ) ) u in the function 2.17 ) for the energy π 32768 4 π ( 5760 u u u ~ ( ( ( ∼ ∼ ∼ ) that 1 + 48 − 1 2 0 a a a 2.17 )] )] )] 2.4 ∼ − ∼ ∼ − u u u 2 2 2 ~ ~ ~ ( ( ( , we arrive at the perturbative expansion ( ) ) )    0 1 2 ~ u u u a a a e e e ( ( ( [ [ [ R R R e e e D 2 D 1 D 0 Similarly, the dual actions can be expanded about the bottom of the wells, 4 2 R R R and i a i a i a ~ ~ − − − this expansion ( Thus the “exact Bohr-Sommerfelddeep condition” inside ( theband. wells expresses Inverting this the Bohr-SommerfeldN perturbative condition, expansion to express forexpression the ( location of the the results of Zinn-Justin and Jentschura [ Comparing with ( next-to-highest powers of each power of Recalling ( sions in powers of The actions can be expanded about the bottom of the wells, JHEP02(2015)160  (3.34) (3.31) (3.32) (3.33) (3.35) (3.36) (3.37) ...  + 6 ... u − asymptotics of 6 ) and the pure ) looks near the  u   u 75075 ) u B ( with the center of 65536 ( ~ a  ...... u D 105 ) + a E, ~ + + ( 4 16384 ) function as follows: u 6 6 B  E, − u u ( + 4 1155 B ... E, g u 1024 ... 1 2 ( + − 75075  15 − + 65536 1 2 6 2297295 131072 ZJJ 2 1024 u   + u A   + 35 − 4 2 ln Γ ~ ~ 32 105 4 u 32 32 2  u − u   16384 1 2 ln Γ 1.
1155 1 + 5005 16 ) − 1024 512 −  + 152759 ) ln ) ln E N ) 4 − 2 + + ~ ~ / ~ 2 u  ~ + 6227 → − E 1 5 2 2 )  ~ u 15  E, E, u u ~ u ( ( + 1 35 u 1 + 32 ) = – 22 – 273 1024 32 64 2 B B u u − ~ 1 + + 1 8(2 2 2 32 ( ( √ −  a − 1 as: u 2 D − − − ( 2 1 + 1 + a  ) ) u D  1   ∼ ) the real period is identified with π π a 2 2 16 u
/ / Im 7 5 πi ~ ) diverge as − ) ) 4 ... u π u u ~ → ∞ 1 ). The exact quantization that identifies 1 1 ( 43260 log 4 − + ln(2 + ln(2 ) is related to Zinn-Justin’s u 1530 log  D n  )+ ( u ) is non-trivial, as it requires using the large ~ a  u u = ~ D π ( 64 (2 16 (2 2 ]. u, a π ) = 2 ( ~ √ a E, ( D 4 92 , a ~ ∼ − ∼ ∼ − A E, + 1) 1, the 1 ( 91 ) ) ) u )+ ≥ ( u u u u ( ( ( ( ZJJ n 2 0 1 23592960 503316480 1 a a a A a − − 2 ~ )+ u ( pansion 0 The dual action a 2 ~ Combining these expansions we find The actions can be expanded for In the electric regionimaginary period (i.e. is still the gap is the ln Γ function. Thesolution subtraction of near these the terms corresponds bottomfrom to of matching the the the exact perturbative solution welltop for to of an the the inverted solution barrier harmonic near [ well, which the is top3.2 how of the barrier, Electric coming region: convergent expansions and the Nekrasov instanton ex- The comparison for Notice that for JHEP02(2015)160   ... (3.45) (3.42) (3.43) (3.44) (3.38) (3.39) (3.40) (3.41) ... ) of the + + 6 ) u 2.26 u 2297295 ... , the transition is 131072  4 2 + + u ...   4 49140 log(8 + u 1 2 ... and D 0 1024 ) ~ − a + 5005 u 1 / + 512 8 ~ ) Im N − / 4 + π u ~ e ) determines the (convergent) 8 2  u ~ 2 − ) ... − ~ 1 2 e u e 127461 u, 4 + + 30 log(8 ] 273 1 2 ( 64 u 2048 0 a + N 1 +  − ( − a 1 [ ) N − − e 420 log(8 u ∂u 1 + 47  − 1024 ! R =  − 2 ~
∂  + 2 1 2 1 as:  / u ~ π ) ··· 7 2 e ) + 883 u u  3 32 u ∼ ! +1) ~ 32 2 1 + N + u 16 180 log(8 )] + N u D 0 – 23 – N )  u a 4( log(8 − ~ ( ... u 32(2 16 2 0 ln Im

− a − − 2 π π [  ~ , and inverting, we obtain the expansion ( 2 2 1 567 ~ e π u 2 u − r N R e 6 log(8 2 16 ) + ∼ ∼ 1 + 1 + u − − ∂u ∂N 1 + 2  13 2 π band 2 4 + / u 3 ∼ ) ∼ − ∼ ∆ 1 + u u ]  2 + log(8 u ∆ ) we obtain the band width estimate (using Stirling’s formula in the D 0 (2 u − 1 a π [ 2  6 ). Thus, the all-orders-WKB action √ m 3.42 1 u ~ π I 2 ]: π π √ 24 45 2 N, 94 ( – u ∼ ∼ ∼ 91 ) ) ) u u u ( ( ( D 2 D 0 D 1 The dual actions can be expanded for ia ia ia − − − Therefore, from ( last step): region is [ In the dyonic region, istic of the electricsmooth, region. but As connecting is thesions. clear regions Of from particular requires the interest careful plots arearise the interpretation in in different the figures of mechanisms different by the physical which regions.tially various non-perturbative narrow For terms expan- example, width the of general a expression for band the deep exponen- in the dyonic region, and of a gap high in the electric spectrum, as discussed in the next subsection. 3.3 Magnetic region:Across duality the and magnetic region, analytic there continuationcharacteristic is of across a the the transition dyonic barrier from region, the and divergent the perturbative convergent behavior perturbative expansions character- These dual actions determine the exponentially narrow width of the gap, high up in the Identifying the left-hand-side with gap energy expansion of the location of the gap high up in the spectrum. JHEP02(2015)160 ) (3.49) (3.50) (3.51) (3.54) (3.46) (3.47) (3.48) ) with ~ ~ ( O N, ( + u # (3.55) ) ... ) respectively: 1. ~ ( (1) − ∼ 4 O O 2.23 u  + ∼ 8 π of the bands/gaps when # N  ...... ) and ( 18 + 33 2 , c edges ... + , but the above fixes the non-  2.8 ~ ... − + 10 / 3 1 2 + 1  ! N  2 4 π N 2 ∼  2) + : 8 π  ). The 2   1 8 ~ 1 4 ~ −  N ) has the correct form in both extreme N 9 e − ) 64 32 ± ~ 3.51 = 14 ~ u 5 u 8 and 2  π + ~ N 3.42  8 6 ··· − π N  N ∼ π  ln 2 (ln(8 ... 2 2 + 2  1 4 ~ 4 π ∼ u 2 0 16  π – 24 – 1 2 a  8 π ± ∼ N π − 1 2 √ 1) + − 5  2 32 N Q u 8 − ∼ ∼ 1 gap 2 +

] 1 is more subtle. In this regime, u u + u ( 2 2 0 D − ∆
∼ 4  1 2 π a [ 1 4 8 π 4 π u N m ∼ ∼ ± ) is approximately linear, with a slope depending inversely 1  c 16 I 2 0 4 1 2 D 0 ~ . It is instructive to evaluate the energy eigenvalue π a N ] − ia + /π 1 ). Thus, the formula ( ≡ ). On the other hand, in the electric region, (3.52) (3.53) 2 − ( − 94 " ) ) , . We also see that in the immediate vicinity of these points, the  ~ ~ ) we obtain the gap width estimate: 1 Q 8 π ( ( 4 π 2.30 93 6 2.11 : ∼ O O  on ) 3.42 N " 1 + ± u ( N 1 2 u ∼ − = 1 + ∼ = 1 + dyonic electric u u In fact, we can refine further the estimate in ( The magnetic region near = 1 are given by [ as shown in figure dependence of quadratically on It is remarkable that these two very different expansionsu coincide at this scaling, in both the dyonic and electric regions, using ( relation gives us the leading scaling between It is clear thattrivial the coefficient barrier to top be is 8 in the vicinity of The latter relation tells us that the exponential behavior becomes of order unity. The first in agreement with ( limits, in one case referring to the width of a band, and in the other to the width of a gap. Therefore, from ( in agreement with ( JHEP02(2015)160 , ) ~ is K N 3.16 (3.58) (3.57) and , where 6 E , where 2 symmetry of
2 1 4 Z ± N 2 16 π . 1, in a way that connects = ! ) ∼ ) Q 1, the bands and gaps are of u u ! u ( ( ) ) ≈ n u D n u ( a ( a u 0 D 0 (3.56) a ) a ~ ! ( 1 O ]. For the subleading terms from ( ! ± 1 ∼ 97 1 0 ± 1 ± 1 0 ) in the region gap 1

u ± u i (

n n ∆ – 25 – i a 1) ∼ − − = 1, we observe equal widths of the band and gap ( = − u band ! ) u ) = u u ∆ ! − − ) ( ) ( ]). This is the manifestation of the residual u 0 u D 0 u a [ a − − ( ( m

= 1. The vertical lines mark the values n D n I a a ( u

sign symmetry of the gauge theory [ = 1. ) off the positive real axis, which effectively gives a small imaginary part g u R . Plots of the bands (shaded) and gaps (unshaded) in the magnetic region, in the vicinity 1 denotes ± Another way to understand the smooth transition across the barrier top is to note that . This then avoids the divergences of u where the broken U(1) we see that this relies on theanalytical analytic continuation continuation properties properties connect of different points these in functions. the moduli More space precisely, as the follows in both the dyonic(or and equivalently electric region,to we must avoid polessmoothly. by analytically Since continuing all the actions are expressed in terms of the elliptic functions equal width, and are notfrom exponentially a narrow. given This intersection canabove be point and seen below at clearly that in intersection figure point. In fact, of the barrier topthe where band label, andintersect which the coincide line very accurately with the points atThis which implies the that band/gap in edges the vicinity of the barrier top, where Figure 6 JHEP02(2015)160 ~ ). 2, / and ~ 2.26 (3.61) (3.59) (3.60) u N ... − = Qu 2 a expansions. 2 signs, in many − Q 2 ~ ]) reflects the ± u [ +4) m N region by invoking ( I ( 1, which governs the 2 . The first few terms − ~ , as written in ( Q /  ~ ≤ ∞ and equating both sides 1 sign and small Qu λ 2 u 1. So far, this is an alter- u  0 ~ as a double series in  and − ≤ K a  2 2 N 4 . These elliptic functions have u ik /a K by using the relation + 2) ∓ N N K ( for high lying states can be obtained 2 and ), directly from the continued fraction 0 http://dlmf.nist.gov/19.7.E3 k u = for E With the above connection formulas, the − ) in the electric region. In the electric re- 3.38
a ... 0 0 5 − E K ). 2 i 1), which governs the dyonic region, using the i expansion. 2 3.34 4) k 2 ~ ± – 26 – as a series in 1 − ( ∓  Q 1). The appearance of N E m ( u K λ  I , which is valid for − k 1 k 2 k expansion ( Qu ∼ − 1, can be related to the 1 − 2 λ = = u u 2 ≤ n − 2 , by making a series ansatz for   Q 2 u 4 ~ 1, this relation can be used to relate the magnetic regime 2 2 1 1 ~ 2) k k / ≤ n,k ≤   c − E K u N n,k ). Next, we substitute = 16 ( ≤ 2 P − 1 Q u 2.23 2 − 1 and we reorganize ]. Qu √ ), complementary to the WKB derivation decribed earlier in this section. 2 99  − is a Stokes line for the small 2 ) = 2 3.61 ~ + λ/ N is determined by the sign of R u, ( ). This expansion is valid for high lying levels with − = ∈ ± a 1) to the dyonic regime ( a ~ In the electric region, the energy eigenvalue Also, the region with 0 Qu 3.61 Note that these analytic continuation properties are incorrectly stated, without the ∼ 2 5 , i.e. u ~ native way of obtaining theexpression large ( books and tables [ of ( of it are givenwhich in is ( the exactgion quantization condition ( Having obtained this double series, we invert it and obtain as an expansion in expansion for the low lyingknowledge modes acquired (i.e. from the “continuedelectric fraction region. expansion” for from the continued fraction expression In this sectionThese we two explain expansions the are very connectionis different divergent. between in nature: the Yet the the large to exact former relate is WKB convergent methods one and described to the in latter the the other. previous section can We be present used an explicit method for generating the small where transition across the barrier is smooth. 3.4 Connecting strong and weak coupling regimes non-trivial connection formulas, with ambiguous non-perturbativewhen imaginary evaluated contributions on the real line. See, for example, ( necessity of analytic continuation mentionedthat above, and is ultimately related with the fact the inversion relations for the elliptic functions In particular, for JHEP02(2015)160 , ) D = n a ( k , by D κ n a 2 (3.63) (3.64) (3.66) (3.65) ~ + and a (3.62) a = 0] via a ]) ), we know ! n ] = ,... n ]) a 0 limit. The [1 [ n , 3.16 e ] , to the gauge ] is a shorthand ) [1 c n n ), and comparing → R c ! , ! ] n . ([1 2  1 ... 2 3.38 − ∆ ) ([1 + 2 1
Q  ) 10∆ + 2 − ∆ 8 , and [1 of ( 2 n |  0 ~ 2 + Q 0 1 ) given in ( − c ,  3  for any u ~ 1 Y  ~ n , we then expand 1
|  2 2 using the exact quantization ∆ 1) u ] in the 2 ),  c u,  6( = 1 ( u Λ a,  − 2 + 7  ( N | ( a 2  8∆ + 30 0 2 . − 2 ) which is the perturbative, small + 2 , 1  Λ Y a  | ~ 2 to N N , 29 inst Nek. ]  =0 5 ∞ = 1 a X n a Z has a rather simple group theoretical [ . Therefore, for any given order 4) ( 8 =0 ) e ∞ 6 = X n n ) and − ( k s. Once we know these coefficients, R u 8Λ

) log ( , c κ 2  ( n ) 4∆(16∆ u 2 ( k D 0 2 N  . The first couple of relevant entries of the satisfying κ ( a log . 0 ,  n 0  1 → 2 +  – 27 – Y 2 lim ) ) to the coefficient of = ; 3.1  4 2 ∂ a |  ~ log Λ [11]) = ( differs from some of the literature on the subject. 1 πi , 4  ) from Y and ∂ (dyonic) expansion for 3.16 + 4 | F 1) 4 F u is the Shapovalov matrix associated with a conformal 2 2 1 ( −  Y   − i πi a N 8Λ 1 ( ) can alternatively be obtained from the conformal block ([11] 0 4 2  1 ∆ | with → − 0 − ∆ ) = N ] − 2 lim ) and switching from ( 1  Y 2 2.23 }   is related to the lowest order coefficient [with a − 1 ) and ; 1

 16 a L  u 100 n , small 2 2.28 , we can solve for ,Q , ( ( 2 8 . ~ 2 ~ n Y − ~  D 2 1 L 1 ,..., a 32 ~ | inst ,  NS = = 1 1 ) = exp , ∆ F . 2∆ 2 31 h 1 ) given in ( { as described in section ,  Λ inst ~ NS 1 ∆ =  ∂ D = ) = F ; a, 0 a ∂ ( are a [1]) = 1. In the dyonic region, the exact quantization condition is ( Y u with Young tableaux , Λ . 0 limit, the prepotential stays finite: ∆ 2). Then we invert back to obtain i Y,Y 2 / Q iπ ( ∼ − ([1] inst ∆ → Nek. | 1 ∆ Z 2 u − ∆ + 1  Q Q N To obtain the small Before closing this section, we also would like to point out that the continued fraction However, from the general form of the WKB period Note that our overall normalization of 6 2( expansion. The non-perturbative, instanton induced part of the trans-series can be / Using Matone’s relation ( we get theory variables: In the inverse of The higher orderrespondence terms maps can the be conformal computed dimension, in ∆, a and straightforward central way. charge, The AGT cor- where primary notation for expansion of expansion obtained frominstanton the part AGT of correspondence theexpansion Nekrasov [ given partition as function [ around ~ ~ obtained from that each coefficient of differential operator, with expansion coefficients applying the differential operator inwith ( the coefficient of we can construct both JHEP02(2015)160 , , 103 (4.1) (4.2) (4.3) (4.4) (4.5) [ 105 , ], (gener- 102 , 75 , 74 101 , 3 ] for the vacuum pair  72 2 – γ worldline instantons 2 70 ), one finds for the function γ ], and Popov [ 1 + ω t  73 )  ) have a very different form from 1 γ  E ( cos( π g 1  − 2 3.48 E E π m   2 E 2 ) we obtained from the continued frac- − ) = m γ t  2 ( − γ E  2.23 ]), that for a monochromatic time dependent E 1 + exp m ω ]: 69 . . . , γ  exp ∼ – 28 – ≡ K . . . , γ 104 γ  ∼ ) + , + γ 2 2 γ expansion of the ABJM matrix model, in which non- γ 103 2 , ln(4 , comes from the perturbative part of the Nekrasov ex- 1 8 γ N 2 1 + 75 − 4 N – p probability π γ 1 ), the usual Schwinger formula [ 72 4 π probability        ω t ∼ , despite the fact that the perturbative expansions in the electric 0, we recover the familiar Heisenberg-Schwinger result, ) = cos( 3 γ ( E → g γ ] for the large ) = t 50 , ( E 49 ) the expression [ ) low in the spectrum. This is analogous to the results of Drukker, Mari˜no ). The leading term, 4.1 3.45 3.61 Recall the semiclassical result of Br´ezinand Itzykson [ ]). Here we], give for an this analogous effect physical in identification the with Mathieu spectrum. Furthermore, we present a simple physical ) in ( γ ( In the static limit, which is obviously non-perturbativeknown in interpretation in the terms strength of of tunneling the from the applied Dirac field, sea and through the has barrier a created well- For this monochromatic time-dependent field g production probability in a background electric field becomes Here the dimensionless “Keldysh adiabaticity parameter” is defined as physical example in whichturns a smoothly into non-perturbative a quantity, multi-photon the expression pair of production a veryalizing probability, different Keldysh’s form. work in atomic ionizationelectric [ field, the significance of complex106 instantons for non-perturbative104 effects see [ interpretation of the transition between differentfrom spectral regions, tunneling in pair terms of production the transition to multi-photon pair production. This analogy gives a region are convergent, there aregap still splittings non-perturbative high in instanton the effects,those spectrum. leading ( to These splittings narrow ( and Putrov [ perturbative effects are related to complex space-time instantons (for other examples of tion ( pansion. 4 Worldline instantons andAs multi-photon discussed vacuum in pair section production which exactly coincides with the expansion ( JHEP02(2015)160 , ) ]. 2 1, 4.1 mc (4.7) (4.8)  (4.6) 104 (4.10) , γ (4.9) 103 )[ dτ = 0 ) high in the 4.3  φ ) µ ˙ 3.48 x x ), ( µ A 4.1 2 cos(2 +  ) has two very different 2 2 2 ˙ m/ω x  4 ω E ]. It’s a probability, so the 4.1 1 2 2 2  E ω  75  –  E Z = m ω 1 2 4 73 , −  ] = 69 x φ [ → # 2 ,Q  ν is the multi-photon number, the number ) x  γ ) ˙ 2 ( x ), implies that the semiclassical result ( E ω g ( ,S m/ω  ]: π # ), the classical trajectories are known, and the 4.4 µν 2 2 1 2 E F – 29 – 75  ω t – m 2 = + ]] ], which are finite action periodic solutions to the E = 0 − ω 73 2 µ  cos( ¨ x φ   104 E 1 2 ω , ! m ) is that 2 exp 2  + instanton  4.6 103 ), shown in ( " ∼ ) = x ) 2 [ [ t γ (  ( + S ω t E g 00 ω − m φ  sin( " required to excite the virtual pair over the “binding energy” 2 exp [ E ω = ω → ∼  probability A + 2 m . Thus we see that the semiclassical expression (

m ) low in the Mathieu spectrum, to power-law narrow gaps ( − worldline instantons  ¨ probability φ 3.45 ω − The relation to the Mathieu equation is the following. The pair production probability First, recall that the above pair production results have a natural interpretation in written now in Mathieu form. So we identify the Mathieu equation parameters: gives the same exponential behavior) [ For the cosine electric field, associated worldline instanton action indeed equals the exponent incan ( alternatively be computed viareflection coefficient a in Bogoliubov the transformation Klein-Gordon astum, equation to a (we take get quantum zero the mechanical electron/positron leading momen- effect, and treat scalar QED instead of spinor QED, since this also world line path integralpression representation for of the the pair production QED probability effective action gives the leading ex- classical Euclidean equationsbackground of electric motion field: for the world line of a chargedFor particle such in worldline the instanton solutions, a semiclassical approximation to the Feynman’s limits, and we show below thatbands this is ( analogous to the transitionMathieu from exponentially spectrum. narrow terms of of photons of energy by a multi-photon process, ratherpower is than twice by this tunneling multi-photon [ of number. the The applied final answer electric iswhere field, perturbative, but as it it is is a a power very high power in the relevant semiclassical limit the logarithmic beahvior of in fact produces a perturbative result: The physical interpretation of ( by the constant electric field. In the opposite limit of a high frequency field, where JHEP02(2015)160 , . 2 E m 105  ) (4.16) (4.17) (4.18) (4.11) (4.12) (4.13) (4.15) (4.19) γ mω/ ( E  g ≡ π 2 γ E m in the Mathieu −  N exp ∼   )] u 2 2 1 γ 2 (4.14) − E 2.4 ) − + 1 1 πm γ u u (  g =  K π 2 E = 1 + 2 2 ) and ( probability E 1, we have ~ − m πγ ) can be re-expressed as 2 Q A 4 2.2 + 1)  2  4 ~ m u γ 1 ∼ ( 3.11 ≡ ≡  1 2 ←→ − + 1 1) 2 2 ←→ u u E E in ( − ω ω −  m ω  D 0 u 1, and   a  ( K u , u   – 30 –   π )  ∼ ~ 2 ) ω − ω m 2 2 m u u ( u 1 ω ( ≡ ∼ D 0 + 1 D 0  2 a  a u  ω E ) E N  4 u E  √  ( 2 i Im = 0 D Im π 2 √ π a i ~ π  ~ 2 Q − 2 2  √  = 2 2 Im 2 ) = 1, this implies ≡ π − ~ u 4  ~ ( , we see that it corresponds to the gap label index  D 0 ω m γ a exp ≡ 7 ∼ N 2 . But this still permits arbitrary values of the adiabaticity parameter m  On the other hand, in the multi-photon limit, ω Note that the pair production probability is related to a bounce, which is a closed-path instanton/anti- ]. 7 (gap width) instanton configuration, referred to assingle-instanton a effect. “worldline instanton”; For while time-dependent theare electric Mathieu important fields band quantum or associated interference gap with effects106 splitting more and is realistic these a laser are pulses, captured by there complex instanton trajectories [ Then defining spectral problem. Indeed, in this limit so we can consistently consider the hierarchy of scales In the static limit, which leads to the standard Schwinger formula for pair production in a static electric field. To see how thisvacuum works pair in production, the the electronand various mass limits, sets recall the dominant that scale, in so the we require semiclassical approach to This implies that Therefore, Further, note that the leading dual action Therefore, we further identify [using the scalings in ( JHEP02(2015)160 ) for (5.1) (5.2) (5.3) (5.4) (5.5) (4.20) ]. The 6= 0, ] is un- 2.30 ~ 45 96 )[ ). After the ~ a, 3.7 ( ) . . Physically this ˆ ~ NS ) from the gauge This follows from ~ ) , F a, ~ ˆ (ˆ ~ 8 2.21 ∂ a, = 0 NS ( ~ ˆ  ) ∂ F ˆ NS ~ ~ ∂ F ˆ ~ ) we obtain ˆ a, a, m/ω ~ 2 ∂ ( 4  ∂ ~ Λ D . 5.4  NS 2 − − ) leads to ˆ ∂a ~ 48 E F ) ) ~ 2 m ω e ~ ˆ ~ − 4 4.11 − a, a, )  ¨ ( Unsal relation ( ˆ (ˆ a ) ~ := Λ ∂a ~ ∂ ∼ NS NS a,  ˆ a, and using ( ( Λ . In addition to the SUSY inspired proof, F F N ( ~ Λ ∂a ∂ 4 ∂ ∂ a D NS , a ˆ  a F 2.3 2 a Λ ~ ∂a ∂ − Λ ¨ e – 31 – Unsal relation and its geometric inter-  a ) ) and the corresponding periods ( N Λ ~ − −  2 NS iπ ˆ a, ) F NS ( 1.13 ∼ ~ 2 ˆ F 2 , the instanton part of the original relation [ NS 2 ) = a, ( ~ F D 2.3 ). a, ¨ a Unsal relation can be identified as the generalization of the ) := Λ ( = 2 = 2Λ ~ ) with respect to u + is due to the perturbative contribution to 1.13 ) a, ) ~ 5.1 ( u ~ a, NS (gap width) a, ( Λ ∂a ( F ∂ NS ∂u F i ], as mentioned in section π ∂ 2 Λ 12 has mass dimension 1 in the gauge theory/QM correspondence. This can easily be seen , ). ~ 11 4.6 [using Stirling’s formula], the identification ( N Our starting point of the proof is the generalization of Matone’s relation for pretation Note that 8 Then, by differentiating ( from the Schr¨odingerequation ( where the werescaling use the the Schr¨odingerequation symbol ( ˆrescaling, to it follows denote that dimensionless quantities. where as mentioned inchanged section and the shiftnext in step is to observe that the the prepotential can be expressed as follows identification is an explicitexpansion example and of the the geometry of connection compact between Riemann the surfaces. resurgent trans-series In this sectiontheory we point present of a view.perturbative proof In fluctuations around terms of the of theinstantons vacuum quantum to [ Dunne- mechanics, the this non-perturbative relation fluctuationswe remarkably also around links show the thatPicard-Fuchs the equation Dunne- for the quantized elliptic curve with nonzero pression ( 5 A simple proof of the Dunne- large which, up to a multiplicative pre-factor, is the multi-photon Br´ezin/Itzykson/Popov ex- in Mathieu language. Thus, from the Mathieu equation gap splitting formula ( JHEP02(2015)160 ) ) is D  (5.9) (5.6) (5.7) (5.8) 5.10 D (5.10) a, a ZJJ ∂u ∂a ). With A ) (which ~ ): ~ ~ ∂a ∂ u, B, − 1.14 ( , and ~ ∂a ∂u ) to ( ZJJ a/ ~ D E ~ . a, ∂ ∂a = 2 1  contribute to the “total ≥ . B 1 ) and on the right-hand i ~ − , π D 0 2 ~ a  a, / i , = , n π 0 2 1) ∂a ∂u a   ∂a ∂u − = = 0 ~ u ZJJ =  0 ! ∂ D k du ~ da ∂A D ~ = ( − ∂ ). on the right hand side does not get any ~ D 0 ∂ n ∂a du a ∂a ~ + da ) in terms of the expansions ( − ZJJ 5.10 − i/π D k B , E ) transformations that act on the pair ( a 2 D 0 D 5.8 ) becomes Z a  du . The final step in order to arrive at the form − , da – 32 – ) transform as follows:  D D 0 k 5.5 ~ 16 a a ∂u − − ∂a 5.5 D n − du = 1 D ¨ Unsal relation with the Picard-Fuchs equation provides − da = ∂u ¨ k ∂a Unsal relation takes the form of Picard-Fuchs equation,  a /∂a  ZJJ

∂a ∂u F ~ ) ∂B ∂  ∂a ∂ k ∂E 2 ~ = − − D . In other words, by using the Riemann bilinear identities on ( a (1 ∂a ~ :  ∂a ~ n =0 k X , ). In these variables, ( 1 3.33 ) is invariant under the SL(2 − ). Then, the Dunne -  ~ 5.8 ) (which encodes the fluctuations around instantons) and ) we deduce that only the zeroth order periods ∂a ∂u u, ~  B, 5.10 ( = It is interesting to note that similar connection formulas between perturbative and The identification of the Dunne- It is also illuminating to switch the independent variables from ( ZJJ ∂a ∂u connects these two independentnon-perturbative periods. expansions Furthermore, take even though veryspectrum, the this different perturbative connection forms and holds in throughout different the spectrum. regions ofnon-perturbative the physics exist energy for other QM potentials, such as the double-well and SUSY contributions from different orders as seen in ( an explicit geometric interpretationstanton of expansion. the The connection perturbative between correctionsby perturbative to the series the energy and quantization eigenvalue in- of arerections characterized the are real characterized period, by while the the dual exponentially pure suppressed imaginary instanton period. cor- The Picard-Fuchs relation the Picard-Fuchs equation, and thecorrections constant at 2 nonzero and ( flux” on the torus,the whereas flux the from higher higher order order terms terms do is not. a result However, the of cancellation rather of intricate cancellations that involve The second line of this equation shows that all the higher order terms contribute as zero to Note that ( as expected. It is further useful to express ( where the independent variablesside are on ( the left-handextended to side nonzero are ( this change of variables, the terms in ( which relates the instanton expansion toA perturbative expansion is to switch toencodes the variables the perturbation expansion)defined as where in ( where we used the fact that JHEP02(2015)160 – 34 ]. and (6.3) → ∞ a 107 ν saddle paths 9 ]. 48 1 spectral region of http://dlmf.nist. (6.2) complex  ) 2 0 limit with the λ k ; (6.1) → x -function as: ( , and then use a completely 2 H 2 P = 0 k 2 8 sn remains finite. This then leads ~ ) exists whenever the spectrum 6.1 ψ 2 2 ) k
k 2 1 + 2.21 +
z, k + 1) ↔ − ν 2 + 1 ( 2 ν sn SUSY gauge theory k ) reduces to the Mathieu system in a special 2 ≡ k ∗ , u 1 3 2 6.1 ]; other papers sometimes use differing conventions [ ] have not taken into account the fact that in this κ – 33 – − 2 + 1) 76 = 2 36 k theory corresponds to the = ν – ( ∗ ]. This is ultimately due to the fact that the Lam´e
N 0 ν 34 ), we must combine the + 1) 48 z K 4 − ( = 2 ν 2 , i ( is related to the Weierstrass H path configurations, the existence of ν N K . 2 ; 0 p + ( K real 6.2 2 i ψ ↔ 2 for the Mathieu equation, a formal expansion of the action may ; 0) = sin dz + ~ d z 3 x ( 2 ). The Lam´eequation ( P 2 ]. In fact, in general a relation like ( k ]. − 12 , (1 120 11 – : K ≡ 118 0 , K 40 , The Jacobian form of the is Lam´eequation conventionally written We use the standard period conventions of [ . For more general cases, where the Riemann surface is of higher genus, the relation will 9 39 , D 36 where scaling limit: since sn limit, in such a wayto that the the natural combination identifications: The Jacobi elliptic function sn potential is doubly-periodic in the complexpath plane, integral and even is though a the sum quantumhas mechanical over a direct influence on the divergent structure ofgov/29.2.i perturbation theory [ the Lam´esystem. Previous analyses [ region the spectral expansionssystem, are divergent and and so non-Borel-summable, as shouldresurgent for structure be the is described Mathieu evenboth by richer real a than and resurgent complex for trans-series. instantons the [ In Mathieu fact, system, due the to associated the existence of different technique to studyKdV the hierarchy: electric see region, section using the relation between WKB6.1 and the Resurgent analysis inThe the dyonic gauge dyonic region region of the As discussed in section be obtained by anlow all-orders energy WKB region. analysis.the Instead novel of This features repeating can that arise these then in steps be the for expanded dyonic the region in in Lam´esystem, the section we high summarize or is such that the associated Riemanna surface is described by justgeneralize the to two include dual actions, multiple pairs of actions whose number is equal6 to the genus Lam´eequation [ and SU(2) double-well [ JHEP02(2015)160 ) ]. − ~ u, (6.5) (6.4) # ( 109 , a ... 108 − ! 3 4 ... + − band is  # 0. 1 2 1 4 th !# 0, recalling that (1 → + + N  2 ! 2 1 2 k N →    1 2 ); for details see [ 2 1 2 + k ), which has a manifest symmetry + + N 2 + 4  N 2 N z, k ) as 5 4 (    × 2 . Approximate expressions for these 1 2  " 3 4 4 ~ + 2 k k ) as: 2.11 3 k + ! + ) in the limit ) by a simple Landen transformation.  1 k − 3 2 + 1) N 1 2 1 + 1   2 2.8  k k 1 2 z, k  k + ( 3 ( 2 ), the splitting of the − 2 ln +

N ~ 16 1 1 + ) k  2 4 6.3 N – 34 –  ~ − k expansion of the all-orders WKB actions

 In particular, it means that the leading large-order

2  u =

k 1 2 1 2 2 10 4 + ) (1 + ]. 2 − inst + N ~ k S 32 48  , these expansions are divergent, with factorially growing N 1 ~ ) 2 2 −  k k 1 (1 + http://dlmf.nist.gov/29.7.E5 ~ " " − 32 3 8 and × 2 ~ 1 + for the Mathieu equation, the electric region of the Lam´espec- (1 ~ ~ − − 3 / 1  2 π ) = http://dlmf.nist.gov/29.7.E1  ~ r ! N . This potential is related to the sn ~ ] uses another form of the elliptic potential, sd N, 2 2 N ( k 48 in this limit. u − ). Here we describe a different, complementary, technique to analyze the elec- ∼ ] ~ 1 /e 1 u, → ( 111 band N 2 , D → k u . This more intricate structure is due to the existence of both real and complex instan- a Associated with this resurgent divergent structure of the perturbative expansions is the For a given 2 The paper [ ∆ /k k 110 1 10 ) under As discussed in section trum can be studiedand by using atric large region, using the work of Gelfand and Dikii concerning the high-energy asymptotic Note that this reducesk to the Mathieu expression ( 6.2 WKB, Gelfand-Dikii expansion, and KdV, in the electric region This band-splitting is a one-instanton effect,is as [ the instanton action for the Lam´epotential instanton action, but also by the complex (‘ghost’) instanton/anti-instantonexistence action. of non-perturbative band splittings, forband any splittings are given inTranslating these ( results to our notation ( coefficients. But the large orderon behavior can be alternatingtons, or and non-alternating, is depending discussed in detailgrowth in of [ the perturbative coefficients is not solely governed by the real instanton/anti- Notice that we recover the Mathieu expression ( Lam´eeigenvalues ( With these identifications, we can express the standard perturbative expansions of the JHEP02(2015)160 (6.9) (6.6) (6.7) (6.8) . For (6.14) (6.12) (6.13) (6.10) (6.11) 2 3  2 2 ~  , ... ), we identify π L − = 6.6
 to the high energy x ... + 15 − N cos 4 2 ∞ ~ / 2 2 4 7 ) of period + ~ 0 ) . For the Mathieu system, x 2 V  0 ( / E 5 dx , . . . 3 → V 7 V ) (4 1  − u 4 − ,I (2 V E 2 ,E / √ 1 64 3 5 2 0 ) E ψ + 5 V  − ,... 2 E 2 2 =
) given by the KdV conserved quantities. 2 2 2 0 ~ (4 ] / x ~ ψ ( V  2 V ) ! +1 − dx [ / V 2 j x π 5 ) 2 (
)  V 1 +1 / 3 j u = E 3 2 V – 35 – 2 , 2 I ) 0 V ~ 0  (2 (4 + V ) with the Gelfand-Dikii form ( E V x ] relates the level number  + 10 1 ψ 16 (4 + 2 =0 ∞ 2 . The KdV integrals are simple to evaluate: = 2.1 + 2 j X ]. The coefficients of this expansion are expressed di- 2
2 cos ]. This is described below. π − − ) 00 d 113 V 2 1 0 4 2 L dx , dx 15 2 2 V ~ V / 2 E 114 − 0 113 ( −

3  ↔ 4 , ) V V dx V  112 1 4 2 √ u E L L L L L  0 0 0 0 112 √ (2 2 − 2 Z Z Z Z ~ ,I 1 4 ∼ E  expansion of the exact result , and − √ ] = ] = ] = ] = π u L u V V V V πN 2 ∞ 2 ∼ 2 [ [ [ [ = 0 = ~ 1 2 3 4 + √   I I I I x x L ↔ Nπ → ∼ E cos cos ~ ] are functionals of the potential E , 2 2 2 x N V 2 2 [ ~ ~   +1 cos 4 1 j I I 2 I 2 ~ For the Schr¨odingerequation with a periodic potential ↔ which leads to the expansion: to compare theV Mathieu equation ( example, for a constant potential, which is the It is an instructivealso exercise for to compute the these Mathieu KdV potential, integrals to for confirm a the constant all-orders potential, WKB and analysis in section where These can be generated by simple recursion relations, and the first few are: the Gelfand-Dikii expansionasymptotics [ as rectly in terms ofparticularly the interesting KdV for integrals, the evaluatedple Lam´eequation on because polynomials, the the QM with expansion potential. interesting coefficientsmultiplies combinatorial the are This properties, elliptic sim- observation potential in is [ the scaling parameter that expansion of the resolvent [ JHEP02(2015)160 . (6.18) (6.19) (6.16) (6.17) 2) k > . , of increas- ( 2 3 2, we recover F g ~ / , ~ 2 k in  ) N K π ≡ 2 τ m 1 ) could be referred to as a  + ∞ ]: 8 6.13 n + 27 ] considered the potential is a multiplicative scaling ( and 114 µ = 0) → can also be written as: , ~ 114 polynomials 6 1. )[ 6 (0 ], we rewrite the coefficients of ) dependence of the Eisenstein \ K E 2 2 (  Z k are ζ ,E 120 ∈ X a ,E ), and (6.15) ) 2 – ) ≡ / and 1 µ ], generalized to elliptic functions associated 1 6.2 2 4 ( n,m u η ( g 118 E 2 , +1 ) 115 ) / 2 ), the high-energy Gelfand-Dikii expan- , j 0 k . is identified as 2 µ  ( F 1 0 ( +1 K 40 E ζ . In our conventions for the periods of the j , (therefore τ K K ) 6 K π 2 ≡ +1 , i i as a function of , with coefficients expressed in terms of the 2 j τ 39 ] E E , where Thus, for this Weierstrassian form of the Lam´e µ K  F u – 36 – ; P ≡ -function ( K (4 0 ) = 3 4 / 12 µ τ P τ K 1 and ]. ( =0 i ∞ η = [2 k j X 4 4 + +1 114 E K [ 1 j theories [ x 2 I , ) is a re-arrangement of the all-orders WKB expansion ( ,E 2 ∗ , and ]. In this sense, the polynomials in ( have interesting combinatorial properties, and were named − ,E 3 P . E g ) to express 6.14 E 114 = 2 11 )) 1 τ ), √ in the electric regime where K η ( ) = N µ and 6.14 η u ( x ∼ 2 2 (  g +1 V K j log( K is the Weierstrass 2 π πN F 2 ) for ∂ P ∂τ  2.26 πi (2) ) with 2 = 3 ζ 2 − 6.6 E ) is the Dedekind eta function, and , where τ ) can be written P ( ). So, inverting ( ) = ] reduce to simple polynomials in η µ τ 6.7 ( V Turning now to the Lam´esystem, Grosset and Veselov [ [ 2 elliptic Faulhaber polynomials 3.38 Not to be confusedThe with name the derives from prepotential a which remarkable connection we between denote the with classical Faulhaber the polynomials calligraphic of letter Number = 2 E 11 12 +1 j series in the following equations. Theory and the KdVwith integrals periodic for soliton-like arrays potentialstrigonometric of Faulhaber [ polynomials solitons [ Weierstrass function, the modular parameter For notational simplicity, we suppress the Thus, all coefficients ofthe the first elliptic few Faulhaber Eisenstein polynomials series are expressed in terms of just where In order to facilitateNekrasov the prepotential comparison in with resultsthe in elliptic the Fauhaber polynomials physics literature in terms concerning of the the Eisenstein series These polynomials, the equation, ( sion ( parameter to beI specified below.Weierstrass Grosset invariants and Veselov showed that the KdV integrals ing order. Thisin expansion ( ( the expansion ( V Note that the coefficients of the inverse powers of JHEP02(2015)160 ]. 114 (6.29) (6.30) (6.21) (6.24) (6.26) (6.27) (6.28) (6.22) (6.23)  2 ) = 0 term µ j (2
 2 6 )
E 2 µ 2 ) µ E ] (2 4 4 3 − ) ] (2 ) E 2 4 4 µ µ 2 ) are discussed in [ E E (2 E (2  2 µ  (  E 6 − 4 to yield ) are +1 6 E ! E ) − j ) 2 µ   2 + 96 13 2 E µ ( 6 F µ )
E ( j 3 E 2 E 2 ) µ 6 [ (2 +1 j ) ( a +1 j µ 2 2 µ j − 3 4 + 7 F 12 [ +1 (2 E + 18 G N ) 3 ) becomes (2 j 6 )  2 µ 4  G − 6 + E µ 2 4 +2 E 2 3 (2 j E ) 6.24 (2 E 2 E 2  =0 ∞ E µ j X 6  2 2  E 3 + 2 E 2 2, ( + 10  ~ + 6 2 − E ] (2 − 4 2 4  K 5 2 E 4 π E 2 4 N/ 2 E 2 8 E – 37 – N E 2 ~ 2 − E 2 =1 ∞ E  5  j X 4 −   E 2 ) can be viewed as a high-energy expansion of an E = −

− ) 2 2 1 2 9 4 E (6.20)   80 2 µ a 2 )  E + 7 − − 2 µ +  E 6.16 (2 ) K 4 2 6 π 2 2 ~ 15  µ +72 2 (6.25) +12  a E 2 E ] (2 2 )  + 3  2 1 2 5 3 µ E [4 35 ] (2 6 ) are simple combinations of the elliptic Faulhaber polyno- 2 1 E 4  ∼ [ µ ∼ E − − ( ] (2 1 3 E ≡ 2 [ 4 ˜ 2 u 2 1 E  2 ˜ u +1 9 135 189 1  E E j  [  6 8 4 G 1 3 1 K    1 π 1 2 540 9072 36 − K K K ]. π π π  2 2 2   −  121 ) = ) = ) = ) = , µ µ µ µ ( ( ( ( 28 3 4 1 2 ) = ) = ) = ) = ): G G G G µ µ µ µ µ ( ( ( ( ( 2 3 4 1 F F F F +1 j K K K K 1 1 1 1 F 2 2 2 2 The Gelfand-Dikii expansion ( Then the first few elliptic Faulhaber polynomials It is interesting to note that the Lagrange inversion of series can be naturally expressed in terms of 13 Young tableaux [ from the sum: Then in terms of the action variable It is convenient to define the rescaled and shifted eigenvalue, absorbing the where the polynomials mials Further properties of these elliptic Faulhaber polynomials all-orders Bohr-Sommerfeld relation, which can be inverted JHEP02(2015)160 , 2  14 k 2 4 ]: 16 2 . E ~ (6.35) (6.37) (6.38) (6.31) (6.32) (6.33) (6.36) (6.39) (6.34) 120 − there – ↔ 6 F E µ πi 1
2 4 limit, which 4 116 , E ~ −  2 0 = 1 2 40 H , j ] → ∞ 2 6 = 39 here 0 2 2 E 6 ) with the expansion j F j a ~ H m E H are directly related to +1 d ] as j j 6.30 dτ ↔ 2 H ), we identify 2 + 180[ 40 µ theory [ , πi 1 2 2 =1 ∗ ∞ 39 ~ 6.2 j X 3 0  , H = 2 πi 1 2 4 0 ] ] 4 4 , and the Nekrasov deformation ~ 6 6 with 2 H
N m theory: E − 2 E  − ) and ( 1 4 6 ~ ∗ , 2 )) − as 2 0 E + 3 = 2 theory in the τ 6.3 − 4 m 2 1 ( H 4 4 = 2 :  ~ 2 ) E ] η  2 0 N E 2 4 µ ~ 2 m + 11 2 = ( N − H E E [ 24 4 E E 2 log( ~ ≡ 3 [ +1 1 3 E 0 j m + 2 15 0 2 πi – 38 – G − ) does indeed reduce to the Mathieu large period ∗ H = E 2 ~ 36 [7 ≡ H 3 0 4 =2 − 0 +2 − H j E 6.30 ↔ N 2 NS H 2 ), required to have a smooth reduction to the Mathieu  finite. + 84 ~ 4 d F µ 2 dτ 3 2 E τ a k 2 +1 ∂ E j 2 6.29 1 2 differs from the normalization in [ ∂τ j πi 1 + 2 2 m 2 F 2 ∼ = 2 0 175 − E ) ˜ u H , 5  ] are polynomials in = theory reduces to the 2  , m  j j 4 ∗ E ~ [ 0 with 1 ], vanishes, as it does for the Mathieu potential. H H E 1 a, V 1 [ 12 360 60480 : ∂ ( = 2 − → ∂τ 1 ∗ I 2 2 2 +1 ). N j πi k 1 =2 E G 2 ) = ) = ) = 0 limit we find that (  N NS 0 0 0 1 2.26 F is just the shift in ( H H H → 12 ( ( ( 2 u 1 2 3 = is expressed in terms of the scalar mass, k H H H 2 0 E H d dτ We can now compare the Bohr-Sommerfeld energy expansion ( Note that our normalizationEffectively, of we are subtracting the average value of the Weierstrass potential over one period, so that πi 1 14 15 2 The shift in ˜ eigenvalue, as the is combined with the first KdV integral, We thus arrive at the Matone relation for the we observe the remarkablethe fact polynomials that the expansion coefficients and the identification Using the differentiation properties of the Eisenstein series The higher coefficients where parameter in the Nekrasov-Shatashvili limit and in the expansion in ( of the Nekrasov-Shatashvili prepotential for the SU(2) It is a non-trivial check that with the scaling of ( JHEP02(2015)160 ]. 111 . We λ Riemann j ] to large gap high in the has an algebraic th 47 gauge theory. But, , dE N ∗ ) regime with the gauge 46 , ) for the energy is only E ( λ = 2 carries a clear geometric ρ 12 , ], with the number of gaps 6.30 N (6.40) 11 dE ) 122 [ . Since the structure is quite E + 1) is related to the Nekrasov defor- ( = 2 SUSY SU(2) gauge theories. j ρ ( m j N R 2, the Bohr-Sommerfeld expression / regime where perturbative expansions = ~ is the spectral level number, we asso- ∼ λ µ N 2 finite gap N πN with ⇒ a – 39 – , where ~ ~  N 1 2 = + j λ , the associated density of states  j = 1 regime with the gauge magnetic regime. We have shown that as m ∼ http://dlmf.nist.gov/29.3.iii ~ λ ≡ 1  regime with the gauge electric regime, the small λ ], and there is also a precise strong/weak band/gap duality symmetry [ 122 , . In this case, much more explicit descriptions of the spectral information can be is an integer the Lam´esystem becomes j j 114 Thus, in the large action spectral region of the Lam´eequation we identify the all-orders It is interesting to note that when the scalar mass The familiar relation between divergentfects must perturbative be expansions generalized to and accommodate non-perturbativeare the large ef- convergent, and non-perturbative effectscients. are This associated generalizes with the poles resurgentprovide trans-series of a analysis expansion physical of coeffi- [ analogy of this change of non-perturbative behavior, in the transition dyonic regime, and the exact WKB provides a completeeffects, and description permits of direct all mappingsorders regimes, between these Bohr-Sommerfeld including sectors. relations all Previous non-perturbative described analysesperturbative only based band part on or all- gap of splittings. theturbative This information, and analysis neglecting non-perturbative also non- shows contributions that is the radically relation different between in per- the different regimes. In this paper we havedescription applied of resurgence and the all-orders differenttheir exact spectral close WKB connection regions to with provide of the aDefining the low complete a energy Mathieu ’t behavior and of Hooftciate Lam´esystems, parameter, the stressing large similar, we do notdeferred to repeat a all future the publication. steps here.7 Further discussion of Conclusions the Lam´esystem is gives a (convergent) perturbativespectrum. expression However, for physically the it isin location clear the of that spectrum, there the are andin also these section non-perturbatively small 3, are gaps these relatedfraction gaps to arise expressions instantons. due at to As poles in in the the expansion Mathieu coefficients: case see discussed the continued WKB Bohr-Sommerfeld expansion with theidentified Gelfand-Dikii with expansion, the and the scalar eigenvalue condensateas is moduli in the parameter Mathieu of equation,part the this of Bohr-Sommerfeld the expression ( story. Identifying the action equal to given [ Notably for integer valuesgeometric of meaning: it is ansurface. abelian differential Therefore of the theinterpretation. quantization second condition kind over a genus- mation parameter then if JHEP02(2015)160 , , 11 39 , (A.1) 32 , 31 + 0 = 2 system. . The series is → N /π 1)! , g − n n ( g 2 ∼
! 1 2 n n c n + 2 n π Γ ) is asymptotic, with non-alternating =0 ∞ theory reduces to the X n ]. ∗ 1 g A.1 e 107  = 2 , from which the resolvent and spectrum can 1 g π/g 1 2 – 40 – Ht  N − 0 ]. We demonstrate a direct relation between the p I e 45 = ∼ x cos 1 g − ], certain features of the divergence of the perturbative expansion dx e π π − 123 , Z π 1 10 2 , 9 ≡ ) g ( ] and Whitham dynamics [ Z 44 , The reinterpretation of the spectral problem in the language of the Nekrasov partition ¨ Unsal and A. Vainshtein for discussions and correspondence. GB thanks the DOE 43 , ] that shows that all non-perturbative information of the trans-series eigenvalues is subtly The relation todimensional partition the function quantumbution gives mechanical to the the spectral resummed heatbe leading problem kernel extracted. derivative expansion arises expansion tr The contri- factorial because perturbative large this order expansion behavior in zero- of ( the expansion coefficients, A Simple analog ofAs uniform discussed in asymptotic [ behavior for the ground statepartition energy function of the Mathieu system are captured by the zero-dimensional grant DE-FG02-13ER41989, and acknowledgesDepartment hospitality at and the support Technion, fromUniversit¨at,Jena, and the where the Physics much Theoretical ofMercator Physics this Professor Institute work Grant, at wasScholar Friedrich-Schiller and done. Award. the US-German GD Fulbright also Commission thanks for the a DFG US for a Senior Acknowledgments We thank C. Bender, O.M. Costin, L. Gallot, A.for Gorsky, A. support Lerda, through M. grant Mari˜no,N. Nekrasov, DE-FG02-93ER-40762. GD acknowledges support from DOE 40 all-orders Bohr-Sommerfeld relation andinvariants, the and Gelfand-Dikii show expansion, explicitly basedFuture work on how will the the develop KdV these ideas further [ function leads naturally to a12 simple proof of a quantumencoded mechanical resurgence within relation perturbation [ theory, whichalso is give an a extreme geometrical form interpretationand of of the resurgence this Picard-Fuchs behavior. fact equation, in We complementary terms to of work the on Riemann quantum bilinear geometry identity [ in a time-dependent electricnaturally field, described as the by adiabaticity worldlinehave of instantons. the the Since field physical the changes. interpretationsignificance Nekrasov This of of deformation physics constant parameters is these graviphoton worldlineshould fields, be instantons, the further and understood. SUSY their gauge associated theory non-perturbative effects, from tunneling pair production to multi-photon pair production in the Schwinger effect JHEP02(2015)160 , . /g 1)! π 1 −   (A.2) (A.4) n ( 2 ]. This )  N 16 1 n ]: N N g limit with 1) g ( 2 0 such that (A.5) 1 2 − N (
! 1 2 → 1 + n ∼ g n + q ) fixed) 2 n (A.6) π N   ( N and n ( Γ n c U n n fixed) 1) + 1 0 N → ∞ − g ( ( → N =0 ∞ ∞ =0 ∞ X n X n (A.3) + ) ∞ 1 g N z + − ( , g →   0 e 2 ) → n 1 (A.7) g i π I ) , g N g π/g ) high in the spectrum where i ) ( 2 ∓ ∞ http://dlmf.nist.gov/10.41.E3 N [ ) ,N p k ( 1 2.30 2 z 1 N g n 1 + n ( g c ± 0 q expansion where 2 n ]. N K =0 ∞ – 41 – g !) X  n 48 2 can be modeled by considering instead k 1 ). But we can also consider the large 1 g 1 + < N g <
k ! ) = 1 2 e e N n   z N g (2 (0 A.1 n + 2 i iπ 2 uniform 2 4 ± 1 =0 π/g n  ∞ g 1 / π e k X 2 1 ) satisfies a second order differential equation [ ∞ (  0 g Γ + . This trans-series expression contains all information about ∼ p + ( 2 πN δ 1 2 1 K g ) , as in ( 2 Z =0 g ) satisfies a third-order differential equation, and this fact is reflected in ∞ ∼ − → N n X n ( g √ + π (  1 g  Z q 3 2 2 Z limit this function has a convergent “strong-coupling” expansion: e 1 g h ∼ 1 N g ≤ )  Ng  g g ( exp N π/g 1 N ,N I 2 N π  ph Z 2 ≡ 1 p N ) √ I g ∼ ≤ ∓ ( = ∼ ) δ N g is a (known) polynomial of degree 3 + Z (  is: 2 π 1 g n Z U −  N g is kept fixed. The uniform expansion valid for all values of the ’t Hooft parameter N For the Lam´epotential, Z ≡ fixed: 16 uniform expression interpolates smoothlythe function between in the the two intermediate extreme region, the limits analog and ofthe describes the appearance “magnetic of region”. threeand different exponential both terms, real which and can complex be (’ghost’) identified instantons with [ the perturbative vacuum where On the other hand, thereNg is also a λ g which has the same form as the gap splitting ( where the series isat again large divergent perturbative and order non-Borel-summable, with Note that in the large The dependence on the level number the function, including thepart Stokes is required phenomenon. in order Infor to be the particular, consistent Bessel the with functions: the second analytic sub-dominant continuation connection formula two exponential terms since where non-Borel-summable and should instead be represented by a trans-series, which has just JHEP02(2015)160 (B.1) , 73 Phys. Rev. 3503554560 , denotes the co- (2008) 349 ]. n 2 ]. ]. (2012) 063 (2010) 351 127 61729 SPIRE 154828800 08 SPIRE IN 11 81749606400 SPIRE IN ][ IN ][ ][ gauge theories, matrix models , that enter into the relation JHEP . ) N , n ) ( k 13 u κ ]. k ( 780341 716800 0 + 81749606400 a n k + du SPIRE n , Chapman & Hall/CRC, U.S.A. (2009). d IN arXiv:1106.5922 k ][ [ u arXiv:1206.6272 ) arXiv:1210.2423 [ 517 Commun. Num. Theor. Phys. n [ ( k Nonperturbative effects and the large-order behavior 8430053 4128768 , – 42 – Annales Henri Poincar´e κ The resurgence of instantons in string theory , 163499212800 n =0 , volumes I–III, Publ. Math. 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