An Introduction to Resurgence, Trans-Series and Alien Calculus
PPrepared for submission to JHEP
DAMTP-2014-44
An Introduction to Resurgence, Trans-Series and AlienCalculus
Daniele Dorigoni
DAMTP, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
E-mail: [email protected]
Abstract:
In these notes we give an overview of different topics in resurgence theoryfrom a physics point of view, but with particular mathematical flavour. After a short reviewof the standard Borel method for the resummation of asymptotic series, we introduce theclass of simple resurgent functions, explaining their importance in physical problems. Wedefine the Stokes automorphism and the alien derivative and discuss these objects in concreteexamples using the notion of trans-series expansion. With all the tools introduced, we see howresurgence and alien calculus allow us to extract non-perturbative physics from perturbationtheory. To conclude, we apply Morse theory to a toy model path integral to understand whyphysical observables should be resurgent functions. a r X i v : . [ h e p - t h ] J a n ontents Glossary 21 Introduction 32 Borel Resummation 53 The Algebra of Simple Resurgent Functions 104 Stokes Automorphism and Alien Derivative 145 Intermezzo on Trans-Series 206 Trans-series Expansion and Bridge Equation 247 Median Resummation and Cancellation of Ambiguities 358 Outlook 439 Acknowledgment 45 – 1 – ist of Mathamatical Symbols and Notations C [[ z − ]] Set of formal power series in 1 /z . C { ζ } Set of convergent power series in ζ , i.e. germs of analytic functions at the origin.˜ φ ( z ) Generic physical observable as a formal power series in 1 /z . B [ ˜ φ ] Borel transform of the formal power series ˜ φ .ˆ φ ( ζ ) Borel transform of a generic formal power series ˜ φ . L θ ˆ φ ( ζ ) Directional Laplace transform of ˆ φ along the complex direction arg = θ .˜ H Multiplicative model of the algebra of resurgent functions.ˆ H Convolutive model of the algebra of resurgent functions. (cid:103)
RES simp
Multiplicative model of the algebra of simple resurgent functions. (cid:100)
RES simp
Convolutive model of the algebra of simple resurgent functions.
Sing ω ˆ φ ( ζ ) Singular part of the simple resurgent function ˆ φ ( ζ ) close to the point ω . S θ ± ˜ φ Lateral Borel sum of the formal power series ˜ φ along the complex direction arg = θ . S θ ˜ φ Stokes automorphism of the formal power series ˜ φ along the complex direction arg = θ .∆ ω ˜ φ Alien derivative of the formal power series ˜ φ at the singular point ω . Disc θ Φ( z ) Discontinuity of the analytic function Φ( z ) across the complex direction arg = θ .– 2 – Introduction
When confronted with computing various physical quantities, i.e. partition functions, vacuumenergies, anomalous dimensions of operators, in different physical systems, we are almostalways facing the problem that, unless something miraculous is coming to help (integrability,supersymmetric localization,...) we will not be able to get an exact answer. So either we sitidle and declare defeat or we try one of the few things that we are almost always able to do:perturbation theory.To perform a perturbative expansion, we first have to find a suitable parameter, say g , that we can tune to be small, and as we dial it from zero to some specific value, weinterpolate between a simpler model, at g = 0, for which we have an exact answer for thephysical observable O under consideration, and the actual model of interest. At this pointwe expect to be able to write our observable as a power series in g O ( g ) = c + c g + c g + ... , (1.1)where c is the same observable we are interested in but computed in the exactly solvablemodel at g = 0 and all the correction c n can be in principle computed within this exact model.We are all familiar with the astonishing precision test of QED perturbation theory used tocompute the anomalous g − g ∼
0. The origin of the asymptotic character of perturbation theory is the rapid growth ofFeynman diagrams [2, 3]: the number of diagrams contributing at order n grows factoriallywith n . This combinatorial argument by itself is not enough to conclude that the perturbativeseries is asymptotic, some magic cancelations might happen when we sum all the diagramscontributing to a certain order. Thanks to Lipatov [4] we know that this is not the case:one can show, via a saddle-point method, that indeed the perturbative coefficients grow athigher orders as c n ∼ n ! . Similarly in quantum mechanics [5–7], matrix models [8, 9] andtopological strings [10], we encounter the same factorial growth of the coefficients c n in (1.1),effectively making our perturbative expansion only an asymptotic series [11] with zero radiusof convergence.It was soon realised that the asymptotic nature of the perturbative expansion was actuallyhiding deep and valuable information about the exact answer. The venerable idea of theBorel summation was introduced as a suitable analytic continuation of our asymptotic seriesby means of a contour integral of the associated Borel transform in the complex plane. Thisprocedure generically gives rise to ambiguities in the resummation process due to the presence The 6 dimensional N = (2 ,
0) is precisely an exception to this. – 3 –f poles in the Borel transform, changing the contour of integration leads to many differentanalytic continuations of the same physical observables O ( g ) (i.e. Stokes phenomena). The“strength” of these ambiguities is related to terms that cannot be possibly captured by anexpansion of the form (1.1), precisely the non-perturbative (NP) physics.Furthermore, even in cases when the Borel sum of the perturbative series alone wouldgive rise to an unambiguous analytic continuation, this might not be the exact answer [12].We have to investigate the analytic properties of the Borel transform in the entire complexBorel plane. We stress that, in general, we do not have a complete argument for why thepoles of the Borel transform of the perturbative expansion should all be associated with newNP physics so it is perhaps surprising that, in all the cases analysed in the literature, it isalways possible to find a suitable weak coupling regime in which these poles can be interpretedas particular non-perturbative objects of the underlying microscopic theory, i.e. instantons,D-branes, quasi-normal modes [13] etc.It is clear then that if we want a unique and well defined resummation procedure for ourobservable O ( g ), we have to use something more general than (1.1). A natural extension ofthe perturbative expansion is the so called trans-series expansion O ( g ) = (cid:88) n ≥ c (0) n g n + (cid:88) i e − S i /g (cid:88) n ≥ c ( i ) n g n . (1.2)We note that the terms e − S i /g are precisely the type of terms that cannot be captured by aperturbative expansion for g small, since they vanish, together with all their derivatives, for g →
0. From a path integral point of view we are just adding to the perturbative expansionaround the vacuum, (cid:80) c (0) n g n , the (multi-)instantons corrections e − S i /g , or other type of non-trivial saddle points, and the perturbative expansions, (cid:80) c ( i ) n g n , on top of them. As for theoriginal perturbative series, all these new perturbative expansions, around all the differentnon-perturbative saddles, will only be asymptotic series. If we Borel transform each one ofthe series (cid:80) c ( i ) n g n and try to apply our resummation procedure, as described before, we willintroduce many more new ambiguities: naively it looks like we only made our original problemworse.Luckily for us there is a systematic mathematical framework, called resurgence theory,to study precisely these kind of trans-series. Resurgence theory was discovered in a differentcontext by Ecalle in the early 80s [14] and since then it has been applied with success toquantum mechanics [6, 15–18], matrix models [19], supersymmetric localizable field theories[20] and topological string theory [21–23]. Only recently Argyres, Dunne and ¨Unsal [24–27]were able to apply resurgence to certain asymptotically free QFT and they were able toobtain, for the first time, a weak coupling interpretation of the IR renormalons [28, 29].Resurgent functions can be describe using a certain class of trans-series with particulardistinctive properties. The key properties, as we will see in more details later on, are that eachperturbative expansion (cid:80) c ( i ) n g n , appearing in the trans-series, will define, through its Borel See that last Section for a possible explanation. We present it here in its most basic form, see later on for a more general and complete discussion. – 4 –ransform, a holomorphic function with “few” singularities in the complex Borel plane andwhose behaviour close to its critical points will be entirely captured by the Borel transformof the perturbative expansion associated to a different perturbative series (cid:80) c ( j ) n g n , i.e. thecoefficients c ( i ) n know about different non-perturbative saddles. To put it in a suggestive waythe perturbative series around the trivial vacuum will know of all the other non-perturbativesaddle points e − S i /g and it will also know about their perturbative series coefficients c ( i ) n andvice versa.The idea behind these notes is to give a short introduction to resurgence, by keeping toa minimum the number of mathematical technicalities and always having in mind concreteexamples. For a more rigorous and mathematical discussion we refer to the comprehensivebooks by Ecalle [14] and the more recent and excellent works by Sauzin [30, 31].In Section 2 we will give an overview of the standard Borel transform and resumma-tion procedure, leading, in Section 3, to the introduction of the algebra of simple resurgentfunctions. The behaviours of this class of resurgent functions close to their singular pointsis the subject of Section 4, where we will introduce the Stokes automorphism and the alienderivative. In Section 5 we will briefly discuss some generic properties of trans-series. Withall the tools introduced we will be able to understand, in Section 6, how the perturbativeand non-perturbative physics are linked together by means of the bridge equations. Finally,in Section 7, we will combine everything together and discuss a path integral toy model:Borel Ecalle theory will tell us how to perform an unambiguous median resummation and, inSection 8, we will conclude with some speculation on why physical observables should takethe form of simple resurgent functions. With a slight change of notation from the physics literature and the Introduction, instead ofdescribing a physical observable as a formal series obtained from a weak coupling expansion g ∼
0, we will write everything in terms of z = 1 /g and work at z ∼ ∞ . By denoting with C [[ z − ]], the set of all the formal power series (generically with infinitely many terms) in 1 /z , we can introduce the algebra of formal power series for z ∼ ∞ z − C [[ z − ]] = (cid:40) ∞ (cid:88) n =0 c n z − n − , c n ∈ C (cid:41) . (2.1)Every formal power series is specified by an infinite list { c n } of complex numbers that can beseen as the coefficients of a infinite order polynomial in 1 /z without constant term . Definition . We can define a linear operator B called Borel Transform B : z − C [[ z − ]] → C [[ ζ ]] , (2.2) B : ˜ φ ( z ) = ∞ (cid:88) n =0 c n z − n − → ˆ φ ( ζ ) = ∞ (cid:88) n =0 c n ζ n n ! . (2.3) The reason to avoid the constant term is just a technicality and we will remove this restriction later on. – 5 –ultiplicative model Convolutive model˜ φ ( z ) ˆ φ ( ζ ) z − α − ζ α / Γ( α + 1) ∂ z ˜ φ ( z ) − ζ ˆ φ ( ζ ) z ˜ φ ( z ) ∂ ζ ˆ φ ( ζ )˜ φ ( λz ) λ ˆ φ ( λ − ζ )˜ φ ( z + λ ) e − λζ ˆ φ ( ζ )˜ ψ ( z ) ˜ φ ( z ) ( ˆ ψ ∗ ˆ φ )( ζ ) = (cid:82) ζ dζ ˆ ψ ( ζ ) ˆ φ ( ζ − ζ ) Table 1 . Mapping of operations from the multiplicative model to the convolutive model.
This operator improves the convergence of the original series, in fact if ˜ φ ( z ) converges forall | z − | < r , its Borel transform ˆ φ is an entire function of exponential type in every direction | ˆ φ ( ζ ) | ≤ Ce R | ζ | (2.4)with R > r . Conversely if ˆ φ = B [ ˜ φ ] has only a finite radius of convergence the radius ofconvergence of ˜ φ will be zero. It is easy to see how basic properties of the formal power series˜ φ ∈ z − C [[ z − ]] are translated into properties of its Borel transform B : ∂ ˜ φ ( z ) → − ζ ˆ φ ( ζ ) , (2.5) B : ˜ φ ( z + 1) → e − ζ ˆ φ ( ζ ) , (2.6) B : ˜ ψ ( z ) ˜ φ ( z ) → ( ˆ ψ ∗ ˆ φ )( ζ ) = (cid:90) ζ dζ ˆ ψ ( ζ ) ˆ φ ( ζ − ζ ) . (2.7)The last property is telling us that when we pass to the Borel transform, the natural multipli-cation of formal power series in the algebra z − C [[ z − ]] becomes a convolution in C [[ ζ ]]. Thisis why sometimes, when we will compute physical observables as asymptotic power series, wewill say that they belong to the formal multiplicative model , while when we will pass to theirBorel transforms we will be working in the convolutive model . The precise mapping betweendifferent type of operations, in the multiplicative model and in the convolutive model, isconcisely presented in Table 1 .Just to clarify better: if we have two observables ˜ φ ( z ) , ˜ φ ( z ) obtained as formal asymp-totic power series, their product will be once again a formal asymptotic power series˜ φ ( z ) ˜ φ ( z ) = (cid:32) ∞ (cid:88) n =0 a n z − n − (cid:33) (cid:32) ∞ (cid:88) n =0 b n z − n − (cid:33) = (cid:32) ∞ (cid:88) n =1 c n z − n − (cid:33) , (2.8) We thank Mithat ¨Unsal for this Table. – 6 –here the coefficients c n are simply the convolution sum of the { a n } and { b n } c n = (cid:88) p + q = n − a p b q , n ≥ . (2.9)Passing to the Borel transforms ˆ φ ( ζ ) , ˆ φ ( ζ ) we can write their convolution product (2.3)as ( ˆ ψ ∗ ˆ φ )( ζ ) = (cid:88) n,m ≥ a n b m n ! m ! (cid:90) ζ dζ ζ n ( ζ − ζ ) m = (cid:88) n,m ≥ a n b m n ! m ! B ( n + 1 , m + 1) ζ n + m +1 , (2.10)where B ( n, m ) is the Euler Beta function. We can rearrange the convolution product by usingthe known identity B ( n, m ) = Γ( n )Γ( m ) / Γ( n + m ), obtaining( ˆ ψ ∗ ˆ φ )( ζ ) = (cid:88) n,m ≥ a n b m ( n + m + 1)! ζ n + m +1 = (cid:88) n ≥ c n n ! ζ n , (2.11)with precisely the same coefficients c n found before in (2.9) from the product of the two formalseries ˜ φ , ˜ φ . Hence the name multiplicative model for the algebra of formal power series andconvolutive model for their Borel transforms.Standard observables in QFT, when computed in a perturbative regime, takes preciselythe form ˜ φ ( z ) = (cid:80) ∞ n =0 c n z − n − , with 1 /z playing the role of the small coupling constant, withthe tree-level contribution subtracted out (see later on in this Section). As already mentionedin the Introduction, thanks to standard arguments [1, 3, 4], we know that, since the numberof Feynman diagrams at order n grows factorially with n , the coefficients c n = O ( C n n !)will diverge and the perturbative expansion will only be an asymptotic expansion, with zeroradius of convergence. Definition . We will say that a formal power series ˜ φ ( z ) = (cid:80) ∞ n =0 c n z − n − is of Gevreyorder- /m , if the large orders asymptotic terms are bounded by | c n | ≤ α C n ( n !) m , (2.12)for some constants α and C .Note that thanks to Stirling formula the Gevrey order of a formal power series with c n ∼ ( n !) m is the same of the power series with d n ∼ ( m · n )!. From the arguments ofDyson and Lipatov, we can deduce that, in standard QFT, physical observables computed byperturbation theory are given by Gevrey-1 formal power series. Proposition 1.
The Borel transform ˆ φ ( ζ ) of a formal power series ˜ φ ( z ) = (cid:80) ∞ n =0 c n z − n − has a finite radius of convergence if and only if ˜ φ ( z ) is of Gevrey-1 type | c n | = O ( C n n !) . (2.13)– 7 –ote that if the c n are growing faster than n !, for example c n ∼ (2 n )!, we can alwaysmake a change of variables z → z ( w ), in this case z → w , so that the new series in w − n − is of Gevrey-1 type, however this change of variables will introduce new monodromies in thecomplex z -plane.From now on we will always assume, unless differently specified, that ˜ φ is of Gevrey-1 type and its Borel transform ˆ φ defines a convergent expansion at the origin, in mathematicallanguage it defines a germ of analytic functions at ζ ∼
0. A germ of analytic functions at z is the set of all the analytic functions with the same Taylor expansion around the point z .We will usually write a germ ˆ φ of analytic functions at the origin as ˆ φ ∈ C { ζ } .After having improved the convergence of the original formal series ˜ φ → B [ ˜ φ ], we needan operator to bring us back to a suitable analytic extension of the original formal series. Definition . We define the directional Laplace transform : L θ [ ˆ φ ]( z ) = (cid:90) e iθ ∞ dζ e − z ζ ˆ φ ( ζ ) . (2.14)This operator is linear and it maps analytic functions on e iθ R + , with rate of growth ofat most exponential type e r | ζ | , into analytic functions L θ ˆ φ in the half plane Re (cid:0) ze iθ (cid:1) > r .In particular, let’s note that we can easily compute L on the real positive line for all themonomials L [ ζ α ] = Γ( α + 1) z α +1 , (2.15)and similarly for the inverse Laplace transform (cid:0) L (cid:1) − [ z − α − ] = ζ α Γ( α + 1) . (2.16)Thank to Table 1, we see that L , when applied to ζ α , acts precisely as the inverse Boreltransform. Example . Euler studied the properties of the series˜ φ ( z ) = ∞ (cid:88) n =0 ( − n n ! z − n − (2.17)for z = 1. He noticed that ˜ φ ( z ) formally solves the ODE φ (cid:48) ( z ) − φ ( z ) = − z . (2.18)From (2.17) it is easy to compute the Borel transform of ˜ φ obtainingˆ φ ( ζ ) = 11 + ζ . (2.19) Note that, generically, the coefficients c n , obtained from the perturbative expansions of our favouritephysical quantity, contain as well power law and logarithmic corrections to the leading factorial growth.Roughly speaking they take the form c n ∼ n ! n α log β n which is clearly of Gevrey-1 type. – 8 – nalytic Functionin the RegionFormal Power Series BorelTransform
Germ of Analytic functionsin the origin
LaplaceTransformAsymptotic Expansion ˜ ( z ) = X n =0 c n z n B [ ˜ ]( ⇣ ) = X n =0 c n ⇣ n n ! < ( z ) > z ! 1L [ B [ ˜ ]]( z ) = Z d⇣ e z⇣ B [ ˜ ]( ⇣ ) Figure 1 . Schematic form of the Borel regularisation procedure.
By applying the Laplace transform in the direction θ = 0, we get the analytic continuationin the half-plane Re ( z ) > φ ( z ) L [ ˆ φ ]( z ) = (cid:90) ∞ dζ e − z ζ
11 + ζ = e z Γ(0; z ) , (2.20)where Γ(0; z ) is the incomplete gamma function. If we expand the above equation for z → ∞ we recover the formal power series defined above, furthermore it is easy to check that L ˆ φ ( z )is a particular solution to Euler’s equation (2.18), while the generic solution takes the form ϕ ( z ) = e z Γ(0; z ) + C e z , (2.21)with C an arbitrary constant. The homogeneous term C e z is non-analytic for z → ∞ andcannot be expanded as a formal power series in z − C [[ z − ]], these kind of terms will beexplored more in details in the context of trans-series, see Section 5. In this case, the Borelsum of the asymptotic power series ˜ φ ( z ) gives us the unique solution, ϕ ( z ), to Euler’s equation(2.18), vanishing at z → ∞ .The idea behind combining Borel transform and Laplace transform comes from the verywell known equation for the gamma function1 = 1 n ! (cid:90) ∞ dζ ζ n e − ζ , (2.22)– 9 –y plugging this identity, or using (2.15)-(2.16), in each term of the formal power series ˜ φ weget (after a trivial change of variables)˜ φ ( z ) = ∞ (cid:88) n =0 (cid:90) ∞ dζ e − z ζ c n n ! ζ n . (2.23)We can commute the sum with the integral and obtain an analytic continuation of our originaldiverging series as the Laplace integral of its Borel transform˜ φ ( z ) = L [ B [ ˜ φ ]]( z ) , (2.24)this defines for us a regularisation procedure for our diverging series, see Figure 1.In the example above we were able to compute exactly the Laplace integral for the Boreltransform of the formal power series ˜ φ , solution to Euler’s equation, but generically, unlessˆ φ is analytic along the contour of integration, we will not be able to do so. The resurgentfunctions are a particular class of formal series for which the singularities in the ζ -plane (alsocalled Borel plane ) will satisfy certain conditions. Just by studying the behaviour of suchfunctions close to their singular points, we will be able to constrain, via the Alien calculusdiscussed in Section 6, the entire structure of the function on the whole Borel plane.
As we have already mentioned, only in very few cases the Borel transform ˆ φ = B [ ˜ φ ] will nothave any singularity along the line of integration, and even in rarer occasions there will beno singularity at all (and actually in this situation there is no need for all this machinery).So, generically, we will be expecting at least some singularity in ˆ φ . The number and typeof such singularities is encoded in the following definitions. First of all we will need to beable to integrate along some path from the origin to infinity, so we cannot have “too many”singularities. Definition . We will say that a germ of analytic functions at the origin ˆ φ ∈ C { ζ } is endlesslycontinuable on C if for all R >
0, there exists a finite set Γ R ( ˆ φ ) ⊂ C of accessible singularities,such that ˆ φ can be analytically continued along all paths γ whose length is less than R ,avoiding the singularities Γ R ( ˆ φ ).Ecalle’s definition is more general than the one just presented, but for the present workthis definition will suffice. Being endlessly continuable means that even if the Borel transformof our formal power series will present possibly infinitely many singularities in the Borel plane,nonetheless it will be possible to consider a suitable deformed path γ , issuing from the originand going to infinity in any direction θ . Endless continuability roughly means that there areno natural boundaries in the, possibly infinitely sheeted, Riemann surface where ˆ φ is defined.We have to assume as well some hypothesis on the type of singularities that ˆ φ can have.– 10 – efinition . A holomorphic function ˆ φ in an open disk D ⊂ C is said to have a simplesingularity at ω , adherent to D , if there exist α ∈ C and two germs of analytic functions atthe origin ˆΦ( ζ ) , reg ( ζ ) ∈ C { ζ } , such thatˆ φ ( ζ ) = α πi ( ζ − ω ) + 12 πi ˆΦ( ζ − ω ) log( ζ − ω ) + reg ( ζ − ω ) , (3.1)for all ζ ∈ D close enough to ω , where reg stands for a regular term close to ω . The constant α is called the residuum and ˆΦ the minor .The holomorphic function ˆΦ associated with the logarithmic singularity can be obtainedby considering ˆΦ( ζ ) = ˆ φ ( ζ + ω ) − ˆ φ ( ζ e − πi + ω ) , (3.2)where with ˆ φ ( ζe − πi + ω ) we mean following the analytic continuation of ˆ φ along the circularpath ζe − πi t + ω with t ∈ [0 , Example . Consider in example the following convolution product1 ∗ ˆ φ ( ζ ) = (cid:90) ζ dζ ˆ φ ( ζ ) , (3.3)with ˆ φ a meromorphic function with poles in Γ ⊂ C ∗ . Then clearly 1 ∗ ˆ φ admits an analyticcontinuation along any path issuing from the origin whose support is not intersecting Γ. Thismeans that 1 ∗ ˆ φ is actually an holomorphic function defined on the universal covering of C \ Γwith only logarithmic singularities located precisely at the poles of ˆ φ . Example . It is easy to see that the convolution of two functions with simple singularitiesgenerates new singular points. We can takeˆ φ ( ζ ) = 1 ζ − ω , (3.4)ˆ ψ ( ζ ) = 1 ζ − ω , (3.5)ˆ φ ∗ ˆ ψ ( ζ ) = 1 ζ − ( ω + ω ) (cid:18)(cid:90) ζ dζ ζ − ω + (cid:90) ζ dζ ζ − ω (cid:19) . (3.6)The product ˆ φ ∗ ˆ ψ has logarithmic singularities at ω , ω and a pole at ω + ω (note that thispole is not on the first sheet), thus we can extend ˆ φ ∗ ˆ ψ to a meromorphic function on theuniversal covering of C \ { ω , ω } with a simple pole at ω = ω + ω whose residue dependson the particular sheet considered. I.e. ω belongs to the closure of D . – 11 –s it will become clearer later in our discussion, for each problem that we wish to solvethrough resurgence, we have to find an infinite discrete subset of points Γ ⊂ C (usually alattice), corresponding to the singular points of all the germs of analytic functions in playfor our particular problem (i.e. Γ = 2 πi Z \ { } for certain difference equations , see [30, 31],while for various 2d QFTs [24, 32] Γ = S Z , with S > ⊂ C , there is a natural Riemann surfaceassociated with the universal covering of C \ Γ. Definition . The Riemann surface R is the set of homotopy classes of paths with fixedextremities, starting from the origin and whose support is contained in C \ Γ. The coveringmap π is a mapping from R back to C \ Γ given by π : R → C \ Γ , (3.7) π [ c ] = γ (1) ∈ C \ Γ , (3.8)where γ ( t ) is a particular representative of the equivalence class c ∈ R , and γ (1) correspondto its end point. By pulling back with π the complex structure of C \ Γ we get R as a Riemannsurface. The origin of R is the unique point corresponding to π − [0], which corresponds tothe homotopy class of the constant path.Without dwelling on too many technical details we can introduce holomorphic functionson R by simply taking a germ of holomorphic functions at the origin which admits an holo-morphic continuation along any path whose support avoids Γ. Since we are dealing only withendlessly continuable functions, this Γ cannot be too “dense”. The key point is that theconvolution of germs induces a commutative and associative law on the space of holomorphicfunctions of R (see the beautiful works of Sauzin for more details [30, 31]). Definition . The space of all holomorphic functions on R endowed with the convolutionproduct is an algebra called convolutive model of the algebra of resurgent functions and usuallydenoted by (cid:98) H ( R ). Considering the inverse Borel transform of these functions we get˜ H = B − (cid:16) (cid:98) H ( R ) (cid:17) (3.9)which is usually called multiplicative model of the algebra of resurgent functions .There is no unity for the convolution product ∗ within (cid:98) H ( R ), for this reason we have tointroduce a new symbol δ and extend the algebra in the following straightforward manner˜ φ ( z ) = C + ∞ (cid:88) n =0 c n z − n − ∈ C [[ z − ]] , (3.10)ˆ φ ( ζ ) = B [ ˜ φ ]( ζ ) = C δ + ∞ (cid:88) n =0 c n n ! ζ n ∈ δ C ⊕ C [[ ζ ]] . (3.11)The convolution product is extended from C [[ ζ ]] to δ C ⊕ C [[ ζ ]] by simply treating δ as a unity δ ∗ ˆ φ = ˆ φ . – 12 –t is possible to show that this algebra behaves nicely under composition as well. Moredetails and proofs for all these statements can be found in the original works by Ecalle [14](see also the more recent [30, 31]).Within the whole algebra (cid:98) H ( R ) we can focus on resurgent functions ˆ φ with a simplesingularity at ω , which meansˆ φ ( ζ ) = α πi ( ζ − ω ) + 12 πi ˆΦ( ζ − ω ) log( ζ − ω ) + reg ( ζ − ω ) , (3.12)where ˆΦ and reg are convergent series close to the origin. We can thus define the operatorSing ω ˆ φ = α δ + ˆΦ ∈ δ C ⊕ C { ζ } . (3.13)Note that a change in the determination of the logarithm gives rise only to a change in theregular part, reg , and not on α or ˆΦ. Definition . A simple resurgent function ˆ ψ is such that ˆ ψ = c δ + ˆ φ ( ζ ) ∈ δ C ⊕ (cid:98) H ( R ) andfor all ω ∈ Γ, i.e. all the accessible singularities, and all the paths γ ( t ), originating from 0,avoiding Γ and whose extremity lies in a disk D close enough to ω (where close enough meansthat ω is the only singularity contained in D , i.e. { ω } = Γ ∩ D ), the determination cont γ ˆ ψ has a simple singularity at ω . Where cont γ ˆ ψ is the determination of ˆ ψ obtained by analyticcontinuation of ˆ ψ along γ . This continuation is clearly analytic at least in any open diskcontaining the extremity of the path γ and avoiding all the singular points in Γ (rememberthat endlessly continuable functions cannot have too dense singular points).This following proposition summarises all the concepts introduced so far in this Section. Proposition 2.
The subspace of all simple resurgent functions, which usually is denoted by (cid:100)
RES simp , is a subalgebra of the convolution algebra δ C ⊕ (cid:98) H ( R ).In the multiplicative model the conjugate through Borel transform of (cid:100) RES simp will bedenoted by (cid:103)
RES simp . The main points to remember about simple resurgent functions arebriefly summarised in what follows: • The singular points are not too dense. We want to be able to integrate from 0 to infinityalong some complex direction by suitably dodging few singular points; • The singular behaviour close to these singular points is captured by yet another simpleresurgent function; • These functions behave nicely under composition, convolution and Borel transform [30,31], i.e. they form a sub-algebra of the convolutive model.– 13 – igure 2 . Lateral Borel summation along the direction θ . In the previous Section we introduced many formal concepts and properties of resurgentfunctions, but we still do not know how to define a suitable resummation procedure whenthe direction θ , along which we compute the Laplace integral L θ , contains singular points.Since we cannot directly integrate along a singular direction, we have to “dodge” the singularpoints. Definition . The lateral Borel summations for ˜ ψ = c + ˜ φ ( ζ ) ∈ C ⊕ ˜ H along the direction θ are given by S θ + ˜ ψ ( z ) = c + (cid:90) e i θ ( ∞ + i (cid:15) )0 dζ e − z ζ ˆ φ ( ζ ) , (4.1) S θ − ˜ ψ ( z ) = c + (cid:90) e i θ ( ∞− i (cid:15) )0 dζ e − z ζ ˆ φ ( ζ ) , (4.2)as schematically depicted in Fig.2: we deform slightly the contour of integration to pass eitherabove ( S θ + ) or belove ( S θ − ) all singular points along the direction θ . Example . A slight modification to Euler’s equation ψ (cid:48) ( z ) + ψ ( z ) = 1 z , (4.3)– 14 –ields to the following formal power series solution ˜ φ ( z ) = (cid:80) ∞ n =0 n ! z − n − . Note the absenceof the alternating factor ( − n present instead in the formal solution to eq. (2.18). The Boreltransform is straightforward to obtain ˆ φ ( ζ ) = 11 − ζ . (4.4)We cannot apply directly the Laplace transform along the direction θ = 0 since we wouldencounter a singularity at ζ = 1 (the direction θ = π for Euler’s equation (2.18) is exactlyconjugated to the direction θ = 0 for the current ODE). The later Borel summations differfrom each others S + ˜ φ ( z ) = (cid:90) ∞ + i (cid:15) dζ e − z ζ − ζ , (4.5) S − ˜ φ ( z ) = (cid:90) ∞− i (cid:15) dζ e − z ζ − ζ , (4.6)and ( S + − S − ) ˜ φ ( z ) = 2 πi e − z . (4.7)Note that the difference between the two lateral summations is non-analytic for z ∼ ∞ and cannot be possibly captured by our formal asymptotic expansion in powers of 1 /z (seeSection 5). Secondly the difference e − z is precisely a solution to the homogenous problem ψ (cid:48) ( z ) + ψ ( z ) = 0.When the direction θ is a singular direction, the Borel summation jumps as we cross this Stokes line , and the full discontinuity across this direction plays a crucial role.
Definition . Consider the lateral Borel summations S θ ± , we define the Stokes automorphism S θ , from (cid:103) RES simp into itself, as S θ + = S θ − ◦ S θ = S θ − ◦ (Id − Disc θ ) , (4.8) S θ + − S θ − = −S θ − ◦ Disc θ . (4.9)Where Disc θ encodes the full discontinuity across θ .If S θ + ( ˜ φ ) = S θ − ( ˜ φ ) we easily obtain S θ ˜ φ = ˜ φ , (4.10)and ˜ φ is called a resurgence constant . This means that the Borel transform of ˜ φ has no singu-larities along the θ direction and is given by a convergent power series. In this case we havealready seen that the Laplace integral of the Borel transform of ˜ φ gives us an unambiguousresummation procedure for the original formal power series.The Stokes automorphism is telling us how the resummed series jumps across a Stokesline, as we will see in more details later the reason for this jump is that our formal power series˜ φ ( z ) = c + c /z + c /z + ... is actually incomplete, we missed non-analytic (non-perturbative)– 15 – igure 3 . The difference between left and right resummation along the singular direction θ as a sumover Hankel contours. terms of the form e − z . These terms are of course exponentially suppressed for z ∼ ∞ butacross a Stokes line, precisely the terms that we have forgotten, become relevant and have tobe taken into account.It is easy to see, by a simply contour deformation, that the difference between the θ + and θ − deformation is nothing but a sum over Hankel’s contours, and the discontinuity of S across θ is given as an infinite sum of contribution coming from each one of the singularpoints, see Figure 3. Definition . The logarithm of the Stokes automorphism defines the
Alien derivative ∆ ω by S θ = exp (cid:88) ω ∈ Γ θ e − ω z ∆ ω , (4.11)where we denoted with Γ θ the set of singular points of the Borel transform along the θ direction.Using the above definition we can rewrite equation (4.8) as S θ + ˜ φ ( z ) = S θ − ˜ φ ( z ) + ∞ (cid:88) k =1 (cid:88) { n ,...n k ≥ } e − ( ω n + ... + ω nk ) z k ! S θ − (cid:16) ∆ ω n ... ∆ ω nk ˜ φ ( z ) (cid:17) . (4.12)The Alien, etranger , derivative can be thought of as the logarithm of the Stokes auto-morphism, but our definition (4.11) is still pretty mysterious and unintelligible. Example . To understand better how this Alien derivative works we can start with the easiertask of understanding the Stokes automorphism when the Borel transform of our formal powerseries ˜ φ ( z ) ∈ (cid:103) RES simp takes the formˆ φ ( ζ ) = α πi ( ζ − ω ) + 12 πi ˆΦ( ζ − ω ) log( ζ − ω ) , (4.13)– 16 –ith arg( ω ) = θ and ˆ φ has no singularities along the direction θ . Note that we are assumingthe simple singularity form (3.1) along the whole direction θ , and not only close to the singularpoint ω . The difference between the two lateral resummations is clearly different from zeroand it gets a first contribution coming from the simple pole and a second one coming fromthe change in the determination of the logarithm. Having assumed that ˆΦ is entire along thedirection θ , after a trivial change of variables ζ → ζ − ω ,we get( S θ + − S θ − ) ˜ φ ( z ) = α e − ω z + e − ω z (cid:90) ∞ dζ e − z ζ ˆΦ( ζ ) . (4.14)Since ω is the only singular point, equation (4.12) simplifies drastically to( S θ + − S θ − ) ˜ φ ( z ) = e − ω z S θ − (cid:16) ∆ ω ˜ φ ( z ) (cid:17) , (4.15)from which we can read how the alien derivative act on a simple resurgent function of theform (4.13) ∆ ω ˜ φ ( z ) = α + ˜Φ( z ) , (4.16)where ˜Φ is the inverse Borel transform of ˆΦ, or equivalently in the convolutive model∆ ω ˆ φ ( ζ ) = α δ + ˆΦ( ζ ) . (4.17) Example . Let’s consider a more concrete example. Take the formal power series˜ φ ( z ) = az ω + a + b − c ( z ω ) + ∞ (cid:88) n =2 n !( z ω ) n +1 (cid:18) a + bn + cn ( n − (cid:19) , (4.18)with a, b, c, ω ∈ C external parameters. This series is clearly of Gevrey type 1 and it is notso difficult to compute its Borel transformˆ φ ( ζ ) = − aζ − ω − c ( ζ − ω ) + b ωω log(1 − ζ/ω ) . (4.19)The residuum at ω is α = − πi a while the minor ˆΦ( ζ ) = − πi ( c ζ + bω ) /ω . In this case itis particularly easy to find the inverse Borel transform of the minor ˜Φ( z ) = − πi ( b/ ( z ω ) + c/ ( z ω ) ), so that the Alien derivative at ω is∆ ω ˜ φ ( z ) = − πi (cid:18) a + b ( ω z ) + c ( ω z ) (cid:19) . (4.20)In the generic case the definition of the Alien derivative is a little bit more complicated,but the main idea are collected in the previous example. Along a singular direction the alienderivative at a singular point will receive contributions from all the singularities it encountersalong its way ∆ ω = (cid:88) n (cid:88) ω + ... + ω n = ω ( − n − n ∆ + ω ... ∆ + ω n , (4.21)– 17 – igure 4 . To obtain ∆ + ω , we have to consider the determination of ˆ φ along the path γ ω , issuing fromthe origin and reaching ω by avoiding all the singularities from the right. where ω i are singular points, ordered along the direction θ and the operator ∆ + ω from (cid:100) RES simp into itself, is defined by ∆ + ω ˆ φ ( ζ ) = α γ ω δ + ˆΦ γ ω ( ζ ) , (4.22)where α γ ω ∈ C and ˆΦ γ ω ∈ (cid:100) RES simp are respectively the residuum and the minor of ˆ φ at thesimple singularity ω , as in equation (3.1), defined following the determination of ˆ φ along thepath γ ω , issuing from the origin in the direction θ = arg( ω ) and arriving in ω by circumventingall the intermediate singularities to the right, see Figure 4. For an equivalent definitionwithout having to introduce ∆ + ω see [30, 31].Thanks to the Borel transform, we can keep on moving from the convolutive model tothe multiplicative formal model, so we can understand ∆ ω as acting on both (cid:100) RES simp and (cid:103)
RES simp . As the name suggest the Alien derivative is indeed a derivative∆ ω (cid:16) ˆ φ ∗ ˆ φ (cid:17) = ∆ ω ˆ φ ∗ ˆ φ + ˆ φ ∗ ∆ ω ˆ φ , (4.23)∆ ω (cid:16) ˜ φ · ˜ φ (cid:17) = ∆ ω ˜ φ · ˜ φ + ˜ φ · ∆ ω ˜ φ . (4.24)Note that however, ∆ + ω is generically not a derivation. The alien derivative does not commutewith the standard derivative but rather∆ ω ∂ z ˜ φ = ∂ z ∆ ω ˜ φ − ω ∆ ω ˜ φ . (4.25)– 18 –e can define the dotted alien derivative˙∆ ω = e − ω z ∆ ω , (4.26)which, thanks to the previous equation, does now commute with the standard derivative (cid:104) ∂ z , ˙∆ ω (cid:105) = 0 . (4.27)Note that e − ω z has to be understood as a new symbol (see Section 5), external to the algebraof simple resurgent functions, this is usually called a simple resurgent symbol . It obeys theusual rules for multiplication and derivation with respect to z . These simple resurgent symbolscan be used to obtain the graded algebra (cid:103) RES simp [[ e − ω z ]], where ω ∈ Γ are all the singularpoints for the particular problem studied. The introduction of these symbols is telling us thatsomehow our formal power series expansion has to be extended to a more general expansion,which goes under the names of trans-series expansion, we refer to Section 5 for more details.For the time being we just need to know that e − ω z has to be understood as an externalsymbol to our algebra of simple resurgent functions, hence˙∆ ω (cid:16) e − ω z ˜ φ (cid:17) = e − ( ω + ω ) z ∆ ω ˜ φ . (4.28) Remark.
We have to stress that there is no operatorial relations between the various ∆ ω :they generate a free Lie algebra. They give a way to encode the entire singular behaviour ofa resurgent function ˆ φ : in fact, given a sequence ω , ..., ω N of singular points, the evaluationof ∆ ω ... ∆ ω N ˆ φ is obtained by many different determinations of ˆ φ at the singularity ω = ω + ... + ω N . Vice versa any possible determination and singularity of ˆ φ could be computedif we knew all these compositions of alien derivatives for all the sequences ω , ...ω N . Example . The full knowledge of a resurgent function is coming only when we know ALL itsalien derivatives. In particular it is not sufficient to know that ∆ ω ˆ φ = 0 to deduce that ˜ φ hasno singularities at ω for all its determinations. In fact, let’s analyse the previous Example 3ˆ φ ( ζ ) = 1 ζ − ω , ˆ ψ ( ζ ) = 1 ζ − ω , ˆ φ ∗ ˆ ψ ( ζ ) = 1 ζ − ( ω + ω ) (cid:18)(cid:90) ζ dζ ζ − ω + (cid:90) ζ dζ ζ − ω (cid:19) . We can easily compute the alien derivatives∆ ω ˆ φ = 2 πi δ , ∆ ω ˆ φ = 0 ∀ ω (cid:54) = ω , (4.29)∆ ω ˆ ψ = 2 πi δ , ∆ ω ˆ ψ = 0 ∀ ω (cid:54) = ω , (4.30)so that ∆ ω + ω ( ˆ φ ∗ ˆ ψ ) = (∆ ω + ω ˆ φ ) ∗ ˆ ψ + ˆ φ ∗ (∆ ω + ω ˆ ψ ) = 0 , (4.31)– 19 –here we used the Jacobi identity (4.24). The vanishing of ∆ ω + ω ( ˆ φ ∗ ˆ ψ ) does not meanthat all the determinations of ˆ φ ∗ ˆ ψ have no singularity at ω + ω as it is manifest from theexplicit form for ˆ φ ∗ ˆ ψ . This fact is encoded in the composition of different alien derivatives∆ ω ( ˆ φ ∗ ˆ ψ ) = 2 πi ˆ ψ , ∆ ω ( ˆ φ ∗ ˆ ψ ) = 2 πi ˆ φ , (4.32)∆ ω ∆ ω ( ˆ φ ∗ ˆ ψ ) = ∆ ω ∆ ω ( ˆ φ ∗ ˆ ψ ) = − π δ . (4.33)It is clear that if we want to find a suitable resummation procedure for a particular formalpower series of interest these objects will play a crucial role, but before being able to applythis machinery we need to understand the following questions: • What kind of generalisation to formal power series in 1 /z do we expect? Trans-SeriesSection 5; • How do we compute in practice ∆ ω ? Bridge equations Section 6; • And finally how do we find the physical resummation procedure ? Median resummationand Stokes phenomenon Section 7.
In this intermezzo we will introduce some basic concepts regarding the trans-series expansion.For a more complete overview of the subject we refer to [33].
Definition . A Log-free trans-monomial is a symbol of the form g = z a e T (5.1)with a ∈ R and T is a purely large log-free trans-series.These trans-monomial are the building blocks of trans-series and they come with an orderrelation denoted by (cid:29) given by the relation z a e T (cid:29) z a e T (5.2)if either T > T (where the symbol > for trans-series will be defined shortly) or if T = T and a > a as real numbers. In example e e z (cid:29) e z (cid:29) z − e z (cid:29) z . Definition . A Log-free trans-series is a formal sum of symbols T = (cid:88) j c j g j (5.3)where the coefficients c j ∈ R and the g j are Log-free trans-monomial.– 20 –he height of a trans-monomial z a e T is defined as the number of times we compose theformal exponential symbol, i.e. z e e z + z has height 2. Usually only finite height trans-monomialand trans-series are considered.We just defined the trans-monomial using the notion of a trans-series and defined thetrans-series starting from trans-monomials, in an Ouroboros manner. It is possible (see[33]) to give a more precise definition of trans-monomials and trans-series in terms of Hahnseries defined on the ordered abelian group of monomials but we will not need this level ofsophistication.We will say that the trans-series T is purely large if g j (cid:29) g j in T . Similarly a trans-series will be small if g j (cid:28) j . Alternatively we can calllarge term inifinite and small term infinitesimal since we are implicitly assuming the limit z → + ∞ . A non-zero trans-series T = (cid:80) j c j g j has a leading term (also called dominance ) dom ( T ) = c g with the leading monomial (also called magnitude of T ) mag ( T ) = g (cid:29) g j for all the other terms present in T .If the coefficient c of the dominant term is positive, we say that the trans-series is positiveand write T >
0. In this way we can define an order relation between trans-series defined by
T > S iff T − S >
0. Similarly if the mag ( T ) (cid:29) mag ( S ) we will write T (cid:29) S , while if theyhave the same behaviour for z → ∞ , meaning that dom ( T ) = dom ( S ), we will say that T is asymptotic to S , T ∼ S . Note that only the zero trans-series can be asymptotic to 0.Trans-series inherit almost all standard properties of usual power series treated as formalsums. In example differentiation of a trans-series is defined by the standard differentiation oftrans-monomial g (cid:48) = (cid:0) z a e T (cid:1) (cid:48) = a z a − e T + z a T (cid:48) e T , (5.4) T (cid:48) = (cid:88) j c j g j (cid:48) = (cid:88) j c j g (cid:48) j . (5.5)A general trans-series is obtained by replacing for some z inside a log-free trans-seriesthe symbol log m z , with the identificationlog m z = log ◦ ... ◦ log z (5.6)where we composed the logarithm m ∈ N times. The integer m is called depth of the trans-series.Finite depth trans-series arise naturally when considering instanton contributions to phys-ical observables. The instanton action plus perturbative corrections on top of that usuallygive rise to an height 1 log-free trans-series, while the integration over the quasi-zero modeslead to the appearance of logarithmic corrections. Hence in a generic theory with only onetype of non-perturbative saddle points, a physical observable will take the form [34, 35] E ( z ) = ∞ (cid:88) n =0 n − (cid:88) k =0 (cid:0) z α e − S z (cid:1) n (cid:104) (log z ) k E + n,k ( z ) + (log( − z )) k E − n,k ( z ) (cid:105) , (5.7)– 21 –ith E ± n,k ( z ) an height 0, log-free trans-series, a.k.a. the asymptotic perturbative expansionaround the n -instantons sector E ± n,k ( z ) = ∞ (cid:88) p =0 c ± n,k,p z − p − . (5.8)Two comments are in order. Firstly we notice that the logarithms start appearing only atlevel n = 2, what we would call two instantons sector. The reason is that in quantum mechan-ics [36–38] and quantum field theories [24, 39], the log sector is coming from the integrationover quasi-zero modes. Quasi-zero modes are not exact zero modes, but nonetheless they areparametrically suppressed compared to genuine gaussian modes. The n − n different instantons are not exact zero modes because of intanton/(anti)instantoniteractions, when we integrate over these separations we will generate precisely between 0 and n − k in (5.7).Secondly, it is striking that generic physical observables can be described by very easytrans-series, without having to use logarithms with depth bigger than one, i.e. log(log( z )), ormore involved exponential terms, i.e. e e /z . A possible explanation might be traced back tothe path integral formulation of the theory. In the last Section we will see in a concrete, finitedimensional, example that the semiclassical decomposition of the functional integral as a sumover steepest descent contours would give rise to precisely only height 1, depth 1 trans-series.Unfortunately a full fledged path integral derivation of this result is still missing.Hyperasymptotic expansions are extremely useful also in the context of linear and non-linear ODEs, where the connection with resurgence is well established, see [40] for the wellknown example of the Airy function. In many cases we do not know an explicit solution to agiven problem and we are forced to exploit asymptotic methods to get a feeling of how theactual solution might behave. In various interesting cases a simple power series expansionis not good enough and one has to use a trans-series expansion. We will not discuss indetails the hyperasymptotic expansion for ODEs, so we refer to the literature [41] for a moredetailed exposition of this interesting subject. It is nonetheless instructive to analyse througha concrete example, based on [42], how to implement this machinery of trans-series expansionin ODEs. Example . Let’s study the non-linear ODE y (cid:48) ( x ) = cos ( πx y ( x )) . (5.9)While for many interesting physical problems [42], a complete knowledge of the space ofsolutions is required, no explicit solution is actually known.If we look for a solution going to 0 for x (cid:29)
1, we can assume that it has an asymptoticexpansion of the form y ( x ) ∼ a x + a x + O ( x − ) . (5.10)By plugging this ansatz in (5.9), we see that y (cid:48) ( x ) vanishes for large x if a = n + 1 / n ∈ Z . After we fix the coefficient a all the remaining coefficients are uniquely fixed in terms– 22 – (cid:72) x (cid:76) Figure 5 . Numerical solutions with asymptotic behaviours of the form 1 / (2 x ) (blue), 5 / (2 x ) (red),9 / (2 x ) (green). of n , i.e. a = 0 , a = ( − n ( n + 1 / /π and so on. Something strange is going on here,we know that the solutions to this first order ODE should come with an arbitrary constant,i.e. y (0), while instead it looks like we have a discrete set of solutions which asymptoticallybehave as y ( x ) ∼ / (2 x ) , / (2 x ) , / (2 x ) , ... , how does the initial condition y (0) enter ourasymptotic expansion?We can solve numerically (5.9), and as we vary the initial condition y (0) = y the differentsolutions fall into disjoint classes, see Figure 5, with discrete asymptotic behaviour of theform y ( x ) ∼ ( n + 1 / /x + ... and n even. From our previous perturbative expansion we wereexpecting solutions with asymptotic form y ( x ) ∼ ( n + 1 / /x + ... for all integers n , but fromour numerics we find only behaviours like 1 / (2 x ) , / (2 x ) , / (2 x ), so what happened to thesolutions with n odd?It turns out that in order to find the solutions with asymptotic form y ( x ) ∼ ( n +1 / /x + ... and n odd, one has to give a precise (and unique) initial condition y (0) = y ( n )0 , all the solutionswith initial data slightly off from this particular value, will fall either into the set with n + 1or n − y ( x ) and y ( x ) are both solu-tions to (5.9) with the exact same a = ( n + 1 / n ∈ Z . Since all the higher ordersterms a i are uniquely fixed once we fix n , tha asymptotic forms for y and y are preciselythe same, which in particular means that y − y is asymptotically smaller than any power of1 /x ! Let’s define u ( x ) = y ( x ) − y ( x ), we know that its asymptotic form for large x cannotbe of the form b /x + b /x + ... , but we also know that y and y differ from each other justbecause y (0) (cid:54) = y (0), so u cannot vanish identically. From (5.9), we can deduce the ODE– 23 –atisfied by u ( x ): u (cid:48) ( x ) = y (cid:48) ( x ) − y (cid:48) ( x ) = − (cid:18) πx ( y ( x ) + y ( x ))2 (cid:19) sin (cid:18) πx u ( x )2 (cid:19) , (5.11)where we used some trigonometric identities to rewrite the difference of two cosines as productof sines. We can use at this point the fact that y ( x ) + y ( x ) ∼ n + 1 / /x , valid for large x , and obtain u (cid:48) ( x ) ∼ ( − n +1 (cid:18) πx u ( x )2 (cid:19) ∼ ( − n +1 πx u ( x ) . (5.12)At this point it is straightforward to obtain the asymptotic form for u ( x ) u ( x ) = y ( x ) − y ( x ) ∼ Const. e − ( − n π x / . (5.13)This equation answers all our previous questions: firstly, the arbitrary constant that we weremissing in the asymptotic expansion (5.10) was actually hiding in the hyperasymptotic part,and secondly we see that for n even the difference between two solutions with different initialvalue, but in the same asymptotic class, is exponentially suppressed, while for n odd theydeviate from one another exponentially fast. Clearly, for n odd, we are not expecting thesolution u ( x ) ∼ e + πx to be valid since we have assumed u small when we expanded ourODE. All we know is that two such putative solutions y , y , with different initial conditionsbut belonging to the same asymptotic class with n odd, will have to deviate from one another.This means that only for a precise initial value y ( n )0 , that can be computed numerically, wefind the unique separatrix solution.The use of a simple trans-series expansion allows us to get a complete understanding ofthe space of solutions for linear and non-linear ODEs, even (and especially) when analyticsolutions are not known!After this brief introduction to trans-series, we can go back to the main story of thiswork. We will now see how to include more generic trans-series in the context of resurgenceand why including such terms is actually the crucial step in obtaining well defined physicalobservables. As we have anticipated in Section 4, the perturbative power series expansion to our favouritephysics or maths problem, is usually insufficient to recover the correct solution. Just bystudying the analytic properties of the Borel transform of the perturbative series, we under-stand that resurgent symbols, i.e. non-perturbative contributions of the form e − S z , have tobe included to obtain a consistent formal solution. This non-analytic, non-perturbative termswill be accompanied by a standard perturbative expansion on top of them, and in many caseswe will also have to include logarithmic corrections (due to resonance in the case of Painlev´e– 24 –DEs [21, 43, 44] or integration over quasi-zero modes for multi-instantons solutions [18]).A general solution to our problem will eventually take the form of a sum of trans-monomial z α log m z e S ( z ) ˜ φ ( z ) (6.1)with S ( z ) possibly a trans-series itself and ˜ φ ( z ) a simple resurgent function.Given the particular linear or non-linear problem to solve, the first step is understandingthe type of trans-monomial that we have to use to obtain the complete solution. For the toymodel we will focus on, our ansatz solution will only contain height-1, log-free trans-monomialof the form e − S n z ˜ φ n ( z ) , (6.2)where S ∈ R + will be our instanton action and ˜ φ n will be the perturbative expansion aroundthe n -instantons solution. This one parameter trans-series mimic a toy model in which wehave only one type of non-perturbative configurations with real and positive action S , this isusually the case when the problem at hand depends on just one single boundary condition. Itis possible to obtain theories where one has multiple instantonic configurations with differentactions S a , S b , ... [21], and even complex valued action for the so called ghost-instantons[45] or more generically non-topological saddle points [32]. For a more complete treatment ofmulti-parameter trans-series with the inclusion of logarithmic sectors we refer to the thoroughworks of Aniceto, Schiappa and Vonk [21, 46].We will give more explicit examples later on, but for the moment we assume that ourperturbative formal solution give rise to a resurgent function ˜ φ ( z ), with singularities locatedat S Z ∗ , for some instanton action S ∈ R . This means that the Stokes automorphisms S and S π will act non-trivially on ˜ φ . We are expecting some non-perturbative effect to modifyour simple formal series ansatz turning it into the trans-series formΦ( z ) = ∞ (cid:88) n =0 (cid:0) e − S z (cid:1) n ˜ φ n ( z ) , (6.3)where we can interpret the various simple resurgent functions ˜ φ n as the perturbative con-tributions on top of the n -instanton configuration. It is useful to introduce an additionalcomplex parameter σ to keep track of the resurgent symbols e − S z , we will discuss then theone-parameter trans-series Φ( z, σ ) = ∞ (cid:88) n =0 σ n (cid:0) e − S z (cid:1) n ˜ φ n ( z ) . (6.4) Example . For concreteness let’s study a Riccati ODE ∂φ∂z − a φ + 1 z φ = − z . (6.5)Without the non-linear term φ this equation is a simple generalisation of Euler’s equation(2.18) ∂ψ∂z − a ψ = − z , (6.6)– 25 –hose solution has the formal asymptotic form ψ = (cid:80) n ≥ ( − n n ! / ( a z ) n +1 and its Boreltransform takes the nice form B [ ψ ] = 1 / ( a + ζ ), which is clearly a simple resurgent functionwith just one singularity at ζ = − a . The non linearity makes things more interesting: thesolution ψ gets modified φ = ∞ (cid:88) n =0 ( − n n !( a z ) n +1 c n ( a ) = 1 a z − a z ) + 2( a z ) − a z ) (cid:16) − a (cid:17) + ... , (6.7)where the coefficient c n ( a ) are polynomials of degree (cid:98) n/ (cid:99) in a , defined by the followingrecursion relation c = c = c = 1 ,c n = c n − − a n − (cid:88) l =0 ( n − l − l ! n ! c n − l − c l , n ≥ . (6.8)It is possible to prove [17] that φ is of Gevrey-1 type and that its Borel transform B [ φ ]has simple singularities for {− a, − a, − a, ... } , hence, for this particular problem, the Rie-mann surface R , defined in Section 3, is simply given by the universal covering of C \{− a, − a, − a, .. } , and B [ φ ] defines a simple resurgent function in (cid:100) RES simp ⊂ δ C ⊕ (cid:98) H ( R ).From the perturbative solution φ , we can build a one parameter family, σ ∈ C , ofsolutions Φ( z, σ ; a ) = ∞ (cid:88) n =0 σ n ( e a z ) n φ n ( z ) , (6.9)where Φ( z, σ ; a ) is an height one, log-free trans-series. Each φ n is computed by identifyingterm by term the different powers of σ when substituing in (6.5) ∂φ ∂z + 2 z φ φ = 0 , (6.10) ∂φ ∂z + a φ + 2 φ φ z = − φ z . (6.11)Generically substituting a trans-series ansatz into a non-linear problem will give us a non-linear equation for φ , that we will have to solve perturbatively, as we just did. The equationfor φ will then be linear and homogeneous, while the one for the higher terms φ n ≥ will belinear but inhomogeneous. It turns out that for the normalization choice φ = 1 + O ( z − )there exists a unique formal solution to our Riccati equation (6.5) of the trans-series form (6.9)where each resurgent symbol e n a z φ n ( z ) gives us a simple resurgent function B [ φ n ] defined onthe universal covering of C \ { ( n − a, ( n − a, ..., , − a, − a, ... } . The Riemann surface R of Section 3 for this problem is simply C \ a Z .So let’s go back to our general discussion and assume that we have constructed ourheight one, log-free trans-series ansatz (6.4) for the particular non-linear problem to solve.As already stated in Section 4, the Stokes automorphisms along some singular direction (in– 26 –his case either θ = 0 or θ = π since we assumed S ∈ R ) is entirely captured by the Alienderivatives along that particular direction. The problem is: we do not now how to computefor a generic trans-series (6.4) its alien derivative, say at the singular point ω = k S , for some k ∈ Z ∗ . This is where we need to relate the alien derivative to the standard derivative throughsome bridge equation , which builds a bridge between alien calculus and standard differentialcalculus.Suppose that Φ( z, σ ) is the solution to some non-linear problem (i.e. finite differenceequations for matrix models, Painlev´e ODE for minimal strings or WKB for energy eigenval-ues) in the variable z , then we know that (cid:104) ∂ z , ˙∆ k S (cid:105) = 0 , (6.12)which means that ˙∆ k S Φ( z, σ ) solves a linear homogeneous differential equation in z . Sim-ilarly, since [ ∂ z , ∂ σ ] = 0, ∂ σ Φ( z, σ ) solves exactly the same linear homogeneous problem(modulo some caveat on the initial data). Since ˙∆ k S Φ( z, σ ) and ∂ σ Φ( z, σ ) are solutions tothe same linear homogeneous ODE, say of order 1 for example, they must be proportional toeach others ˙∆ k S Φ( z, σ ) = A k ( σ ) ∂ σ Φ( z, σ ) . (6.13)This is called Ecalle’s Bridge equation , it gives us a bridge to relate the Alien derivative tothe usual derivative in the trans-series parameter σ . Example . Let’s go back to Riccati ODE (6.5) and our trans-series formal solution Φ( z, σ ; a )(6.9). The trans-series Φ( z, σ ; a ) solves ∂ Φ ∂z − a Φ + 1 z Φ = − z , so let’s apply ˙∆ na , with n ∈ Z ∗ , on both sides ∂ z (cid:16) ˙∆ na Φ (cid:17) − a ˙∆ na Φ + 2 Φ z ˙∆ na Φ = 0 , (6.14)where we used that ˙∆ commutes with ∂ z together with the fact that ∆ acts as a derivation(i.e. Jacobi holds). Note that ˙∆ na /z = 0 and ˙∆ na /z = 0 since their Borel transform areentire function along the real line. If we apply now ∂ σ on both sides of the Riccati equationwe get ∂ z ( ∂ σ Φ) − a ∂ σ Φ + 2 Φ z ∂ σ Φ = 0 . (6.15)As anticipated ˙∆ na Φ and ∂ σ Φ are both solutions to the same homogeneous, order one equationin ∂ z , hence they must be proportional˙∆ na Φ( z, σ ; a ) = A n ( σ ; a ) ∂ σ Φ( z, σ ; a ) , (6.16)which is precisely Ecalle’s Bridge equation.– 27 –calle’s Bridge equation is the crucial missing piece of the puzzle, with this equation wecan relate the mysterious alien derivative to standard calculus. This equation tells us that, atall singular points, the alien derivative gives back the original asymptotic expansion, hencethe name resurgence . Let’s investigate further the Bridge equation (6.13). Focusing on theLHS we get ˙∆ kS Φ = ∞ (cid:88) n =0 σ n e − ( n + k ) S z ∆ kS ˜ φ n , (6.17)while on the RHS A k ( σ ) ∂ σ Φ( z, σ ) = ∞ (cid:88) n =0 A k ( σ ) n σ n − e − nS z ˜ φ n . (6.18)We have to match term by term on the two sides, with exactly the same power of σ m andthe same resurgent symbol e − n S z . Since ˙∆ kS Φ contains only positive powers of σ we canassume that A k ( σ ) = ∞ (cid:88) m =0 A k,m σ m , (6.19)for some complex numbers A k,m . Furthermore in Φ, each resurgent symbol e − n S z is ac-companied by precisely σ n and since ˙∆ kS introduces an additional e − kS z , it means that, torestore the degree between σ and e − S z , we must have A k ( σ ) = A k σ − k .By matching each term in (6.17) with the terms in (6.18), with exactly the same powerof σ and the same resurgent symbol e − n S z , we obtain the set of equations∆ kS ˜ φ n = 0 , k > , (6.20)∆ kS ˜ φ n = A k ( n + k ) ˜ φ n + k , k ≤ , (6.21)with the definition ˜ φ n = 0 for n <
0, and where the complex constants A k ∈ C are called holomorphic or analytic invariants of the problem. In principle we would know all the alienderivatives if we knew all the A , A , A − , ... , needless to say the various A k are really hardto compute. Remark:
Note that the vanishing of ∆ kS ˜ φ n for k > kS is aregular point! As we have already seen before in (4.33) the singular behaviour is knownonce we know all the multiple alien derivatives, in example we have ∆ S ˜ φ = 0 while∆ S ∆ S ˜ φ = 2 A ˜ φ . The singular behaviour close to 2 S of what we would call the pertur-bative series ˜ φ is entirely captured (and vice versa) by the perturbative expansion aroundthe 2-instantons contribution. Once again the perturbative series surges up, or resurges, fromthe non-perturbative physics, furthermore, since ∆ S ˜ φ = 0 while ∆ S ˜ φ (cid:54) = 0, we know thatthis new singular point is not associated with a new non-perturbative object with action 2 S ,but rather it arises from a multi-instanton saddle.From the definition of Alien derivative (4.20), we see that the Bridge equations (6.20)-(6.21) tell us that, close to the singular point kS , the singular behaviour of the simple– 28 –esurgent function ˜ φ n is entirely governed by ˜ φ n + k since (passing to the convolutive modelnow) B [ ˜ φ n ]( ζ + k S ) ∼ A k ( n + k ) B [ ˜ φ n + k ]( ζ ) log ζ/ πi . (6.22)We were a little bit too sketchy here, as we know, the precise definition of Alien derivative(4.21) is more complicated than that, but the main point still remains: the singular part of˜ φ n at kS is entirely captured by ˜ φ n + k .The Bridge equations (6.20)-(6.21) not only allow us to reconstruct the entire behaviourof our trans-series close to a singular point but they also make manifest the appearance ofthe Stokes phenomena along the singular lines θ = 0 and π . To see that, let’s go back to theexpression for the Stokes automorphism in term of Alien derivative (4.11) and specialise it tothe singular direction θ = 0 S = exp (cid:32) ∞ (cid:88) k =1 e − k S z ∆ kS (cid:33) . (6.23)Given our trans-series ansatz and the Bridge equations (6.20)-(6.21), we already know that∆ kS ˜ φ n = 0 for all n as soon as k >
1, for this reason the above equation simplifies drasticallyto S = 1 + e − S z ∆ S + 12 e − S z ∆ S + ... . (6.24)It is easy to compute multiple alien derivatives just by iterating∆ S ˜ φ n = A ( n + 1) ˜ φ n +1 , (6.25)so that ∆ kS ˜ φ n = A k ( n ) k ˜ φ n + k , (6.26)where we used the Pochhammer symbol ( n ) k = (cid:81) ki =1 ( n + i ). We have now all the ingredientsto compute the Stokes automorphism along the positive real line S ˜ φ n = ∞ (cid:88) k =0 k ! ∆ kS ˜ φ n = ∞ (cid:88) k =0 (cid:18) n + kn (cid:19) A k e − k S z ˜ φ n + k . (6.27)We can use the definition (4.8) of the Stokes automorphism to relate the two sectorialsums above, S + , and below, S − , the positive real axis S + Φ( z, σ ) = S − ◦ S Φ( z, σ )= S − (cid:0) e − S z ∆ S + ... (cid:1) (cid:32) ∞ (cid:88) n =0 σ n e − nS z ˜ φ n (cid:33) = S − (cid:34) ∞ (cid:88) n =0 σ n e − nS z (cid:32) ∞ (cid:88) k =0 (cid:18) n + kn (cid:19) A k e − k S z ˜ φ n + k (cid:33)(cid:35) . (6.28) For the direction θ = π the situation is a little bit more involved but as we will show later on the endresults will be the same – 29 –ote that even if the Stokes automorphism, when applied to ˜ φ n , generates an infinite sum(6.27), nonetheless each resurgent symbols e − mS z in S Φ receives contributions only froma finite number of terms, precisely from n, k ∈ N such that n + k = m . We can thus changevariables in the sum from n, k ∈ N to m = n + k ∈ N and p ∈ { , , ..., m } and arrive at S + Φ( z, σ ) = S − ∞ (cid:88) m =0 e − mS z ˜ φ m m (cid:88) p =0 (cid:18) mp (cid:19) σ m − p A p (6.29)= S − (cid:34) ∞ (cid:88) m =0 e − mS z ˜ φ m ( σ + A ) m (cid:35) , (6.30)comparing this to our original expansion (6.4) we have finally found S + Φ( z, σ ) = S − Φ( z, σ + A ) . (6.31)We could have obtained the same result directly from the original Bridge equation written interms of Φ ˙∆ kS Φ( z, σ ) = A k σ − k ∂ Φ ∂σ , (6.32)valid for all k ≤ θ = 0 becomes S Φ( z, σ ) = exp (cid:16) ˙∆ − S (cid:17) Φ( z, σ ) = exp (cid:18) A ∂∂σ (cid:19) Φ( z, σ ) = Φ( z, σ + A ) . (6.33)The equation just obtain is a beautiful summary of all our alien calculus journey: along asingular direction, say the positive real line, the resummed series when θ = 0 + can be obtainedby the resummed series for θ = 0 − plus a jump in the trans-series parameter σ exactly equalto Ecalle’s holomorphic invariant A . The Stokes phenomenon is encoded perfectly in thetrans-series analysis of the Bridge equations, the only thing we are left to understand is howto define a non-ambiguous, unique (and possibly real, depending on the case) sum for ourtrans-series across a singular direction. This will be the aim of the next Section.For completeness, let’s analyse what happens to the trans-series ansatz and the Stokesautomorphism along the singular direction θ = π . The Bridge equations (6.20)-(6.21) tell usthat all the Alien derivatives ∆ − kS , with k = 1 , , .. , will act non-trivially in S π S π = exp (cid:32) ∞ (cid:88) k =1 e kS z ∆ − kS (cid:33) = 1 + e S z ∆ − S + e S z (cid:18) ∆ − S + 12 ∆ − S (cid:19) + ... . (6.34)We have to compute the action of multiple alien derivatives on each simple resurgent function˜ φ n since the contributions to each resurgent symbol e kS z in S π come from∆ − k S ... ∆ − k N S ˜ φ n (6.35)where the { k i } are all the possible integer partitions of k = k + ... + k N , with k i ≥
1. Noteas well that these are ordered partitions since the Alien derivatives at different points do notcommute [∆ − k S , ∆ − k S ] ˜ φ n = A − k A − k ( k − k ) ( n − k − k ) ˜ φ n − k − k . (6.36)– 30 –urthermore, from (6.20)-(6.21), we deduce that the infinite sum in S π ˜ φ n is actually a finitesum (contrary to the θ = 0 case) since as soon as we reach the level e − nS z we will have tocompute some Alien derivative of the form∆ − k S ... ∆ − k N S ˜ φ n , n = N (cid:88) i =1 k i , (6.37)which are all vanishing, together with each subsequent application of the alien derivativeoperator. The generic iteration of multiple derivatives gives us N (cid:89) i =1 ∆ − k ( N +1 − i ) S ˜ φ n = N (cid:89) i =1 A − k i · N (cid:89) i =1 n − i (cid:88) j =1 k j ˜ φ n − (cid:80) i k i , (6.38)which clearly vanishes as soon as (cid:80) Ni =1 k i ≥ n .It is possible to obtain an analytic expression for S π ˜ φ n but it is not particularly illumi-nating [21, 46], the important point to keep in mind is that along all the singular directions,the action of the Stokes automorphism on Φ( z, σ ) can be recast in term of a differential op-erator acting on the trans-series parameter σ , giving rise to the Stokes phenomenon. Fromequation (6.32) and the expression (6.39) for S π written in terms of alien derivatives we get S π Φ( z, σ ) = exp (cid:32) ∞ (cid:88) k =1 ˙∆ − kS (cid:33) Φ( z, σ ) = exp (cid:32) ∞ (cid:88) k =1 A − k σ k +1 ∂∂σ (cid:33) Φ( z, σ ) . (6.39)To get a feeling on how S π acts on Φ( z, σ ), we can assume for the moment, that allthe holomorphic invariants are vanishing except one, say A − k (cid:54) = 0. In this situation, theStokes automorphism will simply be S π = exp( A − k σ k +1 ∂/∂σ ), and its action on Φ( z, σ ) isa simple translation of an associated trans-series parameter σ − k → σ − k − k A − k . This meansthat in this particular case where A − k is the only non-zero holomorphic invariant, the Stokesphenomenon along θ = π takes the form S π Φ( z, σ ) = Φ( z, ( σ − k − kA − k ) − /k ) , (6.40)a generalisation of the θ = 0 case (6.33). Clearly when all the A − k are non vanishing theStokes automorphism will be much more complicated, and given by (6.39).Before concluding this Section, as a concrete example of what just discussed, we canstudy the case in which the trans-series contains only two terms, namely F ( z, σ , σ ) = σ F ( z ) + σ F ( z ) , (6.41)where the trans-monomials F l ( z ), with l = 0 ,
1, take the form F l ( z ) = e − M l z Φ l ( z ) = e − M l z ∞ (cid:88) n =0 a ( l ) n z − n − . (6.42)– 31 –or simplicity we will work with M = 0, which we will call the perturbative vacuum, and M = M ∈ R + , which we will call the NP-saddle, or instanton sector.In this particular example, the only possible singular directions in the Borel plane will be θ = 0 and θ = π . To compute the Stokes automorphism across these two singular directionswe will need the Bridge equation (6.13), which in this case takes the form˙∆ ω F ( z, σ , σ ) = (cid:88) l =0 A [ l ] ω ( σ , σ ) ∂F ( z, σ , σ ) ∂σ l , (6.43)where the undetermined functions A [ l ] ω ( σ , σ ) are related to the Stokes constants (analyticinvariants). We can Taylor expand these unknown functions A [ l ] ω ( σ , σ ) = (cid:88) k,m ≥ A [ l ] ( k,m ) ω σ k σ m , (6.44)where the complex numbers A [ l ] ( k,m ) ω are precisely the Stokes constants, non vanishing onlyfor very few particular values of ω, l, k, m .We can expand the l.h.s. of (6.43)˙∆ ω F ( z, σ , σ ) = (cid:88) l =0 σ l e − ( M l + ω ) z ∆ ω Φ l ( z ) , (6.45)and substitute the Taylor expansion in the r.h.s. of (6.43) to get (cid:88) l =0 σ l e − ( M l + ω ) z ∆ ω Φ l ( z ) = (cid:88) i =0 (cid:88) k,m ≥ A [ i ] ( k,m ) ω σ k σ m e − M i z Φ i ( z ) . (6.46)The crucial point behind the trans-series expansion for the Bridge equation is that by matchingequal powers of σ , σ and e − z , the allowed non vanishing Stokes constants will be enormouslyconstrained. In particular for this two parameters trans-series the only allowed constantsare a subset of T = { A [ i ] (1 , , A [ i ] (0 , } , and these constants can be non-zero if and only if M l + ω = M i for some ω, l, i .We can specialise (6.46) to the singular direction θ = 0 for which we get∆ M Φ ( z ) = A [1] (1 , M Φ ( z ) , (6.47)∆ M Φ ( z ) = 0 , (6.48)with all the other alien derivatives vanishing for all ω ∈ R + and ω (cid:54) = M . Similarly for thesingular direction θ = π , equation (6.46) becomes∆ − M Φ ( z ) = 0 , (6.49)∆ − M Φ ( z ) = A [0] (0 , − M Φ ( z ) , (6.50)– 32 –nd once again all the other alien derivatives are vanishing for all ω ∈ R − and ω (cid:54) = − M . Byrenaming the only non vanishing Stokes constants A [1] (1 , M = A M and A [0] (0 , − M = A − M , wecan rewrite the entire resurgence algebra for this two-terms trans-series in the form∆ M Φ ( z ) = A M Φ ( z ) , ∆ − M Φ ( z ) = 0 , ∆ M Φ ( z ) = 0 , ∆ − M Φ ( z ) = A − M Φ ( z ) . (6.51)Thanks to the above equations, the Stokes automorphism along θ = 0 can be explicitlywritten as S = exp (cid:32)(cid:88) ω e − ωz ∆ ω (cid:33) = 1 + e − Mz ∆ M , (6.52)so that S Φ ( z ) = Φ ( z ) + A M e − Mz Φ ( z ) , S Φ ( z ) = Φ ( z ) . (6.53)This means that, as we approach the singular direction θ = 0, Φ makes no jump while thejump of Φ is entirely dictated by Φ . For the full trans-series the Stokes automorphism alongthis direction is given by S F ( z, σ , σ ) = S (cid:0) σ Φ ( z ) + σ e − Mz Φ ( z ) (cid:1) = σ (cid:0) Φ ( z ) + A M e − Mz Φ ( z ) (cid:1) + σ e − Mz Φ ( z ) , = σ Φ ( z ) + ( σ + σ A M ) e − Mz Φ ( z ) , = F ( z, σ , σ + A M σ ) . (6.54)Had we started with perturbation theory alone F ( z, σ = 1 , σ = 0) = Φ ( z ), the Stokesautomorphism along the singular direction θ would have generated for us a second recessiveterm S F ( z, ,
0) = F ( z, , A M ) = Φ ( z ) + A M e − Mz Φ ( z ) . (6.55)In a similar manner, the Stokes automorphism along the singular direction θ = π is givenby S π = exp (cid:32)(cid:88) ω e − ωz ∆ ω (cid:33) = 1 + e + Mz ∆ − M , (6.56)so that S π Φ ( z ) = Φ ( z ) , S π Φ ( z ) = Φ ( z ) + A − M e + Mz Φ ( z ) . (6.57)The roles are now inverted, Φ makes no jump along the negative real line while the entirejump of Φ is dictated by Φ . The Stokes automorphism along the negative real line for thetwo terms trans-series becomes S π F ( z, σ , σ ) = F ( z, σ + A − M σ , σ ) . (6.58)– 33 –his complete knowledge of the Stokes automorphism allows us to study the large oderbehaviour of the perturbative (and non-perturbative) coefficients a ( l ) n in (6.42). By Cauchytheorem we know that F ( z ) = 12 πi (cid:73) F ( ω ) ω − z = 12 πi (cid:90) ∞ dω Disc F ( ω ) ω − z + 12 πi (cid:90) −∞ dω Disc π F ( ω ) ω − z , (6.59)and by expanding for z → ∞ ω − z = − ∞ (cid:88) n =0 ω n z − n − , (6.60)we get F n ∼ − πi (cid:90) ∞ dω ω n Disc F ( ω ) − πi (cid:90) −∞ dω ω n Disc π F ( ω ) , (6.61)where we schematically wrote F ( z ) ∼ (cid:80) n ≥ F n z − n − .We can specialise the above equations for the large orders behaviour of our perturbativeexpansion Φ ( z ) = ∞ (cid:88) n =0 a (0) n z − n − , (6.62)and thanks to (6.53)-(6.57), we know the full discontinuities Disc Φ ( z ) = (Id − S ) Φ ( z ) = − A M e − Mz Φ ( z ) , (6.63) Disc π Φ ( z ) = (Id − S π ) Φ ( z ) = 0 . (6.64)In (6.61) only the first term contributes and the large orders behaviour of the perturbativeexpansion is entirely controlled by the lower orders of the NP-saddle expansion [47] a (0) n ∼ A M πi (cid:88) k ≥ a (1) k (cid:90) ∞ dωω n − k e − Mω ∼ A M πi (cid:88) k ≥ a (1) k Γ( n + 1 − k ) M n +1 − k ∼ A M πi n ! M n +1 (cid:18) a (1)0 + a (1)1 Mn + a (1)2 M n ( n −
1) + ... (cid:19) . (6.65)Note that since M > n large enough.The story for the large orders of Φ can be repeated verbatim. Again using (6.53)-(6.57),the discontinuities are Disc Φ ( z ) = (Id − S ) Φ ( z ) = 0 , (6.66) Disc π Φ ( z ) = (Id − S π ) Φ ( z ) = − A − M e + Mz Φ ( z ) . (6.67) Note that generically we would get contributions coming from all the discontinuities across all the singulardirections and from all the residues at simple poles in the Borel plane. – 34 –he large orders behaviour for the NP-saddle perturbation theory is entirely captured by thelower orders coefficients of the perturbative vacuum a (1) n ∼ A − M πi (cid:88) k ≥ a (0) k Γ( n + 1 − k )( − M ) n +1 − k ∼ A − M πi n !( − M ) n +1 (cid:18) a (0)0 − a (0)1 Mn + a (0)2 M n ( n −
1) + ... (cid:19) . (6.68)Note that, thanks to the ( − M ) n +1 term, the NP-saddle perturbative coefficients are eventu-ally alternating in sign, in constrast to the vacuum coefficients a (0) n . Remark.
It is worth emphasizing again in words what we just found: the large orderscoefficients of the perturbative expansion do contain explicitly the lower orders coefficients ofthe NP saddle expansion and vice versa!In the next Section we will construct a toy model for path integral calculations in whichwe will be able to check explicitly how the lower orders non-perturbative coefficients areencoded and buried in the higher orders coefficients of the perturbative series.
In this final Section we will put everything together and show how a trans-series expansion forphysical observables, obtained from a path integral, will automatically be free from ambigui-ties coming from the non-Borel summability, along the positive real line, of the perturbativeseries.To be concrete and closest as possible to an actual path integral calculation, we can focuson a particular dimensional reduction, down to 0 dimensions, of the 2 d sine-Gordon model ,see [39]. The partition function is given by Z ( λ ) = 1 √ λ (cid:90) π − π dy e λ (cos y − = 1 √ λ (cid:90) π/ − π/ dx e − λ sin x = π √ λ e − λ I (cid:18) λ (cid:19) , (7.1)where I q is the modified Bessel function of the first kind with index q . We can also consideran other observable, related to the expectation value of the operator e iqy , given by O q ( λ ) = 1 √ λ (cid:90) π − π dy e λ (cos y − e iqy = 1 √ λ (cid:90) π − π dy e λ (cos y − cos( qy )= 1 √ λ (cid:90) π/ − π/ dx e − λ sin x e iqx = π √ λ e − λ I q (cid:18) λ (cid:19) , (7.2) See also [48] for a discussion on the monodromies properties of the model. – 35 –ith q ∈ N , clearly Z ( λ ) = O ( λ ), while O q ( λ ) = O − q ( λ ).So how do we compute these observables in perturbation theory? The way to proceedis first to complexify both the coupling constant, λ = e iθ | λ | , and the field variable, from x ∈ [ − π/ , π/
2] to z ∈ C , and the action, S ( z ), becomes a meromorphic function of z .Secondly, we have to identify all the saddle points of the action in the whole complex plane z ∈ C : S = 12 λ sin z , (7.3)given by the usual Euler-Lagrange equation dS/dz = 0, whose solutions are simply z = 0 → S | o.s. = S = 0 , (7.4) z = π/ → S | o.s. = S = 12 λ . (7.5). The original integration contour I = [ − π/ , π/
2] has real dimension 1, so even aftercomplexifing we still need to integrate over real dimension 1 cycles, Σ( θ ), that will dependon the argument θ of the coupling constant. For each saddle point z i , there is associated aunique integration cycle J i , called Lefschetz thimble or steepest descend path, defined by theflow equations: ∂z∂t = − ∂ z S ( z ) , (7.6) z ( t → −∞ ) = z i , (7.7)where t is the “time” along the thimble. Thanks to the flow equations it is easy to show thatthe phase remains stationary along the thimblesIm S ( z ) | J i = Im S ( z i ) . (7.8)Equivalently the flows equations can be seen as hamiltonian equations for (Re z, Im z ) withIm S as Hamiltonian, hence the stationarity of the phase .The thimbles J i are generically unbounded, even when the original integration contour isbounded. For this reason the convergence of the integral over a thimble is not guaranteed. Wedivide the complex z -plane into “good” and “bad” regions [49–51], good regions correspondsto Re S ( z ) >
0, while in bad regions Re S ( z ) <
0. The set of admissible Lefschetz thimbles,i.e. the ones whose asymptotic tails lay in the good regions, form a linearly independent andcomplete basis of integration cycles.As we dial the argument θ of the complexified coupling constant, every Lefschetz thimble J i will deform smoothly and, for generic values of θ , it will only pass through one saddle, i.e.its associated saddle z i from equation (7.7). For specific values of θ , precisely at Stokes lines,these contours will also pass through a subset of other saddles. A generic integration cycle, All these and all the following results are a direct consequence of the fact that Re S , being the real part ofa holomorphic function, defines a perfect Morse function [49]. – 36 – +J +J zz θ=0 θ=0 + −z −J +J Figure 6 . The original integration cycle as a linear combination of Lefschetz thimbles at θ = 0 − and θ = 0 + . θ = 0 is a Stokes line. Σ( θ ), on which the integral converges, can be written as a sum over thimbles, so that Σ( θ )passes (in principle) from all the critical points:Σ( θ ) = (cid:88) i n i J i , (7.9)where n i are integers.These coefficients n i will jump precisely when θ crosses a Stokes line. For example, in thecase at hand, the original integration contour I = [ − π/ , π/
2] can be written in two differentways, depending how we are approaching the real coupling case θ = 0: I = (cid:104) − π , π (cid:105) −→ Σ = (cid:40) J (0 − ) + J (0 − ) J (0 + ) − J (0 + ) (7.10)as shown in Figure 6.If we were able to compute exactly the integrals along the different cycles, we wouldobtain an exact result for our observables. Unfortunately, as usual, we cannot do that and wecan only approximate each integral using perturbation theory around each different saddle.The contribution to our observables (7.15), coming from perturbation theory around thetrivial vacuum z = 0, is given by the asymptotic series Z ( λ, q ) = e − S Φ ( λ, q ) , (7.11)and Φ ( λ, q ) = √ π ∞ (cid:88) n =0 a (0) n ( q ) λ n = √ π ∞ (cid:88) n =0 ( + q ) n ( − q ) n n ! (2 λ ) n , (7.12)– 37 –here ( x ) n = Γ( x + n ) / Γ( x ) denotes the Pochhammer symbol. As expected the perturbativeseries is diverging of Gevrey-1 type. Furthermore the a n are non-alternating in sign denotingthe presence of a Stokes line for arg( λ ) = 0.Similarly, the perturbative contribution to (7.15), coming from the non-perturbative sad-dle point z = π , is given by Z ( λ, q ) = e − S Φ ( λ, q ) , (7.13)and Φ ( λ, q ) = √ π ∞ (cid:88) n =0 a (1) n ( q ) λ n = √ π ∞ (cid:88) n =0 ( + q ) n ( − q ) n n ! ( − λ ) n , (7.14)which is once again of Gevrey-1 type but with coefficients a (1) n alternating in sign, hallmarkthat arg( λ ) = 0 is not a Stokes line for Φ ( λ, q ), while now arg( λ ) = π becomes the singulardirection.The semiclassical expansion for O q can be represented with a two-terms trans-series O q ( λ, σ , σ ) = σ Z ( λ, q ) + σ Z ( λ, q ) , (7.15)where the σ i are the usual trans-series parameters. We stress that a precise decompositionof the path integral will give us some definite values for the σ i , whose role is exactly thesame as the n i coefficients of the Lefschetz thimbles. It is useful to keep (7.15) with genericparameters σ i and study its resurgence properties and only at the end see the connectionwith the geometric structure of the path integral in terms of thimbles.We can easily obtain the Borel transform of the above seriesˆΦ ( ζ, q ) = √ π F (cid:18) q + 12 , − q, (cid:12)(cid:12)(cid:12) ζ (cid:19) , (7.16)ˆΦ ( ζ, q ) = √ π F (cid:18) q + 12 , − q, (cid:12)(cid:12)(cid:12) − ζ (cid:19) , (7.17)where F is the hypergeometric function. Both ˆΦ ( ζ, q ) and ˆΦ ( ζ, q ) define simple resurgentfunctions, with branch cuts respectively for ζ ∈ [1 / , + ∞ ) and ζ ∈ ( −∞ , − / θ = 0, the discussion for θ = π willbe exactly the same just by replacing Φ with Φ . We know the full discontinuity across thecut for the hypergeometric function [52]: F (cid:16) a, b, c (cid:12)(cid:12)(cid:12) ζ + iε (cid:17) − F (cid:16) a, b, c (cid:12)(cid:12)(cid:12) ζ − iε (cid:17) = 2 πi Γ( c )Γ( a )Γ( b ) F (cid:16) c − a, c − b, (cid:12)(cid:12)(cid:12) − ζ (cid:17) , (7.18)valid for a + b = c . Using this relation, we can obtain the full discontinuity across the cut forˆΦ :ˆΦ ( ζ + i(cid:15), q ) − ˆΦ ( ζ − i(cid:15), q ) = 2 πi Γ( q + )Γ( − q ) √ π F (cid:18) q + 12 , − q, (cid:12)(cid:12)(cid:12) − ζ (cid:19) . (7.19)– 38 – igure 7 . The right Borel resummation can be rewritten as the sum of the left Borel resummationplus the contribution coming from the Hankel contour γ , coming from t → −∞ , circling around thebranch cut starting at t = 1 / ∞ . Since we just obtained the full discontinuity for ˆΦ along the positive real axis, we caneasily compute the Stokes automorphism and the full alien derivative algebra. We proceedas illustrated in Fig. 7: the difference of the right and left Borel resummation can be writtenas an integral over the Hankel contour γ which starts at ∞ below the imaginary axis, thencircles the singular point at t = 1 /
2, and then goes back to ∞ above the imaginary axis:( S + − S − )Φ ( λ, q ) = √ πλ (cid:90) γ dt e − t/λ F (cid:18) q + 12 , − q,
1; 2 t (cid:19) = √ πλ (cid:90) ∞ / dt e − t/λ (cid:20) F (cid:18) q + 12 , − q, , t + iε (cid:19) − ( i(cid:15) → − i(cid:15) ) (cid:21) = √ πλ (cid:90) ∞ / dt e − t/λ πi Γ( q + )Γ( − q ) F (cid:18) q + 12 , − q, , − t (cid:19) = 2 πi Γ( q + )Γ( − q ) √ πe − / (2 λ ) λ (cid:90) ∞ dt e − t/λ F (cid:18) q + 12 , − q, , − t (cid:19) = 2 πi Γ( q + )Γ( − q ) e − / (2 λ ) S Φ ( λ, q ) . (7.20)We can combine our definition of the Stokes automorphism (4.8), represented as differenceof later Borel resummations, with the expression for the alien derivative as the logarithm of S θ (4.11), and thanks to the equations (4.13-4.16), we obtain the only non-trivial alien derivativesalong the Stokes line θ = 0∆ ˆΦ ( ζ, q ) = 2 i sin( π ( q + 1 / ( ζ, q ) , (7.21)where we made use of the known formula Γ( a )Γ(1 − a ) = π/ sin( πa ) to rewrite the Stokesmultiplier A / = 2 πi/ (Γ(1 / q )Γ(1 / − q )) = 2 i sin( π ( q + 1 / θ = π where the only singular point is ζ = − /
2, it is now ˆΦ ( ζ, q ) to have a non-trivial alienderivative ∆ − ˆΦ ( ζ, q ) = 2 i sin( π ( q + 1 / ( ζ, q ) , (7.22)– 39 –nd we note that the two Stokes constants are equal A − / = A / .We can summarise the entire resurgent algebra for the problem at hand∆ ˆΦ ( ζ, q ) = 2 i ( − q ˆΦ ( ζ, q ) , ∆ − ˆΦ ( ζ, q ) = 0 , ∆ ˆΦ ( ζ, q ) = 0 , ∆ − ˆΦ ( ζ, q ) = 2 i ( − q ˆΦ ( ζ, q ) , (7.23)with all the other alien derivatives being zero. This algebra is precisely of the form studiedbefore in (6.51), the Stokes constants for these two-terms trans-series are A / = A − / =2 i ( − q , when q ∈ N .Having the explicit expressions for the perturbative and non-perturbative saddle coeffi-cients, we can check how the low orders coefficients of the NP physics are encoded in the largeorders perturbative ones. Taking (7.12) and changing variable to z = 1 /λ we getΦ ( z, q ) = √ π + √ π ∞ (cid:88) n =0 c (0) n ( q ) z − n − = √ π + √ π ∞ (cid:88) n =0 ( + q ) n +1 ( − q ) n +1 ( n + 1)! (2 /z ) n +1 . (7.24)We can easily extract the high orders behaviour of the perturbative coefficients c (0) n and writeit in the suggestive form c (0) n ∼ πi πi Γ(1 / q )Γ(1 / − q ) n !(1 / n +1 ×× (cid:18) q −
1) 1 / n + 16 q − q + 92 (1 / n ( n −
1) + ... (cid:19) . (7.25)Comparing this expansion for c (0) n with (6.65-6.68) we can read precisely the lower orderscoefficients c (1)0 , c (1)1 , ... of the perturbative expansion for the non-trivial saddle. Furthermorewe can recover the instanton action M = 1 / A / = 2 πi/ (Γ(1 / q )Γ(1 / − q )) just obtained via alien calculus. The perturbative vacuum knows everythingabout the non-perturbative saddle point and vice versa!The action of the Stokes automorphisms S , S π has been computed in (6.54-6.58) sofrom (4.8) we can relate the left and right resummation of the full trans-series S + O q ( λ, σ , σ ) = S − O q ( λ, σ , σ + 2 i ( − q σ ) , (7.26) S π + O q ( λ, σ , σ ) = S π − O q ( λ, σ + 2 i ( − q σ , σ ) . (7.27)Had we had started with the perturbative series Z alone, by crossing the positive real axis (oneof the two Stokes line), we would have automatically generated the recessive (exponentiallysuppressed) term Z , coming from perturbation theory around the non-trivial saddle-point.We have finally come to the question: How do we assign an unambiguous sum to thetrans-series (7.15), corresponding to (7.1) and (7.15)? If we compute a real physical quantityin terms of a trans-series O ( z ) = (cid:88) j c j g j ( z ) , (7.28)– 40 –he reality of this observable translates into a reality condition for both the coefficients c j ∈ R and the trans-monomials g j : ( C O ) ( z ) = O (¯ z ) , (7.29)where C denotes the complex conjugation operator.For the 0 d model observables (7.15) it is manifest that both the partition function Z ( λ )and the conformal primaries O q ( λ ) should be real functions when λ is real. We have alreadyseen that if we resum the trans-series along a Stokes line, using either S + or S − , we will notobtain something real (7.26), focusing for example on the positive real axis: C ( S + O ) ( z ) (cid:54) = ( S + O ) (¯ z ) , (7.30) C ( S − O ) ( z ) (cid:54) = ( S − O ) (¯ z ) . (7.31)This follows from the fact that on a singular direction the Stokes automorphism actsnon-trivial on our trans-series, i.e. S + O (cid:54) = S − O , while instead the complex conjugationswaps the two lateral summations: C ◦ S + = S − ◦ C , (7.32)which translates into C ◦ S − = S ◦ C . (7.33)This means that the conjugation operator has to anti-commute with the alien derivative C ◦ ˙∆ = − ˙∆ ◦ C , (7.34)where ˙∆ = (cid:80) ω ∈ Γ ˙∆ ω . Across a Stokes line neither of the two lateral summations canpossibly give a real resummation of our real observable.To obtain a real resummation procedure we have to introduce what it is called mediansummation. Firstly, since we express the alien derivative as the logarithm of the Stokesautomorphism (4.11), we can define non-integers power of S simply by S νθ . = exp ν (cid:88) ω ∈ Γ θ e − ω z ∆ ω = exp (cid:16) ν ˙∆ θ (cid:17) , (7.35)with ν ∈ C while Γ θ denotes the set of singular points along the θ direction. Thanks to theour discussion above, we know now how complex conjugation acts on the non-integers powerof the Stokes automorphism C ◦ S ν = S − ν ◦ C . (7.36) Definition . We define the median resummation S med = S − ◦ S − / = S + ◦ S / , (7.37)which, in contrast to S ± , resum power series with real coefficients into real analytic functionsof z , for z ∈ R + . – 41 –he median resummation does precisely what we were expecting from our summationprocedure C ( S med O ) ( z ) = ( S med O ) (¯ z ) , (7.38)where we used equation (7.36) for ν = 1 / d example. In this case it is pretty easy to compute S ν since we know the full resurgent algebra (7.23) S ν O q ( λ, σ , σ ) = O q ( λ, σ , σ + 2 i ( − q ν σ ) . (7.39)We can finally notice that the median resummation for our observables leads to( O q ) R ( λ ) = S med O q ( λ, , S + O q ( λ, , − i ( − q ) = S − O q ( λ, , + i ( − q ) , (7.40)which is exactly equivalent to the correct decomposition of the path integral in terms ofLefschetz thimbles, with the precise intersection numbers n i , computed by Morse theory,necessary to write our original integration contour as a sum of steepest descent paths (7.9).As we have seen before the original path integral does receive contribution from boththimbles, let’s consider for example the contribution to the partition function coming fromthe perturbative vacuum thimble. Changing variables from the field variable z to the actionvariable u = S ( z ):1 √ λ (cid:90) J (0 ∓ ) e − λ sin ( z ) = 2 √ λ (cid:90) / du e − u/λ (cid:112) u (1 − u ) ∓ i √ λ (cid:90) ∞ / du e − u/λ (cid:112) u (2 u − √ λ (cid:90) / du e − u/λ (cid:112) u (1 − u ) ∓ ie − λ √ λ (cid:90) ∞ du e − u/λ (cid:112) (2 u + 1)2 u = Re S Φ ∓ ie − λ S Φ (7.41)where Re S Φ is unambiguous. The integral that we identify with Re S Φ is dominated by u (cid:46) λ in the small λ regime. The procedure to obtain the perturbative expansion Φ fromthis expression involves two steps. First, we should extend the integration domain to [0 , ∞ ).Secondly, we Taylor expand √ (1 − u ) around the origin, and, performing the integration termby term, we will obtain the divergent asymptotic expansion Φ . The reason for the divergenceis the use of the Taylor expansion beyond its radius of convergence .We recover in different form the presence of a Stokes phenomenon for the perturbativeseries at θ = 0 together with a non-trivial Stokes automorphism (7.20). The contributionsto the partition function from the perturbative Lefschetz thimbles J (0 ± ) give rise to anambiguity for real coupling, but of course this is not the only term to consider to get the fullanswer. We know that the original domain of integration, I = [ − π/ , π/ One can also obtain the (convergent) strong coupling expansion from the integral representation in the λ (cid:29) λ and performing order by order integration. – 42 –s a linear combination of J and J according to (7.10). It is only after adding the termarising from the J thimble, with precisely the right intersection number n (function of θ ),that we get an exact cancellation between the ambiguity coming from the resummation ofthe perturbation theory and the jump of the J contribution. Rather than coming from ourresummation procedure,the jump of the J contribution is geometric in nature instead, andit is due to the different integration contours decompositions, as we approach θ = 0 ± !To conclude, the ambiguity in the imaginary part of the integration (cid:82) J (0 ± ) is cancelledexactly by the ambiguity in the prefactor of the (cid:82) J (0 ± ) integral. The path integral, decom-posed into thimbles, gives precisely the median resummation prescription from first principles,as a geometric construction, the ambiguity coming from the resummation of the perturbativeexpansion is intertwined with the jumps in the decomposition of the original contour of inte-gration into steepest descent path, in the spirit of Morse theory. In this simple 0d case, themedian resummation is fairly easy to obtain and consists simply in taking the real part ofthe S resummation of the perturbation series around the vacuum alone (7.41). The genericcase when logarithm and branch cuts are present is much more complicated and has beendiscussed in meticulous details in [46]. From the analysis carried out in the previous Section, we can draw a simple analogy betweenfinite dimensional integrals, where the relation between saddle points approximation andtrans-series expansion is well established [53, 54], and the path integral formulation of QM andQFT, to understand why physical observables should be obtained as trans-series expansions ofsimple resurgent functions. In a semiclassical path integral calculation, we should first look forall the finite action, classical solutions to the equation of motion, in a suitable complexificationof our fields space, then, by deforming the original “contour” of integration, we should addup all these exponentially suppressed contributions, together with the remaining fluctuationson top of them, i.e. zero-, quasi-zero- and gaussian-modes.We have to stress that, while in euclidean QM and QFT, it is natural to expect instantonscontributions in an euclidean path integral calculation (simply because we can construct thesefinite action non-perturbative saddles), in real time QM and QFT, on the other hand, it isnot. The “weight” of a classical configuration, in the real time path integral, is given by e iS ,so it is not clear how a saddle points expansion could give rise to exponentially suppressedterms, i.e. the energy splitting in a double-well QM. As it turns out [55, 56], even if wecomplexify all our fields and look for more generic, real time saddle points, we still do notfind instanton-like configurations. To find smooth and exponentially suppressed instantons,it seems that we have to complexify both the fields and the time variable! Note that there isno need to go all the way to imaginary time t → − iτ , i.e. the euclidean formulation, as soonas we work a little bit off the real time t → t e i(cid:15) (very much likely the + i(cid:15) -prescription for thepropagators), regular, finite and real action instantons appear as solutions to the complexifiedclassical equations of motion. – 43 –oing back to the Euclidean case, in QM [36–38] and QFT [39], instanton-anti-instantontype of amplitudes possess unambiguous real parts and two-fold ambiguous imaginary parts,necessary to “cure” the ambiguity coming from the resummation of perturbation theoryaround the perturbative vacuum alone. The cancellation of the imaginary parts in pathintegral examples is essentially the same as for ordinary integrals. However, in QFT thereare also real unambiguous contributions to observables from NP-saddles.Of course, in semiclassically calculable regimes of QFTs and in QM, there are infinitelymany saddle points and, even if a mathematically rigorous definition for the path integralmeasure is lacking, we can still ask the question: is it possible to write our original, infinitedimensional, “contour” of integration, i.e. the space of fields, as an infinite sum of nicer,infinite dimensional thimbles, living in a suitable complexification of the original space offields?We can try to use the (complexified) action as a Morse function on the complexification ofthe space of fields, and, as a generalisation of the flow equations (7.6)-(7.7) described above,we get that a thimble is a solution to ∂φ ( x, t ) ∂t = − δS [ φ ] δφ ( x, t ) , lim t →−∞ φ ( x, t ) = φ cl ( x ) ,δS [ φ ] δφ ( x, t ) (cid:12)(cid:12)(cid:12)(cid:12) φ ( x,t )= φ cl ( x ) = 0 . The flow originates, at t → −∞ , from a classical solution, φ cl ( x ), to the complexifiedequation of motion and, as we increase a little the flow time t , we start moving in the space offields accordingly to the eigenvectors and eigenvalues of the quadratic fluctuations operator δ S/δφ | φ cl . The thimbles are infinite dimensional algebraic varieties and when the theoryis regularised on a finite lattice with a finite size, they become finite dimensional algebraicvarieties [57].In few lucky cases, namely 3d Chern-Simons [49] and QM in phase space [51], these flowequations become elliptic PDEs and can be solved. The original “contour” of integration inthe path integral can be then rewritten in terms of thimbles living in a complexification of theoriginal fields space, and each thimble is associated to one (away from Stokes lines) classicalsolution to the complexified equations of motion in the same spirit of Section 7.For generic QFTs we do not know if the original “contour” of integration of the pathintegral can be decomposed as a sum of thimbles, but both Floer homology for parabolicflows [58] and our trans-series expansion for physical observables seems to suggest that this isindeed the case. The infinite dimensional analog of Lefschetz thimbles in QFT and QM is farfrom being completely understood [59], but it looks exactly the right setup to understand whyphysical observables can be obtained as trans-series expansions for simple resurgent functions. Generically these flow equations are parabolic PDEs so the flow is actually only a semi-flow. – 44 –
Acknowledgment
I am very grateful to Mithat ¨Unsal for introducing me to the fascinating subject of resurgenceand for countless discussions, and to Gerald Dunne for his invaluable help, careful proofreading of these notes and for being a continuous source of new ideas. I would also like to thankInes Aniceto, Gokce Basar, Aleksey Cherman, Hugh Osborn, Slava Rychkov, David Sauzinand Kenny Wong for stimulating and enjoyable discussions. I am grateful for the support ofEuropean Research Council Advanced Grant No. 247252, Properties and Applications of theGauge/Gravity Correspondence.
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