Resurgence of the large-charge expansion
Nicola Dondi, Ioannis Kalogerakis, Domenico Orlando, Susanne Reffert
RResurgence of the large-chargeexpansion
Nicola Dondi (cid:73) , Ioannis Kalogerakis (cid:73) , Domenico Orlando (cid:72)(cid:73) , and
Susanne Reffert (cid:73) (cid:73)
Albert Einstein Center for Fundamental PhysicsInstitute for Theoretical Physics, University of Bern,Sidlerstrasse 5, CH-3012 Bern, Switzerland (cid:72)
INFN sezione di Torino | Arnold–Regge Centervia Pietro Giuria 1, 10125 Torino, Italy
Abstract
We study the O ( N ) model at criticality in three dimensions in the double scalinglimit of large N and large charge. We show that the large-charge expansionis an asymptotic series, and we use resurgence techniques to study the non-perturbative corrections and to extend the validity of the eft to any value of thecharge. We conjecture the general form of the non-perturbative behavior of theconformal dimensions for any value of N and find very good agreement withprevious lattice data. a r X i v : . [ h e p - t h ] F e b ontents Working in sectors of large global charge leads to important simplifications in stronglycoupled and otherwise inaccessible conformal field theories (cfts) [1–3].One striking feature of the large-charge expansion of the O(2N) model at theWilson–Fisher (wf) fixed point in 2+1 dimensions is that it appears to also work forsmall charges, which taken as such is quite astounding: in general we expect thesemiclassical expansion to work for systems with a very large number of degrees offreedom. This fact came to light first in the comparison of the large-charge predictionfor the scaling dimension of the lowest operator of charge Q with lattice results forthe O(2) [4] and the O(4) [5] models. It was noted that very few terms in the effectiveaction were sufficient to reproduce the lattice results with very high accuracy.1rom the effective field theory (eft) point of view, there seems to no reason for thelarge-charge predictions to keep working for small charges. But if we add anothercontrolling parameter to the mix, we can go beyond the reach of the eft and try tounderstand this behavior. This is for example the case when we study the large- N limit of the O(2N) model at large charge [6, 7]. In the double-scaling limit Q → ∞ , N → ∞ , with Q/ ( N ) = ^ q constant it is possible to solve the problem exactly atleading order in N for any value of ^ q . Building on these results, in this paper weshow that the large-charge expansion in the double-scaling limit is asymptotic, andis closely related to the asymptotic Seeley–DeWitt expansion of heat kernels andrelated ζ -functions on spheres [8, 9].Asymptotic series are a common feature of perturbative solutions to quantumproblems as originally argued by Dyson [10]. This feature signals the presence ofnon-perturbative phenomena in the theory, as was first quantitatively discussed in aseries of papers by Bender and Wu starting with [11] in the context of anaharmonicoscillators. The modern approach to the subject goes under the name of resurgentasymptotics , or simply resurgence, and originates from Écalle’s works [12]. Modernreviews on the subject and on applications in physics and mathematics are [13, 14] towhich we refer for a complete list of references. We will use resurgent methods toshow which non-perturbative contributions are present in the double-scaling limit ofthe large-charge expansion and how non-perturbative ambiguities cancel. The finalresult is an extrapolation to small charge which matches the small-charge expansionwith excellent precision.We develop a geometric picture interpreting the exponential corrections to theperturbative series as worldline instantons describing a particle of mass equal to thechemical potential µ moving along geodesics. We also find a geometric interpretationof the lateral Borel resummation in terms of unstable worldline modes.We conjecture this picture to be robust enough to carry over to the general caseof finite N , lending credibility to the general validity of our observations made inthe double-scaling limit. Assuming that the qualitative features of the worldlineinstanton persist also for finite N , we can use the knowledge of the leading exponentialeffects to derive constraints on the perturbative expansion, even though in this case,the coefficients are not accessible within the eft. We find that a few terms (of order N ∗ ≈ π √ Q ) are enough to describe the theory also at small charge with a precisionof order 1%, in agreement with the lattice estimates. The same geometric picture alsoappears in the case of supersymmetric systems. For example, in N = ( n ) !. They scale as e −^ q / , while more traditional instantonsassociated to Feynamn diagram proliferation are controlled by e −^ q / and are thussuppressed. The same ( n ) ! factorial growth is found also in computations of effectiveactions of the Euler–Heisenberg type [22], see also [23] for a comprehensive review.In those instances, it has been shown [24, 25] that this faster-than-factorial growth isdriven by worldline instantons, as it turns out to be also in the present work. Thisis also reminiscent of the situation in string theory, where D-instantons genericallydominate over NS5 and gravitational instantons.The plan of this note is as follows. In Section two, we study the asymptoticsof the O(2N) model in the double-scaling limit with resurgence methods. Asspecific examples, we discuss the model on the torus (Section 2.2) and on the sphere(Section 2.3), giving both the perturbation series and the exponentially suppressednon-perturbative corrections. These examples have special features due to theirgeometry — the perturbation series of the torus consists of a single term due to thevanishing of the curvature. The case of the sphere is more involved — in fact, we needto supplement our resurgence calculation with a physical input on how to resolvethe ambiguities. This input is provided in Section three, where we reformulate ourproblem as the quantum mechanics of a particle moving on closed geodesics. Werecast the heat-kernel as a path integral (Section 3.1) and discuss again the torus(Section 3.2) and sphere (Section 3.3) examples. In Section four, we combine theresults of the previous two sections and obtain the exact form of the grand potentialfor any value of the charge. This is substantiated numerically against the small-chargeexpansion. In Section five, we go beyond the double-scaling limit by arguing that thegeometric interpretation in terms of worldline geodesics works for the general case.Assuming the large-charge expansion to always be an asymptotic series for any N ,we can give the optimal truncation as a function of the charge. This result is borneout by the observations made on the lattice. In Section six, we summarize our resultand point out future lines of investigations.In the appendix we summarize the prerequisites of large charge at large N(Appendix A), introduce some basic facts about resurgence (Appendix B), discussLipatov’s instantons (Appendix C), discuss the optimal truncation in the double-scaling limit (Appendix D), and present a simple worked-out example of resurgentanalysis which is relevant for Section two (Appendix E).3 Asymptotics at large charge
We start with the Landau–Ginzburg model for 2 N real scalar fields in the vectorrepresentation of O ( N ) (which we encode as N complex fields) in ( + ) dimensionswith Euclidean signature on ( R × M ) , where M is a Riemann surface. Including allterms up to mass dimension three, we have S [ ϕ i ] = N (cid:88) i = (cid:90) d t d M (cid:104) g µν ( ∂ µ ϕ i ) ∗ ( ∂ ν ϕ i ) + rϕ ∗ i ϕ i + u ( ϕ ∗ i ϕ i ) + v ( ϕ ∗ i ϕ i ) (cid:105) . (2.1)This model flows to the wf fixed point in the infrared (ir) limit u → ∞ when r isfine-tuned to the value of the conformal coupling. We are interested in the free energy of the O(2N) vector model at criticality. We wantto work in a sector of fixed charge Q which corresponds to considering the completelysymmetric representation of rank Q of the global symmetry group O(2N) [5–7, 26].We take the double-scaling limit N → ∞ , Q → ∞ , ^ q = Q/ ( N ) fixed. (2.2)The main result of [6] is that the free energy can be expressed as the Legendretransform of a zeta function (see Appendix A for details). Let f (^ q ) = F/ ( N ) bethe free energy per degree of freedom (dof), ω ( µ ) the grand potential, and µ thechemical potential. We have f (^ q ) = sup µ ( µ ^ q − ω ( µ )) , (2.3) ^ q = d ω ( µ ) d µ , (2.4) ω ( µ ) = − ζ (− |M , µ ) , (2.5)where ζ ( s |M , µ ) is the zeta function for the operator − (cid:52) + µ , i.e. if spec (− (cid:52) ) = { λ j } , ζ ( s |M , µ ) = (cid:88) j ( λ j + µ ) − s . (2.6)The large- ^ q regime corresponds to large chemical potential µ . In this case it isconvenient to write the zeta function in the Mellin representation ζ ( s |M , µ ) = Γ ( s ) (cid:90) ∞ d tt t s e − µ t Tr (cid:16) e (cid:52) t (cid:17) . (2.7)4f µ is large, the integral localizes around t =
0. This reduces the large-chargeproblem to the classical problem of the Weyl asymptotic expansion of the heat kernel,which can be written in terms of Seeley–DeWitt coefficients [27, 28]:Tr (cid:16) e (cid:52) t (cid:17) ∼ V πt (cid:18) + R t + . . . (cid:19) . (2.8)In the following we will concentrate on the examples of the torus M = T and thesphere M = S . We choose the former because of its simplicity and the latter becausewe can identify the free energy on the sphere with the conformal dimension of thelowest operator of charge Q using the state-operator correspondence. In the caseof T , the Weyl expansion contains only the leading term because all the othersare proportional to the curvature. In the case of S we will see that the series isasymptotic and can be studied with the methods of resurgence theory. Our first example is the torus M = T . This case is particularly simple becausethe free energy can be written exactly, but it is interesting because it shows somequalitative properties that are general.Consider a square torus of side L . All but the first Seeley–DeWitt coefficientsvanish, so the leading asymptotic of the heat trace and ζ -function areTr (cid:16) e (cid:52) t (cid:17) ∼ L πt + O (cid:16) e − /t (cid:17) , ζ ( s | T , µ ) = L µ ( − s ) π ( s − ) + O (cid:0) e − µ (cid:1) . (2.9)One can readily derive all the quantities appearing in the system Eq. (2.3)-(2.5): f (^ q ) = √ π L ^ q / , (2.10) ^ q = L µ π , (2.11) ω ( µ ) = − ζ (− | T , µ ) = L µ π . (2.12)These expressions are perturbatively exact at leading order in large µ (respectively ^ q ), but one can do much better since the O (cid:0) e − /t (cid:1) corrections in Eq. (2.9) are knownin closed form. From the spectrum of the T Laplacian,spec ( (cid:52) T ) = (cid:12) − π L ( k + k ) (cid:12)(cid:12)(cid:12)(cid:12) k , k ∈ Z (cid:13) , (2.13)5ollows that the heat kernel trace is the square of a theta function:Tr (cid:16) e (cid:52) t (cid:17) = (cid:88) k , k ∈ Z e − π L ( k + k ) t = (cid:20) θ ( e − π tL ) (cid:21) . (2.14)After Poisson resummation, we find the appropriate expansion around t → + ,Tr (cid:16) e (cid:52) t (cid:17) = L πt + (cid:88) (cid:48) k ∈ Z e − (cid:107) k (cid:107) L t , (2.15)where (cid:107) k (cid:107) = k + k , and the prime indicates that the sum does not include the zeromode. This expression is exact and valid also for finite t , which allows us to find thesubleading contribution for µ → ∞ of the ζ -function: ζ ( s | T , µ ) = L µ − s π ( s − ) + L π (cid:88) (cid:48) k − s Γ ( s ) (cid:18) (cid:107) k (cid:107) Lµ (cid:19) s − K − s ( (cid:107) k (cid:107) µL ) , (2.16)where K n ( z ) is the modified Bessel function of the second kind. The subleadingterms in the grand potential and the free energy are then given in closed form by ω ( µ ) = − ζ (− | T , µ ) = L µ π (cid:32) + (cid:88) (cid:48) k e − (cid:107) k (cid:107) µL (cid:107) k (cid:107) µ L (cid:18) + (cid:107) k (cid:107) µL (cid:19)(cid:33) , (2.17) f (^ q ) = sup µ ( µ ^ q − ω ( µ )) = √ π L ^ q / (cid:32) − (cid:88) (cid:48) k e − (cid:107) k (cid:107)√ π ^ q (cid:107) k (cid:107) π ^ q + . . . (cid:33) . (2.18)The relation between the Legendre-conjugate variables ^ q and µ is computed recur-sively, so that further corrections to the free energy are present. However, evenextrapolating to small ^ q up to ^ q =
1, the contribution of the further exponentially-suppressed terms is of order O (cid:0) − (cid:1) .This is the generic form that we will encounter: a perturbative expansion in µ (which here contains a single term) plus exponentially suppressed terms controlledby the dimensionless parameter µL , where L is the typical scale of the manifold M .Equivalently, the free energy is written as a double expansion in the two parameters1 / ^ q and e − √ π ^ q . These structures are known as trans-series and appear naturallyin perturbative problems. In general, given a problem with a small parameter z , a Note the square root in the exponential, suggesting that there are non-perturbative effects moreimportant than the “usual” quantum field theory (qft) instantons that would be generically of order O (cid:16) e −^ q / (cid:17) , see Appendix C. z → Φ ( σ k , z ) = Φ ( ) ( z ) + (cid:88) k (cid:54) = σ k e − A k /z /βk z − b k /β k Φ ( k ) ( z ) . (2.19)Generically, all Φ ( k ) ( z ) are asymptotic series, in particular Φ ( ) is the (formal) solutionof the perturbative problem. For this reason, these expressions make sense only oncea prescription for the summation of these series is given. We will discuss this issue inthe next section. Resurgent trans-series are characterized by a specific set of relationsbetween the numbers A k , β k , b k and the series Φ ( k ) . Thanks to these relations, it ispossible to fix the value of the trans-series parameters σ k ∈ C such that an appropriatesummation of the trans-series produces an unambiguous full function which has Φ ( ) as leading perturbative asymptotic expansion. Such a function is known as resurgent function . For a more complete treatment of the subject we refer to [13, 29].The heat trace on T indeed has this form, but trivially all Φ ( k ) are one-loop exact.No ambiguities related to asymptotic series are present, so the resurgent functionsolution coincides with its trans-series representation, which is just the function θ .As we will see, this simple result does not carry over to the case of M = S . Next we study the case of the sphere of radius r , M = S . Now the small- t behavior ofthe heat kernel can be represented in terms of an asymptotic series, whose coefficientsare the well-known Seeley–DeWitt coefficients for S . We will show how the heat tracecan be recast into a trans-series form. This series is not Borel resummable, and we willneed to supplement the perturbative expansion with non-perturbative exponentialsin order to make sense of it. These exponentials have a clear interpretation in terms ofworldline instantons, and the non-perturbative ambiguities are related to tachyonicinstabilities as we will see in Section 3.1. Seeley–DeWitt coefficients.
The spectrum of the Laplacian on the two-sphere isgiven by spec ( (cid:52) S ) = (cid:12) − (cid:96) ( (cid:96) + ) r (cid:12)(cid:12)(cid:12)(cid:12) (cid:96) ∈ N (cid:13) (2.20)and each eigenvalue has multiplicity ( (cid:96) + ) . Like in the case of the torus, the tracesum can be rewritten using Poisson resummation after some massaging (see e.g. [30]).7t is convenient to consider the trace of the conformal Laplacian, which readsTr (cid:20) e (cid:16) (cid:52) S − r (cid:17) t (cid:21) = ∞ (cid:88) (cid:96) = ( (cid:96) + ) e − tr ( (cid:96) + ) = ∞ (cid:88) (cid:96) =− ∞ (cid:12)(cid:12)(cid:12)(cid:12) (cid:96) + (cid:12)(cid:12)(cid:12)(cid:12) e − tr ( (cid:96) + ) = (cid:90) R d ρ | ρ | e − ρ t/r + (cid:88) (cid:48) k ∈ Z (− ) k (cid:90) R d ρ | ρ | e − ρ t/r + πikρ = r t + (cid:88) (cid:48) k ∈ Z (− ) k (cid:20) r t − | k | πr t / F (cid:16) πr | k | √ t (cid:17)(cid:21) . (2.21)In the last line we have introduced the Dawson’s function F ( z ) which is related tothe imaginary error function: F ( z ) = e − z (cid:90) z d t e − t = √ π e − z erfi ( z ) . (2.22)For small values of its argument, we can use the asymptotic expansion of F ( z ) , F ( z ) ∼ ∞ (cid:88) n = ( n − ) !!2 n + (cid:18) z (cid:19) n + . (2.23)After some formal manipulations one obtains the leading asymptotic of the heattrace:Tr (cid:20) e (cid:16) (cid:52) − r (cid:17) t (cid:21) ∼ r t − ∞ (cid:88) n = (− ) n ( − − n ) n ! r n − B n t n − ≡ r t ∞ (cid:88) n = a n (cid:18) tr (cid:19) n , (2.24)where B n are the Bernoulli numbers. This expression was already discussedin [31], based on previous work in [32]. The series is asymptotic since at large n theSeeley–DeWitt coefficients diverge like n !: B n = (− ) n + ( n ) ! ( π ) n ζ ( n ) = ⇒ a n = (− ) n + ( − − n ) n ! B n ∼ √ π n − / π n n !.(2.25)This divergence is a direct consequence of the expansion of the Dawson’s functionEq. (2.23), which is itself asymptotic.This expansion is only valid formally and needs a summing prescription. Weassume that this series can be completed into a resurgent trans-series, and that anappropriate summation procedure leads to an unambiguous solution in terms of aresurgent function. The first step is to identify the correct form of the non-perturbativecorrections in the generic trans-series of Eq.(2.19), where the perturbative expansion See Appendix E for a summary of the properties of F ( z ) and on the construction of its trans-seriesrepresentation.
8n Eq. (2.24) plays the role of Φ ( ) . We will take them to have the general form (cid:88) k (cid:54) = e − A k /z /βk z − b k /β k Φ ( k ) ( z ) , Φ ( k ) ( z ) ∼ ∞ (cid:88) (cid:96) = a ( k ) (cid:96) z (cid:96)/β k , (2.26)where we employed z = t/r . One of the main results concerning resurgent functionsis that the coefficients a ( k ) (cid:96) of the series Φ ( k> ) together with the numbers β , A k , b k are encoded in the large-order behavior of the perturbative series [13]: a n ∼ (cid:88) k S k πi β k A nβ k + b k k ∞ (cid:88) (cid:96) = a ( k ) (cid:96) A (cid:96)k Γ ( β k n + b k − (cid:96) ) , (2.27)where the S k are Stokes constants. In our case we have complete knowledge of the a n . Upon using the identity (cid:88) k (cid:54) = (− ) k k n = ( − n + − )(− ) n + ( π ) n ( n ) ! B n , (2.28)we can write them in the suggestive form a n = − √ π (cid:88) k (cid:54) = (− ) k Γ ( n + )( πk ) n . (2.29)Comparing the two expressions we find β = b k =
12 , A k = ( πk ) , S k πi a ( k ) = (− ) k + | k | √ π , a ( k ) > =
0. (2.30)The series around each exponential are truncated to only one term. This shows thata trans-series representation of the heat trace has to contain the termsTr (cid:20) e (cid:16) (cid:52) − r (cid:17) t (cid:21) ⊃ i (cid:18) πr t (cid:19) (− ) k + | k | e −( kπr ) /t . (2.31)These contributions to the trans-series representations are defined up to a k -dependentcomplex constant (the trans-series parameters σ k in Eq. (2.19)). The large-orderanalysis of Φ ( ) cannot fix these constants, which is a reflection of the fact that forany choice of σ k , the resulting trans-series has Φ ( ) as perturbative asymptotics. Grand potential and free energy.
The grand potential and the free energy arethemselves asymptotic series. Being related to the Mellin transform of the heat trace,these quantities are higher factorially divergent. This seems to be a feature of the The large-order relation is trivially realized in the case of T . the large-charge expansion of the conformaldimension is asymptotic and the coefficients in the series diverge like ( n ) !. We will arguein Section 5 that this feature is a general feature of large-charge limits.A similar large-order analysis can be carried out for the large- µ expansion of thezeta function, ζ ( s | S , µ ) = Γ ( s ) (cid:90) ∞ d t t s − e − µ t Tr (cid:104) e (cid:52) t (cid:105) . (2.32)It is convenient to use the conformal Laplacian, which amounts to a shift µ → µ − / ( r ) ≡ m . One obtains [6] ζ ( s | S , m ) = r m ( − s ) ∞ (cid:88) n = a n Γ ( n + s − ) Γ ( s ) ( mr ) n . (2.33)As expected, this is an expansion around m ∼ µ → ∞ , with coefficients related tothe Seeley–DeWitt coefficients on S that we have computed in Eq. (2.25). Note thatthe extra gamma function gives rise to a further n ! enhancement of the large-orderdivergence. For s = − / ω ( m ) = − ζ (cid:18) − (cid:12)(cid:12)(cid:12)(cid:12) S , m (cid:19) = r m ∞ (cid:88) n = ω n ( mr ) n = r m − m + mr + . . .(2.34)The coefficients of the grand potential can be written in closed form as ω n = − π (cid:88) k (cid:54) = (− ) k ( kπ ) n Γ (cid:18) n + (cid:19) Γ (cid:18) n − (cid:19) . (2.35)The double gamma function renders the matching with the general behavior of thehigher-order trans-series coefficients in Eq.(2.27) slightly less immediate. We canmake use of the identity2 n Γ ( n + ) Γ ( n − ) = (cid:114) π ∞ (cid:88) k = γ k Γ (cid:18) n − − k (cid:19) = (cid:114) π (cid:20) Γ ( n − ) + Γ ( n − ) + Γ ( n − ) + . . . (cid:21) ,(2.36)where the coefficients γ k can be computed recursively.This relation allows us to match with the generic large-order behavior Eq. (2.27),where z = / ( mr ) , obtaining β = b k = −
32 , A k = πk , S k πi a ( k ) = (− ) k + √ π γ ( π | k | ) , a ( k ) (cid:96)> = γ (cid:96) γ ( π | k | ) (cid:96) γ ,(2.37)10o that the non-perturbative corrections to the grand potential have the form ω ( m ) ⊃ √ rm (− ) k ( π | k | ) e −( π | k | ) rm ∞ (cid:88) (cid:96) = (cid:18) γ (cid:96) γ (cid:19) ( π | k | mr ) (cid:96) . (2.38)We find again the same structure of non-perturbative corrections that we had seenon the torus. They are controlled by the exponential of the typical length of themanifold 2 πr and the chemical potential µ .The coefficients γ (cid:96) are factorially growing and alternating in sign. They can beshown to appear in Henkel’s expansion of the modified Bessel function: K ( z ) ∼ (cid:114) π z ea − z ∞ (cid:88) (cid:96) = (cid:18) γ (cid:96) γ (cid:19) z (cid:96) as z → ∞ . (2.39)This fact will become relevant when we discuss Borel resummation.Now that we have understood the behavior of the grand potential, we can moveto the free energy. The Legendre transform can be computed order by order in ^ q starting from the perturbative part: ^ q = dd µ ω ( µ ) = ⇒ rm (^ q ) = ^ q / − ^ q − / + ^ q − / + . . . (2.40) f (^ q ) = µ ^ q − ω ( µ ) = ⇒ f (^ q ) = r ^ q / + r ^ q / − r ^ q − / + . . . (2.41)This is clearly an asymptotic series. For the scope of this paper, it is sufficient toconsider just the leading non-perturbative terms appearing in f (^ q ) and thus in thecritical exponents. These already give a high level of precision when matching to thesmall-charge result, and are obtained by using the leading-order approximation ofEq. (2.40). We find f (^ q ) ⊃ ^ q / r (− ) k ( π | k | ) e −( π | k | ) √ ^ q + . . . (2.42)This corresponds to a 2 n ! factorial divergence of the perturbative series of f (^ q ) , orequivalently of the critical exponents of the model (see Figure 1).From the point of view of the eft, the ( n ) ! divergence is a tree-level effect. Ifwe identify the grand potential with an effective action (see Section 5 for details),the Wilsonian coefficients form a divergent series. This is to be compared with the n ! divergence that we generically expect in qft from the proliferation of Feynmandiagrams (see Appendix C for a discussion of Lipatov’s instantons). In the regimeunder discussion, the classical divergence is more important than the quantum one.Having the explicit expressions for all the terms in the perturbative series, it ispossible to extrapolate the result to arbitrarily small charge if we manage to resum11 n f n + n f n S ( r n ) S ( r n ) S ( r n ) � �� �� �� �� ������������������������������������� Figure 1 – Ratio r n = n − f n + /f n for the coefficients in the expansion of the freeenergy, together with the first three Shank transforms [33] as function of n . The con-vergence at large n to a constant value of O ( ) indicates a double-factorial leadingbehavior f n ∼ ( n ! ) , as expected from the form of the non-perturbative contributions. the associated trans-series into a resurgent function. This can be done using Borelresummation, as shown in the next paragraph. Borel resummation.
In the first part of this section we have constructed the generalform of non-perturbative terms associated to our factorially divergent series that wehave assumed to be the asymptotics of a resurgent function. We are however still leftwith the problem of giving a meaning to the factorially divergent series we startedwith. The Borel resummation is a prescription that achieves this goal, systematicallyincorporating the non-perturbative terms that we have found (see Appendix B for ashort discussion).Let us start with the Borel transform of the heat trace expansion in Eq. (2.24). Inthe case of M = S we have the luxury of having a closed-form expression for theBorel transform using the definition B{ Φ ( ) } ( ζ ) = ∞ (cid:88) n = a n Γ ( n + / ) ζ n = √ π ζ sin ζ , (2.43)where we have used the Taylor expansion of1sin ( z ) = ∞ (cid:88) n = B n (− ) n ( − n − )( n ) ! z n − . (2.44) We divide out the prefactor r /t in order to work with power series with positive powers only. It can bere-introduced at the end of the analysis. This definition is equivalent to the one given in Appendix B up to the mapping ζ → ζ . In this way wehave a Borel transform without branch cuts. − π π − π π − π C + C − ζ Figure 2 – Pole structure and integration contours C ± for the lateral Borel transformof the trace of the heat kernel on S . The two choices differ by the residues at y = kπ , k =
1, 2, . . . . The appropriate Borel resummation in the direction θ = S{ Φ ( ) } ( z ) = √ z (cid:90) ∞ d ζ e − ζ /z B{ Φ ( ) } ( ζ ) = √ πz (cid:90) ∞ d ζ ζ e − ζ /z sin ζ . (2.45)This is the integral representation of the S heat trace originally found in [34] andrecovered here as a Borel integral.The Borel integral however is ill-defined: the integrand has simple poles onthe integration path for ζ = kπ , k = ∈ Z + , which indicate that the series is notBorel summable and θ = S ± { Φ ( ) } ( t ) , whose integration contours pass over or under the poles (see Figure 2).This introduces the following ambiguity in the summation: [ S + − S − ] { Φ ( ) } ( z ) = −( πi ) ∞ (cid:88) k = Res ζ = kπ (cid:32) √ πz ζ e − ζ /z sin ζ (cid:33) = iz (cid:16) πz (cid:17) / ∞ (cid:88) k (cid:54) = (−) k + | k | e − k π /z , (2.46)which matches exactly with Eq. (2.31). Each term corresponds to a pole on thepositive real axis in the Borel plane. For either choice of the contour, the path movesaway from the real axis and the integral picks up an imaginary contribution. Thus,the (ambiguous) trans-series associated to the heat trace isTr (cid:20) e (cid:16) (cid:52) − r (cid:17) t (cid:21) = r √ πt / (cid:90) C ± d ζ ζ e − ζ r /t sin ζ + i (cid:18) πr t (cid:19) (cid:88) k (cid:54) = σ ± k (− ) k + | k | e − k π r t .(2.47)An analogous treatment can be carried out for the grand potential starting fromthe perturbative coefficients in Eq.(2.35). However, the Borel transform of ω ( ) doesnot have a neat closed form. Instead, we apply the Mellin transform to the Borel13esummation of the heat kernel in Eq. (2.45) to find ζ ± ( s | S , m ) = r s Γ ( s ) (cid:90) ∞ d z z s − e − m r z (cid:104) S ± { Φ ( ) } ( z ) − − z (cid:105) + r m − s s − + m − s
12 .(2.48)The Mellin integral has been analytically continued so that it is convergent for s = − /
2. This allows us to exchange the integration order and obtain ω ± ( m ) = − ζ (cid:18) − (cid:12)(cid:12)(cid:12)(cid:12) S , m (cid:19) = r m − m + m rπ (cid:90) C ± d ζζ (cid:18) ζ sin ζ − − ζ (cid:19) K ( mrζ ) . (2.49)From this expression we can identify the (non-standard) Borel resummation of theseries ω ( ) with the integral S ± { ω ( ) } = (cid:90) C ± d ζζ (cid:18) ζ sin ζ − − ζ (cid:19) K ( mrζ ) . (2.50)It has a discontinuity, [ S + − S − ] { ω ( ) } = ∞ (cid:88) k = (− ) k k π K ( πkmr ) , (2.51)which turns out to be in perfect agreement with the result that we had found above inEq. (2.38). In the analysis of the heat kernel we had found that the non-perturbativecontributions are semiclassically exact, i.e. consist of only one term. The same appliesalso to the grand potential: we still have only one term but the non-perturbativecorrections contain a Bessel function instead of the usual instanton-like exponentialstypical of qft problems.The non-perturbative ambiguities from the Borel summation in the grand potentialand the free energy are related to the ones appearing in the heat trace in Eq. (2.47).They can be fixed in different ways: • Imposing the reality of the heat trace in Eq. (2.47) for t ∈ R + . In general this is notguaranteed to fix completely the parameters σ k , but in our case it turns out to besufficient (see Appendix E) and one finds σ ± k = ± /
2, which implies S k = k . The heat trace then readsTr (cid:20) e (cid:16) (cid:52) − r (cid:17) t (cid:21) = √ π (cid:18) r t (cid:19) (cid:90) C ± d ζ ζ e − ζ r /t sin ζ ± i (cid:18) πr t (cid:19) (cid:88) k (cid:54) = (− ) k + | k | e − k π r t = √ π (cid:18) r t (cid:19) P . V . (cid:34) (cid:90) C ± d ζ ζ e − ζ r /t sin ζ (cid:35) , (2.52)which is unambiguous and real, despite the appearances. This usually holds forvarious systems involving ordinary differential equations (odes) [35]. Even thoughthe heat trace is the solution of a non-linear partial differential equation (pde) (the heatequation) at coincident points, it is interesting to see how its trans-series structurecan be deduced from a linear Dawson’s ode. • Finding a (path-)integral definition of the heat trace, where a trans-series structurearises automatically from the semiclassical expansion around non-trivial saddlepoints. For ordinary integrals it has been shown in [36] that such a reality prescriptionis sufficient as it yields the Lefschetz thimble decomposition of the integral. Thesame authors have shown that in the case of path integrals, also unstable saddlesplay a role in the cancellation of ambiguities. We will pursue this direction furtherin the next section and show that a similar phenomenon arises in a path integralformulation of the heat trace.
It has been known since the work of Feynman that Green’s functions of ellipticoperators can admit a representation in terms of quantum mechanical path integrals.In the worldline formalism, see [37] and references therein, one constructs anappropriate quantum mechanics path integral which computes the determinant of agiven operator. It has been successfully applied to the computation of qft amplitudesand effective actions on classical field backgrounds. Heat kernels can be representedas the worldline integral of a free particle moving on the curved manifold of interest.However, quantum mechanics on curved space is not a trivial subject, and has beenknown since DeWitt [38] to be plagued with ambiguities related to the problemof defining path-integral measures on curved space. Most of the difficulties havebeen solved in the past years and have lead to a perturbative definition of suchpath integrals, see for example [39], which have been shown to match the first fewSeeley–DeWitt coefficients on general manifolds [40].In this section we will use the worldline approach to show that the trans-seriesEq. (2.52) can be obtained as a saddle-point approximation as t → + of an appropriatequantum mechanical path integral computing the heat trace. This results in an entirely15eometric interpretation of the non-perturbative terms and ambiguities that appearin the resurgent analysis. This same structure carries over from the heat trace to thegrand potential ω , and ultimately to the large-charge expansion of the conformaldimension in the double-scaling limit of the O ( N ) model. The starting point of the worldline approach to the calculation of functional determi-nants is Schwinger’s representation:log (cid:0) det (cid:0) − ∂ − (cid:52) + µ (cid:1)(cid:1) = − (cid:90) ∞ d tt e − µ t Tr (cid:16) e ( ∂ + (cid:52) ) t (cid:17) . (3.1)This functional determinant computes the grand potential ω ( µ ) , see Appendix A.On a product manifold such as R × M the trace in the Schwinger integral factorizes,so that one can study directly the heat trace on M :log (cid:0) det (cid:0) − ∂ − (cid:52) + µ (cid:1)(cid:1) = − (cid:90) ∞ d tt e − µ t √ πt Tr (cid:16) e (cid:52) t (cid:17) . (3.2)The idea is to interpret the heat trace as the partition function for a particle at inversetemperature t and Hamiltonian H = − (cid:52) , i.e. a free quantum particle moving on M [40–42]. If we take a coordinate system x µ on M then the classical action of the freeparticle is S [ X ] = (cid:90) t d τ g µν ( x ) ˙ x µ ( τ ) ˙ x ν ( τ ) , (3.3)where g µν is the metric on M and x µ : ( t ) → M is the worldline described by themotion of the free particle. The heat trace is then related to a path integral over closedloops (Feynman–Kac formula) of the form Tr (cid:104) e (cid:52) t (cid:105) ≡ (cid:90) x ( t )= x ( ) D x µ e − S [ x ] . (3.4)We will take this path integral as the definition of the heat trace. Note howeverthat the application of the Feynman–Kac formula is quite involved because of theintrinsic diffeomorphism invariance and of the ordering ambiguities introduced bythe curvature terms in the quantization of the Hamiltonian. All these issues havebeen resolved for the semiclassical expansion of Eq. (3.4) around the trivial loop x µ cl ( τ ) = t → + . This is in fact a semiclassical expansion: The action has a reparametrization invariance which we have fixed, which appears as a gauge-invarianceof the heat trace. It will not play a role in our computation. τ → τt , the action can be rewritten as S [ x ] = t (cid:90) d τ g µν ( x ) ˙ x µ ˙ x ν , (3.5)and the small- t expansion of the heat kernel corresponds to the expansion in (cid:32) h inthis quantum mechanical system. In this limit, the path integral localizes around thesaddle points of S [ x ] and we can expand it perturbatively in powers of t .The Euler–Lagrange equations for our action are the geodesic equations¨ x µ cl + Γ µνρ ( x ) ˙ x ν cl ˙ x ρ cl =
0, (3.6)so that the heat trace path integral localizes on a sum over all the closed geodesics γ on M . These non-trivial geodesics are the equivalent of the worldline instantonsin [24, 25] which govern the non-perturbative contributions to Euler–Heisenberg-typeLagrangians [23].Ordinary instanton calculus shows that, in general, each of these saddles willcome with its own perturbative series in t , weighted by e − (cid:96) ( γ ) / ( t ) , where (cid:96) ( γ ) isthe length of γ , so that the semi-classical expansion has the general formTr (cid:104) e (cid:52) t (cid:105) = t − b ∞ (cid:88) n = a ( ) n t n + (cid:88) (cid:48) γ ∈ closed geodesics e − (cid:96) ( γ ) t t − b γ ∞ (cid:88) n = a ( γ ) n t n , (3.7)where the sum runs over the non-trivial geodesics, and the b γ depend on thegeometry. The sequences a ( γ ) n are generally expected to be factorially growing byusual Feynman graph proliferation arguments.The similarity with the structure of the generic trans-series in Eq. (2.19) is not acoincidence: the latter were introduced to match semiclassical expansions, wherethey appear naturally. However, there is a conceptual difference with respect to theresurgent analysis carried on in the previous section. Resurgence does not rely onthe existence of a non-perturbative definition of the observable we want to compute(such as a path integral definition). For this reason there is in general no geometricinterpretation of the trans-series structure and there are ambiguities which cannot befixed a priori. In our case we have shown that the ambiguities can be removed byjust imposing the reality of the end result. In the following we will see that the pathintegral definition reproduces the result Eq. (2.52) without further input. In this section we review the derivation of the heat trace on T that we have found inEq. (2.15) via a path integral prescription.17onsider a square torus of side L , with metricd s = g ij d x i d x j = ( d x ) + ( d x ) . (3.8)Since the torus is flat, no subtleties related to curved space arise and our workingdefinition in Eq. (3.4) is correct. The heat trace is defined via the path integralTr (cid:16) e (cid:52) t (cid:17) = (cid:90) x ( t )= x ( ) D x e − t (cid:82) d τ (( ˙ x ) +( ˙ x ) ) . (3.9)In the limit t → + it can be computed semiclassically via saddle-point approximation.We represent the torus as R with the identifications x i (cid:39) x i + L . If we fix apoint, say the origin, we obtain a lattice Z of equivalent points. The closed geodesicspassing through this point are straight lines joining the origin with another point ofthe lattice (see Figure 3). This means that closed geodesics can be labeled by pairs ofintegers ( k , k ) (this includes the trivial geodesic of zero length). The correspondinglength is simply (cid:96) ( k , k ) = L (cid:113) k + k . (3.10)The field in the path integral can be decomposed as the sum of the classicalsolutions X i cl and fluctuations h i ( τ ) , X i ( τ ) = X i cl ( τ ) + h i ( τ ) = k i Lτ + h i ( τ ) . (3.11)The action is just quadratic since T is flat, and we can separate the two contributions: S [ X ] = S [ X cl ] + S [ h ] = L ( k + k ) t + t (cid:90) d τ (cid:104) ( ˙ h ) + ( ˙ h ) (cid:105) . (3.12)Then the path integral becomesTr (cid:16) e (cid:52) t (cid:17) = (cid:90) T d x (cid:90) X ( )= X ( )= x D X e S [ X ] = L (cid:88) k ∈ Z e − L ( k + k ) t (cid:90) h ( t )= h ( )= D h e − S [ h ] ,(3.13)where we have added the integral over the fixed point x through which each geodesicpasses. The remaining path integral is Gaussian and can be defined, for examplein zeta-function regularization, up to a normalization constant which is chosen toreproduce the standard worldline normalization (cid:90) h ( t )= h ( )= D h e − t (cid:82) d τ ˙ h = √ πt . (3.14)18 igure 3 – Closed geodesics on the torus labeled by the integers (
1, 3 ) (blue) and (
5, 2 ) (red) as segments in R and on a three-dimensional rendering. The final result is Tr (cid:16) e (cid:52) t (cid:17) = L πt (cid:88) k ∈ Z e − L (cid:107) k (cid:107) t , (3.15)which is precisely the right-hand side of the Poisson resummation formula used inEq. (2.15).This correspondence is known in the literature as spectrum-geodesic duality oncompact manifolds. For each eigenvalue of the Laplacian there is a correspondingclosed geodesic, see [42, 43] for a review. There exist (Selberg-like) trace formulaswhich realize this duality for the heat trace, and the right hand-side can be interpretedas a saddle-point approximation of an appropriate path integral. In the torus case,the duality is realized via Poisson resummation. In the case of M = S the subtleties related to curved space play a role. As mentionedat the beginning of the section, we will employ a “natural” generalization of flatspace path integrals and define (cid:104) y | e (cid:52) t | x (cid:105) ≡ (cid:90) x ( t )= yx ( )= x D x µ e − t (cid:82) d τ g µν ( x ) ˙ x µ ˙ x ν . (3.16)We will use standard polar coordinates on the sphere x µ = ( θ , φ ) so that the volumeelement is D x µ = sin ( θ ) D θ D φ . These will be our worldline fields, with action andequations of motion (eom) given by: S = r t (cid:90) d τ (cid:2) ˙ θ + sin θ ˙ φ (cid:3) , (3.17) ¨ φ + ( θ ) ˙ θ ˙ φ = θ − ˙ φ sin ( θ ) =
0. (3.18)19s we are ultimately interested in the heat trace we consider coincident endpoints x µ ( ) = x µ ( ) = ( π/
2, 0 ) . There are infinitely many winding geodesics which solve the eom which areparametrized as θ cl ( τ ) = π/ φ cl ( τ ) = πkτ , k ∈ Z , (3.19)where we allow φ to extend beyond its domain with the identification φ ∼ φ + π .If we introduce the fluctuations h θ , h φ around these solutions, satisfying Dirichletboundary conditions, we can formally rewrite the heat kernel at coincident points as (cid:104) x | e ( (cid:52) ) t | x (cid:105) = e − ( πkr ) t (cid:90) h i ( t )= h i ( )= D h θ D h φ e − r t (cid:82) d τ [ ˙ h θ −( πk ) h θ + ˙ h φ + O ( h )] .(3.20)Standard loop-diagram counting arguments show that the O (cid:0) h (cid:1) (the “interactionterms”) gives contributions that are higher order in t . The leading order is obtainedby computing the appropriate functional determinant from the quadratic action ofthe fluctuations.The integral in h φ is massless and reproduces our normalization convention (3.14)with the substitution t → t/r .The integral over h θ , on the other hand, contains a zero-mode and multiplenegative modes. To see that, we expand the fluctuations in an orthonormal basis ofmodes satisfying h θ ( τ ) = (cid:88) n = c n h nθ ( τ ) , − r t (cid:2) ∂ τ + ( πk ) (cid:3) h nθ ( τ ) = λ n h nθ ( τ ) , (3.21)which can be written explicitly as h nθ = √ ( πnτ ) , λ n = π r t (cid:0) n − k (cid:1) . (3.22)The integration measure is then written in terms of Fourier modes as (cid:90) D h θ ≡ ∞ (cid:89) n = (cid:90) d c n √ π . (3.23)For n = k there is a zero mode that needs to be treated separately, while the ( k − ) modes h n< kθ are tachyonic. The presence of negative modes is not surprising,since π ( S ) =
0. These winding geodesics are not stable as they can be contracted to Note that the action Eq. (3.16) in these coordinates is not rotationally invariant in θ as there are twosingular points (the poles). However, this appears only at higher order in t , so that the leading order in t → + is fine. igure 4 – Unstable mode n = (left), zero mode n = (middle) and massive mode n = (right) for the fluctuations around a geodesic winding once around the sphere(red). a point (see Figure 4), in contrast to what happened in the torus case, where windinggeodesics are topologically stable saddles.The zero mode corresponds to a rigid rotation of the sphere, which is a symmetryof the quadratic action. In fact, it is easy to see that θ α cl = π + α sin ( πkτ ) , φ cl = πkτ , (3.24)is a family of solutions of the eom (3.18) at leading order in the modulus α ∈ ( π ) .This is exactly the fluctuation h kθ , and using the rules of instanton calculus we cantrade the integral over the mode amplitude c k for an integral over the modulus α ,which parametrizes the family of solutions: (cid:90) d c k √ π = (cid:114) (cid:90) d α √ π = √ π (cid:48) (cid:18) − r t (cid:0) ∂ τ + ( πk ) (cid:1)(cid:19) − = √ π | k | r √ t det (cid:18) − r t ∂ τ (cid:19) − det (cid:48) (cid:18) Id + π k ∂ τ (cid:19) − ,(3.26)where we divided and multiplied by the n = k eigenvalue of ∂ τ in order to recoverour initial normalization. The remaining determinant does not need regularizationand readsdet (cid:48) (cid:18) Id + π k ∂ τ (cid:19) − = e iπνq ∞ (cid:89) n = n (cid:54) = k (cid:12)(cid:12)(cid:12)(cid:12) − k n (cid:12)(cid:12)(cid:12)(cid:12) − = √ e iπνq , (3.27) No multiplicative anomaly arises in this determinant splitting [44]. k > (cid:89) n (cid:54) = k (cid:12)(cid:12)(cid:12)(cid:12) − k n (cid:12)(cid:12)(cid:12)(cid:12) = k − (cid:89) n = (cid:18) k n − (cid:19) ∞ (cid:89) n = k + (cid:18) − k n (cid:19) = Γ ( k ) kΓ ( k ) Γ ( k + ) Γ ( k + ) =
12 .(3.28)We have introduced the Morse index ν q which arises naturally for functionaldeterminants with q negative modes [45], and can be interpreted as an analog of theintersection numbers arising in the Lefschetz thimble decomposition of ordinaryintegrals. In our computation, it is related to the fact that Gaussian integrals fornegative modes have a two-fold ambiguity in their analytic continuation: (cid:90) d c n √ π e λ n c n = e ± i π √ λ n . (3.29)In the computation of the determinant of Eq. (3.28) we have factored out these q = k − e iπνq = ( ± i ) k − = ∓ i (− ) k . (3.30)Putting all these results together, we obtain the saddle-point approximation ofthe heat trace:Tr (cid:104) e (cid:52) t (cid:105) = r t ( + O ( t )) ± i (cid:18) πr t (cid:19) (cid:88) (cid:48) k ∈ Z (− ) k + | k | e − k π r t ( + O ( t )) . (3.31)This expression matches exactly the unambiguous trans-series representation inEq. (2.52).Our prescription does not allow us to compute next-to-leading order contributionswithin each sector, which would also include the conformal coupling. However, thereis a prescription that computes the Seeley–DeWitt coefficients in the perturbativesector via a diagram expansion, see for example [46]. To our knowledge, there hasbeen no attempt to reproduce the fluctuations around non-trivial sectors. Whileno fundamental obstructions are expected, to obtain a one-loop exact structure ofthe form suggested by the non-perturbative corrections in Eq. (2.52) would requirehighly non-trivial cancellations in the absence of supersymmetry.While this is beyond the scope of this work, the matching we have obtainedpermits us to draw interesting conclusions: • The trans-series structure of the heat trace on S , and consequently of the scalingdimensions of large-charge operators of the O ( N ) model, is entirely determined by22eometric considerations. • The saddles driving factorial growth in the large-charge expansion do not needto be stable. While topological arguments are usually taken as a guideline to thenon-perturbative structure of the model, resurgent asymptotics can be driven also bysaddles in the same topological class, such as great circles on S . This was alreadyobserved for two-dimensional field theories in [36]. • The presence of negative modes and the choice of the continuation in Eq. (3.29)giving rise to the non-trivial Morse index are a geometric realization of the Borelambiguity in the resurgence computation of Eq. (2.52). Different choices of thesephases correspond to the different paths that avoid singularities in the Borel plane inall possible ways. This is ultimately related to the Lefschetz thimble decompositionof (path-)integrals [47].
Up to this point we have discussed the large- ^ q limit and its resurgence properties. Infact, in the double-scaling limit, the small-charge regime is directly accessible: onecan write a convergent expansion of the grand potential as a function of µ . Usingthe definition of the zeta function, the grand potential on the sphere (see Eq. (2.5))becomes [6] ω ( µ ) = − ζ (− | S , µ ) = − ∞ (cid:88) l = ( l + ) (cid:18) l ( l + ) r + µ (cid:19) − s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s =− / = − r − s ∞ (cid:88) k = (cid:18) − sk (cid:19) ζ ( s + k − ) (cid:18) µ r − (cid:19) k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s =− / , (4.1)where ζ ( s , a ) is the Hurwitz zeta function ζ ( s , a ) = ∞ (cid:88) n = ( n + a ) − s . (4.2)Once again we notice that the correct expansion variable is m = µ − r . Theformula can be rewritten in terms of Bernoulli numbers: ω ( m ) = rm ∞ (cid:88) k = (− ) k (cid:18) / k + (cid:19) ( π ) k ( k − ) B k ( k ) ! ( rm ) k . (4.3)This expansion is convergent rather than asymptotic, with radius of convergence | rm | < /
2. This corresponds to the value µ =
0, which turns the (cid:96) = m = − / ( r ) . For thisreason the expansions for m → + and m → ∞ admit a smooth interpolation forall m > µ , ^ q has to be solved order by order. We give herethe first few orders, which can be computed analytically in closed form: rf (^ q ) = ^ q + ^ q π + (cid:0) π − (cid:1) ^ q π + (cid:0) − π + π (cid:1) ^ q π . . . (4.4) rµ = + ^ qπ + (cid:0) π − (cid:1) ^ q π + (cid:0) − π + π (cid:1) ^ q π + . . . (4.5)The free energy is also a convergent expansion, with estimated radius of convergence | ^ q | (cid:47) ^ q so thatthe small and large-charge regimes can be connected without obstructions. Thisexpansion can be in principle computed within perturbation theory around theuncharged vacuum of the O ( N ) cft. It turns out to be in general asymptotic, butbecomes convergent at leading order in large N .This result is to compared with our expression for the grand potential in Eq. (2.49)that we can put into the form ω ( m ) = P . V . (cid:20) rm π (cid:90) ∞ d ζ K ( mrζ ) ζ sin ( ζ ) (cid:21) . (4.6)In Figure 5 we compare the real and imaginary part of the lateral Borel summationof the grand potential to the convergent small-charge expansion and the worldlinecomputation of the exponential correction. The two approaches agree completely atthe level of resolution of our numerical computation. For example at mr = ^ q (cid:39) rω ( mr = ) (cid:12)(cid:12)(cid:12)(cid:12) small charge = rω ( mr = ) (cid:12)(cid:12)(cid:12)(cid:12) resurgence = × − , which is six orders of magnitude smaller than theexponential correction e − π × ≈ × − . This is strong evidence that we havetaken all non-perturbative corrections into account.24 igure 5 – Real and imaginary part of the lateral Borel summation for the grand po-tential (dots) compared to the small-charge expansion and the exponential correctionsfrom worldline instantons (continuous line) as function of mr . The two approachesagree completely at the level of resolution of our numerical computation. The small-charge expansion is not convergent in the shaded region. In this section we would like to use the exact results that we have obtained in thedouble-scaling limit [6] to derive some general features of the large-charge expansion.In the eft approach to strongly coupled cfts at large charge, one studies the systemon a manifold M of typical length L and considers sectors of fixed charge Q . Thisallows us to write an eft with a cutoff Λ that is bounded by the energy scales Λ low = /L fixed by the geometry, and Λ high = Q /d /L fixed by the charge density.If the hierarchy 1 L (cid:28) Λ (cid:28) Q /d L (5.1)is satisfied, the eft is weakly coupled and controlled by the ratio of the two scales1 /Q /d . Lorentz invariance fixes the actual coupling to be 1 /Q /d . Scale invariancefixes the dimension of the terms in the action: for a class of systems that includesthe wf point, the eft is written in terms of a dimensionless field χ to be expandedaround the ground state at fixed charge χ = µt . The action takes the form L eft = ω ( ∂ µ χ ∂ µ χ ) / + ω ( ∂ µ χ ∂ µ χ ) / + . . . , (5.2)where we have specialized to the case of 2 + /Q . The energy on the two-sphere for example, which viathe state-operator correspondence is proportional to the conformal dimension of thelowest operator of charge Q , takes the form E = r (cid:16) f Q / + f Q / + . . . (cid:17) , (5.3)where the coefficients f n cannot be computed within the eft: they enter as an inputand have to be computed independently, e.g. in a double-scaling limit [26, 48–59] or25n the lattice [4, 5].In our double-scaling limit, the quantities that we have studied above correspondto the minimal energy configuration for this action. The grand potential is the actionevaluated at the minimum: ω ( µ ) = S eft (cid:12)(cid:12)(cid:12)(cid:12) χ = µt (5.4)and the free energy is the corresponding ground state energy. From the formula forthe free energy in Eq. (2.41) we can infer that in the large-N limit of the O ( N ) wfpoint, the coefficients f n are given by f = ( N ) / , f = ( N ) / f = − ( N ) / , (5.5)and so on.Even if these coefficients cannot be computed within the eft, we can still use ournon-perturbative analysis to say something about their generic large-order behavior.To do so, we make the following assumptions for any N :i. the large-charge expansion is asymptotic;ii. the leading singularity in the Borel plane can be obtained as the first non-trivialsaddle of a worldline integral for a particle of mass µ .This second assumption relies on the fact that we are describing a cft which hasno intrinsic scales. The only dimensionful parameter is related to the fixed chargedensity. This means that we expect the conformal dimensions to take the form of a doubleexpansion in 1 /Q and e − πrµ : ∆ ( Q ) = Q / (cid:88) n f n Q n + C Q κ e − πf √ Q (cid:88) n f ( ) n Q n/ + . . . , (5.6)where C and κ are constants, and we have used the fact that µ and Q are (Legendre)dual variables: µ = r δ∆ ( Q ) δQ = r f Q / + . . . (5.7)Note that the second series is an expansion in 1 /Q / , in agreement with the explicitexpression found in Eq. (2.38). This would still leave room for the possibility of the particle mass being proportional to µ . In the large- N limit, this proportionality factor is one. If we look at the system that we have discussed in the previoussection as the limit of a set of finite- N efts on the sphere, the consistency of the large- N expansionsuggests that the proportionality factor remains one for any N . A first step in verifying this conjecturewould be to compute the 1 /N corrections in the double-scaling limit. This trans-series structure, with expansion in Q / in the non-perturbative sector, is also found in theLarge- N asymptotics in matrix models [60, 61]. We thank Gerald Dunne for bringing this fact to ourattention. /Q is related to theparameters appearing in the non-perturbative part. For this reason we can turn ourconjecture on the form of the exponential term into a universal prediction for thelarge-order behavior of the f n . In general, if the f n diverge at large n as f n ∼ ( βn ) ! A n (5.8)one obtains an optimal truncation of the perturbative series for the value N ∗ corre-sponding to a minimum of f n Q − n , which in this case is N ∗ ≈ β | AQ | /β . (5.9)The error is of order (cid:15) ( Q ) ∼ e −( AQ ) /β . (5.10)In our problem we can invert the usual logic. As we assume the form of the(leading) non-perturbative terms to be the same for any N , from Eq. (5.6) we read β = A = π f , (5.11)which preserves the asymptotic growth as ( n ) !: f n ∼ ( n ) ! ( πf ) n , (5.12)which is optimally truncated at N ∗ ≈ πf Q / . (5.13)The ( n ) ! divergence is semi-classical and is the leading contribution to thenonperturbative effects. The usual instantons associated to quantum correctionsgrow as n ! and are of order e − Q / , see Appendix C. Note that there is an interplaybetween the small- n and large- n coefficients. The non-perturbative expansion isassociated to the large- n behavior of the f n via resurgence and to the small- n coefficients via the eom in Eq. (5.7). This is the reason why we can write the optimaltruncation in terms of the lowest coefficient f .This analysis helps to shed some light on the lattice results concerning the O ( ) and O ( ) model [4, 5]. It was remarked in these papers that the large-chargeexpansion remains very good also for small values of Q and that a few terms aresufficient to predict the conformal dimensions of the lowest operators. Genericallywe expect f to be of order one. In fact, lattice estimates for the O ( ) and O ( ) model27ive, respectively, f ∼ ( ) and f ∼ ( ) . Under the previous assumptions,our formula predicts that the optimal truncation is for N ∗ = O (cid:0) √ Q (cid:1) with an error oforder O (cid:16) e − π √ Q (cid:17) . This is perfectly consistent with the numerical results. In [4, 5] ithas been observed that the first two or three terms in the expansion are sufficientto reproduce the lattice results with great accuracy for charges up to Q = O ( ) .At Q =
1, the error is of order O (cid:0) − (cid:1) and at Q =
11, it is of order O (cid:0) − (cid:1) , to becompared to e − π ≈ × − and e − π √ ≈ × − . In this note, we have used resurgence methods to analyze and extend the large-chargeexpansion of the O(2N) vector model in three dimensions at the Wilson–Fisher pointusing results from the double-scaling limit Q → ∞ , N → ∞ , with Q/ ( N ) = ^ q constant. We have studied in detail the cases of the model compactified both onthe two-torus and the two-sphere which, via the state-operator correspondenceallows to compute the conformal dimension of the lowest operator of charge ^ q . Inboth cases we have computed both the perturbation series and the exponentiallysuppressed non-perturbative corrections. In the case of the sphere, resurgence leavesus with an ambiguity in the non-perturbative contribution, which we are able toresolve in two ways: via a (simpler) resurgent analysis of the Dawson’s function,and using a geometric interpretation in terms of quantum mechanics of particlespropagating along geodesics on the compactification manifold. The latter methodprovides an interesting geometric interpretation of the non-perturbative correctionsof the problem as well of its Borel ambiguities and allows us to propose an exact formof the grand potential, valid for any value of ^ q , which we verify to high precisionnumerically.The fact that the non-perturbative corrections are finite-volume effects relatedto the geometry of the compactification manifold seems to be a robust featureindependent of the double-scaling limit. This motivates us further to extend ourresults to the regime of large charge but finite N, conjecturing that the large-chargeexpansion is always asymptotic and giving an optimal truncation of N ∗ ≈ πf Q / with an error of order (cid:15) ( Q ) = O (cid:16) e − πf √ Q (cid:17) — a result consistent with the latticeresults of [4, 5]. The nonperturbative contributions that we find are due to the eftitself being an asymptotic expansion and dominate over ordinary qft instantons.Our observations lead the way to a number of further applications of resurgencein the context of large charge. The first natural extension is to d (cid:54) = ε -expansion [26, 49–59]. In the same spirit, it would be interesting toperform a resurgence analysis of supersymmetric systems, both at large R-chargeand in the double-scaling limit g → Q → ∞ [18–21].Probably the most interesting issue is the one of the behavior of the large-chargeexpansion of a generic model. Here we have conjectured a precise relationshipbetween the perturbative eft and the leading non-perturbative corrections. It wouldbe most interesting to prove (or falsify) this and to explore the applicability of thistype of reasoning to more general efts. Acknowledgments
We would like to thank Luis Álvarez-Gaumé, Daniele Dorigoni, Gerald Dunne,Simeon Hellerman, Marcos Mariño, and Donald Youmans for illuminating discus-sions and comments on the manuscript.The work of S.R. is supported by the Swiss National Science Foundation undergrants number 200021 192137 and PP00P2183718/1. D.O. acknowledges partialsupport by the nccr 51nf40–141869 “The Mathematics of Physics” (Swissmap).
A Large charge at large N
In this appendix, we briefly summarize the results of [6] (for an extended discussion,see the original paper). Our starting point is the action for the O ( N ) vector modelin 2 + S [ ϕ , λ ] = N (cid:88) i = (cid:90) d t d M [ ∂ µ ϕ ∗ i ∂ µ ϕ i + ( ξR + λ ) ϕ ∗ i ϕ i ] , (A.1)where M is the compact two-dimensional manifold of volume V on which weare working, R is its Ricci-scalar, ξ = / λ is theLagrange-multiplier that was promoted to a field with a Hubbard–Stratonovichtransformation [62, 63].We want to calculate the canonical partition function for the case that we fixedthe charge corresponding to the Cartan generator which rotates the field ϕ N , whichcorresponds to restricting our attention to the completely symmetric representationof rank Q (for an extended discussion of charge fixing in the O(2N) vector model, seeSection 4.1 in [3], see also [26]). We have Z ( Q ) = Tr (cid:16) e − T H δ ( Q − ^ Q ) (cid:17) = (cid:90) d θ π e iθQ D ϕ i D ϕ ∗ i D λ e − S θ [ ϕ , λ ] , (A.2)29here S θ [ ϕ , λ ] = (cid:90) d t d M (cid:34) N − (cid:88) i = ∂ µ ϕ ∗ i ∂ µ ϕ i + ( D µ ϕ N ) ∗ D µ ϕ N + ( ξR + λ ) N (cid:88) i = ϕ ∗ i ϕ i (cid:35) (A.3)with D µ ϕ = ( ∂ + iT θ ) ϕ∂ i ϕ , (A.4)where T is the inverse length of the time circle. The integral of the first N − ϕ i is quadratic and can be performed to obtain an effective action for the fields ϕ N and λ : Z ( Q ) = (cid:90) d θ π e iθQ D ϕ N D ϕ ∗ N D λ e − S θ [ ϕ N , λ ] , (A.5)where S [ ϕ N , λ ] = ( N − ) Tr (cid:2) log (cid:0) − ∂ − (cid:52) + ξR + λ (cid:1)(cid:3) ++ (cid:90) d t d M [( D µ ϕ N ) ∗ D µ ϕ N + ( ξR + λ ) ϕ ∗ N ϕ N ] . (A.6)The path integral localizes around the saddle point obtained from minimizing theaction w.r.t. θ and the zero modes of ϕ N and λ . We expand the fields into vev andfluctuations: ϕ N = A √ + u , λ = µ − ξR + ^ λ = m + ^ λ . (A.7)The parameter µ is the mass with respect to the Laplace–Beltrami operator (cid:52) , while m is the mass with respect to the conformal Laplacian (cid:52) − ξR .The free energy is obtained by minimizing S Q = − iθQ + ( N − ) Tr (cid:104) log (cid:16) − ∂ − (cid:52) + µ + ^ λ (cid:17)(cid:105) ++ (cid:90) d t d M (cid:34) ( D µ u ) ∗ D µ u + A θ T + ( µ + ^ λ ) (cid:12)(cid:12)(cid:12)(cid:12) A √ + u (cid:12)(cid:12)(cid:12)(cid:12) (cid:35) , (A.8)setting the fluctuations to zero: S saddle Q = − iθQ + ( N − ) Tr (cid:2) log (cid:0) − ∂ − (cid:52) + µ (cid:1)(cid:3) + T V A (cid:0) θ T + µ (cid:1) . (A.9)30his last expression needs to be minimized: ∂ θ : − iQ + VA θT = ∂ µ : ( N − ) ∂∂µ Tr (cid:2) log (cid:0) − ∂ − (cid:52) + µ (cid:1)(cid:3) + T VA µ = ∂ A : θ T + µ =
0, (A.10)which we can rearrange as θT = iµ , Q = − iVA θT = − VA µ = ( N − ) ∂ µ Tr (cid:2) log (cid:0) − ∂ − (cid:52) + µ (cid:1)(cid:3) . (A.11)If we take the double scaling limit Q → ∞ , N → ∞ , Q N = ^ q fixed, (A.12)the first line of Eq. (A.8) dominates and the path integral localizes around the saddle.The free energy at the saddle takes the form F (^ q ) = − T log ( Z ( Q )) = N (cid:20) µ ^ q + T (cid:0) det (cid:0) − ∂ − (cid:52) + µ (cid:1)(cid:1)(cid:21) + O (cid:0) N (cid:1) , (A.13)where µ is to be understood as a function of ^ q via ^ q = ∂∂µ (cid:18) T (cid:0) det (cid:0) − ∂ − (cid:52) + µ (cid:1)(cid:1)(cid:19) . (A.14)One can read these equations as the free energy being the Legendre transform ofthe functional determinant, which is naturally identified with the grand potential(Landau free energy) f (^ q ) = F (^ q ) N = sup µ ( µ ^ q − ω ( µ )) , (A.15)where ω ( µ ) = − T (cid:0) det (cid:0) − ∂ − (cid:52) + µ (cid:1)(cid:1) . (A.16)We are interested in studying the theory on different manifolds, so it is convenientto write the functional determinant in terms of zeta functionslog (cid:0) det (cid:0) − ∂ − (cid:52) + µ (cid:1)(cid:1) = − dd s ζ ( s | S × M , µ ) (cid:12)(cid:12)(cid:12)(cid:12) s = , (A.17) Since we are working at leading order in N it is convenient to use the free energy and grand potentialper degree of freedom. The corresponding extensive quantities are related by a similar Legendretransformation F ( Q ) = sup µ ( µQ − Ω ( µ )) . ζ ( s | S × M , µ ) is the zeta function that we can write as a Mellin integral: ζ ( s | M , µ ) = Γ ( s ) (cid:90) ∞ d tt t s Tr (cid:16) e ( ∂ + (cid:52) − µ ) t (cid:17) (A.18)so that log (cid:0) det (cid:0) − ∂ − (cid:52) + µ (cid:1)(cid:1) = − (cid:90) ∞ d tt Tr (cid:16) e ( ∂ + (cid:52) − µ ) t (cid:17) . (A.19)Using the fact that our geometry is a product S /T × M , we can separate the S part.The corresponding heat kernel trace is a theta function:Tr (cid:16) e ∂ t (cid:17) = (cid:88) n ∈ Z e − π n T t = θ ( e − π T t ) = T √ πt (cid:32) + (cid:88) (cid:48) k ∈ Z e − k T t (cid:33) , (A.20)where the prime indicates that the sum runs over the non-zero modes. At zerotemperature T → (cid:16) e ∂ t (cid:17) ∼ T √ πt . (A.21)Then we can write the functional determinant in terms of the Laplace transform ofthe heat kernel on M :log (cid:0) det (cid:0) − ∂ − (cid:52) + µ (cid:1)(cid:1) = − T √ π (cid:90) ∞ d tt t − / e − µ t Tr (cid:16) e (cid:52) t (cid:17) = T ζ (− |M , µ ) .(A.22)The grand potential is ω ( µ ) = − ζ (− |M , µ ) , (A.23)and the free energy is F (^ q ) = N (cid:20) µ ^ q + ζ (− |M , µ ) (cid:21) . (A.24) µ is the solution to ^ q = − µζ ( |M , µ ) , (A.25)where we have used the difference-differential equationdd µ ζ ( s |M , µ ) = − µζ ( s + |M , µ ) . (A.26)These expressions are exact in the limit of N → ∞ for any value of ^ q = Q/N and canbe interpreted as a semiclassical resummation of infinite 1 /N corrections.32 The Borel transform
The
Borel transform is an operation acting in the space of power series as follows: Φ ( z ) ∼ ∞ (cid:88) n = a n z n −→ B { Φ } ( ζ ) = ∞ (cid:88) n = a n Γ ( βn + b ) ζ n , (B.1)where we assumed a generic large order behavior of a k as in Eq. (2.27) with b ≡ max { b k } , β ≡ max { β k } . This ensures that the series defined by B { Φ } isconvergent in a disc centered at the origin of the ζ -plane, also denoted as Borel plane .The analytic properties of the Borel transforms can be directly inferred from Eq. (2.27).For example, if we have a n Γ ( βn + b ) ∼ A βn + b then B{ Φ } ( ζ ) ζ → A β −−−−→ A − b − ζ /A β + regular. (B.2)It is then possible to define the Borel resummation of Φ as S{ Φ } ( z ) = β (cid:90) ∞ d ζζ (cid:18) ζz (cid:19) bβ e −( ζ/z ) /β B{ Φ } ( ζ ) . (B.3)Using the definition of Gamma function, it is evident that S { Φ } ( z ) ∼ Φ ( z ) as z → + .However, if the integral is well-defined, it defines a function computable for allvalues of z , which represents the “sum” of the divergent series Φ . This function isunambiguous unless B{ Φ } presents singularities along the integration path. Inthis case one needs to define a directional summation S θ by integrating along theray with angle θ in the Borel plane: the formula in Eq. (B.3) corresponds to the case θ = θ along which B{ Φ } has singularities is a Stokes ray and the Borel re-summation becomes ambiguous there. One then defines lateral summations S θ ± by deforming the contour of integration to avoid the singularities. This indicatesthe emergence of a branch cut for Arg ( z ) = θ for the Borel resummation, with adiscontinuity computed as [ S θ + − S θ − ] { Φ } = −[ S θ − ◦ Disc θ ] { Φ } . (B.4)This discontinuity is purely non-perturbative, as one can see from the example inEq. (B.2) where θ = : [ S + − S − ] { Φ } = − πiβ z − b/β e − A/z /β = − Disc { Φ } . (B.5) The behavior at ζ → ∞ is relevant as well, but always regular enough for the cases we study. Notice that S acts formally as an operator on power series with integer positive powers only, so thathere we use S − { } = Φ ( ) , the quantity Disc { Φ ( ) } pro-vides the structure of non-perturbative terms that one has to add to the trans-seriescompletion in Eq. (2.19), and the large-order behavior in Eq. (2.27) follows apply-ing Cauchy’s integral representation. The lateral Borel summation of the generaltrans-series is defined as S θ ± { Φ } ( σ k , z ) = S θ ± { Φ ( ) } ( z ) + (cid:88) k σ ± k e − A k /z /βk z − b k /β k S θ ± { Φ ( k ) } ( z ) . (B.6)This will not define a unique resurgent function for Arg ( z ) = θ unless we somehowfix the (non-perturbative) ambiguity related to the integration path. This can bedone by imposing extra conditions on the lateral Borel sums, which determine thetrans-series parameters σ ± k in such a way that the ambiguity is removed. When thisis possible, we achieved “semiclassical decoding” in the terminology of [64]. C Lipatov’s instantons
The non-perturbative effects that we have discussed in this work come from thefact that the effective action is by itself an asymptotic expansion, meaning that itsWilson coefficients are responsible for the factorial growth. This is different fromthe ordinary factorial growth related to the presence of instantons, and manifestthemselves in the diagram proliferation in the quantum perturbative expansion. Herewe follow the discussion of Lipatov [65] to show that these effects are subleadingwith respect to the ones we have discussed.Consider the leading term in the effective action in Eq. (5.2), L = ω (cid:107) d χ (cid:107) . (C.1)To quantize we expand around the fixed-charge solution χ = µt . It is convenient toexpand χ into ground state and massless fluctuations ^ χ , rescaled as χ ( t , x ) = µt + µ ^ χ ( t , x ) . (C.2)The action is then L ( χ ) = ω µ (cid:2) ( + ∂ ^ χ ) − ( ∂ i ^ χ ) (cid:3) / ≡ ω µ L (^ χ ) . (C.3)Perturbation theory around the solution ^ χ = g ≡ / ( ω µ ) sincepropagators are proportional to 1 /g and any vertex carries a factor of g . The34grand-canonical) partition function is a series in g : Z ( g ) = (cid:90) D ^ χ e − g S [^ χ ] = ∞ (cid:88) n = Z n g n . (C.4)We can estimate the leading behavior of the Z n for large n the expression in terms ofa contour integral around g = Z n = πi (cid:73) d gg Z ( g ) g n . (C.5)For n → ∞ the integral can be solved via saddle point approximation for both thepath integral and the contour integral: Z n = πi (cid:73) d gg (cid:90) D ^ χ exp (cid:20) − g S [^ χ ] − n log ( g ) (cid:21) . (C.6)The saddle point equations are δS [^ χ ] δ ^ χ = g S [^ χ ] − ng =
0. (C.7)Assuming the existence of a finite-action solution to these equations, the value ¯ S of S [^ χ ] at the saddle does not depend on n or on g , so the second equation implies thatat the saddle g ∼ /n . Then one can see that Z n g n ≈ exp (cid:20) − g S [^ χ ] − n log ( g ) (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) saddle g n = e n log ( n )− n ( log ( ¯ S ) + ) ≈ n ! √ πn ¯ S n g n ≈ n ! µ − n ≈ n ! Q − n/ . (C.8)We find that the quantum effects lead to a divergent series in g whose coefficientsgrow like n ! (as opposed to the ( n ) ! that we have encountered in Section 2.3). Thisis the growth related to diagram proliferation when one takes the leading orderof Eq. (5.2) as an action. By the usual resurge arguments, this corresponds to anon-perturbative effect of order ω ( µ ) ≈ O (cid:16) e − µ (cid:17) ≈ O (cid:16) e − Q / (cid:17) (C.9)which is parametrically smaller than the O (cid:16) e − Q / (cid:17) semiclassical effects.35 Optimal truncation in the double-scaling limit
In the double-scaling limit we know all the series coefficients of ω , so it is possibleto compute the Borel transform and evaluate the free energy to any given precision.This is not possible in the generic eft arising at finite N , where we need to resortto numerical calculations on the lattice to find the first few coefficients. In thisappendix, we show what kind of results to expect for the optimal truncation using theresults in the double-scaling limit. More sophisticated analysis such as Borel-Padéand conformal mappings [66] can lead to significant improvement on the optimaltruncation estimates.Consider the grand potential ω ( m ) in Eq. (2.34): ω ( m ) = r m ∞ (cid:88) n = (− ) n (cid:0) − − n (cid:1) B n Γ ( − n ) √ πΓ ( n + ) ( rm ) n = r m ∞ (cid:88) n = ω n ( rm ) n .(D.1)In the main text we show that if for large values of n the coefficients ω n grow like ω n ∼ ( βn ) ! A − n , (D.2)then the series has an optimal truncation given by the value of n for which ω n ( rm ) − n has a saddle: N ∗ ≈ β (cid:12)(cid:12) A ( rm ) (cid:12)(cid:12) /β , (D.3)and the error in the truncation is of the order (cid:15) ( m ) ∼ e −( Ar m ) /β . (D.4)In our case, for large n we have ω n ∼ ( n ) ! ( π ) − n √ πn / , (D.5)so A = π , β = N ∗ ≈ πrm . (D.6)Using the fact that for large values of ^ q , the coefficient m scales as rm ∼ √ ^ q (seeEq. (2.40)), we find that in the expansion of grand potential, and of the free energy,the optimal truncation is found at N ∗ ≈ π (cid:112) ^ q . (D.7)In Figure 6 we compare the asymptotic expansion truncated at the N -th term with36 ��������������������������������� ����������������������� Figure 6 – Left: Grand potential ω as function of m in the small charge expansion andfor different numbers of terms. The small-charge expansion breaks down in the red-shaded region. Right: Error in the truncation w.r.t. the exact small-charge expansion. the convergent small-charge expansion. Empirically we observe that the optimaltruncation for m < ^ q (cid:39) N = N ∗ (cid:47) E Trans-series representation of the Dawson’s function
In this appendix we construct the trans-series representation of the Dawson’s functionappearing in Section 2.3. We will show that no non-perturbative ambiguities are leftto fix once we impose the reality condition on the heat trace of S . The treatment isstandard and follows the one used in Euler’s and Riccati’s ode [13]. The Dawson’sfunction is the unique solution of the following Cauchy problem:d F d z + zF ( z ) = F ( ) =
0. (E.1)It is evident that z = ∞ is a critical point of the ode. One may attempt to find asolution in terms of an asymptotic series around this point, which turns out to be F ( z ) ∼ ∞ (cid:88) k = a n z k + = ∞ (cid:88) k = ( k − ) !!2 k + z k + for z → ∞ . (E.2)This solution has the following problems: (a) it is an asymptotic series, and it holdsonly at the formal level; (b) there is no constant of integration one can fix via theinitial condition.These problems have a common solution. Define the following Borel transform; B{ Φ } ( ζ ) = ∞ (cid:88) k = a n Γ (cid:0) k + (cid:1) ζ k + = √ π ζ − ζ , (E.3)The series is not Borel summable because of the singularity at ζ =
1, and the37ppropriate lateral Borel summations are S ± { Φ } ( z ) = (cid:90) ± d ζ ζ − e − z ζ B{ Φ } ( ζ ) . (E.4)The discontinuity between the lateral summations reads [ S + − S − ] { Φ } ( z ) = − Disc { Φ } ( z ) = − πi √ π e − z . (E.5)This expression is a solution to the homogeneous ode. We can then write the generalsolution to the ode as a trans-series centered at z = ∞ as with a single trans-seriesparameter σ as follows: Ξ ( z , σ ) = ∞ (cid:88) k = ( k + ) !!2 k + z k + + σe − z . (E.6)Its lateral Borel sum reads S ± { Ξ } ( z , σ ± ) = √ π (cid:90) ± d ζ √ ζ e − z ζ − ζ + σ ± e − z , (E.7)which is easily proven to be a solution of the inhomogeneous ode.To fix the trans-series parameter we can impose the result to be real for z real,which sets Im σ ± = ± π √ π , (E.8)so that the coefficients of the non-perturbative exponentials are purely imaginary.The real part of σ ± is left unfixed and corresponds to the integration constant whichhad disappeared when we had attempted a power series ansatz for the solution. Ageneral boundary condition sets its value as F ( ) = c , = ⇒ Re σ ∓ = √ π e − c erfi ( c ) . (E.9)For F ( ) = σ ± =
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