Borel and Stokes Nonperturbative Phenomena in Topological String Theory and Matrix Models
We address the nonperturbative structure of topological strings and matrix models, focusing on understanding the nature of instanton effects alongside with exploring their relation to the large–order behavior of the expansion. We consider the Gaussian, Penner and Chern–Simons matrix models, together with their holographic duals, the minimal string at self–dual radius and topological string theory on the resolved conifold. We employ Borel analysis to obtain the exact all–loop multi–instanton corrections to the free energies of the aforementioned models, and show that the leading poles in the Borel plane control the large–order behavior of perturbation theory. We understand the nonperturbative effects in terms of the Schwinger effect and provide a semiclassical picture in terms of eigenvalue tunneling between critical points of the multi–sheeted matrix model effective potentials. In particular, we relate instantons to Stokes phenomena via a hyperasymptotic analysis, providing a smoothing of the nonperturbative ambiguity. Our predictions for the multi–instanton expansions are confirmed within the trans–series set–up, which in the double–scaling limit describes nonperturbative corrections to the Toda equation. Finally, we provide a spacetime realization of our nonperturbative corrections in terms of toric D–brane instantons which, in the double–scaling limit, precisely match D–instanton contributions to minimal strings.
1 Introduction and Summary
The nonperturbative realm of quantum field and string theories has often been a source of many new results and surprises. Another recurrent topic of great interest over the past decades has been the large approximation, lately relating gauge and string theories in a nonperturbative fashion. Of particular interest to us in this work is the case of the perturbative expansion of hermitian matrix models, whose nonperturbative corrections are exponentially suppressed as . In the double–scaling limit these models describe noncritical or minimal (super)string theories, and the nonperturbative structure of the matrix model is related to that of the corresponding string theory : the contributions are instanton effects in the matrix model [2, 3] and they are interpreted as D–brane configurations in the string theoretic description [4, 5, 6]. Of course the study of matrix models is not confined to the vicinity of their critical points and one may also study nonperturbative effects away from the double–scaling limit. The interesting point is that off–critical matrix models may be dual to topological string theories. For instance, this happens in the case first suggested by , where some off–critical matrix models describe the topological string B–model on certain non–compact Calabi–Yau (CY) backgrounds, with the string genus expansion (in powers of the string coupling, ) being identified with the matrix model expansion; and it is also the case for topological strings with a Chern–Simons dual, first studied in [8, 9]. This turns out to be a more general statement, as it was later shown in [10, 11, 12] that topological string theory on mirrors of toric manifolds also enjoys a dual holographic description in terms of off–critical matrix models. It is thus evident that fully understanding the nonperturbative structure of matrix models, both at and off criticality, will have many applications in both minimal and topological string theories.
Recently [13, 14, 15] there has been significative progress in understanding and in quantitatively computing nonperturbative effects in matrix models away from criticality. In , and building upon double–scaled results [2, 3, 16, 17], off–critical saddle–point techniques were developed in order to compute instanton amplitudes (up to two loops) in terms of spectral curve geometrical data. This work focused upon one–instanton contributions in one–cut models, and in  an extension to multi–instanton contributions, again in one–cut models, was obtained, starting from a two–cut analysis. Extensive checks of the nonperturbative proposals in these papers were also performed, by matching against the large–order behavior of the expansion. Another approach to multi–instanton amplitudes was developed in , this time around based on orthogonal polynomial methods, via the use of trans–series solutions to the string equations . Further progress along these lines recently led to the proposal of , where a proper nonperturbative definition of a modular–invariant holomorphic partition function was presented, which was also shown to be manifestly background independent. Remarkably, many of the results uncovered in [13, 14, 15] appear to extend beyond the context of matrix models; e.g., in cases where the theory is controlled by a finite–difference equation—such as the string equation  for matrix models—it is possible to compute nonperturbative effects and relate them to the large–order behavior of the theory. This is the case of Hurwitz theory , which is controlled by a Toda–like equation, and also the case of topological strings on the background considered in 111A local CY threefold given by a bundle over a two–sphere, , , which may be regarded as a quantum group deformation of Hurwitz theory; see  for further details.. However, all models considered in the aforementioned articles lie in the universality class of two–dimensional gravity, with , and methods that have been worked out in this case cannot be applied in a straightforward fashion to the case of topological strings in the universality class of . In view of this, it is necessary to develop new techniques in order to approach nonperturbative effects in models which belong to the universality class of the string at the self–dual radius.
Let us be a bit more specific about the nature of the string perturbative expansion and the type of nonperturbative contributions we shall be looking for. Topological strings, much like physical string theory, are perturbatively defined in terms of two couplings, and , as222Recall that in the A–model the are Kähler parameters while in the B–model they are complex parameters.
where is the free energy and the partition function, and where the fixed genus free energies are themselves perturbatively expanded in . In some sense the expansion is the milder one: it has finite convergence radius, with this radius given by the critical value of the Kähler parameters where one reaches a conifold point in moduli space. As it turns out, the problem of finding a nonperturbative formulation of the A–model free energy, in , may be reduced to that of solving the mirror B–model description, where topological string amplitudes become exact in . In this way, the A–model solution is found by translating B–model amplitudes back to the A–model, by means of the mirror map. This topic has been extensively studied in the literature and we refer the reader to the recent developments [11, 12] and references therein.
The situation gets more complicated as one tries to go beyond perturbation theory in . In this case, one is immediately faced with the familiar string theoretic large–order behavior rendering (1.1) as an asymptotic expansion . In this case, one expects nonperturbative corrections of order , and an adequate nonperturbative formulation of the theory must encode all these corrections. As described above, there are certain cases—such as the backgrounds considered by Dijkgraaf and Vafa , or models with a dual Chern–Simons interpretation—where topological strings have a holographic matrix model description, with the matrix model large expansion reproducing the topological string genus expansion. In these set of backgrounds one would be tempted to use the finite matrix model free energy as the nonperturbative definition of topological string theory333Other nonperturbative completions, provided by a holographic dual, have been proposed in .. In order to establish this result, one must first understand how the finite matrix model would encompass all nonperturbative contributions . This situation is clear for minimal strings, realized in the double–scaling limit of hermitian matrix models: the nonperturbative effects associated to the asymptotic nature of the genus expansion are implemented via eigenvalue tunneling effects in the dual matrix model, and are interpreted in the continuum formulation in terms of Liouville branes in spacetime [1, 6]. For topological strings, a similar understanding has been achieved in the case of the local curve [10, 13], where a matrix model description is available . In this case, the nonperturbative effects associated to the asymptotic behavior, or large–order behavior, have again been matched to instantons arising from matrix eigenvalue tunneling, and a spacetime interpretation in terms of domain walls has been provided .
However, there are several cases where this paradigm seems not to apply, at least not in a straightforward fashion. It is our goal to address such issues in the present work in the prototypical example of the resolved conifold, but also encompassing matrix models in the universality class. Topological strings on the resolved conifold are holographically described by the Chern–Simons matrix model, but there are now no obvious instantons associated to eigenvalue tunneling as the Chern–Simons potential has no local maxima outside of the cut, where the eigenvalue instantons could tunnel to. This problem, which was not an issue in any of the previously mentioned examples, also appears in other matrix models, such as the Gaussian and Penner models; all of them in the universality class. One may then ask where do nonperturbative corrections arise from, or what exactly controls the large–order behavior of the perturbative expansion in these models. We shall answer these questions in this paper. One way out is to directly compute the (would–be) instanton action that controls the large–order behavior of the perturbative expansion, by means of a standard Borel analysis (see, e.g., ). At first this may look like a formidable task, as one may expect the topological string genus expansion to be rather complicated, not amenable to a Borel transform. Happily, the free energies of all cases we consider enjoy a Gopakumar–Vafa (GV) integral representation [23, 24] which allows for an exact location of the singularities in the Borel complex plane controlling the divergence of the asymptotic perturbative series, i.e., the instanton action . This is the topological string generalization of a celebrated string result . Interestingly enough, this integral representation may also be regarded as an one–loop Schwinger integral , thus providing a spacetime interpretation of these nonperturbative effects; as already pointed out in  they control the pair–production rate of BPS bound states. As we shall later see, these results—which we further identify as Stokes phenomena of the finite partition function—will also allow us to explain the nonperturbative contributions as one–eigenvalue effects in the matrix model picture. We find, from a saddle–point analysis, that the nonperturbative effects arise due to the multi–valued structure of the effective potential (as preliminarily suggested in ); a different picture from that of matrix models in the universality class of 2d gravity plus matter, where the one–eigenvalue tunneling occurs from a metastable minimum to the most stable one.
This paper is organized as follows. We begin in section 2 by reviewing the main ideas behind our subsequent work. This includes the definition of the Borel transform and the relation between instantons and the large–order behavior of perturbation theory, both related to the existence of a nonperturbative ambiguity in the calculation of the free energy. In this section we also discuss the Schwinger effect, where one actually has a physical prescription to define the inverse Borel transform, which will turn out to be the case for topological strings and matrix models via the GV integral representation of the topological string free energy. In section 3 we then move on to presenting the matrix models we shall be focusing upon. We review some of their properties, such as their spectral curves and their perturbative genus expansions, and also obtain expressions for their exact, finite partition functions and holomorphic effective potentials, both of which play important roles in sections to come. In this section we also discuss the double–scaling limit of these models and show how they relate to FZZT branes. Section 4 presents one of the main topics in this paper, the Borel analysis of the Gaussian, Penner and Chern–Simons matrix models. We show how to obtain Schwinger–like integral representations of the free energy, via Borel resummation, and how the correct identification of the leading poles in the complex Borel plane leads to the one–instanton action in all our examples. We further show in this section that while the all–loop multi–instanton amplitudes precisely reconstruct the perturbative series, the one–instanton results control the large–order behavior of perturbation theory. We then move on to another of our main topics in section 5, namely the issue of Stokes phenomena. We recall how to obtain Stokes phenomena for integrals with saddles via hyperasymptotic analysis, and perform a detailed calculation for the Gamma function. This extends to the Barnes function and, in this way, allows us to identify instantons with Stokes phenomena as we reproduce the results we have previously found in section 4, out of hyperasymptotic analysis. In section 6 we provide a semiclassical interpretation of our instantons via eigenvalue tunneling, where this tunneling is now associated to the existence of a branched multi–sheeted structure in the relevant holomorphic effective potentials. Indeed, simple monodromy calculations reproduce our results for the multi–instanton action straight out of this interpretation. We further show in this section how to interpret our instantons in spacetime, from the point of view of ZZ branes. In section 7, we discuss the trans–series approach to matrix models and how it further validates our results. Finally, we conclude in section 8 with an outlook and future prospects. We also include two appendices, one dedicated to the study of the monodromy structure of the polylogarithm, and the other dedicated to the Cauchy dispersion relation, in the case of more general topological string theories than the ones we address in this paper.
2 Asymptotic Series, Large Order and Topological Strings
We start by reviewing some useful facts concerning asymptotic series, the relation of their large–order behavior to nonperturbative effects, as described by instantons or by the Schwinger effect, and put them in the context of topological string theory as we wish to study in the present work. For an introduction to these topics with applications in quantum mechanics and quantum field theory, we refer the reader to  and references therein.
Let us consider the perturbative expansion of some function, , with the specific perturbative expansion parameter,
In many interesting examples one may infer that, at large , the coefficients behave as , thus rendering the series divergent. As an approximation to the function , the asymptotic series (2.2) must necessarily be truncated. As such, one is faced with an obvious problem: how to deal with the fact that the perturbative expansion has zero convergence radius? In particular, if we do not know the function , but only its asymptotic series expansion, how do we associate a value to the divergent sum? The best framework to address issues related to asymptotic series is Borel analysis. One starts by introducing the Borel transform of the asymptotic series (2.2) as
which removes the divergent part of the coefficients and renders with finite convergence radius. In particular, if originally had a finite radius of convergence (i.e., if it was not an asymptotic series), would be an entire function in the Borel complex –plane. In general, however, will have singularities and it is crucial to locate them in the complex plane. The reason for this is simple to understand: if has no singularities for real positive one may analytically continue this function on and thus define the inverse Borel transform by means of a Laplace transform as444For simplicity, we are assuming in this expression.
The function has, by construction, the same asymptotic expansion as and may thus provide a solution to our original question; it associates a value to the divergent sum (2.2). If, however, the function has poles or branch cuts on the real axis, things get a bit more subtle: in order to perform the integral (2.4) one needs to choose a contour which avoids such singularities. This choice of contour naturally introduces an ambiguity (as we shall see next, a nonperturbative ambiguity) in the reconstruction of the original function, which renders non–Borel summable555Strictly speaking, the function is said not to be Borel summable if different integration contours yield different results. It may still be the case that, in spite of having singularities in the real axis, all alternative integration contours yield the same result.. As it turns out, different integration paths produce functions with the same asymptotic behavior, but differing by exponentially suppressed terms. For instance, in the presence of a singularity at a distance from the origin, on the real axis, one may define the integral (2.4) on contours , either avoiding the singularity from above, and leading to , or from below, and leading to . One finds that these two functions differ by a nonperturbative term 
In certain cases, e.g., when one has a Schwinger representation for the function [27, 28], there is a natural and rigorous way to define the integral (2.4) on a contour which avoids the singularities, and which also allows for a physical interpretation of the nonperturbative contributions.
So far our discussion has been rather general. However, it takes no effort to figure out the physical relevance of our discussion: divergent series are almost ubiquitous in physics and appear basically each time we approach an interesting problem in perturbation theory . A typical and extensively studied case in quantum mechanics is the anharmonic oscillator (see, e.g., [29, 30, 22]). Herein, the ground state energy may be computed in perturbation theory—as a power series in the quartic coupling—and one finds that it is analytic in all the (coupling constant) complex plane except for a branch cut on the negative real axis, associated to the instability of the potential which becomes unbounded for negative values of the coupling. This instability is reflected by the fact that the series is, as expected, asymptotic. In particular, one can perform a Borel analysis as above and discover that the Borel transform of the ground state energy has singularities on the positive real axis, leading to an ambiguity of order , with the quartic coupling constant. In this simple quantum mechanical example the nonperturbative ambiguity has a clear physical interpretation: it signals the presence—at negative —of instantons mediating the decay from the unstable to the true vacuum, via tunneling under the local maximum of the potential.
What these ideas illustrate is that by means of a purely perturbative analysis, i.e., finding the singularities of the Borel transform of the original perturbative series, it is possible to learn about nonperturbative effects—at least the intensity of the nonperturbative ambiguity (but we shall say more on this in the following). In some examples it is possible to independently compute these nonperturbative terms directly, e.g., using WKB methods or computing the path integral around non–trivial (subdominant) saddle points . In these examples one may then proceed in the opposite way from above and obtain information on the large–order behavior of the perturbative expansion out of the nonperturbative data. This is what we shall illustrate next.
2.1 From Instantons to Large–Order and Back
In physical applications, the factor appearing in (2.5) is the one–instanton action (see, e.g., ). Let us make this relation between instantons and the large–order behavior of perturbation theory a bit more precise, as it will play a crucial role in our later analysis. Consider a quantum system whose free energy is expressed as a perturbative expansion in , the coupling constant666Here and below the index labels the –instanton sector, so that labels the perturbative expansion.,
The series (2.6) will generically be asymptotic, with zero radius of convergence. This is naturally associated to a branch cut of in the complex –plane, located in the negative real axis and associated to instanton effects (just like in the anharmonic oscillator example above). The function is expected to be analytic otherwise. In fact this is saying that our quantum system should actually be thought of as an asymptotic formal power series in two expansion parameters, and , see  for a discussion in the matrix model context. The appropriate expansion of the free energy is thus 
Here, is a parameter corresponding to the nonperturbative ambiguity. Also, is the one–instanton action, a characteristic exponent and is the –loop contribution around the –instanton configuration. Typically, the coefficients are factorially divergent for any , in which case we may think about the –instanton sector as the nonperturbative contribution related to the asymptotic nature of the loop expansion around the –instanton sector.
A standard procedure then relates the coefficients of the perturbative expansion around the zero–instanton sector, , with the one–instanton free energy as follows. The discontinuity of the free energy across the branch cut (associated to the instability of the theory for negative ) is expressed, at first order, in terms of the leading instanton expansion (2.7)
At the same time, we may use the the Cauchy formula to write
In certain situations, e.g., in the aforementioned anharmonic oscillator example , it is possible to show by scaling arguments that the last integral in the expression above does not contribute. In such cases, (2.9) provides a remarkable connection between perturbative and nonperturbative expansions. Using the perturbative expansion (2.6) and the leading one–instanton contribution to the discontinuity , one may obtain from the Cauchy formula (2.9) the following large order (or large ) relation
This explicitly shows that the computation of the one–loop one–instanton partition function determines the leading order of the asymptotic expansion for the perturbative coefficients of the zero–instanton partition function. Higher loop corrections then yield the successive corrections. Furthermore, instanton corrections with action , where we have in mind multi–instanton corrections with action , , will yield corrections to the asymptotics of the coefficients which are exponentially suppressed in .
For the cases we shall consider in this work, namely matrix models and string theory, one finds genus expansions as in (1.1), with , so that the relation (2.10) gets slightly re–written as follows (see, e.g., ). Begin with the free energy in the zero–instanton sector, . Setting , the one–instanton path integral then yields a series of the form
Following a procedure analogous to the one above, where one further assumes that the standard dispersion relation (2.9) still holds, it follows for the zero–instanton sector perturbative coefficients
Again, the computation of the one–loop one–instanton free energy determines the leading order of the asymptotic expansion for the perturbative coefficients of the zero–instanton free energy. Higher loop corrections then yield the successive corrections. One should further notice that recently, in [13, 14, 15], the relation (2.12) has been tested in several models and rather conclusive numerical checks have confirmed its validity.
2.2 The Schwinger Effect and a Semiclassical Interpretation
As discussed above, a nonperturbative ambiguity—typically associated to instantons, from a physical point of view—can arise when defining the integration contour for the inverse Borel transform. However, this is not always the case: we shall now review an example where a prescription to define the inverse Borel transform naturally arises, together with a physical interpretation for the nonperturbative contributions . This is the Schwinger effect .
The one–loop effective Lagrangian describing a charged scalar particle, of charge and mass , in a constant electric field, , has an integral representation given by  (see, e.g.,  for a recent review)
which admits the weak coupling expansion
In here we used the shorthand , with the Bernoulli numbers. Since one may further relate the Bernoulli numbers to the Riemann zeta function via
it becomes evident that diverges factorially fast. In this case, if one first writes the expansion (2.14) as
with , it follows that at large one has rendering this perturbative expansion asymptotic—and actually non–Borel summable as we shall see next. Indeed, computing the Borel transform it follows
from where one immediately notices that the Schwinger integral representation of the effective Lagrangian (2.13) is essentially the inverse Borel transform
Of course so far we still have a nonperturbative ambiguity to deal with: in order to perform the integration on the real axis one still needs to specify a prescription in order to avoid the poles at , . This introduces the usual ambiguities leading to exponentially suppressed contributions to the effective Lagrangian. The novelty in this case is that there is now a natural way to address the integration avoiding the singularities in an unambiguous way .
As it turns out, the contour of integration needs to be deformed in such a way that the integral picks up the contributions of all the poles as if the real axis is approached from above, tantamount to a prescription; and this is the requirement which is dictated by unitarity . As such, one has a physical principle behind the unambiguous choice of contour. Furthermore, the Lagrangian develops an imaginary part which is simple to compute by summing residues777Notice that the residue at precisely vanishes., and which cannot be seen to any finite order in perturbation theory,
an expression with an evident multi–instanton flavor , as in (2.7). Besides the appropriate, physical prescription to perform the integration, and as such unambiguously compute the nonperturbative contributions to the Lagrangian, the Schwinger effect gives us something else: a physical interpretation of this imaginary part. Indeed, the imaginary part of the effective Lagrangian (2.19) is precisely the pair–production rate, or probability per unit volume for pair creation, for scalar electrodynamics in a constant electric field . In other words, the above unitary prescription for the integration contour guarantees that this probability is a positive number between zero and one (which basically demands (2.19) to be real and positive).
Another interesting illustration of the Schwinger effect, which will be of particular relevance in our subsequent discussion on topological strings and matrix models, is the case of a constant (euclidean) self–dual electromagnetic background [33, 34, 35], satisfying
admitting the weak coupling expansion
Notice that there are two possible self–dual backgrounds [33, 34, 35]: a magnetic–like background with real, in which case (2.22) has an alternating sign; and an electric–like background with imaginary, in which case (2.22) is not alternating. If one carries through a Borel analysis similar to the previous one, where we studied the case of constant electric field, one may further notice that while the alternating (magnetic) series has Borel poles on the positive imaginary axis, the non–alternating (electric) series has the Borel poles on the positive real axis, making both situations rather distinct on what respects evaluating the inverse Borel transforms. For the non–alternating (electric) series it is also possible to see that the aforementioned unitarity prescription will pick a Borel contour leading to a nonperturbative imaginary contribution to the Lagrangian, unambiguously given by
This expression can similarly be obtained by first considering the magnetic series, reflecting the integrand in (2.21) to the negative real axis in order to obtain an integral over the entire real line, and then deforming this integration contour such that it just incloses all the poles in the positive imaginary axis (a contour surrounding ). The resulting integral will then produce a sum over residues which, upon “Wick rotation” of the dimensionless coupling , leads to the same expression as above, (2.23). As we shall see in the course of this work, this expression is also at the basis of the nonperturbative structure of topological strings and matrix models.
Another important feature of the Schwinger effect, that we shall further explore later on, is the fact that in the presence of a constant electric field the pair–production process can be given a semiclassical interpretation in terms of a tunneling process, where electrons of negative energy are extracted from the Dirac background by the application of the external field . The motion under the potential barrier, classically forbidden, is considered for imaginary values of time, allowing for a computation of the tunneling probability corresponding to the pair–production rate as
where is the imaginary part of the action developed during motion under the barrier. In here, a crucial point is that a particle in a sub–barrier trajectory satisfies the classical equations of motion. One may then use standard classical mechanics of a relativistic particle in order to describe this process. Indeed, energy conservation
together with the equation of motion , allow for an immediate re–writing of the action (for ) as:
Notice that, because of the logarithm, the action is a multi–valued function.
The spectrum of possible values for the energy is displayed in Figure 1. A potential barrier separates the lower continuum of negative–energy states (the minus sign of ) from the upper continuum of positive–energy states (the plus sign of ). Sub–barrier motion between points and will start at , where , and corresponds to the variation of the imaginary time/momentum along the path , while the real part of the energy remains constant. Indeed, at the classical turning point we will have and , which corresponds to a square–root branch point of the function . The motion ends back at in point , as shown in Figure 1. In this case, we see that the sub–barrier trajectory correspond to an increment of the imaginary part of the action as
which we may compute as the shift of the multi–valued function as we move in–between the sheets of the logarithm. In fact, it is rather simple to realize that the value of on a generic sheet differs from its value on the principal sheet, , by
In the illustration above we went (once) “half way” around the branch cut in which case the shift in the action is given by
For a generic sub–barrier motion, corresponding to a repeated wandering of the particle between the turning points and we may write
Naturally, this semiclassical argument may be refined in order to reproduce the pre–factors of the exponential term, and also so as to include the effect of a magnetic field. The point we wanted to make is that, from a semiclassical perspective, the instanton action describing the pair–production rate as a tunneling process may be computed via the branch cut discontinuities of the multi–valued function . This is a technique that we shall deploy later on in order to provide for a semiclassical interpretation of nonperturbative effects in matrix models in terms of eigenvalue tunneling.
In this paper we shall apply the techniques we have just described, Borel analysis, instanton calculus, and the Schwinger effect, in order to study the nonperturbative structure of topological strings and matrix models. As such, we now turn to topological string theory with emphasis towards the integral representation of its free energy.
2.3 The Topological String Free Energy
Asymptotic series and the Schwinger integral representation also appear in the context of topological string theory (see, e.g.,  for an introduction). Let us start by describing the free energy of the A–model. The closed string sector of the A–model is a theory of maps from a genus– Riemann surface, , into a CY threefold , which may be topologically classified by their homology class . One may expand on a basis of , with associated complexified Kähler parameters .
The topological string free energy has a standard genus expansion in powers of the string coupling , as in (1.1), which in the large–radius phase (i.e., for large values of the Kähler parameters, in units of ) becomes
Here, the sum over is a sum over topological sectors or, equivalently, over world–sheet instantons. We have further introduced , with denoting , and we have chosen units in which . The coefficients are the Gromov–Witten invariants of , counting world–sheet instantons, i.e., the number of curves of genus in the two–homology class .
The expansion in world–sheet instantons in (2.31), regarded as a power series in , generically has a finite convergence radius, , that can be estimated from the asymptotic large behavior of Gromov–Witten invariants 
In here, is a critical exponent. At the critical value of the Kähler parameter, , the so–called conifold point, the geometric interpretation of the A–model large–radius phase breaks down and the topological string free energy undergoes a phase transition to a non–geometric phase, nonperturbative in . One may characterize the theory by its critical behavior at the conifold point. In particular, one can consider the following double–scaling limit
In this case, the double–scaled free energy is universal, as first noticed in , and reads
where is the all–genus free energy of the string at the self–dual radius (for a review on these issues see, e.g., ). The critical behavior (2.34) has been checked in many examples, such as [41, 42]. Furthermore, in , it has been shown that certain local CYs have a critical behavior which is in the universality class of 2d quantum gravity, i.e., they have . Another feature to notice is that the above genus expansion depends on the alternating Bernoulli numbers and, thus, is alternating for real .
which coincides, after an appropriate identification of the parameters, with the one–loop effective Lagrangian for a charged particle in a constant self–dual background, (2.21). This means that enjoys an asymptotic weak coupling expansion as in (2.22) and further develops a nonperturbative imaginary contribution akin to (2.23). In this line of thought, the exploration of Schwinger–like integral representations for the free energies of topological strings and matrix models is one of the main topics in this paper.
2.4 A Schwinger–Gopakumar–Vafa Integral Representation
As should be clear by now, Schwinger–like integral representations for the free energy are bound to play a critical role in our analysis. Happily, for topological string theory, such representations have been provided by Gopakumar and Vafa in [23, 24, 44]. These works explored both the connection of topological strings to the physical IIA string, as well as the duality between type IIA compactified on a CY threefold, at strong coupling, and M–theory compactified on the same CY times a circle, in order to relate topological string amplitudes to the BPS structure of wrapped M2–branes and thus re–write the topological string free energy in terms of an integral representation. The final result in  for the all–genus topological string free energy, on a CY threefold , is
Let us explain the diverse quantities in this expression. The integers are the GV invariants of the threefold . They depend on the Kähler class and on a spin label . Later on we shall be focusing on the case where is the resolved conifold, for which there is only one non–vanishing integer, . The combination represents the central charge of four–dimensional BPS states obtained in the following fashion . Start with M–theory compactified on and consider the BPS spectrum of M2–branes wrapped on cycles of the CY threefold with fixed central charge , with as above and the complexified Kähler parameters. The mass of the wrapped M2–branes is . Upon reduction on each BPS state may have in addition an arbitrary (quantized) momentum around the circle, leading to BPS states of central charge and mass . Notice that these four–dimensional BPS states contributing to the topological string free energy may be understood, from a IIA point of view, as bound states of D2 and D0–branes, and it is the physics of this system which can be related to a Schwinger–type computation and thus to the above integral representation (in fact, thanks to the supersymmetry in the problem, the Schwinger calculation one has to perform in this context turns out to be equivalent to that of a vacuum amplitude for a charged scalar field in the presence of a self–dual electromagnetic field strength, as in (2.21)). Furthermore, the integer , associated to the winding around , counts the number of D0 branes in the D2D0 BPS bound state. This should make (2.36) clear.
One may also recover the perturbative genus expansion from this integral representation. Using the familiar identity
with the Dirac delta function, one may explicitly evaluate the sum over in (2.36) and thus obtain, after the trivial integration over ,
This result expresses the topological string free energy, on a CY threefold , in terms of the GV integer invariants [23, 24, 44]. To be completely precise, it is important to notice that in order to recover the full topological string free energy one still has to add to (2.38) the (alternating) constant map contribution [45, 46]
where is the Euler characteristic of . This term can also be given a Schwinger–like integral representation. From the point of view of the duality between type IIA and M–theory, this amounts to considering only the contribution arising from the D0–branes, or Kaluza–Klein modes. The result is 
In this paper we shall mainly consider the resolved conifold, a toric CY threefold for which and thus the only non–vanishing integer GV invariant is . In this case, the GV integral representation (2.36) immediately yields
an expression which carries a Schwinger flavor, as we have seen above. It is also important to point out that the case of is the only one in which the integrand of the GV integral representation will have “interesting” poles, i.e., poles of the sine function on the real axis. When the only poles of the integrand will be at zero and in the Borel complex plane. So, in particular, when studying more complicated CY threefolds where there is a sum over , it will always be the contribution from GV invariants with which will be the most relevant for the Schwinger analysis we shall carry through later in the paper and, as such, the case of the resolved conifold is a prototypical example for those situations. From the previous expression it is also simple to obtain the perturbative expansion, by summing over as previously described, and one obtains
By carrying through this sum, expanding in powers of , and adding the constant map contribution, one finally obtains the resolved conifold genus expansion as
with the polylogarithm function. We shall later see how a Borel analysis allows for a nonperturbative completion of this expansion and moreover how to relate this nonperturbative completion to the large–order behavior of the above genus expansion.
One final word pertains to the matrix model description of strings on the resolved conifold via a large duality. It was shown in  that there is a duality between closed and open topological A–model string theory on, respectively, the resolved and the deformed conifold; two smooth manifolds related to the same singular geometry. In the resolved conifold case the conifold singularity is removed by blowing up a two–sphere around the singularity; while in the deformed conifold case the conifold singularity is removed by growing a three–sphere around it, which is also a Lagrangian sub–manifold thus providing boundary conditions for open strings. As it turns out, the full open topological string field theory in this latter background, , where we wrap D–branes on the Lagrangian sub–manifold base, , reduces to SU Chern–Simons gauge theory on , whose partition function further admits a matrix model description . The matrix model in question, which we shall review in the next section, has a potential with a single minimum and no local maxima. In this paper we refer to this type of matrix models (which will also include the Gaussian and Penner cases) as matrix models since, as we shall see, they all admit a very natural double–scaling limit to the string at self–dual radius. Notice that matrix models do not belong to the class of matrix models for which the off–critical instanton analysis has been carried out so far. Because understanding nonperturbative corrections to the topological string free energy on the resolved conifold is undissociated from understanding nonperturbative corrections to matrix models, we shall consider this latter case more broadly in order to shed full light on this class of instanton phenomena. As such, matrix models is the subject we shall turn to next.
3 Matrix Models and Topological String Theory
We shall now introduce three distinct matrix models, all in the universality class of the string, and which will be the main focus of our subsequent discussion. As mentioned in the previous section, one of these models is the one describing Chern–Simons gauge theory on , known as the Stieltjes–Wigert matrix model. Another interesting, and rather elementary, matrix model is the Gaussian model. Yet, we shall find that it already displays many features that will also appear for the resolved conifold. Finally, we also address the Penner matrix model, first introduced to study the orbifold Euler characteristic of the moduli space of punctured Riemann surfaces. These three models have been extensively studied in the literature and in the present section we will mostly gather some general facts necessary to obtain their topological large expansions and their holomorphic effective potentials. Then, in the following section, we shall analyze their large asymptotic expansions from the point of view of Borel analysis.
with the usual volume factor of the gauge group. In the eigenvalue diagonal gauge this becomes
where is the Vandermonde determinant. The free energy of the matrix model is then defined as usual and, in the large limit, it has a perturbative genus expansion
with the ’t Hooft coupling. Multi–trace correlation functions in the matrix model may be obtained from their generating functions, the connected correlation functions defined by
In particular, the generator of single–trace correlation functions is where is the resolvent, i.e., the Hilbert transform of the eigenvalue density characterizing the saddle–point associated to the matrix model large limit. In the most general case, this saddle–point is such that has support , with a multi–cut region given by an union of intervals . At large , the eigenvalues condense on these intervals in the complex plane and one may interpret them geometrically as branch cuts of a spectral curve which, in the hermitian one–matrix model, would be a hyperelliptic Riemann surface corresponding to a double–sheet covering of the complex plane , with the two sheets sewed together by the cuts .
The spectral curve, to be denoted by , may be written in terms of the genus zero resolvent which, for a generic one–cut solution with , is given by the ansätz
where one still has to impose that as , in order to fix the position of the cut endpoints888This boundary condition states that the eigenvalue density is normalized to one in the cut, .. The spectral curve is then defined as
For future reference, it is also useful to define the holomorphic effective potential, defined as the line integral of the one–form along the spectral curve,
which appears at leading order in the large expansion of the matrix integral as
Because the real part of the spectral curve relates to the force exerted on a given eigenvalue, it turns out that the effective potential is constant inside the cut , i.e., inside the cut the eigenvalues are free. The imaginary part of the spectral curve, on the other hand, relates to the eigenvalue density as , thus implying that the imaginary part of is zero outside the cut and monotonic inside. These two conditions guarantee that the eigenvalue density is real with support on . Furthermore, as should be clear from the expression above, if the matrix integral is to be convergent a careful choice of integration contour for the eigenvalues has to be made based also on the properties of the holomorphic effective potential [2, 3]. In particular, this contour may be analytically continued to any contour which includes the cut and does not cross any region where , thus guaranteeing global stability of the saddle–point configuration and convergence of the matrix integral (as at the endpoints of the integration contour). These properties of ensure that, in the large limit, the matrix integral can be evaluated with the steepest–descendent method [2, 3].
There are many ways to solve matrix models. In particular,  proposed a recursive method for computing the connected correlation functions (3.47) and the genus– free energies, , entirely in terms of the spectral curve. This recursive method, sometimes denoted by the topological recursion, appears to be extremely general and applies beyond the context of matrix models; see  for a review. For our purposes of computing the genus expansion of the free energy one of the most efficient and simple methods is that of orthogonal polynomials , which we now briefly introduce. If one regards
as a positive–definite measure in , it is immediate to introduce orthogonal polynomials, , with respect to this measure as
where one further normalizes such that . Further noticing that the Vandermonde determinant is , the one–matrix model partition function may be computed as
where we have defined for . These coefficients also appear in the recursion relations of the orthogonal polynomials,
In the large limit the recursion coefficients approach a continuous function , with , and one may proceed to compute the genus expansion of by making use of the Euler–MacLaurin formula; see [18, 50] for details.
3.1 The Gaussian Matrix Model
Let us first focus on the Gaussian matrix model, defined by the potential . This case is rather simple as the matrix integral can be straightforwardly evaluated via gaussian integration, and the volume of the compact unitary group follows by a theorem of Macdonald  as
where is the Barnes function, . The Gaussian partition function thus reads
The same result can be obtained with orthogonal polynomials. With respect to the Gaussian measure one finds Hermite polynomials, , and for the Gaussian matrix model it follows
indeed reproducing the expected result for the partition function as
where we have also used that . The asymptotic genus expansion of the Gaussian free energy simply follows from the asymptotic expansion of the logarithm of the Barnes function and one obtains
where is the Riemann zeta function. One immediately notices that all free energies with diverge when . It is then quite obvious to consider the double–scaling limit, approaching the critical point , as
in order to obtain the string at self–dual radius behavior
Finally, it is very simple to compute the one–form on the spectral curve of the Gaussian model
as well as the holomorphic effective potential
where we have normalized the result such that . In Figures 2 and 3 we plot the Gaussian algebraic curve for different values of , as well as the real value of the holomorphic effective potential in the complex plane. We notice that, with an appropriate identification of parameters, the Gaussian holomorphic effective potential coincides with the action associated to the semiclassical Schwinger effect, (2.26). In the following we shall further comment about this interesting coincidence.
3.2 The Penner Matrix Model
The second example we wish to address is the Penner matrix model . First introduced to study the orbifold Euler characteristic of the moduli space of Riemann surfaces at genus , with punctures, it turns out that in the double–scaling limit this model is actually related to the usual noncritical string theory, its free energy being a Legendre transform of the free energy of the string compactified at self–dual radius [55, 56]. The Penner matrix model is defined by the potential and one may simply compute its partition function again using orthogonal polynomials. Indeed, one may write the Penner measure as
which is, up to normalization, the measure for the generalized, or associated, Laguerre polynomials . It thus follows for the Penner matrix model
This immediately leads to the calculation of the partition function in this model as
where we made use of the Barnes function, satisfying
The normalized Penner free energy