Instanton effects and quantum spectral curves
We study a spectral problem associated to the quantization of a spectral curve arising in local mirror symmetry. The perturbative WKB quantization condition is determined by the quantum periods, or equivalently by the refined topological string in the Nekrasov–Shatashvili (NS) limit. We show that the information encoded in the quantum periods is radically insufficient to determine the spectrum: there is an infinite series of instanton corrections, which are non-perturbative in , and lead to an exact WKB quantization condition. Moreover, we conjecture the precise form of the instanton corrections: they are determined by the standard or un-refined topological string free energy, and we test our conjecture successfully against numerical calculations of the spectrum. This suggests that the non-perturbative sector of the NS refined topological string contains information about the standard topological string. As an application of the WKB quantization condition, we explain some recent observations relating membrane instanton corrections in ABJM theory to the refined topological string.
This paper is motivated by two different, but related problems. The first problem is the non-perturbative structure of topological string theory. Topological strings, like many other string models, are only defined perturbatively, and it is natural to ask whether one can define them non-perturbatively or find new non-perturbative sectors. In the last years, there have been many different proposals addressing this problem, but none of them seems to be conclusive. A bona fide non-perturbative definition must be based on a manifestly well-defined quantity, at least for a certain range of the relevant parameters of the model. This quantity should have an asymptotic expansion, for small values of the string coupling constant, which reproduces the original perturbative expansion. The asymptotic expansion and the perturbative expansion can only differ in quantities which are non-analytic at the origin (like for example instanton effects). Of course, one might find different non-perturbative definitions of the same quantity, all of them differing in non-analytic terms. This is for example what happens in two-dimensional gravity . In some cases, a reasonable physical criterium might single out one non-perturbative definition.
In this paper we will analyze the non-perturbative structure of refined topological strings in the Nekrasov–Shatashvili (NS) limit . This theory depends on the Calabi–Yau (CY) moduli and on a string coupling constant which is usually denoted by . In the original proposal of , this refined string was related, for some special geometries, to quantum integrable systems. It was later pointed out in [3, 4] that the perturbative free energy of the NS topological string can be computed by quantizing a spectral curve given by the mirror Calabi–Yau or a limit thereof. Using the quantum spectral curve one can construct quantum periods, depending on , which define the free energy by an -deformed version of special geometry. The quantum periods have a nice interpretation in terms of one-dimensional Quantum Mechanics: the quantized spectral curve defines a spectral problem, and the quantum periods are quantum-corrected WKB periods, which lead to a perturbative quantization condition, at all orders in .
In this paper we study a spectral problem appearing in the quantization of the curve describing
the mirror of local . We show
that the perturbative quantum periods are radically insufficient to solve the spectral problem: there is an infinite series of
non-perturbative corrections in , of the instanton type
A first-principle calculation of these instanton corrections is difficult, but we conjecture their precise form: they involve the standard (i.e. un-refined) topological string free energy. In particular, we write down an exact WKB quantization condition involving both the perturbative quantum periods and the instanton corrections. We perform a very precise test of this conjecture by comparing the exact quantization condition to the numerical calculation of the spectrum. The agreement is excellent.
Our conjecture gives a novel realization of the Gopakumar–Vafa invariants of local in terms of a spectral problem. More generally, it suggests that the spectral problems associated to quantum spectral curves in local mirror symmetry involve the standard topological string, non-perturbatively. It also suggests that we should define non-perturbative topological strings on local CYs through a well-defined spectral problem associated to the quantization of the mirror curve. For example, for some local CY geometries, the results of  provide a description of the refined topological string in terms of a quantum integrable system, and this should lead naturally to the sought-for spectral problem. If the structure we find in our particular example generalizes to other cases, this definition leads in a single strike to the refined NS string (as the perturbative sector) and the conventional topological string (as the non-perturbative sector). Notice that this non-perturbative definition, in the example of local , satisfies the above two criteria: it is based on a well-defined quantity (the spectrum), and it reproduces perturbatively the quantum B-period (which is the derivative of the free energy). Of course, the most interesting aspect of this definition is the appearance of the un-refined topological string free energy in the non-perturbative sector.
The second motivation for our work comes from ABJM theory . In , the partition function of this theory on a three-sphere was obtained by using localization techniques, and written as a matrix integral. Its full ’t Hooft expansion was obtained in  by using large techniques. In the paper , this matrix integral was written as the thermal partition function of an ideal, one-dimensional Fermi gas. One advantage of this approach, as compared to the standard large techniques, is that one can also compute non-perturbative effects due to membrane instantons, which go beyond the ’t Hooft expansion. In , building on previous work [13, 14, 15, 16], it was conjectured that these non-perturbative effects are encoded in the quantum periods of local . In this paper we prove this conjecture to a large extent. The basic idea is simple: the spectral problem associated to the one-particle Hamiltonian of the Fermi gas is nothing but the spectral problem studied in this paper, i.e. it is a specialization of the spectral problem appearing in the quantization of the spectral curve of local . The WKB analysis of the ABJM spectral problem leads immediately to the connection with these quantum periods. The grand potential of ABJM theory can be computed once we know the spectrum from the WKB quantization condition, and this makes it possible to derive many aspects of this grand potential which were conjectured in .
Many of our results on the WKB approach to the spectral problem are dual to the results on the grand potential of ABJM theory. For example, the fact that quantum periods are divergent for some values of and should get non-perturbative corrections which cancel these divergences is a dual version of the HMO cancellation mechanism of . Our conjecture on the instanton corrections to the spectral problem was motivated to a large extent by the known worldsheet instanton corrections to the grand potential of ABJM theory.
The organization of this paper is as follows: in section 2 we present the spectral problem we will focus on. In section 3 we do a WKB analysis of this problem. We first show that quantum periods are insufficient, we conjecture the form of the instanton corrections, and we perform a detailed test against the numerical calculation of the spectrum. In section 4 we derive from the results in section 3 the structure of the grand potential of ABJM theory, proving in this way some of the conjectures in . Finally, in section 5 we state our conclusions and directions for further research. An Appendix contains some results on Mellin transforms which are used in section 4.
2 The spectral problem
We will consider a spectral problem arising in the quantization of the spectral curve describing the local CY known as local . This is a spectral problem for a difference equation, with an appropriate and natural choice of analyticity and boundary conditions. The resulting problem can be equivalently formulated in terms of an integral equation which plays a crucial role in the Fermi gas approach to ABJM theory . In this section we will first consider the integral equation formulation, and then the formulation in terms of a difference equation.
Let us consider the following integral kernel
The kernel (2.1) defines a non-negative, Hermitian, Hilbert–Schmidt operator, therefore it has a discrete, positive spectrum
This operator can be written as
In this equation, are canonically conjugate operators,
In this paper we will use and interchangeably. The spectral problem (2.2) can now be written as
Let us now define
It follows that
or, equivalently, in the coordinate representation,
This difference equation is only equivalent to the original problem (2.2) provided some analyticity and boundary conditions are imposed on the function . Following , let us denote by the strip in the complex -plane defined by
Let us also denote by those functions which are bounded and analytic in the strip, continuous on its closure, and for which as through real values, when is fixed and satisfies . It can be seen, by using the results in , that the equivalence of (2.12) and (2.2) requires that belongs to the space .
The operator can be used to define a quantum Hamiltonian in the usual way ,
whose classical limit is simply
In  it was noticed that the curve
defining the classical limit of the spectral problem, is a specialization of the curve describing the mirror of the Calabi–Yau known as local . Let us write this curve as in 
and the specialization
where we have denoted, for convenience,
Notice that the above change of variables is essentially a canonical transformation, since it preserves the symplectic form, up to an overall constant,
The curve (2.17) can be quantized, leading to a quantum spectral curve. One simply promotes to quantum operators , satisfying canonical commutation relations. The equation satisfied by wavefunctions is
while the specialization (2.19) has now the quantum correction
In terms of as defined in (2.7), we have
The difference equation (2.12) has the structure of Baxter’s TQ equation, which determines the spectrum of a quantum integrable system and can be regarded as a quantization of the spectral curve of the classical system. This similarity is not surprising: as it is well-known, the curve (2.17), describing the mirror of local , can be regarded as a relativistic deformation of the spectral curve of the periodic Toda chain . The Baxter equations for Toda and relativistic Toda have been studied in  and , respectively. Similar difference equations also appear in the study of supersymmetric gauge theories in the NS limit [22, 23]. (2.12) is also a close cousin of the difference equation studied in , which has its origin in the integral equation of the ’t Hooft model.
Unfortunately, the spectral problem (2.2) does not seem to be exactly solvable, and one has to use numerical or approximate methods.
3 The exact WKB quantization condition
3.1 Perturbative WKB quantization
We will now analyze the spectral problem (2.12) by using the WKB method. At leading order in , the WKB method for bound states is the Bohr–Sommerfeld quantization condition. In this approximation, one calculates the classical volume of phase space as a function of the energy, (here, the subscript means that we are working at zero order in the expansion). The quantization condition says that this volume should be a half-integer multiple
The classical volume of phase space was already determined in . It is given by a period integral on the curve (2.16). Since this is an elliptic curve, it has two periods, the and the periods. The spectral problem we are looking at involves the period, and we find
In the WKB method we are interested in large energies as compared to , i.e. in large quantum numbers. It is then useful to have a basis of classical periods of the curve which is appropriate for the regime. Since the curve (2.16) is a specialization of the mirror of local , the relevant periods are nothing but the large radius periods of this CY. Let us now review some basic facts about these periods.
In the full CY (2.17), i.e. for generic values of , it is useful to consider two different periods and two different periods. The periods are given by
There are two independent -periods, , , which are related by the exchange of and ,
The period is given by
In our spectral problem we have classically (see (2.19)). In this limit, one has
The classical B-period becomes
and one finds that the volume of phase space can be written in terms of this period as
It is well-known that the Bohr–Sommerfeld quantization condition has perturbative corrections in . These can be obtained in a straightforward way by solving the equations (2.12) with a WKB ansatz,
and interpreting as a “quantum” differential. The leading order approximation gives
and reproduces the Bohr–Sommerfeld quantization condition. Using this quantum differential, we can define the perturbative, “quantum” volume of phase space as
We could calculate these corrections directly in the equation (2.12). However, it is more illuminating to obtain them as particular cases of the quantum corrections for the spectral curve (2.22). Indeed, as explained in [3, 4] and reviewed in , these corrections promote the classical periods , to quantum A-periods
and quantum B-periods , . As in the classical case, there are two of them, but they are related by the exchange of the moduli,
The quantum counterpart of (3.7) is
where is given in (2.25).
Let us now come back to the problem of calculating (3.16). This is a quantum period for the spectral curve defined by (2.12), but this curve is just a specialization of (2.22) with the dictionary (2.24) and after a canonical transformation. Therefore, (3.16) should be a combination of the quantum periods of local , specialized to the “slice” (2.24). Let us denote
where , have the -expansion,
Requiring the combination of quantum periods to have the correct classical limit, and that only even powers of appear, we find,
The third term in the last line of (3.23) is an -independent correction to the quantum period which was already computed in . Notice that the perturbative quantum volume has an -expansion of the form,
The first term in this expansion is the function of given in (3.3). By the WKB expansion of the wavefunction (3.13), (3.14), it is possible to find an exact expression for the first quantum correction, which reads
where , are elliptic integrals of the first and second kind, respectively, and their modulus is given by
The derivation of this result is sketched in appendix B.
The equation (3.16) gives the full series of perturbative corrections to the classical phase-space volume. The perturbatively exact quantization condition involves the quantum B-periods of the spectral curve, and it reads
We can now use (3.27) to compute the quantum-corrected spectrum. For example, we can use the explicit expressions (3.3) and (3.25) to compute the energies perturbatively, as a power series expansion around , i.e.
The leading order term corresponds to the zero of , and as found in  this is,
3.2 Non-perturbative WKB quantization
As we have seen, the perturbative WKB quantization condition (3.27) makes it possible to compute the energies as a power series in . However, when we consider finite values of , we find a key problem: the coefficients appearing in the expansion (3.23) diverge for any integer , and lead to a non-sensical WKB expansion when approaches times an integer. At the same time, there is no physical source for this divergence in the spectral problem itself: the eigenvalues for appearing in (2.12) and in the associated integral equation (2.2) are perfectly well-defined for any real value of , and in particular for integer . As a matter of fact they can be computed numerically, as we will see in the next subsection. We stress that the divergence problem in (3.27) is not an artifact of the large expansion used to obtain (3.23). This leads to an asymptotic expansion for the energy levels valid for large quantum numbers, which should be well-defined. It can be shown, by using the BPS structure of the refined topological string free energy [27, 28], that the quantum B-periods of any local CY manifold are divergent for an infinite number of values of . In particular, the WKB quantization condition for B-periods written down in  has an infinite number of poles in the complex -plane.
We conclude that the expression (3.23) is incomplete, and there must be an extra correction which makes the quantum volume of phase space finite and leads to a reasonable quantization condition. Since (3.27) already incorporates all the perturbative information available, this correction must be non-perturbative in .
The possibility of having instanton corrections to the quantum volume of phase space was already anticipated in . As pointed out there, in order to understand these corrections we need the geometric approach to the WKB method developed in for example [29, 30, 31, 32]. In this approach, the perturbative WKB quantization condition is associated to a classical periodic orbit of energy and the quantum fluctuations around it. The classical action of this trajectory is the classical B-period, and the perturbative corrections promote it to a quantum period. In our example, this perturbative analysis leads to the result (3.23). In fact, the classical periodic orbits can be described in detail. The classical Hamiltonian (2.15) leads to the equations of motion
For an orbit of energy , we find
which can be integrated in terms of Jacobi’s elliptic sine,
where the modulus is now given by
The function is doubly periodic in . It has a real period given by
where is the complete elliptic integral of the first kind. The action around this periodic trajectory is
where we took into account a relative factor of coming from .
However, as pointed out in , in order to obtain a non-perturbative quantization condition one should take into account as well complexified trajectories. These trajectories are allowed because the curve (2.16) has genus one and therefore it has an imaginary period, on top of the real period associated to the real periodic orbit (an example of such a situation was discussed recently in ). In our case this is just the imaginary period of the Jacobi sine function,
In the complexified orbit, time is imaginary, as in : . If we now use the relation
where , we find that is also imaginary: . The equation for the complex trajectory becomes
Geometrically, the trajectory (3.33) describes a closed orbit in phase space, along the hypersurface of constant energy . This is the Fermi surface of the ideal Fermi gas introduced in . After complexifying the exponentiated variable , this closed orbit becomes a torus. The imaginary trajectory (3.40) is a closed orbit around the -cycle of this torus, while (3.33) is now regarded as a closed orbit around the -cycle. We depict both orbits in Fig. 2.
In the original Hamiltonian, has a periodicity of , therefore the relevant action is
where we have denoted
The contribution of such a complex trajectory to the quantization condition is of the form
Including the quantum corrections simply promotes the classical period to its quantum counterpart, as already noticed in . As in [31, 32], we will call the exponentiated quantum period associated to a cycle a Voros multiplier. The quantum A-period, specialized to the “slice” (2.24), is
where the sign corresponds to , respectively. We conclude from (3.43) that the appropriate Voros multiplier for the A-period in this theory is
This result was also obtained in 
This is clearly non-perturbative in (or equivalently, in ), and it is invisible in the standard perturbative correction to the WKB condition.
The calculation of from first principles is difficult. In the case of the Schrödinger equation studied in [29, 30, 31, 32], the instanton corrections are determined by the vanishing of the so-called Jost function, and their calculation requires a detailed analysis of the spectral problem and of the WKB wavefunction. We will now present a conjecture for the form of the instanton corrections for the spectral problem (2.12). To write down our formula, let us consider the worldsheet instanton corrections to the un-refined topological string free energy on local , in the Gopakumar–Vafa form :
and is the topological string coupling constant. In (3.48), , are the complexified Kähler classes, corresponding to the two compact s in the geometry, and are the Gopakumar–Vafa invariants of local for genus and degrees . Along the “slice”
the natural invariant is the diagonal one,
We are now ready to state our conjecture about the form of the instanton corrections. We claim that the coefficients in (3.47) are given by
The coefficients have a simple interpretation: they are the coefficients of in the topological string free energy of local , along the slice (3.50), and for
We then conjecture that the non-perturbative contribution to the quantum volume of phase space is
and the total quantum volume is
Notice that the non-perturbative volume (3.55) is also divergent for integer values of . We will now show that the total volume (3.56) is well-defined, i.e. the divergences in cancel against the divergences in .
Before showing this, let us give some indications on the origin of this conjecture. As we will see in the next section, the spectral problem (2.12) appears in the Fermi gas approach to ABJM theory. It turns out that the grand potential of ABJM theory is closely related to the volume of phase space. The perturbative part (3.23) leads to the non-perturbative membrane corrections to the grand potential, while the non-perturbative part (3.55) leads to the worldsheet-instanton corrections to the grand potential. The conjecture (3.55) is inspired by the known form of these corrections [33, 10, 14]. The requirement that divergences should cancel in the total volume (3.56) is a dual manifestation of the HMO cancellation mechanism discovered in . According to this mechanism, the divergences in the worldsheet instanton part of the grand potential of ABJM theory should cancel against the divergences in the membrane instanton part, since the total grand potential is well-defined and finite for any . We have here a similar mechanism, which is based this time on the fact that the spectral problem is well-defined for any value of . The cancellation mechanism in the quantum volume is simpler however than the HMO mechanism, since it only involves simple poles, while the HMO mechanism involves double poles.
Let us now verify that is well-defined for any value of . Since the non-perturbative contribution is defined in terms of , instead of , let us re-express the perturbative part in terms of this variable. This defines a new set of coefficients as follows,
These coefficients were first introduced in , in the context of ABJM theory, and their geometric meaning was uncovered in : they can be expressed in terms of the refined BPS invariants of local as
This formula is based on the fact that the combination of B-periods appearing in (3.23) is a derivative of the refined topological string free energy, which in turn can be written in terms of refined BPS invariants (see for example  for a summary of these facts, and a list of values of the refined invariants for low degrees). On the other hand, the coefficient (3.53) can be expressed in terms of the same invariants by the formula,
see  for a derivation. We can now use a simplified version of the argument appearing in  to check that the singularities in (3.56) cancel. First of all, notice that the singularities appear when takes the form
The singularities are simple poles. The poles appearing in are of the form
The corresponding poles appearing in (3.57) are of the form
By using (3.60), one notices that
This can be seen to be the case by a geometric argument explained in . We conclude that is well-defined and finite for any real value of , as a series in .
The exact WKB quantization condition reads now
where is a sum of the perturbative part and the non-perturbative part . This condition determines the energy levels as functions of and . As in similar examples of exact WKB quantization conditions, the total is a trans-series involving various small parameters, on top of itself. On one hand we have of course , but we also have exponentially small quantities in , and non-analytic functions of at , like the trigonometric functions of appearing in (3.55). We can solve this quantization quantization at small , as one does for example in the case of the double-well potential in Quantum Mechanics . To give a flavour of the type of expressions one finds, let us calculate the first non-perturbative correction to the perturbative series in (3.28). In order to do this calculation, we need the leading term of in an expansion around . We find, after using (3.10),
After expanding it around , we find,
where is the Catalan number. A simple calculation shows that
where, as ,
It seems that is a trans-series involving the small parameter , and trigonometric functions of .
Before closing this subsection, let us summarize the two most important consequences of our proposal for the exact quantization condition:
In solving the spectral problem (2.12) associated to the quantum curve of local