Topological Strings from Quantum Mechanics
We propose a general correspondence which associates a non-perturbative quantum-mechanical operator to a toric Calabi–Yau manifold, and we conjecture an explicit formula for its spectral determinant in terms of an M-theoretic version of the topological string free energy. As a consequence, we derive an exact quantization condition for the operator spectrum, in terms of the vanishing of a generalized theta function. The perturbative part of this quantization condition is given by the Nekrasov–Shatashvili limit of the refined topological string, but there are non-perturbative corrections determined by the conventional topological string. We analyze in detail the cases of local , local and local . In all these cases, the predictions for the spectrum agree with the existing numerical results. We also show explicitly that our conjectured spectral determinant leads to the correct spectral traces of the corresponding operators. Physically, our results provide a non-perturbative formulation of topological strings on toric Calabi–Yau manifolds, in which the genus expansion emerges as a ’t Hooft limit of the spectral traces. Since the spectral determinant is an entire function on moduli space, it leads to a background independent formulation of the theory. Mathematically, our results lead to precise, surprising conjectures relating the spectral theory of functional difference operators to enumerative geometry.
a]and Marcos Mariño
\affiliation[a]Département de Physique Théorique et section de Mathématiques
Université de Genève, Genève, CH-1211 Switzerland \affiliation[b]DESY Theory Group, DESY Hamburg,
Notkestrasse 85, D-22603 Hamburg, Germany \emailAddalba.firstname.lastname@example.org, email@example.com, firstname.lastname@example.org \preprint
As it is well-known, string theory is in principle only defined perturbatively. In the last years, thanks to the AdS/CFT correspondence, non-perturbative formulations have been found in certain backgrounds, in terms of a dual gauge theory. The combination of this duality with localization and integrability techniques have provided us with concrete non-perturbative expressions for many quantities. In general, these quantities have a perturbative genus expansion determined by string perturbation theory, but they involve additional non-perturbative contributions. A particularly interesting example of such a quantity is the partition function of ABJM theory  on the three-sphere. Using localization, this partition function can be expressed in terms of a matrix integral . The ’t Hooft expansion of this integral, fully determined in , gives the genus expansion of the dual type IIA superstring. However, there are additional non-perturbative corrections which were first pointed out in  and then uncovered in a series of papers [5, 6, 7, 8, 9, 10, 11]. One key idea in the study of the non-perturbative structure beyond the genus expansion is the formulation of the matrix model in terms of an ideal Fermi gas , which can be in turn reduced to the spectral problem of an integral operator.
The study of the ABJM matrix model indicated a close connection to topological string theory: its ’t Hooft expansion is identical to the genus expansion of the topological string on the Calabi–Yau (CY) manifold known as local [12, 3]. In additon, the WKB analysis of the spectral problem of the Fermi gas is related to the refined topological string on the same manifold [10, 11], in the so-called Nekrasov–Shatashvili (NS) limit . It is then natural to speculate that similar structures could be found in topological string theory on other local CY manifolds. This had been already pointed out in [5, 14]. In  a concrete proposal was made for a non-perturbative topological string free energy, inspired by the results on ABJM theory. This proposal has two pieces: the perturbative piece is given by the standard genus expansion of the topological string, while the non-perturbative piece involves the refined topological string in the NS limit. A crucial rôle in the proposal was played by the HMO cancellation mechanism , which guaranteed that the total free energy was smooth.
A dual point of view on the problem has been proposed in , where the starting point is the spectral problem associated to the quantization of the mirror curve. Let a toric CY manifold, and let be the curve or Riemann surface encoding its local mirror. The equation describing this curve (sometimes called the spectral curve of ) is of the form
This curve can be “quantized”, and various aspects of this quantization have been studied over the last years, starting with . The quantization of the curve promotes it to a functional difference operator, which can then be studied in the WKB approximation. Inspired by the work of , it was found in [16, 17, 18] that the perturbative WKB quantization condition for the spectrum of these operators is closely related to the NS limit of the refined topological string on . However, it was pointed out in  that, if one looks at the actual spectrum of these operators, this perturbative quantization condition can not be the whole story, and additional non-perturbative information is needed. Moreover,  proposed a non-perturbative quantization condition, based on the results of , in which the perturbative result is complemented by instanton effects coming from the standard topological string. This condition turned out to lead to the correct spectrum in some special cases [11, 19]. Although  focused on the case of local , relevant for ABJM theory, it was suggested there that a similar story should apply to more general toric CY manifolds. This suggestion was pursued in , where the spectrum of the operators associated to some other toric CYs was studied numerically in full detail. The results of [20, 21] indicated that, in general, the quantization condition suggested in  required additional corrections.
In this paper we will propose a detailed conjecture on the relation between non-perturbative quantum operators and local mirror symmetry. We will associate to each spectral curve (1.1) an operator with a positive, discrete spectrum, such that all the traces , , are well-defined (technically, is a positive-definite, trace class operator.) A natural question is then: what is the exact spectrum of this operator? This is a sharp and concrete question, since as it was first noted in  and further studied in , it is possible to calculate this spectrum numerically. Our proposal is that the spectral determinant of is encoded in the non-perturbative topological string free energy constructed in . As we will explain, this free energy (which we will call the modified grand potential of ) defines a generalized theta function. The zeros of the spectral determinant are the zeros of this generalized theta function, and this leads to an exact quantization condition for the spectrum that agrees with all existing numerical results for these operators. In particular, the proposal of  is a natural first approximation to our full quantization condition, and our conjecture explains naturally why it predicts the right spectrum in some special cases. In the general case, we can compute analytically the corrections to the quantization condition of , and we find that they perfectly agree with the numerical results for the spectrum found in . The proposal we make in this paper clarifies the rôle of the non-perturbative free energy of , and its precise relation to the exact quantization condition. But it also gives more information on the spectrum than just the quantization condition, since it provides in principle an exact expression for the spectral determinant of the corresponding operators. In addition, the spectral traces of the operators can be obtained from the behavior of topological string theory near the orbifold point.
As it was already emphasized in [10, 11], our proposal can be regarded as a non-perturbative completion of the topological string, in which the topological string and the refined topological string complement each other non-perturbatively. There have been many proposals for a non-perturbative definition of the topological string, and in a sense this is not a well-posed problem, since there might be many different non-perturbative completions (as it happens for example in 2d gravity.) In fact, there is strong evidence [3, 22] that in many cases the genus expansion of the topological string is Borel summable, so one could take the Borel resummation of this series as a non-perturbative definition. We believe that our proposal is an interesting solution to this problem for three reasons.
First of all, our starting point is the spectral determinant of the operator , which is well-defined and an entire function on the moduli space of . This means in particular that our starting point is background independent. At the same time, different approximation schemes for the computation of this spectral determinant are encoded in different perturbative topological string amplitudes. For example, given the operator , we can define a partition function , which is well-defined for any integer and any real coupling . In the ’t Hooft limit,
this partition function has a ’t Hooft expansion which is determined by the standard genus expansion of the topological string on .
Second, our proposal can be regarded as a concrete M-theoretic version of the topological string, in the spirit of the M-theory expansion of Chern–Simons–matter theories [23, 5]. For example, the partition function has an M-theory expansion at large but fixed which involves in a crucial way the Gopakumar–Vafa invariants of . However, it also includes additional non-perturbative corrections which in particular cure the singularities of the Gopakumar–Vafa free energy, as in the HMO mechanism. We also have, naturally, that
as in a theory of M2-branes . This suggests that the physical theory underlying the spectral theory of the operator might be a theory of M2-branes. It should be noted as well that what our proposal can be understood as a Fermi gas formulation of topological string theory, similar to the Fermi gas formulation of ABJM theory in : the spectrum of the operator gives the energy levels of the fermions, and the spectral determinant is naturally interpreted as the grand canonical partition function of this gas.
Third, our proposal has a surprising mathematical counterpart: it leads to precise and testable predictions for the spectral determinant and the spectrum of non-trivial functional difference operators. According to our conjecture, the answer to these questions involves the refined BPS invariants of local CYs. In this way, we link two mathematically well-posed problems (the spectral theory of these operators, and the generalized enumerative geometry of CYs) in a novel way.
Although we believe that our proposal will hold for very general toric CY manifolds, in this paper we will focus for simplicity on those geometries whose mirror curve has genus one. In that case, the theory is simpler and we can make precision, non-trivial checks of our proposal. The details of the generalization to higher genus will be studied in a forthcoming publication.
This paper is organized as follows. In section 2 we present the correspondence between mirror curves and quantum operators. In section 3 we state our conjecture for the spectral determinant of these operators, we derive the quantization condition implied by our conjecture, we comment on the physical implications of our results, and we study the simplest cases of our theory, which we call the “maximally supersymmetric cases.” Section 4 presents a detailed illustration of our claims in the case of local . Section 5 presents additional evidence for our conjecture by looking at two other geometries: local , and local , which was the original testing ground due to its relationship to ABJM theory. Finally, in section 6 we conclude and list various open problems. In appendix A, we give a derivation of the first quantum correction to the grand potential of local .
2 From mirror curves to quantum operators
In this section we will present a correspondence between mirror curves and quantum operators. Aspects of this correspondence have been explored in various papers, starting in  and, more relevant to our purposes, in [16, 17], building on the work of  for gauge theories. However, our interest will be in defining a non-perturbative spectral problem, from which one can compute a well-defined spectrum. This was first proposed in  and then pursued in .
Let us start by reminding some basic notions of local mirror symmetry [25, 26]. We consider the A-model topological string on a (non-compact) toric CY threefold, which can be described as a symplectic quotient
where . Alternatively, may be viewed physically as the moduli space of vacua for the complex scalars , of chiral superfields in a 2d gauged linear, supersymmetric -model . These fields transform as
under the gauge group . Therefore, is determined by the -term constraints
modulo the action of . The correspond to the Kähler parameters. The CY condition holds if and only if the charges satisfy 
Here, the are moduli parametrizing the complex structures of the mirror , which is given by
The constraints (2.5) have a three-dimensional family of solutions. One of the parameters correspond to a translation of all the fields
which can be used for example to set one of the s to zero. The remaining fields can be expressed in terms of two variables which we will denote by , . The resulting parametrization has a group of symmetries given by transformations of the form ,
After solving for the variables in terms of the variables , , one finds a function
Note that, due to the translation invariance (2.8) and the symmetry (2.9), the function in (2.11) is only well-defined up to an overall factor of the form , , and a transformation of the form (2.9). It turns out [31, 32] that all the perturbative information about the B-model topological string on is encoded in the equation
which can be regarded as the equation for a Riemann surface embedded in .
In this paper we will focus for simplicity on toric CY manifolds in which has genus one, i.e. it is an
These manifolds can be classified by reflexive polyhedra in two dimensions (see for example [26, 33] for a review of this and other facts on these geometries). The polyhedron associated to a surface is the convex hull of a set of two-dimensional vectors
The extended vectors
satisfy the relations
where is the vector of charges characterizing the geometry in (2.3). Note that the two-dimensional vectors satisfy,
It turns out that the complex moduli of the mirror are of two types: one of them, which we will denote as in [33, 34], is a “true” complex modulus for the elliptic curve , and it is associated to the compact four-cycle in . The remaining moduli, which will be denoted as , should be regarded as parameters. For local del Pezzos, there is a canonical parametrization of the curve (2.11), as follows. Let
which can be used to solve for the functions , up to reparametrizations. We then find the equation for the curve,
Let and be standard quantum-mechanical operators satisfying the canonical commutation relation
In this paper, will be a real parameter. We need to consider as well the exponentiated operators
These operators are self-adjoint and they satisfy the Weyl algebra
However, the domains of , should be defined appropriately, since they lead to difference or displacement operators acting on wavefunctions (for example, if we work in the representation, is a difference operator.) The domain of the operator , , consists of wavefunctions such that
Similarly, the domain of , , consists of functions such that
is the wavefunction in the representation, which is essentially given by a Fourier transform. The condition (2.26) can be translated into a condition on (see for example ): this is a function which admits an analytic continuation into the strip
such that for all , and the limit
exists in the sense of convergence in .
We want now to associate a self-adjoint quantum operator of the form
to each toric del Pezzo , in such a way that we have a well-defined eigenvalue problem
i.e. we want to have a discrete and positive spectrum, so that the energies are real. It is convenient to consider the inverse operator
The spectral traces of are defined by
and we will require them to be well-defined (i.e. finite). The semiclassical limit of these traces is given by,
and denotes the classical function underlying (2.30), or more formally, the Wigner transform of the operator (2.30) (this classical function is simply given by the expression (2.30) where we replace , by the corresponding classical variables.) If the semiclassical limit is smooth, as we will assume here, we should have
This leads to useful constraints on the form of .
Let us explain how to associate a quantum operator to a given local del Pezzo. We have seen in (2.19) that, for local del Pezzo’s, the function can always be written in the form (2.19). The operator is obtained by promoting the classical function in (2.20) to a quantum operator. In this promotion, we use Weyl’s prescription for ordering ambiguities. This associates
so that the resulting operator is Hermitian. Clearly, if the parameters satisfy appropriate reality and positivity conditions, the resulting quantum operator will be of the form (2.30). Since this operator is a sum of operators of the form , its domain is given by the intersection of the domains of all the operators of this type appearing in the sum in (2.30).
In order to illustrate this procedure, let us consider the well-known example of local . In this case, we have and the toric CY is defined by a single charge vector . The corresponding polyhedron for is obtained as the convex hull of the vectors
In the mirror, the variables satisfy
and the canonical parametrization is given by
after changing . Therefore, the quantum operator is given by
Following the procedure in the previous example, we can write down operators for other local del Pezzo CYs. A list with some useful examples can be found in table 1, where we used for convenience the classical version . The conventions for the parametrization of the curves (in particular, for the parameters , appearing in the equations) are those of [34, 33]. Note that a transformation of the form (2.9) corresponds to a canonical transformation, and will not change the spectrum of the operator. Note as well that, after changing , the spectral problem (2.31) can be written as
where we use the form (2.19). The spectral problem leads then to a quantization of the modulus , which after the change of sign above, can be interpreted as the exponential of the energy:
We can regard as the exponential of a Hamiltonian , while can be interpreted as the canonical density matrix,
The operator has a complicated Wigner transform (as in the closely related examples of ). Its explicit form will not be needed in this paper, but it might be useful to test some of our statements in a semiclassical analysis, as in .
In order to gain some insight into these operators, and to verify that the requirement (2.36) holds for them, we can consider their semiclassical limit and the corresponding Bohr–Sommerfeld quantization condition. The region of phase space with energy less or equal than is defined by the equation,
As is well-known, in the semiclassical limit each cell of volume in will lead to a quantum state. Therefore, if we want the spectrum of to be discrete, we should require to have a finite volume. The geometry of the region at large energies is easy to understand (and very similar to the situations considered in [5, 37]): for large , we should consider the tropical limit of the curve (2.20), which in the canonical parametrization (2.19) reads
The boundary of the region is the polygon limited by the lines (2.47). This polygon is nothing but the boundary of the dual polyhedron defining the toric del Pezzo, see for example Fig. 1 and Fig. 2 for nice illustrations involving local and local , respectively. Therefore, the region (2.46) has a finite volume. This also guarantees that the classical function
decays exponentially at infinity, so that (2.36) is verified.
We expect the difference operators constructed in this way to have a positive and discrete spectrum. Specifically,
we expect their inverses to be positive-definite and trace class operators. This is clearly indicated by the
behavior of the semiclassical limit, but it would be important to prove it from first principles, in order
to make sure that the spectral problem and the spectral traces are defined rigorously
In practice, one can calculate the spectrum of the operators as in 
can be diagonalized numerically: one first truncates it to an dimensional matrix, computes the eigenvalues , , and observes numerical convergence as grows,
3 Spectral determinants and topological strings
In this section we state our main conjecture, which gives a conjectural expression for the spectral determinant of the operator introduced in the previous section. We also discuss the quantization condition for the spectrum derived from our conjecture, as well as its physical meaning.
3.1 The spectral determinant
The spectral information about the operators and can be encoded in various useful ways. Given a trace class operator with eigenvalues , , and depending on a real parameter , its spectral determinant (also called Fredholm determinant) is defined by
We will refer to as the fugacity, and we will often write it as
where is called the chemical potential. We will use the arguments and interchangeably. The reason for this terminology is that can be physically interpreted as the grand canonical partition function of an ideal Fermi gas where the one-particle problem has energy levels . Note that our spectral determinant is different from the one usually studied in Quantum Mechanics [39, 40, 41]:
Our definition (3.1) uses instead the canonical density matrix. It has better convergence properties and does not need to be regularized, in contrast to (3.3). For example, in the case of the quantum harmonic oscillator, the spectral determinant (3.3) leads, after regularization, to
while with our definition we would obtain
which is the quantum dilogarithm .
The spectral determinant has two important properties: first, it is an entire function of the fugacity (see for example , chapter 3, for a proof of this fact). Second, after setting
it has simple zeros, as a function of , at the energies of the spectrum . This means that one can in principle read the spectrum of the operator by looking at the zeros of the spectral determinant. The grand potential is defined as
and it has the following useful expression in terms of the spectral traces defined in (2.33):
There are certain special combinations of the traces which appear when one expands the spectral determinant around :
We will call the , for , the (canonical) partition functions associated to the operator . We can obtain by taking an appropriate residue at the origin,
If we denote by
then the can be interpreted as the canonical partition functions of an ideal Fermi gas of particles with energy levels :
In this equation, is the permutation group of elements and is the signature of a permutation . The canonical partition functions encode the information in the spectral traces in a slightly different way, as one can see by combining (3.9) with (3.8), and they are related by
where the means that the sum is over the integers satisfying the constraint
We note that the grand potential has a well-defined classical limit: when , one has
where the leading contribution
involves the classical limit of the spectral traces (2.35). As first noted in , the study of this limit for the operators appearing in Chern–Simons–matter theories leads to many insights on their behavior, see for example [44, 45].
We will now make a proposal for the spectral determinant of the operators that we associated to toric CY manifolds. We will focus on the case in which the mirror curve has genus one, i.e. on the case of toric (almost) del Pezzo. We sketch the generalization to higher genus in the final section of the paper. For simplicity, we will first write down our formulae in the case in which the parameters appearing in the operator take their most symmetric value. This value is obtained as follows: the parameters are linear sigma model parameters, and they are related to their corresponding Kähler parameters or flat coordinates by an algebraic mirror map. The most symmetric value of the corresponds to setting . For example, in the case of local and local , the most symmetric value is . We will consider the more general case in section 3.5.
Once we restrict ourselves to the value for the parameters , the del Pezzo surfaces considered in the previous section have a single modulus , which is related to the modulus introduced before as
Here, the value of is determined by the geometry of (in particular, by the canonical class of .) For example, for local we have , while for local we have (see Table 1 for other cases). For each of these geometries, there is also a quantum mirror map  relating the modulus to a flat coordinate , and of the form
We will now introduce, in analogy with ABJM theory , an “effective” parameter
where is defined by a series expansion
and has the form
The coefficient is given as follows. Let us consider the volume of the region defined in (2.46), which we will denote as . At large , the region becomes polygonal, and its volume will behave as
Let us consider again the case of local . At large , the region becomes the triangle whose boundaries are appropriate segments of the lines
which are read immediately from the tropical limit of the mirror curve. The area of this triangle is , so we conclude that
We will verify this value with other techniques later on. ∎
We are now ready to introduce the crucial quantity determining the spectral determinant. In analogy with [7, 10, 19], we will call it the modified grand potential. It is essentially the non-perturbative topological string free energy introduced in , and it has the structure
In this equation, the perturbative piece is given by
The coefficient has the structure
where is the coefficient appearing in the sub-leading asymptotics of , in (3.22). The coefficient can be determined from the first quantum correction to the B-period, as in the calculations of [5, 11, 20]. The coefficient is more difficult to determine, although in some special cases it can be guessed and/or computed numerically. It can be also fixed by a normalization condition, as we will see in a moment. However, since it is independent of , it plays a relatively minor rôle. In particular, it does not enter into the quantization condition. The function has the structure
where and are given by,
The coefficients are determined by the so-called refined BPS invariants of [46, 47, 48], which we will denote by . Here, is a positive integer which denotes the degree w.r.t. the flat coordinate or Kähler modulus in (3.18), and , are the spins of the corresponding BPS multiplets. We have the following expression,
Finally, the worldsheet instanton part of the modified grand potential is defined by
where is also determined by the BPS invariants,
and the -field in (3.34) is such that
for all the values of , , which lead to a non zero BPS invariant . There is a geometric argument, explained in , which shows that there is a natural choice of field which guarantees (3.36). In the toric del Pezzo’s that we are considering, we can set , since they are both determined by the canonical class of . It is important to notice that the combinations of BPS invariants which enter into the modified grand potential are very specific. Namely, the combination entering in (3.34) involves only the Gopakumar–Vafa invariants appearing in the standard topological string ,
while (3.32) involves the combination of the invariants appearing in the NS limit of the refined topological string. Indeed, in this limit, the instanton
part of the topological string free energy can be written as
and we conclude that
i.e. is essentially the quantum B-period of .
One of the most important aspects of the grand potential (3.27) is the following: the worldsheet instanton piece has double poles when is of the form times a rational number. The functions and have poles at the same values. However, in the total function these poles cancel. The proof of this statement is a trivial generalization of the proof offerered in , but we present it here for the convenience of the reader, since it is an important point of the construction. The coefficient (3.35) has double poles when . The coefficient (3.32) has a simple pole when , and due to (3.33) the coefficient will have a double pole at the same values of . These poles contribute to terms of the same order in precisely when takes the form
We have then to examine the pole structure of (3.27) at these values of . Since both (3.35) and (3.32) involve a sum over BPS multiplets with quantum numbers , , we can look at the contribution to the pole structure of each multiplet. In the worldsheet instanton contribution, the singular part associated to a BPS multiplet around is given by
The singular part in associated to a BPS multiplet is given by
Using (3.33), we find that the corresponding singular part in is given by
By using (3.36), it is easy to see that all poles in (3.41) cancel against the poles in (3.42) and (3.43), for any value of . This cancellation phenomenon was of course one of the guiding principles for the proposal of  and it generalizes the HMO cancellation mechanism for the modified grand potential of ABJM theory .
We are now ready to make our main proposal: we conjecture that, given a toric del Pezzo CY , the spectral determinant of the operator associated to it is given by
where is the modified grand potential (3.27), and is given by