Matrix models from operators and topological strings
We propose a new family of matrix models whose expansion captures the all-genus topological string on toric Calabi–Yau threefolds. These matrix models are constructed from the trace class operators appearing in the quantization of the corresponding mirror curves. The fact that they provide a non-perturbative realization of the (standard) topological string follows from a recent conjecture connecting the spectral properties of these operators, to the enumerative invariants of the underlying Calabi–Yau threefolds. We study in detail the resulting matrix models for some simple geometries, like local and local , and we verify that their weak ’t Hooft coupling expansion reproduces the topological string free energies near the conifold singularity. These matrix models are formally similar to those appearing in the Fermi-gas formulation of Chern–Simons–matter theories, and their expansion receives non-perturbative corrections determined by the Nekrasov–Shatashvili limit of the refined topological string.
One of the most surprising aspects of string theory is that, in some circumstances, it can be described by very simple quantum systems. For example, non-critical (super)strings can be formulated in terms of double-scaled matrix models or matrix quantum mechanics. In some cases, these equivalent descriptions provide as well a non-perturbative definition of the corresponding string theory models. More recently, some simple quantities in fully-fledged superstring theory, like partition functions, have been also expressed in terms of matrix integrals, by combining the AdS/CFT correspondence with supersymmetric localization.
It has been suspected for a long time that topological strings on Calabi–Yau (CY) manifolds should be also described by simple quantum models. It was found in  that the type B topological string on a special class of non-compact B-model geometries has such a description, in terms of conventional matrix models. The CY backgrounds considered in  are useful from the point of view of engineering supersymmetric gauge theories, but they have no mirror geometries and no enumerative content. It was later found that bona fide topological strings on fibrations over can be described by Chern–Simons matrix models , as a consequence of the Gopakumar–Vafa large duality  and its generalizations .
Recently, a correspondence has been proposed between topological strings on toric CY threefolds, and the spectral theory of operators arising in the quantization of their mirror curves . The idea that the topological string free energies could emerge from the quantization of mirror curves was first proposed in . In [7, 8], building upon the work of , it was shown that a perturbative treatment of the quantum mirror curve leads to the Nekrasov–Shatashvili (NS) limit of the refined topological string. However, building on the study of matrix models for Chern–Simons–matter theories  and their AdS/CFT duals  (see  for a review), it was pointed out in  that the standard topological string also emerges from the quantum curve, once non-perturbative corrections are taken into account.
The proposal of  incorporates all these ingredients in an exact treatment of the quantum curve. According to , to each mirror curve of a toric Calabi–Yau threefold, one can associate a positive, trace class operator on . This was rigorously proved for a large number of geometries in . Therefore, these operators have a positive, discrete spectrum, and their Fredholm or spectral determinants are well defined. In , an explicit formula for these spectral determinants was conjectured, involving both the NS limit of the refined topological string and the conventional topological string. The conjectural, exact formula of  leads in addition to an exact quantization condition determining the spectrum, which generalizes previous studies of the spectral problem [13, 15, 16, 17] and is conceptually similar to other exact quantization conditions appearing in Quantum Mechanics (see for example ). This establishes a novel and precise connection between spectral theory and mirror symmetry.
One of the consequences of the correspondence of  is that the conventional topological string free energy (at all genera) appears as a ’t Hooft limit of the spectral determinant. A very useful way to encode the information in the spectral determinant is in terms of the so-called fermionic spectral traces (see section 2.1 for precise definitions). As we will show in detail, these traces have a natural matrix model representation. It then follows from the conjecture of  that the ’t Hooft limit of this matrix model,
is given by the asymptotic expansion
where are the standard topological string free energies, and the ’t Hooft parameter
is a flat coordinate for the CY moduli space
As explained in , the correspondence between spectral theory and mirror symmetry provides a rigorous, non-perturbative completion of the topological string, in the sense that the genus expansion of its free energy, which is known to be a divergent series, is realized as the asymptotic expansion of a well-defined function. The implementation of the correspondence of  that we are presenting here, in terms of matrix models, makes this point particularly clear: the quantity is manifestly well-defined for any positive integer and any real , since it is defined by the spectral theory of a trace class operator (it can be also analytically continued to complex , as first explained in , and it is likely that it has an analytic continuation to complex values of ). The ’t Hooft expansion of is given exactly by the genus expansion of the topological string free energy. However, there are non-perturbative corrections to the ’t Hooft expansion, due to large instantons, which are also predicted by the conjecture in , and they are encoded in the NS limit of the refined topological string.
The fact that the matrix model representing leads to the topological string free energies is not at all obvious. On the contrary, it is a highly non-trivial prediction of the conjecture of . The ’t Hooft limit of probes the strong coupling limit of the spectral problem (since is large), and in particular the non-perturbative instanton corrections to the perturbative WKB expansion. For this reason, in this paper we will perform detailed calculations in some examples to verify that, indeed, the’t Hooft expansion of the matrix model defined by the trace class operator gives the topological string free energies. This constitutes an analytic test of the instanton corrections postulated in .
As we mentioned above, the large duality of Gopakumar–Vafa  and its generalizations  provide a matrix model representation of the free energies of topological strings in certain geometries, as it was tested in [4, 20, 21]. Although in this paper we focus on local del Pezzo geometries, our matrix models are potentially valid for any toric geometry, in contrast to the duality of [3, 4]. For example, we will study in detail a matrix model for local , which has no counterpart in the framework of [3, 4]. It would be interesting to understand the relationship between the matrix models for topological strings obtained in  and the ones described here.
Matrix models describing topological strings on more general backgrounds were also proposed in for example [22, 23, 24, 25, 26]. We should note that these models are very different from the ones we construct here: first of all, they are engineered ab initio to reproduce formally the topological string free energies; in contrast, our matrix models are defined by the trace class operators obtained in the quantization of the mirror curve, and the fact that they lead to the correct topological string free energies is a consequence of the non-trivial conjecture of . Second, in the models of [22, 23, 24, 25, 26], the rank of the matrix plays an auxiliary role, while in our case it is a flat coordinate for the Calabi–Yau, as in other large dualities. Third, the matrix models in [22, 23, 24, 25, 26] are often formal (i.e. not convergent) and therefore can not define a non-perturbative completion of the theory; our matrix models are convergent and lead to a non-perturbative completion.
This paper is organized as follows. In section 2 we review elementary aspects of trace class operators on and we note that their fermionic traces have matrix model-like representations. We then focus on the operators coming from quantized mirror curves, and write down explicit matrix models for some of them, including the ones relevant for local and local . These models can be studied in the ’t Hooft expansion, and we compute their weakly coupled expansion to the very first orders. In section 3 we review the conjecture of  and we spell out in detail its prediction for the ’t Hooft expansion of the fermionic traces. Then, we test the conjecture in detail for the matrix models describing local and local . We conclude in section 4 and we list some open problems for the future. In the Appendix, we list some results for the weakly coupled expansion of the matrix model free energies.
2 From operators to matrix models
2.1 General aspects
Let us then begin with some general aspects of operator theory. Let be a positive-definite, trace class operator on the Hilbert space , depending on a real parameter . Since its eigenvalues are discrete and positive, we will denote them by , where . Due to the trace class property, all the spectral traces of
exist. One can also define the fermionic spectral traces as
where the means that the sum is over the integers satisfying the constraint
The Fredholm or spectral determinant of can be defined as
or, equivalently, by the expansion around ,
and is an entire function of . Note that, if is interpreted as a one-particle thermal density operator, the fermionic trace is the canonical partition function for an ideal Fermi gas of particles. It then follows that has the matrix-model-like representation
In this equation, is the kernel of the operator ,
is the permutation group of elements, and is the signature of a permutation . The above formula encapsulates the fermionic nature of .
At this stage, calling (2.7) a matrix model might seem excessive. However, the integrand of (2.7) has an essential property, typical of the integrand of a matrix model in the eigenvalue representation: it vanishes whenever . In standard matrix models, this is an indication of eigenvalue repulsion. To further understand this parallelism, note that, when solved in terms of orthogonal polynomials, the partition function of a Hermitian matrix model is given by the expression
where the kernel can be written in terms of the first orthogonal polynomials. This is very similar to (2.7). A crucial difference though between (2.7) and the more familiar expression (2.9) is that, in (2.7), the kernel does not depend on , and the resulting matrix models are quite special. Matrix models of the form (2.7), coming from integral kernels of operators, have been considered before in for example [28, 29], and they have been recently studied from a more general point of view in , where the notion of “M-theoretic matrix model” was introduced.
As we will see in the next sections, in the case of trace class operators obtained by quantization of mirror curves, the matrix model (2.7) is closely related to Chern–Simons matrix models  and to the matrix models appearing in the localization of Chern–Simons–matter theories [10, 31, 32] (see  for a review). In particular, the resulting matrix models are M-theoretic, in the sense of : they have a well-defined ’t Hooft expansion in the limit (1.1), but they also have a well-defined M-theory limit, in which is large but is fixed.
2.2 Operators from mirror curves and matrix models
In , building on previous work on the quantization of mirror curves, it was postulated that, given a mirror curve to a toric CY threefold, one can quantize it to obtain a trace class operator . As in [5, 14], we will focus on toric (almost) del Pezzo CY threefolds, which are defined as the total space of the anti-canonical bundle on a toric (almost) del Pezzo surface ,
The mirror curve depends on complex moduli , and is of genus one. This means that the complex moduli involve a “true” geometric modulus and a set of “mass” parameters , , where depends on the geometry under consideration [33, 34]. The mirror curves can be put in the form
where has the form
and are suitable functions of the parameters . The vectors can be obtained from the fan defining the toric CY threefold.
In this paper we will be particularly interested in local , where the function is given by,
and local , where it is given by,
To quantize the mirror curve (2.11), we promote , to self-adjoint Heisenberg operators , satisfying the commutation relation
This promotes to an operator, which will be denoted by (possible ordering ambiguities are resolved by requiring the resulting operator to be self-adjoint). As conjectured in  and proved in  for many geometries, the inverse operator
is positive-definite and of trace class. By the construction explained in the previous section, the fermionic traces of this operator, which we will denote by , have an integral representation in terms of the kernel of the operator . Therefore, in order to write down the matrix model (2.7), we need an explicit expression for this kernel. In general, obtaining such an expression is not easy. However, in , this problem was solved for three-term operators of the form
Geometrically, this operator corresponds to the anti-canonical bundle of the weigthed projective space . When , these geometries can be constructed as partial blow-ups of the orbifold , where the orbifold action has the weights . In principle, more general geometries can be obtained by perturbing the operator appropriately. Note that the case gives local , while , gives the limit of local , i.e. a partial blowdown of the local geometry. Let us define
which behaves at large as
In  it was shown that, in terms of an appropriate variable , related to by a linear canonical transformation, one has
In this equation, the parameter is related to as
while , are given by
Since we have an explicit formula for the kernel of , we can write down an explicit expression for the integral calculating the fermionic trace of this operator, which we will denote as . This expression can be put in a very convenient form if we use Cauchy’s identity, as in [38, 39],
In this way one obtains,
The above integral is real, since the kernel (2.21) is Hermitian. Although it is not manifest, it can be also checked that it is symmetric under the exchange .
We are now interested in studying the matrix integral (2.25) in the ’t Hooft limit (1.1). Therefore, we should understand what happens to the integrand in (2.25) when (or equivalently ) is large. To do this, we first change variables to
and introduce the parameter
Note that the strong coupling regime of is the weak coupling regime of . The crucial property to understand this regime is the self-duality of Faddeev’s quantum dilogarithm,
Then, we can write
where is the Bernoulli polynomial. We define the potential of the matrix model as,
The leading contribution as is given by the “classical” potential,
By using the asymptotics of the dilogarithm,
we find that
Therefore, this is a linearly confining potential at infinity, similar to the potentials appearing in matrix models for Chern–Simons–matter theories [39, 30]. The classical potentials for the cases (relevant for local ) and for , (relevant for local ) are shown in Fig. 2.
We can now write the matrix integral as
In the regime in which is large (in particular, in the ’t Hooft limit), we can use the asymptotic expansion (2.33) of the potential to study this matrix integral.
The expression (2.37), computing the fermionic traces of the operators , is very similar to matrix models that
have been studied before in the literature. The interaction between eigenvalues is identical to the one appearing in the generalized models
appearing in , and in some matrix models
for Chern–Simons–matter theories studied in for example . The parameter corresponds to the string coupling constant,
and the potential depends itself on . Note however that, in the planar limit,
only the classical part of the potential (2.34) contributes
2.3 Perturbative expansion
The matrix model (2.37) admits a standard ’t Hooft expansion, of the form
This can be easily seen by noting that, if we just keep the classical part of the potential, we have a generalized matrix model, of the type considered in . The corrections to the potential involve even powers of , therefore they lead to corrections which preserve the form of (2.38).
We would like to compute the genus free energies . Ideally, one would like to obtain them in closed form, as functions of the ’t Hooft parameter , and this might feasible via a suitable generalization of the techniques of . In this paper we are interested in testing whether the above free energies reproduce the genus expansion of the topological string, and we will perform a more pedestrian calculation, by doing perturbation theory in to the very first orders. This means that we regard (2.37) as a Gaussian Hermitian matrix model, perturbed by single and double trace operators. The resulting perturbative expansion can then be converted, by standard means, into a weak coupling expansion around of the very first . This is very similar to the calculations done in  for the lens space matrix model.
In order to work out the expansion of the matrix model, we first have to expand the “classical” potential around its minimum. Let us introduce the parameter
Then, the minimum of the classical potential occurs at
The value of the potential at the minimum is given by
This can be written in a more compact form by using the Bloch–Wigner function
where arg denotes the branch of the argument between and . One finds,
One can use the properties of the Bloch–Wigner function
to verify that (2.42) is symmetric under the exchange of and . For example, we have
is related to the volume of the figure-eight knot. Similarly, we find
where is the Catalan number. As we will see, in order for the conjecture of  to be true, (2.42) has to be related in a precise way to a natural constant appearing in special geometry, namely the value of the large radius Kähler parameter at the conifold point.
We can now expand the full potential appearing in (2.37) around , and write the interaction term as a “deformed” Vandermonde term,
is the usual squared Vandermonde, and
At leading order, we find a Hermitian Gaussian matrix model, and by including the corrections coming from the deformed Vandermonde and the potential (2.33), we can compute systematically the ’t Hooft expansion of around . By using the standard formula for the partition function of the Gaussian matrix model,
where is Barnes’ function, as well as its asymptotic expansion at large , we obtain the following results for the expansion of the genus free energies appearing in (2.38). For the planar free energy, we find
In this equation, is given in (2.51), is given by
and the values of the coefficients can be calculated explicitly as functions of . The results for the very first can be found in Appendix A. Similarly, one finds
The values of for the very first , , for general , can be also found in Appendix A. We now list some results in the case of , relevant as we will see for local . We find,
For , , which is relevant for local , we obtain,
3 Testing the matrix model
3.1 What should we expect?
The conjecture of  gives a very precise prediction for the ’t Hooft expansion (1.2) of the matrix models arising from the trace class operators. To understand this prediction, we have to summarize some of the results of . According to the conjecture of , the basic quantity determining the spectral properties of the operator is the modified grand potential . This grand potential depends on the “fugacity” , which is related to the variable entering in (2.5) as
as well as on the parameters appearing in the operator , which we will collect in a vector . The modified grand potential is determined by the enumerative geometry of the Calabi–Yau . We first need a dictionary between the parameters , , and the parameters appearing in the enumerative geometry of . First of all, we remember that the mirror curve (2.11) involves a modulus . Typically, in mirror symmetry one uses the related modulus
where, the value of is determined by the geometry of . There is a corresponding flat coordinate , determined by
Let us now consider the Kähler parameters of a general local del Pezzo, , in an arbitrary basis. They can be always be written as a linear combination of the Kähler parameter corresponding to , and the Kähler parameters , which are algebraic functions of the parameters appearing in the mirror curve. We have then,
The “effective” parameter is defined by an expansion similar to this one,
The Kähler parameters are then related to the parameters , by the following equation,
We can now write down the general expression for the modified grand potential. It is of the form,
Here, is the perturbative part, which is a cubic polynomial in :
The constants appearing in this expression can be obtained from a semiclassical analysis of the operator (see  for details). The function is not known in closed form, but in some examples there are educated guesses for it. The “membrane” part of the potential has the form,
In this equation, are the refined BPS invariants of the CY , is the vector of Kähler parameters, is the vector of constants appearing in (3.4), and is the vector of degrees. Finally, the worldsheet part of the modified grand potential is
In this equation, are the Gopakumar–Vafa invariants of , while is a -field, which is related to the anticanonical class of , and necessary for the cancellation of poles in .
According to the conjecture of , the spectral determinant of the operator is given by
As first shown in , in the context of ABJM theory, this representation leads to a very convenient formula for the fermionic trace as an integral transform of the modified grand potential,
The contour appearing in this integral is shown in Fig. 3, and in view of the cubic behavior of , it leads to a convergent integral (this is the contour used to define the Airy function).
As already pointed out in , the above conjectural results lead to a very precise formula for the ’t Hooft expansion of . To obtain this expansion, we follow a procedure similar to what was done in . We first note that the function has itself a ’t Hooft limit in which
It is clear that this double scaling is needed if we want the integral in the r.h.s. of (3.14) to be non-trivial as . What is the effect of this limit on ? First of all, we have that
and all the exponential corrections to (3.6) vanish. Similarly, the membrane corrections in (3.10) vanish as well, since they are exponentially small as . The only surviving terms come from the perturbative and the worldsheet parts of the modified grand potential. As we will see in examples, the function has an expansion as of the form
where includes as well a logarithmic dependence on . We conclude that, in the ’t Hooft limit (3.15), the modified grand potential has the expansion,
Here, we have introduced the variable
and is the standard genus topological string free energy, expressed in terms of the conventional Kähler parameters , (these correspond to the parameters , , up to a rescaling by ). We now note that, in order to perform the ’t Hooft limit, we must also make a choice for the scaling of the parameters . If these scale with when , the ’t Hooft limit of reproduces the standard topological string free energy at genus . However, we might also want to keep the fixed. In that case, in order to obtain the large expansion, we have to re-expand the functions , and one finds
where, for the very first orders,