Wavefunctions, integrability, and open strings
It has been recently conjectured that the exact eigenfunctions of quantum mirror curves can be obtained by combining their WKB expansion with the open topological string wavefunction. In this paper we give further evidence for this conjecture. We present closed expressions for the wavefunctions in the so-called maximally supersymmetric case, in various geometries. In the higher genus case, our conjecture provides a solution to the quantum Baxter equation of the corresponding cluster integrable system, and we argue that the quantization conditions of the integrable system follow from imposing appropriate asymptotic conditions on the wavefunction. We also present checks of the conjecture for general values of the Planck constant.
Université de Genève, Genève, CH-1211 Switzerland
There is by now strong evidence that topological strings on toric Calabi–Yau (CY) manifolds are closely related to spectral problems in one dimension, obtained by an appropriate quantization of the mirror curve. Building on previous insights in topological string theory adkmv (); acdkv (); km (); hw (), supersymmetric gauge theory ns (); mirmor (), and on developments in ABJM theory mp (); hmo (); hmo2 (); calvo-m (); hmo3 (); hmmo (); cgm8 (), a precise formulation of this correspondence was put forward in ghm (); cgm () (see mmrev () for a review). The construction developed in ghm (); cgm () associates a set of trace class operators to a given mirror curve. Exact quantization conditions and Fredholm determinants for these operators are then conjecturally encoded in the enumerative geometry of the CY. This provides a correspondence between spectral theory and topological strings, or ST/TS correspondence, which has been further developed in kasm (); mz (); hatsuda-spectral (); kmz (); wzh (); gkmr (); hatsuda-comments (); lst (); hm (); fhm (); oz (); bgt (); grassi (); swh (); butterfly (); sugimoto (); cgum (); grassi-gu (); cms (); ghkk (); hsx (); bst-2 (). So far, in spite of very stringent tests, no counterexample has been found for the conjectures put forward in ghm (); cgm ().
Most of the work which has been done on the ST/TS correspondence focuses on its closed string side, which relates closed string invariants to the eigenvalue spectrum of the operators. However, in order to fully solve the spectral problem, one should also find the eigenfunctions. From the point of view of the ST/TS correspondence, this involves the open string sector. In fact, in the works adkmv (); acdkv (), the central object is the D-brane wavefunction, which is the generating functional of certain open BPS invariants.
A detailed study of wavefunctions in the ST/TS correspondence was made in mz-open (), focusing for simplicity on the local geometry (see amir (); sciarappa () for other attempts to write down the wavefunctions111While this paper was being typed, a very interesting paper appeared antonio-ta () which makes a concrete proposal for the eigenfunctions of the relativistic Toda lattice by using instanton partition functions in the presence of defects. This corresponds to the family of toric CYs engineering pure gauge theories.). Building on calculations performed in different limits, mz-open () conjectured that the exact wavefunctions of the spectral problem can be obtained by combining the WKB solution for the wavefunction with the so-called topological string wavefunction (which encodes open BPS invariants associated to symmetric Young tableaux). This is a direct extension of the exact results of ghm () for the spectral determinant, in which one combines the WKB grand potential with the topological string free energy. The conjectural wavefunctions of mz-open () are quantum generalizations of the Baker–Akhiezer function on the mirror curve, akin to (but different from) the construction of be (). However, there is a new twist in the story: as shown in mz-open (), one has also to consider different copies of the resulting wavefunction, corresponding to the different sheets of the Riemann surface. In the (hyper)elliptic example considered in mz-open (), the contribution of one of the two sheets can be easily calculated from the open BPS invariants, and then one applies an appropriate transformation to obtain the contribution of the second sheet. The total wavefunction is the sum of both contributions. Each contribution is afflicted with WKB-type singularities, which cancel in the sum. This prescription is conceptually similar to the mechanism described in mmss () in the context of non-critical strings. The total wavefunction can be written down very explicitly in the so-called maximally supersymmetric case or self-dual case, when . The result of mz-open () for the local geometry has been verified by Kashaev and Sergeev in ks ().
The conjecture put forward in mz-open () was only developed in detail in the case of local (and for a fixed value of its mass parameter), since this is the simplest and most symmetric example. A deeper understanding of the open string sector for the TS/ST correspondence requires further testing of the conjecture in mz-open (). In this paper we make various steps in this direction, by extending the results of mz-open () in various ways. First, we test the conjecture in the maximally supersymmetric case for two different geometries: local , which has genus one, and more importantly, the resolved orbifold studied in cgm (). This is a genus two geometry, which is technically more challenging. We manage however to obtain an exact expression for the wavefunctions on this genus two geometry, in the self-dual case, and for generic moduli. This result, as well as the conjectural result for local , have been successfully checked against numerical calculations of the wavefunctions.
Our explicit result for a higher genus geometry allows us to explore under a new angle the relation between the quantization of mirror curves put forward in cgm (), and the cluster integrable system of Goncharov and Kenyon gk (). As it turns out, a toric CY leads to two different spectral problems: the spectral problem in one dimension considered in cgm (), based on the quantization of the mirror curve , and the spectral problem in dimensions considered in gk (), based on mutually commuting Hamiltonians (here, is the genus of the mirror curve). The two spectral problems are however closely related. Based on the conjectural exact solution for the spectrum of the cluster integrable system proposed in fhm (), it has been noted that the spectral problem of cgm () is more general than the one associated to the cluster integrable system. In particular, the quantization condition put forward in cgm () leads to a codimension one submanifold of the moduli space. This submanifold contains the spectrum of the integrable system, which is a discrete set of points, as a subset fhm (); swh (); cgm (). It is then important to ask what is the physical mechanism which further restricts the submanifold cgm () to the discrete spectrum of the cluster integrable system. A natural answer is that the one-dimensional operator of cgm () is the Baxter operator for the cluster integrable system. The spectrum of the cluster integrable system should follow from the spectrum of the Baxter operator by requiring appropriate boundary conditions on its solutions, as it happens in the standard Toda lattice gutzwiller (); gp (); kl (). In this paper we give some evidence that this is the case in the example of the resolved orbifold. Namely, we show that the wavefunctions of the Baxter operator, which we find explicitly when , decay more rapidly at infinity precisely when the values of the moduli correspond to the spectrum of the cluster integrable system. This provides a physical realization of the additional quantization conditions found in fhm (); swh ().
Finally, we explore the validity of the conjecture in mz-open () when takes arbitrary values. In this case, the information provided by the open topological string amplitudes is in principle more limited: the generating functions of BPS invariants are given by expansions at large , so we do not have closed formulae for the dependence on the wavefunctions. This leads to important limitations in obtaining the contributions to the wavefunction from the different sheets of the Riemann surface. However, as noted in mz-open (), when the Riemann surface is hyperelliptic, the contributions of the two Riemann sheets can be calculated separately on the spectral theory side. It is then possible to compare the results for the contribution of the first Riemann sheet, which can be obtained from standard open BPS invariants, and we do so in the example of local and for different values of . We find perfect agreement.
This paper is organized as follows. In section 2 we present the conjecture of mz-open () for the exact eigenfunctions in a general setting, we work out in detail the maximally supersymmetric or self-dual case, and we illustrate it with a new example, namely local . In section 3, we study the genus two example of the resolved orbifold and we make a connection between the integrable system and the decay at infinity of the wavefunctions. In section 4 we consider the conjectural eigenfunctions for arbitrary values of , and we study then in detail in the example of local . Finally, in section 5 we present some conclusions and open problems.
2 The exact eigenfunctions: a conjecture
2.1 The closed string sector
We will now summarize some basic ingredients of the TS/ST correspondence. We refer the reader to ghm (); cgm (); mmrev () for more details and extensive references to the background results on topological string theory and local mirror symmetry.
Let be a toric Calabi–Yau manifold, with “true” moduli denoted by , . It also has mass parameters, , hkp (); kpsw (). We will denote by the total number of moduli of . Its mirror curve has genus and it is given by an equation of the form
It is convenient to write this curve in a “canonical” form, by picking up one of the geometric moduli, say , so that (2.1) can be written as
The function is a sum of monomials of the form , with coefficients that depend on the moduli and the mass parameters. We can write
where . We can obtain an operator by Weyl quantization of the mirror curve: we promote , to self-adjoint Heisenberg operators , satisfying the commutation relation
Under Weyl quantization, we have that ,
so that the function becomes a self-adjoint operator, which will be denoted by . If the mass parameters and geometric moduli satisfy appropriate positivity conditions, the operator
Since there are canonical forms for the curve, there are operators that one can consider. However, these operators are related by a similarity transformation
where is the operator corresponding to the monomial . In particular, the eigenfunctions associated to the operators are related as cgm ()
The conjectures of ghm (); cgm (); mz-open () provide an answer for this spectral problem, based on the (refined) BPS invariants of the toric CY . Therefore, in order to write down explicit formulae for these quantities, we have to introduce some generating functionals of BPS invariants for . In doing this, we will mostly follow the conventions of cgum (). As discussed above, the CY has “true moduli” denoted by , . We will introduce the associated “chemical potentials” by
The true moduli and the mass parameters are encoded in the Batyrev coordinates defined by
One can choose the Batyrev coordinates in such a way that, for , the ’s correspond to true moduli, while for , they correspond to mass parameters. For such a choice, the non-vanishing coefficients in (2.11)
form an invertible matrix, which agrees (up to an overall sign) with the charge matrix appearing in kpsw (). The mirror map expresses the Kähler moduli of the CY in terms of the Batyrev coordinates :
where is a power series in . Together with (2.11), this implies that
By using the quantized mirror curve, one can promote the classical mirror map to a quantum mirror map depending on acdkv ():
The enumerative invariants of are encoded in various important functions. The topological string genus free energies encode the information about the Gromov–Witten invariants of . In the so-called large radius frame, they have the structure
In these formulae, are the Gromov–Witten invariants of at genus and multi-degree . The coefficients , are cubic and linear couplings characterizing the perturbative genus zero and genus one free energies. Finally, is the so-called constant map contribution bcov (). The constants usually appear in the linear term of (see below, (2.23)). The total free energy of the topological string is the formal series,
and is the topological string coupling constant.
Note that, as formal power series, we have
In the case of toric CYs, the Gopakumar–Vafa invariants are special cases of the refined BPS invariants ikv (); ckk (); no (). These refined invariants depend on the degrees and on two non-negative half-integers or “spins”, , . We will denote them by . We now define the Nekrasov–Shatahsvili (NS) free energy as
The first term in this series, , is equal to , the standard genus zero free energy.
Following hmmo (), we now define the grand potential of the CY 222In some papers, this is also called the modified grand potential since it does not agree with the grand potential of the corresponding Fermi gas. In this paper we shorten the name to grand potential tout court.. It is the sum of two functions. The first one is
The function is only known in a closed form in some simple geometries. The second function is the “worldsheet” grand potential, which is obtained from the generating functional (2.19),
It involves a constant integer vector (or “B-field”) which depends on the geometry under consideration. This vector satisfies the following requirement: for all , and such that is non-vanishing, we must have
The total grand potential is the sum of the above two functions,
In practice, the total grand potential can be computed by using the (refined) topological vertex akmv (); ikv (), which can be used to compute and by taking the standard and the NS limit of the refined topological string free energy, respectively.
The central quantity determining the spectral properties of the operator is the (generalized) spectral determinant of . To define it, we write the quantized mirror curve as
The spectral determinant of is given by
Although in defining this operator we have singled out one particular canonical form of the mirror curve (i.e. made a particular choice of ), it is shown in cgm () that the above definition is independent of this choice, so the spectral determinant is associated to the mirror curve itself, and not to any particular parametrization of it. The zero locus of defines a codimension one submanifold in the -dimensional space of “true” moduli. This submanifold gives the spectrum of the operator (and of the other operators obtained from it by similarity transformations). For example, if we fix the values of the moduli , , we find a discrete set of values of in , , , which are identified as (minus) the eigenvalues of (see cgm () for a detailed discussion and illustration in the case of the resolved orbifold).
In particular, this conjecture solves completely the problem of determining the spectrum of the operator(s) associated to the mirror curve.
2.2 A conjecture for the exact eigenfunctions
The total grand potential corresponds to the closed string sector of the topological string on , and it solves the problem of calculating the eigenvalues of the quantum mirror curve. In order to extract the exact eigenfunctions, we have to find its open string theory counterpart. The spectral problem (2.7) has a WKB solution for the eigenfunction which is a formal power series expansion in ,
It turns out that this expansion can be resummed, order by order in an expansion at and at large radius. When expressed in terms of flat coordinates for both the open and the closed string moduli, this resummation has the following structure. Let us introduce the vector of quantum corrected Kähler parameters, obtained from the quantum mirror map (2.15)
and the exponentiated Planck constant,
We will use very often the exponentiated coordinate, which plays the rôle of the open string modulus,
as well as its rescaled version,
where is a vector of rational entries which depends on the geometry. Then, the open string WKB grand potential is given by
In this equation, is a perturbative part, which is a polynomial in , and are integer invariants which depend on a spin , a winding number , and the multi-degrees as (); amir (). The minus sign in in this equation is due to the fact that, in the WKB solution, the sign of is the opposite one to what is required by integrality of the invariants. The total WKB grand potential is obtained by adding (2.25) and (2.37), i.e.
We note that, although the closed WKB grand potential can be computed from the refined topological vertex in the NS limit, we have not found a clear relationship between the refined vertex and the generating function in (2.37). In practice, we calculate (2.37) directly from the WKB solution for the eigenfunction. In principle it should be possible to calculate it also from the instanton partition function with defects (see sciarappa () and references therein, and antonio-ta () for very recent progress in this direction).
As in the closed string case, the open string grand potential also has a contribution from the standard open topological string. We recall that the open topological string free energy of a toric CY manifold depends on a choice of Lagrangian D-brane. For each choice of Lagrangian brane, one can define open BPS invariants ov (); lmv () which generalize the Gopakumar–Vafa invariants of the closed topological strings. They depend on a quantum number or “genus,” the multi-degree , and winding numbers of the boundaries. The topological string wavefunction is a particular case of the open string free energy, depending on a single open modulus (see mz-open () for more details on this relation). It can be written in terms of the open BPS invariants as lmv ()
The topological string wavefunction can be computed for example by using the topological vertex akmv (). In the topological vertex formalism, D-brane amplitudes are given by partition functions labelled by Young tableaux. The topological string wavefunction involves only tableaux with a single row. We now introduce the worldsheet contribution to the open string grand potential,
The first term in the r.h.s. is the worldsheet grand potential (2.26), while
We will sometimes use the dual Planck constant,
The total, -dependent grand potential is
The first term in the r.h.s. of this equation is a resummation of the WKB expansion, while the second term is a non-perturbative correction in to the perturbative WKB result. Note that both terms have poles when is a rational number. However, as shown in mz-open (), they cancel when we add both functions, provided that amir ()
for all and such that .
As we just mentioned, the open topological string wavefunction depends on a choice of Lagrangian D-brane in the geometry. What is then the right choice of D-brane to solve the spectral problem? It turns out that the wavefunctions associated to different branes are related by a linear canonical transformation akv (), therefore they are physically equivalent and give different representations of the same wavefunction. However, one should make a choice of the Lagrangian brane which is compatible with the choice of coordinate in the wavefunction. We will see some non-trivial examples of this in the genus two case of section 3.
In writing the open string grand potential we have made another implicit choice, namely a choice of sheet for the Riemann surface defining the mirror curve. For example, when the mirror curve is hyperelliptic, in the exponent of (2.32) there is an implicit choice of sign, just as in the standard WKB method. We will denote the choice of sheet by a subindex in the open grand potential. When the mirror curve is hyperelliptic, and there are only two sheets, we have . The conjecture of mz-open (), slightly generalized to the higher genus case, states that the wavefunction is given by the sum over the different sheets,
After summing over the different sheets, we expect to find an entire function on the complex plane, as pointed out in mmss () in the context of non-critical strings, and as illustrated in mz-open () in the case of local .
There are various observations that can be made on (2.45). First of all, the wavefunction can be defined for any value of the moduli . However, it will not be square integrable unless the values of belong to the zero locus of the spectral determinant. In this case, we will say that the wavefunction is “on-shell.” If, for example, we consider the eigenvalue equation (2.7) for fixed values of the moduli , , we obtain a sequence of eigenvalues . The expression (2.45), evaluated on these values, provides the exact eigenfunctions corresponding to the eigenvalues. We can however keep the wavefunction (2.45) “off-shell.” In this case, the expression (2.45) gives an -dependent generalization of the spectral determinant that can be calculated from the Fredholm theory of the operator . This was shown in detail in mz-open () in the case of local . In this and the next section, we will focus on on-shell wavefunctions, while in section 4 we will consider the theory off-shell.
We should mention that the implementation of the sum over the different sheets turns out to be quite subtle for general values of . In the hyperelliptic case, one of the sheets (which we will take to be ) involves standard BPS invariants, as obtained from the WKB expansion and the topological vertex. The wavefunction with is obtained by transforming to the second sheet of the Riemann surface. This can be done in detail in the maximally supersymmetric case, as discussed in mz-open () and in the next section, but for general values of the transformation is more difficult to implement.
2.3 The maximally supersymmetric case
This is the “self-dual” value for the Planck constant, in which . For this value, the expressions for the spectral determinant and for the wavefunctions become exact at one-loop in the topological string expansion and in the WKB expansion. We will now write down explicit and general expressions for the wavefunctions in the maximally supersymmetric case and for any toric geometry. For simplicity, we will assume in the following that there are no mass parameters in the model, so the matrix reduces to the invertible matrix (2.12) (the inclusion of mass parameters is straightforward but it requires some additional ingredients and notation).
In the self-dual case , the only contribution from the topological string wavefunction involves the disk amplitude , and the annulus amplitude . Let us introduce the functions,
Here, we use the “classical” Kähler parameters . Up to a change of sign in the exponentiated open string moduli, these functions are, respectively, the disk amplitude and the annulus amplitude for . In order to proceed, we define two constant vectors and by the equality,
Using these two vectors, we can define the following transformations in the closed and open moduli,
We can use this transformation to obtain new functions , from the standard disk and annulus amplitudes (2.48):
The remaining ingredient is the exponentially small part of the next-to-leading term in the WKB expansion,
This is essentially the one-loop correction to the WKB wavefunction. After transforming the closed and open moduli as in (2.50), we obtain the function . A simple calculation by using all the above ingredients leads to the following expression
All the quantities appearing here can be computed explicitly in terms of geometric ingredients on the mirror curve. First of all, since the theory at the self-dual point involves the shift of the moduli given in (2.50), we implement this transformation directly in the equation for the mirror curve. We will denote by the corresponding solution to the transformed equation. At large , this solution goes as , where is a polynomial in and . Let us now define the following set of differentials,
and the associated matrix of A-periods,
By using the normalized differentials
we define the Abel-Jacobi map as
where is the matrix appearing in (2.11). Using the above information, we can write
In order to obtain the wavefunction (2.45), we have to sum over all the shifts of by . Only terms with explicit factors of inherit the shift:
To proceed, we have to be more explicit about the structure of the closed string contribution to the grand potential. Let us denote by , the free energies (2.16), (2.24) in which has been shifted by the B-field in the worldsheet instanton part. The resulting free energies have the following structure
where the instanton contributions, labelled by “inst”, are invariant under the shift (2.61). Also, the quantity is totally symmetric in its labels. We then obtain
where repeated indices are now summed over, and
In all the examples that have been considered, the cubic term in in (2.63) could always be absorbed into constant linear and quadratic terms, thus introducing shifts in and . We will call these shifted quantities and . To write down the final answer for the wavefunction, we have to use the Riemann theta function with characteristics , :
It is an odd function when odd. For definiteness, we call the theta function with . The Riemann theta function with will be denoted simply by . The normalized B-periods of the (transformed) mirror curve can be written as
where is a matrix of constants. According to the theory of the B-model presented in mmopen (); bkmp (), the annulus amplitude can be written in terms of the Bergman kernel of the mirror curve (see mz-open () for details of a similar computation), and one finds,
is a dependant constant, and is the derivative of the Abel-Jacobi map with respect to (not ). Our final expression for is then,
This wavefunction is very similar to a classical Baker–Akhiezer function on the mirror curve akhiezer () (see for example dubrovin (); bbt ()), although there are also some important differences (for example, the term is not part of the standard Baker–Akhiezer function).
So far we have not been explicit about the multi-covering structure of the mirror curve. When the mirror curve is hyperelliptic, so that the Riemann surface is a two–sheeted covering of the complex plane, the wavefunction (2.69) corresponds to the contribution of the first sheet , and it involves the standard open BPS invariants. The second contribution is obtained by considering the transformation of (2.69) to the second sheet. This involves a detailed analysis of the covering structure, but in the maximally supersymmetric case its calculation is in principle straightforward. Such a transformation was successfully implemented in the case of local in mz-open (), and we will see more examples in the next subsection and in section 3. One intriguing aspect of this transformation is that the contribution of the second sheet seems to involve a different realization of the open string BPS invariants. We will see an illustration of this in the example of local .
2.4 An application: eigenfunctions for local
In mz-open () we used the conjecture (2.45) to write down an exact expression for the wavefunctions in the maximally supersymmetric case and for local . We now apply this to another important example, namely the local geometry, also for , where we can write a fully closed expression.
The mirror curve for local is
The corresponding spectral problem is
In order to write down the wavefunction, we have to consider the relation (2.49). By looking at the quantum mirror map of local adkmv (); ghm (), we find that , , . In addition, in (2.36) we have . The transformation (2.50) reads then,
We can now write down the ingredients appearing in (2.60). The function is given by
The Abel–Jacobi map is
The perturbative WKB piece is given by
For the annulus amplitude, one finds
where the elliptic modulus is given by
Our conventions for the genus one theta functions are as in akhiezer (). Finally, the function is given by
This can be easily found by a standard WKB expansion.
Using all these data, one finds, by specializing (2.69),
and the closed string grand potential has been calculated in ghm (). As the subindex indicates, the expression (2.81) gives just the contribution of the first sheet. The condition that decays at large is satisfied if the ratio of theta functions goes to a constant in the large limit. This happens if
which is precisely the quantization condition in the maximally supersymmetric case found in ghm (). This condition determines a discrete set of values for , giving the spectrum of the operator in (2.71) when .
The wavefunction has singularities at the “turning points” defined by . In order to remove these singularities, we have to add to this function the wavefunction living in the second sheet of the Riemann surface. The transformation to the second sheet is similar to what was done in mz-open () in the case of local . Since we want to eventually use these results to write down the actual eigenfunctions, we will assume that , with . The transformation of the Abel–Jacobi map turns out to be given by
By integrating this relation and fixing the integration constant carefully, one finds