Exact quantization conditions for the elliptic Ruijsenaars-Schneider model
We propose and test exact quantization conditions for the -particle quantum elliptic Ruijsenaars-Schneider integrable system, as well as its Calogero-Moser limit, based on the conjectural correspondence to the five-dimensional gauge theory in the Nekrasov-Shatashvili limit. We discuss two natural sets of quantization conditions, related by the electro-magnetic duality, and the importance of non-perturbative corrections in the Planck constant. We also comment on the eigenfunction problem, by reinterpreting the Separation of Variables approach in gauge theory terms.
1 Introduction and summary
Among all quantum integrable systems, elliptic ones are the hardest to analyse due to the doubly-periodic nature of their potential. They are however also the most fundamental ones, since many other integrable systems can be obtained from them by taking opportune limits. The most famous example is the -particle elliptic Calogero-Moser system, which in various special limits reduces to its trigonometric or hyperbolic version, or even to the -particle closed Toda chain Olshanetsky:1983wh (): although much progress has been made in the last few years, its analytic solution is not known if not for very special cases. The situation is even worse if we consider its relativistic generalization, also known as -particle elliptic Ruijsenaars-Schneider system RUIJSENAARS1986370 (); ruijsenaars1987 (); Ruijsenaars:1992ua (); Ruijsenaars1999 (): in that case not only the analytic solution is not known, but also determining its Hilbert integrability (i.e. its domain of self-adjointness) is a rather non-trivial problem since the Hamiltonian is not a differential but a finite-difference operator 0305-4470-32-9-018 (); Ruijsenaars1999 (); 2015SIGMA..11..004R (). It would therefore be quite helpful to develop new techniques to approach these elliptic problems.
As it became clear recently, supersymmetric gauge theories and topological string theory can provide an alternative, non-conventional framework to study quantum integrable systems. This was first pointed out in 2010maph.conf..265N (), where it was conjectured that all-orders WKB quantization conditions and energy spectrum for a particular class of integrable systems (such as closed Toda chains, elliptic Calogero-Moser and their relativistic versions) can be obtained by considering certain observables in -background deformed four or five-dimensional gauge theories with at least eight supercharges and unitary gauge group, in a special limit known as Nekrasov-Shatashvili (NS) limit. In this limit only one (say, ) of the two -background parameters and is kept, and plays the role of the Planck constant of the quantum integrable system; other gauge theory parameters can also be identified with parameters of the quantum Hamiltonians. The most important advantage of the gauge theory approach is that it naturally gives exact WKB quantization conditions and energy spectrum expressed not as an asymptotic series in , but as a convergent power series in an auxiliary parameter (the instanton counting parameter , associated to some parameter in the Hamiltonians) whose coefficients are exact in .
The conjecture by 2010maph.conf..265N () was tested in Mironov:2009uv (); Mironov:2009dv (); Kozlowski:2010tv () for the case of the -particle closed Toda chain, associated to four-dimensional pure super Yang-Mills theory in the NS limit, and was shown to reproduce the known Toda solution studied in GUTZWILLER1980347 (); GUTZWILLER1981304 (); 0305-4470-25-20-007 (); Kharchev:1999bh (); Kharchev:2000ug (); Kharchev:2000yj (); An2009 (). More recently other tests were performed in 2015arXiv151102860H () for the case of the -particle relativistic closed Toda chain ruijsenaars1990 (), associated to the NS limit of five-dimensional pure super Yang-Mills theory compactified on a circle, and in Franco:2015rnr () for the case of other relativistic quantum integrable systems of cluster type 2011arXiv1107.5588G (), which in general can only be associated to a topological string theory on a local toric Calabi-Yau 3-fold (or -brane web) rather than a five-dimensional gauge theory. Quite interestingly, the conjecture by 2010maph.conf..265N () turns out to be inconsistent for all the relativistic systems considered in 2015arXiv151102860H (); Franco:2015rnr (): this is because even if the exact WKB quantization conditions obtained from gauge theory/topological string theory are exact in and convergent as a series in , they present an infinitely dense set of poles as functions of , which is inconsistent with the fact that the quantum integrable systems are perfectly well-defined for any .
This is however not too surprising: in principle in any quantum mechanical system complex instantons may generate important non-perturbative contributions in , but these will be missed if we only consider the WKB solution. One can therefore imagine that inconsistencies of the conjecture by 2010maph.conf..265N () are due to the fact that complex instantons are present in all relativistic integrable systems, but the gauge theory approach does not take them into account. We thus expect the poles in should disappear when non-perturbative effects are considered. This is indeed what seems to happen: as conjectured and tested in 2015arXiv151102860H (); Franco:2015rnr (), if we add the relevant non-perturbative contributions to the exact WKB quantization conditions we obtain pole-free quantization conditions for relativistic Toda chains and relativistic cluster integrable systems, and the resulting energy spectrum matches the one we can compute numerically by standard quantum mechanical arguments.
Of course, the biggest problem is to precisely determine the non-perturbative contributions. In 2015arXiv151102860H (); Franco:2015rnr () (based on previous works Hatsuda:2012dt (); 2013arXiv1308.6485K (); Huang:2014eha (); Grassi:2014zfa (); Wang:2015wdy (); Gu:2015pda (); Codesido:2015dia (); 2015arXiv150704799H ()) it is suggested that these are completely fixed by requiring the full (WKB + non-perturbative) exact quantization conditions to be invariant under the exchange . Although the effectiveness of such a simple proposal may seem surprising at first, this is not completely unexpected, being in line with the quantum group and modular double structure underlying the relativistic Toda chain Kharchev:2001rs () and, more in general, any relativistic cluster integrable system 2011arXiv1107.5588G ().
The purpose of the present work is to further test the conjecture by 2010maph.conf..265N (), or its non-perturbative refinement à la 2015arXiv151102860H (); Franco:2015rnr (), in the case of -particle elliptic quantum integrable systems of Calogero-Moser and Ruijsenaars-Schneider type, related to four-dimensional and five-dimensional gauge theories respectively, mostly focussing on the 2-particle case for simplicity. In order to test these conjectures against numerical results, we will restrict our attention to the region in parameter space for which the elliptic quantum integrable systems admit a discrete set of energy levels.
As we will see, the situation is pretty much similar to what happens for the closed Toda chain, relativistic and not: the exact WKB quantization conditions conjectured by 2010maph.conf..265N () seem to be sufficient to reproduce the Calogero-Moser spectrum, while in the Ruijsenaars-Schneider case we also need to include non-perturbative corrections, fixed by the requirement of invariance under along the lines of 2015arXiv151102860H (); Franco:2015rnr (). It is important to remark that although the elliptic Ruijsenaars-Schneider system is not of cluster type, i.e. it is not related to a toric Calabi-Yau 3-fold, the symmetry is still expected due to its underlying modular double structure 0305-4470-32-9-018 (); Ruijsenaars1999 (); 2015SIGMA..11..004R ().111Because of different conventions for the parameters, in this work the invariance will be replaced by invariance under with real period of the torus on which the integrable system is defined.
There is however a new ingredient to the story, which is special to elliptic system. When the potential is doubly-periodic with real and imaginary periods , we can either consider the quantum mechanical problem relative to the real period or the one relative to the imaginary period. Clearly the two problems are the same since they are simply related by the exchange , but as already pointed out in 2010maph.conf..265N () the gauge theory approach treats them differently: quantization conditions for the first and second case correspond to quantize the -deformed Seiberg-Witten periods or the -deformed dual periods respectively. This is not unexpected: since the complex structure of the torus is identified with the complexified gauge coupling, exchanging corresponds to electro-magnetic duality, which leads to the exchange .
When this was already made clear in Seiberg:1994aj () using the fact that are related by the Seiberg-Witten prepotential; we will see that this remains true even at if we replace the Seiberg-Witten prepotential by its appropriate -deformed version (usually called twisted effective superpotential), which however in the five-dimensional case needs to include contributions which are non-perturbative in as we already discussed: we therefore conclude that away from the Seiberg-Witten limit, realizing electro-magnetic duality in five dimensions can only be achieved by also considering non-perturbative effects in the Omega background parameters.
The plan of the paper is as follows. Section 2 will be dedicated to studying the 2-particle elliptic Calogero-Moser system. We first introduce the system as well as its trigonometric and hyperbolic limits. After review what is known about its analytic solution, both in these special limits and in general, we explain how to study the problem numerically; we then move to discuss how gauge theory can be used to determine the energy spectrum analytically. Finally, we comment on what gauge theory can tell us about the problem of constructing eigenfunctions. Section 3 will follow the same steps, but for the 2-particle elliptic Ruijsenaars-Schneider system; Section 4 instead is devoted to the study of the -particle case. Useful formulae are collected in Appendices A, B.
2 2-particle systems of Calogero-Moser type
Let us consider a complex coordinate () on a rectangular torus of half-periods with , ; this means that
With gauge theory applications in mind, we will sometimes find it more convenient to reparametrize the periods as
The 2-particle elliptic Calogero-Moser system (2-eCM) Hamiltonian is given by the second-order ordinary differential equation (see Appendix A.1 for our conventions on elliptic functions)
Here all parameters . One possible problem one could study is to look for the two linearly independent solutions to this equation for generic complex values of the parameters. We are however interested in quantum mechanical applications of (4), so we should only be considering values of parameters for which (4) realizes a well-defined quantum mechanical problem with self-adjoint Hamiltonian. Among various other possibilities,222For example the choice of parameters considered in He:2011zk (); Piatek:2013ifa (); 2015JHEP…02..160B (); Beccaria:2016wop (), which leads to a quantum mechanical problem with continuous energy spectrum and a band/gap structure, will not be studied here. this happens in the following two cases:
Focussing on the real slice and requiring , , the Hamiltonian (4) reduces to
or alternatively, by introducing the sometimes more familiar parameter ,
For (or ) the potential is confining, and (5) defines a quantum mechanical problem on whose energy spectrum is real and discrete; we will refer to (5) as the B-type 2-particle elliptic Calogero-Moser quantum integrable system (2-eCM).
In the limit (i.e. ) this reduces (modulo constants) to the 2-particle trigonometric Calogero-Moser quantum integrable system (2-tCM)
this is still a quantum mechanical problem on with a discrete set of energy levels for , and its solution is known analytically.
Focussing instead on the slice and requiring , , the Hamiltonian (4) reduces to
or alternatively, by introducing the parameter ,
For (or ) the potential is confining, and (8) defines a quantum mechanical problem on with energy spectrum real and discrete; we will refer to (8) as the A-type 2-particle elliptic Calogero-Moser quantum integrable system (2-eCM).
In the limit (i.e. ) this reduces (modulo constants) to the 2-particle hyperbolic Calogero-Moser quantum integrable system (2-hCM)
this is still a well-defined quantum mechanical problem for and whose solution is known analytically, however its energy spectrum is continuous.
Although we introduced them as different, it is important to remark that the two problems 2-eCM and 2-eCM are actually the same: in fact since lawden1989elliptic ()
the 2-eCM problem with half-periods is equivalent to the 2-eCM problem with half-periods if we also identify , with , , that is
Despite this fact, we prefer to keep distinguishing them in order to facilitate later comparison with gauge theory. In fact, we will see that the gauge theory approach treats B- and A-model differently; verifying that these different treatments respect (12) will therefore be a non-trivial consistency check for the validity of the gauge theory approach.
2.1 Trigonometric and hyperbolic cases
Before discussing the 2-eCM and 2-eCM systems, we will first review the known analytic solution to their trigonometric (2-tCM) and hyperbolic (2-hCM) limits. This will be of help later in Section 2.2 to understand how to numerically evaluate the spectrum of the elliptic system.
Let us start by considering the 2-tCM system. For generic half-period , the 2-tCM problem reads
where for later convenience we reparametrized the energy in terms of an auxiliary variable as
Equation (13) will in general have two linearly independent solutions; however, since we will ultimately be interested in eigenfunctions, we start by considering only a particular linear combination of them which vanishes at , given by (modulo normalization factors)
for . Expression (15) is actually a solution only in the interval ; from the symmetry of the problem, the solution in the remaining interval should be of the same form of (15), possibly up to an overall sign. However, the and solutions are not connected smoothly at for generic values of (or, equivalently, ). In order for them to be smoothly connected, i.e. for the total solution to be in , we have to tune (i.e. the energy ) carefully; this is why the energy spectrum of the 2-tCM system is quantized. More concretely, the smoothness conditions we must impose are
these conditions determine the only allowed values and , which are simply
Quantization conditions (17) give rise to parity-even (with respect to ) or parity-odd eigenfunctions for even or odd respectively:
By using the identity
these (unnormalized) eigenfunctions can be rewritten in the more compact form
where are the Gegenbauer polynomials
Thanks to the orthogonality properties of the Gegenbauer polynomials we can then construct a normalized basis of 2-tCM eigenfunctions
which satisfy the orthonormality condition
We may also replace by via (17) and write
Let us now briefly discuss the 2-hCM problem
where again we reparametrized the energy in terms of as
Let us stress again that in principle there are two linearly independent solutions to (25), but since we are interested in solutions vanishing at (due to the divergence of the potential) we only consider their particular linear combination (27). Since this combination also vanishes fast enough at , (27) is actually the complete solution to the 2-hCM system; given that there is no need to impose further conditions on the solution, the energy (26) is continuous and positive.
2.2 Elliptic case: B-model
We are now ready to consider the 2-eCM quantum mechanical problem, defined as
where for convenience we made the explicit dependence on , that is on , of the eigenfunction (which of course also depends on all the other parameters of the problem). This problem can be studied either analytically as a perturbation series in or numerically, as we are going to illustrate now.
In order to solve the 2-eCM problem analytically on we can use perturbation theory around (or ), that is around the trigonometric limit.333This is the same approach followed in Langmann:2004sj (). We will therefore consider the -series expansion of the Hamiltonian
The discrete energy levels and eigenfunctions will admit similar expansions:
With these considerations, we can now look for a solution to the 2-eCM problem
order by order in the expansion. This can be done if we further consider expanding all coefficients , in terms of the 2-tCM orthonormal basis , i.e.
where the coefficients should be fixed by the normalization condition . By doing this we find for example
as well as
where for , while for
given more explicitly by
Similar results can be obtained for higher orders in the expansion.
To compute the higher order corrections more efficiently, one can use the idea in PhysRev.184.1231 (). The perturbative expansion of the wave function takes the following form:
where is an irrelevant normalization constant. The important point is that the functions are polynomials in . Of course, we have . One can fix unknown coefficients in as well as in order to satisfy the eigenvalue equation.
The 2-eCM problem (28) can also be studied numerically. For doing so, we simply have to numerically diagonalize the matrix constructed out of the matrix elements
at fixed and , where we are using the 2-tCM orthonormal basis (22) to perform diagonalization. Clearly, this procedure would in principle require diagonalizing an matrix; at the practical level what we do instead is to compute the eigenvalues of a truncated version of this matrix, and then check convergence of the eigenvalues as we increase the matrix size. In this way we can obtain both the numerical spectrum and the numerical eigenfunctions of the 2-eCM system.
2.3 Elliptic case: A-model
Let us finally comment on the 2-eCM quantum mechanical problem
Trying to solve this problem analytically on in terms of perturbation theory around (or ) appears to be more complicated than in the 2-eCM case: this is because in the limit we reduce to the 2-hCM system, which has continuous spectrum and eigenfunctions in rather than . It should however be possible to do so, by using the quantum Separation of Variables approach which involves the construction of entire solutions to the associated Baxter equation.444A similar quantum Separation of Variables approach was followed in Gerasimov:2002cf () to construct the eigenfunction for a special case of the 2-particle hyperbolic Calogero-Moser system.
We will outline this procedure, reinterpreted in gauge theory language, in Section 2.5.
For our purposes, it will however be sufficient to study the solution to (42) numerically. As for the case of the B-model, we just have to diagonalize the matrix constructed out of the matrix elements
at fixed and , where this time diagonalization is performed by using the orthonormal basis with half-period rather than . Clearly, because of (12) diagonalizing with half-periods is equivalent to diagonalizing with half-periods and parameters , , that is
In fact, as we already mentioned the 2-eCM and 2-eCM problems are related by S-duality (in ), i.e. by the exchange of half-periods , which also implies the exchange
An alternative option to compute the 2-eCM spectrum would therefore be to study the 2-eCM problem in terms of perturbation theory around (or ), i.e. around the 2-tCM system of period , along the lines of what we did in Section 2.2.
A special case with analytic solution
We should also mention that there are special values of the parameter for which the solution to the 2-eCM and 2-eCM problems is known analytically. This happens for example when , in which case the generic elliptic problem (4) reduces to
The solution in this case can be found for example Section 1 of Olshanetsky:1983wh (): if we define , as
then the linear combination