# Generalized \(q\)-deformed correlation functions as spectral functions of hyperbolic geometry

- 359 Downloads
- 1 Citations

## Abstract

We analyze the role of vertex operator algebra and 2d amplitudes from the point of view of the representation theory of infinite-dimensional Lie algebras, MacMahon and Ruelle functions. By definition p-dimensional MacMahon function, with \(p\le 3\), is the generating function of *p*-dimensional partitions of integers. These functions can be represented as amplitudes of a two-dimensional *c* = 1 CFT, and, as such, they can be generalized to \(p>3\). With some abuse of language we call the latter amplitudes generalized MacMahon functions. In this paper we show that generalized p-dimensional MacMahon functions can be rewritten in terms of Ruelle spectral functions, whose spectrum is encoded in the Patterson–Selberg function of three-dimensional hyperbolic geometry.

### Keywords

Partition Function Vertex Operator Elliptic Genus Vertex Operator Algebra Hyperbolic Geometry## 1 Introduction

This paper, whose main focus is the relation among CFT correlators, MacMahon and Ruelle functions, is motivated by the steady, if not growing, interest in the application of symmetric functions, in particular of the two-dimensional MacMahon function and its higher-dimensional generalizations to physical systems. This occurs in many areas of statistical physics [1, 2, 3] and topological string theory [4, 5, 6], BPS black holes, models of branes wrapping collapsed cycles in Calabi–Yau orbifolds, and quiver gauge theories [7, 8, 9]. Generalized MacMahon functions are used, in particular, in the computation of amplitudes of the A-model topological string [10, 11, 12, 13], more specifically as regards the so-called topological vertex.

A \(p\)-dimensional MacMahon function for \(p \le 3\) is the generating function of a \(p\)-dimensional partition of integers, which is the number of different ways in which we can split an integer using distinct *p*-dimensional arrays of other nonnegative integers. As we shall see these functions can be represented as amplitudes of a two-dimensional CFT. We extend this representation as a correlator to generic \(p\) and with some abuse of language we call the resulting objects *generalized MacMahon functions*. For \(p\ge 4\) these functions do not coincide precisely with the generating functions of \(p\)-dimensional partition of integers, but they have all the same remarkable properties. In particular we will show in this paper that generalized \(p\)-dimensional MacMahon functions can be rewritten in terms of Ruelle spectral functions, whose spectrum is encoded in the Patterson–Selberg function of three-dimensional hyperbolic geometry. This may lead to an interpretation of the above results in terms of the ADS/CFT correspondence, an attractive possibility which we leave for future investigation.

There is also another side of our work we would like to recall. We have remarked elsewhere the important connection between quantum generating functions in physics and formal power series associated with dimensions of chains and homologies of suitable infinite-dimensional Lie algebras. MacMahon and symmetric functions play an important role in the homological aspects of this connection; its application to partition functions of minimal three-dimensional gravities in the space-time asymptotic to AdS\(_3\), which also describe the three-dimensional Euclidean black holes, pure supergravity, elliptic genera, and associated \(q\)-series were studied in [14, 15]. On the other hand special applications of symmetric functions appear in the representation theory of infinite-dimensional Lie algebras [14, 15, 16, 17]. The usefulness of symmetric function techniques can be demonstrated in providing concrete realizations of the (quantum) affine algebra, for instance in calculating the trace of products of currents of this algebra. These functions are, respectively, the appropriate character \(\mathrm{ch}_{\mathbb R}\) of the basic representations of the \({\mathfrak s}{\mathfrak l}(\infty ) \) and the affine algebra \(\widehat{{\mathfrak s}{\mathfrak l}(\infty )}\) at large central charge \(c\) [18]. Note that all simple (twisted and untwisted) Kac–Moody algebras can be embedded in the \({\mathfrak s}{\mathfrak l}(\infty )\) algebra of infinite matrices with a finite number of non-zero entries, which has a realization in terms of the generators of a Clifford algebra. It has been observed that in the limit \(c\rightarrow \infty \) the basic representation of \(\widehat{{\mathfrak s}{\mathfrak l}(\infty )}\) is related to the partition function of a three-dimensional field theory [7, 19].

Needless to say, although these links are suggestive, the general panorama looks still inarticulate, and more models and examples are needed to accommodate them into a precise scheme. One of the purposes of the present paper is to better understand the role of vertex algebra and 2d amplitudes from the point of view of the representation theory of infinite-dimensional Lie algebras, generalized MacMahon and Ruelle functions. In this regard particularly important is the correspondence between Ruelle spectral functions and the Poincaré \(q\)-series associated with conformal structure in two dimensions.

The organization of the paper is as follows. In Sect. 2 we introduce the algebra of \(q\)-deformed vertex operators (of the \(c=1\) 2d conformal model) and consider their generalizations and the properties essential for the next sections. In Sect. 3 we reformulate the generalized MacMahon functions in terms of the Ruelle spectral functions of hyperbolic geometry. We also broach and briefly discuss the topic of higher-dimensional partitions, as such, originally introduced by MacMahon. We analyze correlation functions of vertex operators, the MacMahon’s conjecture (see (3.15)) and their possible interpretation as \(p\)-dimensional partition functions.

In Sect. 4 we consider multipartite (vector valued) generating functions and utilize well-known formulas for Bell polynomials. We derive the infinite hierarchy of \(q\)-deformed vertex operators and factorized partition functions and represent them by means of spectral functions.

Finally in Sect. 5 we conclude with a summary of the main results accompanied by discussions and suggestions.

In the appendix we give a few formulas involving Ruelle and Patterson–Selberg spectral functions of hyperbolic three-geometry.

## 2 Algebra of vertex operators

*in-state*) and the slice at \(t=\infty \) (

*out-state*). The algebra of these vertex operators takes the following form:

## 3 MacMahon, partitions, and Ruelle spectral functions

In this section, using the formulas in the appendix, we transcribe the generalized MacMahon partition functions in terms of spectral functions of hyperbolic geometry.

**p****-dimensional partition functions.**The structure of the \(p\)-dimensional partition function \({Z}_{pd}\) can be analyzed in terms of the vertex operators \(\Gamma _{\pm }^{(p)}(z, q)\) introduced before. As we have seen above, (2.9), the latter can be interpreted as the level \(p\) generalization of \(\Gamma _{-} (z)\), and they obey the relations \( \Gamma _{-}^{\left( p\right) }\left( z, q\right) =q^{L_{0}}\Gamma _{-}^{\left( p\right) }\left( 1, q\right) q^{-L_{0}},\,p\ge 0. \) The \(p\)-dimensional partition functions \(Z_{pd}\) can be defined as [20]

**Plane partitions.**So far we have called \(Z_{pd}\) a p-dimensional partition function without any comment. Here we would like to motivate this term at least for the cases \(p \le 3\). It comes from the fact that correlation functions of the corresponding vertex operators admit a presentation that can be associated with higher-dimensional partitions. Recall that a higher-dimensional partition of \(n\) is an array of numbers whose sum is \(n\):

Concluding this section, the term MacMahon partition function for \(Z_{pd}\) is fully justified for \(p\le 3\). Following [20] we call \(Z_{pd}\) for \(p\ge 4\) *generalized MacMahon functions* due to the straightforward way they are obtained by generalizing the definition for \(p\le 3\). The relation with the original MacMahon’s definition is still an intriguing open problem: its solution may shed light also into the corresponding 2d CFT mentioned above.

## 4 Multipartite generating functions and infinite hierarchy of \(q\)-deformed vertex operators

**Multipartite generating functions.**Let consider, for any ordered \(\ell \)-tuple of nonnegative integers not all zeros, \((k_1, k_2, \ldots ,k_\ell )=\mathbf{k}\) (referred to as “\(\ell \)-partite” or

*multipartite*numbers), the (multi)partitions, i.e. distinct representations of \((k_1, k_2, \ldots ,k_\ell )\) as sums of multipartite numbers. Let us call \({\mathcal C}_-^{(u,\ell )}(\mathbf{k}) = {\mathcal C}_-^{(\ell )}(u;k_1, k_2 , \ldots , k_\ell )\) the number of such multipartitions, and introduce in addition the symbol \({\mathcal C}_+^{(u,\ell )} (\mathbf{k})= {\mathcal C}_+^{(\ell )}(u;k_1, k_2 , \ldots , k_\ell )\). Their generating functions are defined by

**The infinite hierarchy.**Let us consider again the hierarchy \(\Gamma _{-}^{\left( p\right) }(z, q)\) of \(q\)-deformed vertex operators. We have \( \Gamma _{+}\left( 1\right) \Gamma _{-}^{\left( p\right) }\left( z, q\right) \!=\! M_{p}\left( q\right) \Gamma _{-}^{\left( p\right) }\left( z, q\right) \Gamma _{+}\left( 1\right) \!, \) where \(M_{p}(q)\) is precisely the generalized \(p\)-dimensional MacMahon function. The general relation is the following:

## 5 Conclusions

We have shown above that all \(p\)-dimensional partition functions that have been considered in this paper can be written in terms of Ruelle functions, a spectral function related to hyperbolic geometry in three dimensions (see the appendix). Thus they cannot only be interpreted as correlators in a 2d CFT, but they suggest a possible interpretation in terms of three-dimensional physics. This relation is, however, still to be unveiled. In this last section we would like nevertheless to recall that in some specific cases the interpretation in terms three-dimensional physics has been possible.

This is an example of the fact that the Ruelle function represents a bridge between two-dimensional CFT and three-dimensional physics. This might be the meaning also of the formulas we have derived in the previous sections.

In this light it is important to recall that a particular example of MacMahon function, \(Z_{3d}\), is directly linked to the topological vertex [10] in topological string theory. This is an open topological amplitude in a Calabi–Yau background. Analysis of this vertex, \({\mathcal C}_{\lambda \mu \nu }\), and open string partition function leads to a relation \(Z_{3d}\sim {\mathcal C}_{\lambda \mu \nu }\) [12, 13, 20, 23]. This can be achieved as \(\langle \nu ^t\vert {\mathcal O}_+(\lambda ){\mathcal O}_-(\lambda ^t)\vert \mu \rangle \) where the operators \({\mathcal O}_+\) and \({\mathcal O}_-\) play the role of composite local vertex operators of two-dimensional \(c=1\) conformal theory, and \(\lambda , \mu , \nu \) represent boundary states described by 2d Young diagrams. Applications of the topological vertex [24] suggest a connection with Chern–Simons theory. This may be the clue for our future investigation.

## Notes

### Acknowledgments

AAB would like thank ICTP and SISSA (Italy), and CNPq (Brazil) for financial support and SISSA for hospitality during this research.

### References

- 1.A. Okounkov, N. Reshetikhin, C. Vafa, Quantum Calabi-Yau and classical crystals. Prog. Math.
**244**, 597–618 (2006). hep-th/0309208 - 2.R. Kenyon, A. Okounkov, S. Sheffield, Dimers Amoebae. arXiv:math-ph/0311005
- 3.R.W. Kenyon, D.B. Wilson, Boundary partitions in trees and dimers. Trans. Am. Math. Soc.
**363**, 1325–1364 (2011). [ arXiv:math-CO/0608422] - 4.D. Ghoshal, C. Vafa, c =1 string as the topological theory of the conifold. Nucl. Phys. B
**453**, 121–128 (1995). hep-th/9506122]MathSciNetCrossRefMATHADSGoogle Scholar - 5.E. Witten, Ground ring of two-dimensional string theory. Nucl. Phys. B
**373**, 187–213 (1992). hep-th/9108004]MathSciNetCrossRefADSGoogle Scholar - 6.E.H. Saidi, M.B. Sedra, Topological string in harmonic space and correlation functions in \(S^{3}\) stringy cosmology. Nucl. Phys. B
**748**, 380–457 (2006). hep-th/0604204]MathSciNetCrossRefMATHADSGoogle Scholar - 7.M.R. Douglas, G. Moore, D-branes, Quivers, and ALE instantons. arXiv:hep-th/9603167v1
- 8.M.A. Benhaddou, E.H. Saidi, Explicit analysis of Kahler deformations in 4D N=1 supersymmetric Quiver theories. Phys. Lett. B
**575**, 100–110 (2003). hep-th/0307103]MathSciNetCrossRefADSGoogle Scholar - 9.K. Saraikin, C. Vafa, Non-supersymmetric black holes and topological strings. Class. Quant. Grav.
**25**, 095007 (2008). hep-th/0703214]MathSciNetCrossRefADSGoogle Scholar - 10.M. Aganagic, A. Klemm, M. Marino, C. Vafa, The topological vertex. Commun. Math. Phys.
**254**, 425–478 (2005). hep-th/0305132]MathSciNetCrossRefMATHADSGoogle Scholar - 11.A. Iqbal, N. Nekrasov, A. Okounkov, C. Vafa, Quantum foam and topological strings. JHEP
**0804**, 011 (2008). hep-th/0312022]MathSciNetGoogle Scholar - 12.A. Iqbal, C. Kozcaz, C. Vafa, The refined topological vertex. JHEP
**0910**, 069 (2009). hep-th/0701156]MathSciNetCrossRefADSGoogle Scholar - 13.L.B. Drissi, J. Houda, E.H. Saidi, Refining the shifted topological vertex. J. Math. Phys.
**50**, 013509 (2009). arXiv:0812.0513 [hep-th] - 14.L. Bonora, A.A. Bytsenko, Partition functions for quantum gravity, black holes, elliptic genera and Lie algebra homologies. Nucl. Phys. B
**852**, 508–537 (2011). arXiv:1105.4571 [hep-th] - 15.L. Bonora, A.A. Bytsenko, E. Elizalde, String partition functions, Hilbert schemes and affine Lie algebra representations on homology groups. J. Phys. A
**45**, 374002 (2012). arXiv:1206.0664 [hep-th] - 16.V.G. Kac,
*Infinite dimensional lie algebras*, 3rd edn. (Cambridge University Press, Cambridge, 1990)Google Scholar - 17.H. Awata, M. Fukuma, Y. Matsuo, S. Odake, Representation theory of the \(W_{1+\infty }\) Algebra. Prog. Theor. Phys. Suppl.
**118**, 343–374 (1995). hep-th/9408158]MathSciNetCrossRefADSGoogle Scholar - 18.E. Frenkel, V. Kac, A. Radul, W.-Q. Wang, \(W_{1+\infty }\) and \(W(gl_{N})\) with central charge N. Commun. Math. Phys.
**170**, 337–358 (1995). hep-th/9405121]MathSciNetCrossRefMATHADSGoogle Scholar - 19.J.J. Heckman, C. Vafa, Crystal melting and black holes. JHEP
**0709**, 011 (2007). hep-th/0610005]MathSciNetCrossRefADSGoogle Scholar - 20.L.B. Drissi, J. Houda, E.H. Saidi, Generalized MacMahon \(G_d(q)\) as \(q\) -deformed CFT\(_2\) correlation function. Nucl. Phys. B
**801**, 316–345 (2008). arXiv:0801.2661v2 [hep-th] - 21.G.E. Andrews, The theory of partitions. Encyclopedia of mathematics, vol. 2. (Addison-Wesley Publishing Company, Reading, 1976)Google Scholar
- 22.A. Maloney, E. Witten, Quantum gravity partition function in three dimensions. JHEP
**1002**, 029 (2010). arXiv:0712.0155 [hep-th] - 23.J.-F. Wu, J. Yang, Vertex operators, \({\mathbb{C}}^{3}\) curve, and topological vertex, arXiv:1403.0181v1 [hep-th]
- 24.M. Marino, Chern-Simons theory and topological strings. Rev. Mod. Phys.
**77**, 675–720 (2005). hep-th/0406005]CrossRefADSGoogle Scholar - 25.A.A. Bytsenko, M.E.X. Guimarães, Truncated heat Kernel and one-loop determinants for the BTZ geometry. Eur. Phys. J. C
**58**, 511–516 (2008). arXiv:0809.1416 [hep-th] - 26.A.A. Bytsenko, M. Chaichian, R.J. Szabo, A. Tureanu, Quantum black holes, elliptic genera and spectral partition functions. IJGMMP (2014) (to appear). arXiv:1308.2177 [hep-th]
- 27.M.E.X. Guimarães, R.M. Luna, T.O. Rosa, Topological vertex, string amplitudes and spectral functions of hyperbolic geometry. Eur. Phys. J. C (2014) (to appear). arXiv:1403.7139 [hep-th]
- 28.S.J. Patterson, P.A. Perry, The divisor of the Selberg zeta function for Kleinian groups, with an appendix by Charles Epstein. Duke Math. J.
**106**, 321–390 (2001)MathSciNetCrossRefMATHGoogle Scholar

## Copyright information

**Open Access**This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

Funded by SCOAP^{3} / License Version CC BY 4.0.