Abstract
We construct the (\(\beta \)-deformed) partition function hierarchies with W-representations. Based on the W-representations, we analyze the superintegrability property and derive their character expansions with respect to the Schur functions and Jack polynomials, respectively. Some well known superintegrable matrix models such as the Gaussian hermitian one-matrix model (in the external field), \(N\times N\) complex matrix model, \(\beta \)-deformed Gaussian hermitian and rectangular complex matrix models are contained in the constructed hierarchies.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
Recently there has been increasing interest in the superintegrability for matrix models [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18]. The superintegrability means that for the character expansions of the matrix models, the average of a properly chosen symmetric function is proportional to ratios of symmetric functions on a proper locus, i.e., \(< character > \sim character\). A wide range of matrix models are known to be superintegrable, such as the (deformed) Gaussian hermitian and complex matrix models [1,2,3,4], (Hurwitz-)Kontsevich matrix models [5, 6], unitary matrix models [7], fermionic matrix models [8, 9], and even some non-Gaussian matrix models [10, 11]. The constraints for matrix models are useful to analyze the structures of matrix models. For the Gaussian hermitian one-matrix model, its character expansion with respect to the Schur functions can be derived recursively from a single w-constraint [13]. There are the Virasoro constraints (with higher algebraic structures) for matrix models. They can be applied to analyze the character expansions of the matrix models as well, such as the Gaussian hermitian one-matrix, complex matrix and fermionic matrix models [8, 14].
W-representations of matrix models give the dual expressions for the partition functions through differentiation rather than integration [5]. More precisely, the partition functions are realized by acting on elementary functions with exponents of the given W-operators. For the Gaussian tensor model [19] and (fermionic) rainbow tensor models [8, 20], they can still be expressed as the W-representations. Recently it was shown that the superintegrability for (\(\beta \)-deformed) matrix models can be analyzed from their W-representations [15, 16]. In this paper, we will construct the partition function hierarchies with W-representations and analyze the superintegrability property.
2 Partition function hierarchies with W-representations
The Hurwitz–Kontsevich matrix model is a deformation of the Kontsevich model, which can be used to describe the Hurwitz numbers and Hodge integrals over the moduli space of complex curves [5, 21, 22]. It is the special case of the more general Hurwitz partition functions [23,24,25]. The Hurwitz–Kontsevich model is generated by the exponent of the Hurwitz operator \(W_0\) acting on the function \(e^{p_1/e^{tN}}\),
where t is a deformation parameter, the Hurwitz operator \(W_0\) is given by
and \(c_{\lambda }=\sum _{(i,j)\in \lambda }(N-i+j)\).
The partition function (1) possesses the matrix model representation [5]
where \(\psi \) is an \(N\times N\) matrix and the time variables \(p_k=\mathrm{Tr}\psi ^k\).
Let us define the operator
Using the actions
and
we have
where \(\lambda +\square \) are the Young diagrams obtained by adding one square \(\square =(i_\square , j_\square )\) to \(\lambda \).
Let us set
The action of \(W_{-1}\) on the Schur functions is
In terms of \(W_{-1}\) and \(E_{1}\), we introduce a series of operators
The actions of \(W_{-n}\) with \(n\ge 1\) on the Schur functions are given by
where we denote \(\alpha =1+\delta _{n,1}\) in this paper for later convenience, \(\lambda +\square _1+\cdots +\square _n\) are the Young diagrams obtained by adding n squares to \(\lambda \), \(A_{\lambda }^{\lambda +\square _1+\cdots +\square _n}\) are the coefficients in the actions
It is known that \(A_{\lambda }^{\lambda +\square _1+\cdots +\square _n}=\sum _{k=1}^n (-1)^{k}a^{\lambda +\square _1+\cdots +\square _n}_{\lambda ,(k,1^{n-k})}\), with the Littlewood-Richardson coefficients \(a^{\lambda }_{\mu ,\nu }\) defined by \(S_{\mu }S_{\nu }=\sum _{\lambda }a^{\lambda }_{\mu ,\nu }S_{\lambda }\) [26]. The actions (11) can be proved inductively by using the relations \(p_n=\frac{1}{n-1}[E_{1},p_{n-1}],\ \ n\ge 2\).
Let us introduce the partition function hierarchy with W-representations
It is straightforward to calculate the powers of \(W_{-n}\) acting on \(S_{\lambda }\) with \(\lambda =\varnothing \), leading to the explicit results
where we have used the hook length formula \(\frac{S_{\lambda }\{p_k=N\}}{S_{\lambda }\{p_k=\delta _{k,1}\}}=\prod _{(i,j)\in \lambda } (j-i+N)\).
Then we have
We see that \(Z_{-1}\{p\}\) and \(Z_{-2}\{p\}\) are the \(N\times N\) complex matrix model [1, 27] and Gaussian hermitian one-matrix model [1, 5], respectively,
Note that for any partition function of the form [25]
with the arbitrary function f and parameters \(\bar{p}_k\), it is a \(\tau \)-function of the KP hierarchy. It is clear that the partition function hierarchy (15) gives the \(\tau \)-functions of the KP hierarchy.
Similarly, we define the operator
and
where \(\lambda -\square \) are the Young diagrams obtained by removing one square \(\square =(i_\square , j_\square )\) from \(\lambda \).
We set
There is the action
In terms of \(W_{1}\) and \(E_{-1}\), we introduce a series of operators
The actions of \(W_{n}\) on the Schur functions are given by
where \(\lambda -\square _1-\cdots -\square _n\) are the Young diagrams obtained by removing n squares from \(\lambda \), \(A_{\lambda }^{\lambda -\square _1-\cdots -\square _n}=\sum _{k=1}^{n} (-1)^{k}a^{\lambda }_{\lambda -\square _1-\cdots -\square _n,(k,1^{n-k})}\) are the coefficients in the actions
The actions (23) can be proved inductively by using the relations \(\frac{\partial }{\partial p_n}=\frac{1}{n}[E_{-1},\frac{\partial }{\partial p_{n-1}}],\ \ n\ge 2\). It is interesting to note that there are the same actions as (23) for the operators \(W^-_n\) given by [18]
where H is an \(N\times N\) matrix.
Taking \(p_k=\mathrm{Tr}H^k\), we can rewrite the operators (18), (20) and \(W_2\) in (22) as
where \(H^{T}\) is the transpose of the matrix H.
Since (22) can be expressed as \(W_{n}=\frac{1}{n-1}\left[ W_{n-1},W_{1}\right] , \ n\ge 3,\) it is clear that the operators \(W_n\) (22) and \(W^-_n\) (25) are equivalent for \(n\ge 2\).
Let us introduce the partition function hierarchy with W-representations
Since there are the actions
where \(S_{\lambda /\mu }\) are the skew Schur functions, we obtain the character expansions for the partition function hierarchy (27)
Here we have used the Cauchy formula \(e^{\sum _{k=1}^{\infty }\frac{1}{k}p_k\bar{p}_k}=\sum _{\lambda }S_{\lambda }\{p\}S_{\lambda }\{\bar{p}\}\).
When particularized to the \(n=2\) case in (29), it gives the character expansion [15] of Gaussian hermitian one-matrix model in the external field [5]
where \(p_k=\mathrm{Tr}M_2^k\).
3 \(\beta \)-deformed partition function hierarchies with W-representations
Let us extend the Hurwitz–Kontsevich matrix model (1) to the \(\beta \)-deformed case,
where
Taking \(\bar{p}_k=e^{-tN}\delta _{k,1}\) in the Cauchy formula
we have
Here \(\langle J_{\lambda }, J_{\lambda }\rangle _{\beta }=\frac{h_{\lambda }}{h^{'}_{\lambda }}\), \(h_{\lambda }=\prod _{(i,j)\in \lambda }(1+\lambda _i-j+\beta (\lambda _j^{'}-i))\) and \(h^{'}_{\lambda }=\prod _{(i,j)\in \lambda }(\lambda _i-j+\beta (\lambda _j^{'}-i+1))\) are the deformed hook length, in which \(\lambda ^{'}=(\lambda _1^{'},\lambda _2^{'},\ldots )\) is the conjugate partition of \(\lambda \).
Using the expansion (34) and the action [28]
where \(\bar{c}_{\lambda }=\sum _{(i,j)\in \lambda }c(i,j)\) and \(c(i,j)=j-1+\beta (N-i+1)\), we then obtain
Let us define the operator
We have
where \(B_{\lambda }^{\lambda +\square }\) are the coefficients in the action
Since \(p_2=[\bar{E}_1, p_1]\), from the actions (38) and (39), we obtain
where \(B_{\lambda }^{\lambda +\square _1+\square _2}= (c(i_{\square _2},j_{\square _2})- c(i_{\square _1},j_{\square _1}))B_{\lambda }^{\lambda +\square _1} B_{\lambda +\square _1}^{\lambda +\square _1+\square _2}\).
For the operator
we have
Note that the combination of \({\mathcal {W}}_{-1}\) and \(\bar{E}_{1}\), i.e., \({{\mathcal {W}}}_{-1}+(1-\beta )\bar{E}_{1}\), gives the W-representation of the \(\beta \)-deformed \(N\times N\) complex matrix model [4, 29].
For the operator
it gives the W-operator in the W-representations of \(\beta \)-deformed Gaussian hermitian matrix model [3]. By the actions (38) and (42), we have
where \(B_{\lambda }^{\lambda +\square _1+\square _2}\) are the coefficients in (40).
Let us introduce a series of operators
There are the actions
where \(B_{\lambda } ^{\lambda +\square _1+\cdots +\square _n}\) are the coefficients in
The actions (46) can be proved inductively by using the relations \(p_n=\frac{1}{n-1}[\bar{E}_{1},p_{n-1}],\ \ n\ge 2\).
Let us introduce the partition function hierarchy with W-representations
It is straightforward to calculate the power of \({\mathcal {W}}_{-n}\) acting on \(J_{\lambda }\) with \(\lambda =\varnothing \), leading to the explicit result
where
Using the hook length formula
we further obtain
When particularized to the \(n=1, 2\) cases in (52), it gives the \(\beta \)-deformed rectangular complex (with \(N_1=N_2\)) and Gaussian hermitian matrix models [3], respectively,
Let us turn to construct the operator
where the operator \(\bar{E}_{-1}\) is given by
We have the action
where \(B_\lambda ^{\lambda -\square }\) are the coefficients in
Let us introduce the operators
We have the actions
where \(B_{\lambda }^{\lambda -\square _1-\cdots -\square _n}\) are the coefficients in
We introduce the partition function hierarchy with W-representations
Due to
and the Cauchy formula (33), there are the character expansions for the partition function hierarchy (61)
4 Summary
It was known that the Hurwitz–Kontsevich matrix model (1) can be expressed as the exponent of the Hurwitz operator \(W_0\) acting on the function \(e^{p_1/e^{tN}}\). In terms of the Hurwitz operator \(W_0\), \(p_1\) and \(\frac{\partial }{\partial p_1}\), we have constructed the partition function hierarchies with W-representations. Based on the W-representations, we showed that these partition functions can be expressed as the character expansions with respect to the Schur functions. It was noted that the character expansions of hierarchy (15) give the \(\tau \)-functions of the KP hierarchy, and the \(N\times N\) complex matrix and Gaussian hermitian one-matrix models are contained in the hierarchy (15). For the constructed partition function hierarchy (29), it contains the Gaussian hermitian one-matrix model in the external field. We have also extended the Hurwitz–Kontsevich matrix model (1) to the \(\beta \)-deformed case. Similarly, the \(\beta \)-deformed partition function hierarchies with W-representations were constructed and their character expansions with respect to the Jack polynomials were presented as well. The \(\beta \)-deformed rectangular complex and Gaussian hermitian matrix models are contained in the hierarchy (52). Searching for the matrix model representations of the partition functions in the hierarchies presented in this paper would merit further investigations. Furthermore, it would be interesting to construct q, t-deformed partition function hierarchies with W-representations and study their character expansions with respect to the Macdonald polynomials.
Data Availability
This manuscript has no associated data or the data will not be deposited. [Authors’ comment: This is a theoretical study and no experimental data has been listed.]
References
A. Mironov, A. Morozov, On the complete perturbative solution of one-matrix models. Phys. Lett. B 05, 94 (2017). arXiv:1705.00976
A. Mironov, A. Morozov, Sum rules for characters from character-preservation property of matrix models. J. High Energy Phys. 08, 163 (2018). arXiv:1807.02409
A. Morozov, On \(W\)-representations of \(\beta \)- and \(q, t\)-deformed matrix models. Phys. Lett. B 792, 205 (2019). arXiv:1901.02811
L. Cassia, R. Lodin, M. Zabzine, On matrix models and their \(q\)-deformations. J. High Energy Phys. 10, 126 (2020). arXiv:2007.10354
A. Morozov, Sh. Shakirov, Generation of matrix models by \({\hat{W}}\)-operators. J. High Energy Phys. 04, 064 (2009). arXiv:0902.2627
A. Mironov, A. Morozov, Superintegrability of Kontsevich matrix model. Eur. Phys. J. C 81, 270 (2021). arXiv:2011.12917
A. Mironov, A. Morozov, Z. Zakirova, New insights into superintegrability from unitary matrix models. Phys. Lett. B 831, 137178 (2022). arXiv:2203.03869
L.Y. Wang, R. Wang, K. Wu, W.Z. Zhao, \(W\)-representations of the fermionic matrix and Aristotelian tensor models. Nucl. Phys. B 973, 115612 (2021). arXiv:2110.14269
A. Mironov, V. Mishnyakov, A. Morozov, A. Zhabin, Natanzon-Orlov model and refined superintegrability. Phys. Lett. B 829, 137041 (2022). arXiv:2112.11371
C. Cordova, B. Heidenreich, A. Popolitov, S. Shakirov, Orbifolds and exact solutions of strongly-coupled matrix models. Commun. Math. Phys. 361, 1235 (2018). arXiv:1611.03142
A. Mironov, A. Morozov, A. Popolitov, Superintegrability of matrix Student’s distribution. Phys. Lett. B 824, 136833 (2022). arXiv:2107.13381
A. Mironov, A. Morozov, Superintegrability summary. arXiv:2201.12917
A. Mironov, V. Mishnyakov, A. Morozov, R. Rashkov, Matrix model partition function by a single constraint. Eur. Phys. J. C 81, 1140 (2021). arXiv:2105.09920
A. Mironov, V. Mishnyakov, A. Morozov, R. Rashkov, Virasoro versus superintegrability. Gaussian Hermitian model. JETP Lett. 113, 728 (2021). arXiv:2104.11550
R. Wang, C.H. Zhang, F.H. Zhang, W.Z. Zhao, CFT approach to constraint operators for (\(\beta \)-deformed) hermitian one-matrix models. arXiv:2203.14578
V. Mishnyakov, A. Oreshina, Superintegrability in \(\beta \)-deformed Gaussian Hermitian matrix model from \(W\)-operators. Eur. Phys. J. C 82, 548 (2022). arXiv:2203.15675
S. Barseghyan, A. Popolitov, The ”Null-A” superintegrability for monomial matrix models. arXiv:2204.14074
A. Mironov, A. Morozov, Bilinear character correlators in superintegrable theory. arXiv:2206.02045
H. Itoyama, A. Mironov, A. Morozov, Complete solution to Gaussian tensor model and its integrable properties. Phys. Lett. B 802, 135237 (2020). arXiv:1910.03261
B. Kang, L.Y. Wang, K. Wu, J. Yang, W.Z. Zhao, \(W\)-representation of rainbow tensor model. J. High Energy Phys. 05, 228 (2021). arXiv:2104.01332
I. Goulden, D. Jackson, Transitive factorization into transpositions and holomorphic mappings on the sphere. Proc. Am. Math. Soc. 125, 51 (1997)
A. Mironov, A. Morozov, Virasoro constraints for Kontsevich-Hurwitz partition function. J. High Energy Phys. 02, 024 (2009). arXiv:0807.2843
A. Mironov, A. Morozov, S. Natanzon, Complete set of cut-and-join operators in the Hurwitz-Kontsevich theory. Theor. Math. Phys. 166, 1 (2011). arXiv:0904.4227
A. Mironov, A. Morozov, S. Natanzon, Algebra of differential operators associated with Young diagrams. J. Geom. Phys. 62, 148 (2012). arXiv:1012.0433
A. Alexandrov, A. Mironov, A. Morozov, S. Natanzon, On KP-integrable Hurwitz functions. J. High Energy Phys. 11, 080 (2014). arXiv:1405.1395
I.G. Macdonald, Symmetric functions and Hall polynomials, 2nd edn. (Oxford University Press, Oxford, 1995)
A. Alexandrov, A. Mironov, A. Morozov, BGWM as second constituent of complex matrix model. J. High Energy Phys. 12, 053 (2009). arXiv:0906.3305
M. Lassalle, Jack polynomials and free cumulants. Adv. Math. 222, 2227 (2009). arXiv:0802.0448
Y. Chen, B. Kang, M.L. Li, L.F. Wang, C.H. Zhang, Correlators in the \(\beta \)-deformed Gaussian Hermitian and complex matrix models. Int. J. Mod. Phys. A 34, 1950221 (2019)
Acknowledgements
We are grateful to A. Morozov and A. Mironov for their helpful comments. This work is supported by the National Natural Science Foundation of China (Nos. 11875194 and 12105104) and the Fundamental Research Funds for the Central Universities, China (No. 2022XJLX01).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
Funded by SCOAP3. SCOAP3 supports the goals of the International Year of Basic Sciences for Sustainable Development.
About this article
Cite this article
Wang, R., Liu, F., Zhang, CH. et al. Superintegrability for (\(\beta \)-deformed) partition function hierarchies with W-representations. Eur. Phys. J. C 82, 902 (2022). https://doi.org/10.1140/epjc/s10052-022-10875-z
Received:
Accepted:
Published:
DOI: https://doi.org/10.1140/epjc/s10052-022-10875-z