Abstract
We consider six-vertex model configurations on an (n × N) lattice, n ≤ N, that satisfy a variation on domain wall boundary conditions that we define and call partial domain wall boundary conditions. We obtain two expressions for the corresponding partial domain wall partition function, as an (N × N)-determinant and as an (n × n)-determinant. The latter was first obtained by I Kostov. We show that the two determinants are equal, as expected from the fact that they are partition functions of the same object, that each is a discrete KP τ-function, and, recalling that these determinants represent tree-level structure constants in \( \mathcal{N} = 4\;{\text{SYM}} \), we show that introducing 1-loop corrections, as proposed by N Gromov and P Vieira, preserves the determinant structure.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
I. Bena, J. Polchinski and R. Roiban, Hidden symmetries of the AdS 5 × S 5 superstring, Phys. Rev. D 69 (2004) 046002 [hep-th/0305116] [INSPIRE].
J. Minahan and K. Zarembo, The Bethe ansatz for N = 4 super Yang-Mills, JHEP 03 (2003) 013 [hep-th/0212208] [INSPIRE].
N. Beisert et al., Review of AdS/CFT Integrability: An Overview, Lett. Math. Phys. 99 (2012) 3 [arXiv:1012.3982] [INSPIRE].
V. Korepin, Calculation of norms of Bethe wave functions, Commun. Math. Phys. 86 (1982) 391 [INSPIRE].
A.G. Izergin, Partition function of the six-vertex model in a finite volume, Sov. Phys. Dokl. 32 (1987) 878.
V.E. Korepin, N.M. Bogoliubov and A.G. Izergin, Quantum inverse scattering method and correlation functions, Cambridge University Press, Cambridge U.K. (1993).
N.M. Bogoliubov, A.G. Pronko and M.B. Zvonarev, Boundary correlation functions of the six-vertex model, J. Phys. A 35 (2002) 5525 [math-ph/0203025].
O. Foda and I. Preston, On the correlation functions of the domain wall six vertex model, J. Stat. Mech. 0411 (2004) P11001 [math-ph/0409067] [INSPIRE].
F. Colomo and A. Pronko, On the partition function of the six vertex model with domain wall boundary conditions, J. Phys. A 37 (2004) 1987 [math-ph/0309064] [INSPIRE].
F. Colomo and A.G. Pronko, On two-point boundary correlations in the six-vertex model with domain wall boundary conditions, J. Stat. Mech. 5 (2005) 10 [math-ph/0503049].
F. Colomo and A.G. Pronko, An approach for calculating correlation functions in the six-vertex model with domain wall boundary conditions, Theor. Math. Phys. 171 (2012) 641 [arXiv:1111.4353].
J. Escobedo, N. Gromov, A. Sever and P. Vieira, Tailoring three-point functions and integrability, JHEP 09 (2011) 028 [arXiv:1012.2475] [INSPIRE].
J. Escobedo, N. Gromov, A. Sever and P. Vieira, Tailoring three-point functions and integrability II. Weak/strong coupling match, JHEP 09 (2011) 029 [arXiv:1104.5501] [INSPIRE].
N. Gromov, A. Sever and P. Vieira, Tailoring three-point functions and integrability III. Classical tunneling, JHEP 07 (2012) 044 [arXiv:1111.2349] [INSPIRE].
O. Foda, N = 4 SYM structure constants as determinants, JHEP 03 (2012) 096 [arXiv:1111.4663] [INSPIRE].
M. Wheeler, An Izergin-Korepin procedure for calculating scalar products in six-vertex models, Nucl. Phys. B 852 (2011) 468 [arXiv:1104.2113] [INSPIRE].
I. Kostov, private communication.
I. Kostov, Classical limit of the three-point function from integrability, arXiv:1203.6180 [INSPIRE].
I. Kostov, Three-point function of semiclassical states at weak coupling, arXiv:1205.4412 [INSPIRE].
N. Gromov and P. Vieira, Quantum integrability for three-point functions, arXiv:1202.4103 [INSPIRE].
N. Gromov and P. Vieira, Tailoring three-point functions and integrability IV. Theta-morphism, arXiv:1205.5288 [INSPIRE].
N.A. Slavnov, Calculation of scalar products of wave functions and form factors in the framework of the algebraic Bethe Ansatz, Theor. Math. Phys. 79 (1989) 502.
N. Kitanine, J.M. Maillet and V. Terras, Form factors of the XXZ Heisenberg spin-1/2 finite chain, Nucl. Phys. B 554 (1999) 647 [math-ph/9807020].
O. Foda and G. Schrader, XXZ scalar products, Miwa variables and discrete KP, in New Trends in Quantum Integrable Systems, B. Feigin, M. Jimbo and M. Okado eds., World Scientific, Singapore (2010), pg. 61-80 [arXiv:1003.2524].
Y. Ohta, R. Hirota, S. Tsujimoto and T. Inami, Casorati and discrete Gram type determinant representations of solutions to the discrete KP hierarchy, J. Phys. Soc. Japan 62 (1993) 1872.
I.G. Macdonald, Symmetric functions and Hall polynomials, Oxford University Press, Oxford U.K. (1995).
D. Serban, A note on the eigenvectors of long-range spin chains and their scalar products, arXiv:1203.5842 [INSPIRE].
B. Sutherland, Low-Lying Eigenstates of the One-Dimensional Heisenberg Ferromagnet for any Magnetization and Momentum, Phys. Rev. Lett. 74 (1995) 816 [INSPIRE].
A. Dhar and B. Sriram Shastry, Bloch Walls and Macroscopic String States in Bethe’s Solution of the Heisenberg Ferromagnetic Linear Chain, Phys. Rev. Lett. 85 (2000) 2813 [INSPIRE].
J. Caetano and P. Vieira, private communication.
M. Wheeler, Scalar products in generalized models with SU(3)-symmetry, arXiv:1204.2089 [INSPIRE].
G. Kuperberg, Another proof of the alternating sign matrix conjecture, Int. Math. Res. Notices 3 (1996) 139 [math/9712207].
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1205.4400
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Foda, O., Wheeler, M. Partial domain wall partition functions. J. High Energ. Phys. 2012, 186 (2012). https://doi.org/10.1007/JHEP07(2012)186
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP07(2012)186