Abstract
The goal of the present paper is to calculate the determinant of the Dirac operator with a mass in the cylindrical geometry. The domain of this operator consists of functions that realize a unitary one-dimensional representation of the fundamental group of the cylinder with n marked points. The determinant represents a version of the isomonodromic τ-function, introduced by M. Sato, T. Miwa and M. Jimbo. It is calculated by comparison of two sections of the det*-bundle over an infinite-dimensional grassmannian. The latter is composed of the spaces of boundary values of some local solutions to the Dirac equation. The principal ingredients of the computation are the formulae for the Green function of the singular Dirac operator and for the so-called canonical basis of global solutions on the 1-punctured cylinder. We also derive a set of deformation equations satisfied by the expansion coefficients of the canonical basis in the general case and find a more explicit expression for the τ-function in the simplest case n=2.
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Bugrij, A.I.: Correlation function of the two-dimensional Ising model on the finite lattice. I. Theor. & Math. Phys. 127, 528–548 (2001)
Bugrij, A.I.: Form factor representation of the correlation functions of the two-dimensional Ising model on a cylinder. In: S. Pakuliak, G. von Gehlen (eds.), Integrable structures of exactly solvable two-dimensional models of quantum field theory, NATO Sci. Ser. II Math. Phys. Chem. 35, Dordrecht Kluwer Acad. Publ. 2001, pp. 65–93
Bugrij, A.I., Lisovyy, O.: Spin matrix elements in 2D Ising model on the finite lattice. Phys. Letts. A319, 390–394 (2003)
Bugrij, A.I., Shadura, V.N.: Asymptotic expression for the correlation function of twisted fields in the two-dimensional Dirac model on a lattice. Theor. & Math. Phys. 121, 1535–1549 (1999)
Doyon, B.: Two-point correlation functions of scaling fields in the Dirac theory on the Poincaré disk. Nucl. Phys. B 675, 607–630 (2003)
Fonseca, P., Zamolodchikov, A.: Ising field theory in a magnetic field: analytic properties of the free energy. J. Stat. Phys. 110, 527–590 (2003)
Lisovyy, O.: Nonlinear differential equations for the correlation functions of the 2D Ising model on the cylinder. Adv. Theor. Math. Phys. 5, 909–922 (2001)
Narayanan, R., Tracy, C.A.: Holonomic quantum field theory of bosons in the Poincaré disk and the zero curvature limit. Nucl. Phys. B340, 568–594 (1990)
Palmer, J.: Determinants of Cauchy-Riemann operators as τ-functions. Acta Appl. Math. 18, 199–223 (1990)
Palmer, J.: Tau functions for the Dirac operator in the Euclidean plane. Pacific J. Math. 160, 259–342 (1993)
Palmer, J.: Ising model scaling functions at short distance. http://arxiv.org/abs/nlin.SI/0107013, 2001 Palmer, J.: Short distance asymptotics of Ising correlations. http://arxiv.org/abs/nlin.SI/0107014, 2001
Palmer, J., Tracy, C.A.: Monodromy preserving deformation of the Dirac operator acting on the hyperbolic plane. In: M. S. Berger (ed.), Mathematics of Nonlinear Science: Proceedings of an AMS special session held January 11–14, 1989, Contemporary Mathematics 108, Providence, RI: AMS, 1990, pp. 119–131
Palmer, J., Beatty, M., Tracy, C.A.: Tau functions for the Dirac operator on the Poincaré disk. Commun. Math. Phys. 165, 97–173 (1994)
Pressley, A., Segal, G.: Loop groups Oxford: Clarendon Press, 1986
Quillen, D.: Determinants of Cauchy-Riemann operators on a Riemann surface. Funct. Anal. Appl. 19, 37–41 (1985)
Ray, D.B., Singer, I.M.: Analytic torsion for complex manifolds. Ann. Math. 98, 154–177 (1973)
Sato, M., Miwa, T., Jimbo, M.: Holonomic quantum fields I–V. Publ. RIMS, Kyoto Univ. 14, 223–267 (1978); 15, 201–278 (1979); 15, 577–629 (1979); 15, 871–972 (1979); 16, 531–584 (1980)
Segal, G., Wilson, G.: Loop groups and equations of KdV type. Publ. Math. I.H.E.S. 61, 5–65 (1985)
Witten, E.: Quantum field theory, Grassmannians and algebraic curves. Commun. Math. Phys. 113, 529–600 (1988)
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Communicated by L. Takhtajan
Acknowledgement I would like to thank A. I. Bugrij and V. N. Roubtsov for constant support and numerous stimulating discussions. I am grateful to S. Pakuliak for his lectures on the infinite-dimensional grassmannians and boson-fermion correspondence. I would also like to express my gratitude to J. Palmer, whose clear ideas made this work possible.
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Lisovyy, O. Tau Functions for the Dirac Operator on the Cylinder. Commun. Math. Phys. 255, 61–95 (2005). https://doi.org/10.1007/s00220-004-1252-y
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DOI: https://doi.org/10.1007/s00220-004-1252-y