@article{ac03fe6282794bee9b119e6487ecaa1b,

title = "Theory of strings with boundaries: Fluctuations, topology and quantum geometry",

abstract = "We discuss Polyakov's quantization of the string in the presence of a boundary allowing for an arbitrary topology for the world sheet. In addition to the dynamical conformal factor discovered by Polyakov, there are a finite number of new degrees of freedom if the surface is more complicated than a sphere or a disc. The quantization of the Liouville theory in an arbitrary topology is discussed. A one-loop calculation shows that the model is renormalizable if one performs a mass renormalization and an additive field renormalization. The renormalization group equations have a perturbative infrared unstable fixed point in all topologies.",

author = "Orlando Alvarez",

note = "Funding Information: Throughout this paper we concentrated on compact manifolds with boundary. There are two dimensional manifolds which are compact without boundary, and others non-compact. We did not discuss these since they are not related to the Wilson loop. The compact manifolds without boundary are related to the closed string, and the non-compact ones are related to the open string. There is a very mysterious phenomenon related to these two cases. The Liouville theory in both those cases seems to be independent of the physical world. There is no boundary which brings in the physical scale. One has to introduce sources which couple to the physical coordinates x ~. These sources will couple to the conformal factor and introduce the external physical scale into the Liouville theory. The mass parameter of the LiouviUe theory appears not to be related to the string tension. There is a scale that always has to be introduced into the Liouville theory. It is the scale of the fiducial metric. ~. For example, in the case of a sphere ~b = ~ab(1 q-clzl2) -2 is the standard constant curvature metric. The dimensional constant c sets the scale. In the case of a manifold with boundary, the boundary conditions set the scale. Unless one introduces external sources, we do not know how to set the scale for the fiducial metrics gt{"} I would like to acknowledge the many colleagues who shared their insight with me: K. Bardakci, W. Bardeen, B. Durhuus, D. Gross, M. Halpern, S. Mandelstam, J. L. Petersen, J. Schonfeld, I. Singer, H. Thacker, H. Tye, K. Wilson. I would specially like to thank C. Earle, D. Friedan, E. Martinec, M. Peskin, and B. Svetitsky. I would like to thank M. Chanowitz for the invitation to visit LBL during which this research was begun. I would like to thank the referee for pointing out some references which were unfamiliar to me. I am appreciative of Jenni Morris for her preparation of this manuscript. This paper was written at the insistence of many friends and colleagues. Their persistence is responsible for this manuscript. This research is supported in part by NSF grant PHY77-22336.",

year = "1983",

month = apr,

day = "25",

doi = "10.1016/0550-3213(83)90490-X",

language = "English (US)",

volume = "216",

pages = "125--184",

journal = "Nuclear Physics B",

issn = "0550-3213",

publisher = "Elsevier",

number = "1",

}