Matrix models and non-perturbative two-dimensional quantum gravity
Matrix models and non-perturbative two-dimensional quantum gravity
We investigate soluble toy models of fluctuating random surfaces which arise through the topological expansion of two-dimensional quantum gravity or non-critical string theory. A regularisation in terms of large N matrix field theories is used. In the continuum limit the partition function is obtained through the solution of a non-linear ordinary differential equation and correlators of local operators are determined by the flows of the KdV hierarchy. It is demonstrated how previous attempts to define these models non-perturbatively in topology suffer from instabilities which rule them out as sensible theories.
We study the university of the models, through their representation as a Dyson gas of charged particles in one space dimension, in order to obtain a pictographic understanding of their critical behaviour. This enables us
to identify a valid non-pcrturbativc stabilisation that does not affect the pcrturbativc physics, which we then realise in terms of a specific matrix model. The continuum limit is taken and then exact partition function is found to be well-defined, as arc all correlators of local operators. We then formulate the theory directly in the continuum limit in terms of the requirements of scaling and renormalisation flow. This allows us to show that our stabilisation is in a sense unique. Our differential equation which determines the partition function is the most general that one can obtain from the Dyson gas representation for cut-off independent physics. Equiv-alcntly, for such physics, it is the most general equation compatible with the KdV flows and only the KdV flows.
University of Southampton
1991
Dalley, Simon
(1991)
Matrix models and non-perturbative two-dimensional quantum gravity.
University of Southampton, Doctoral Thesis.
Record type:
Thesis
(Doctoral)
Abstract
We investigate soluble toy models of fluctuating random surfaces which arise through the topological expansion of two-dimensional quantum gravity or non-critical string theory. A regularisation in terms of large N matrix field theories is used. In the continuum limit the partition function is obtained through the solution of a non-linear ordinary differential equation and correlators of local operators are determined by the flows of the KdV hierarchy. It is demonstrated how previous attempts to define these models non-perturbatively in topology suffer from instabilities which rule them out as sensible theories.
We study the university of the models, through their representation as a Dyson gas of charged particles in one space dimension, in order to obtain a pictographic understanding of their critical behaviour. This enables us
to identify a valid non-pcrturbativc stabilisation that does not affect the pcrturbativc physics, which we then realise in terms of a specific matrix model. The continuum limit is taken and then exact partition function is found to be well-defined, as arc all correlators of local operators. We then formulate the theory directly in the continuum limit in terms of the requirements of scaling and renormalisation flow. This allows us to show that our stabilisation is in a sense unique. Our differential equation which determines the partition function is the most general that one can obtain from the Dyson gas representation for cut-off independent physics. Equiv-alcntly, for such physics, it is the most general equation compatible with the KdV flows and only the KdV flows.
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Published date: 1991
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Local EPrints ID: 460485
URI: http://eprints.soton.ac.uk/id/eprint/460485
PURE UUID: b6960ce1-2ee9-4ffb-8336-13fd68425c4b
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Date deposited: 04 Jul 2022 18:23
Last modified: 04 Jul 2022 18:23
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Author:
Simon Dalley
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