TY - JOUR

T1 - Loop equations and Virasoro constraints in non-perturbative two-dimensional quantum gravity

AU - Dijkgraaf, Robbert

AU - Verlinde, Herman

AU - Verlinde, Erik

N1 - Funding Information:
We are grateful to T. Banks for valuable discussions and for first suggesting that our original results (eqs. (3.4) and (3.5)) should be interpreted as loop equations. We would furthermore like to acknowledge discussions with F. David, J. Distler, P . Di Francesco, W. Fischler, D. Gross, D. Kutasov, A. AMidgal, A. Polyakov, S . Shenker, L . Susskind and E. Witten. The research of R.D. is supported by DOE grant DE-AC02-76WRO3072, that of E.V. by NSF Waterman grant 88-17521 and that of H.V. by NSF grant PHY80-19754.

PY - 1991/1/21

Y1 - 1991/1/21

N2 - We give a derivation of the loop equation for two-dimensional gravity from the KdV equations and the string equation of the one-matrix model. We find that the loop equation is equivalent to an infinite set of linear constraints on the square root of the partition function satisfying the Virasoro algebra. We give an interpretation of these equations in topological gravity and discuss their extension to multi-matrix models. For the multi-critical models the loop equation naturally singles out the operators corresponding to the primary fields of the minimal models.

AB - We give a derivation of the loop equation for two-dimensional gravity from the KdV equations and the string equation of the one-matrix model. We find that the loop equation is equivalent to an infinite set of linear constraints on the square root of the partition function satisfying the Virasoro algebra. We give an interpretation of these equations in topological gravity and discuss their extension to multi-matrix models. For the multi-critical models the loop equation naturally singles out the operators corresponding to the primary fields of the minimal models.

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U2 - 10.1016/0550-3213(91)90199-8

DO - 10.1016/0550-3213(91)90199-8

M3 - Article

AN - SCOPUS:0000817931

SN - 0550-3213

VL - 348

SP - 435

EP - 456

JO - Nuclear Physics, Section B

JF - Nuclear Physics, Section B

IS - 3

ER -