Matrix Models of 2D Gravity and Isomonodromic Deformation

One of the principal goals of the theory of 2D gravity is making sense of the formal expression Z( μ ,κ ;tᵢ ) = ∑ₕ ∫_METₕ dge ^μ ∫ √ g + k∫ √ g + k Z_QFT( tᵢ )[ g ] (1.1) where we integrate over metrics g on surfaces with h handles with a weight defined by the Einstein-Hilbert action (μ is the cosmological constant and κ is Newton’s constant, or, equivalently, the string coupling) together with the partition function of some 2D quantum field theory, QFT(t i ). The parameters t i are coordinates on a subspace of the space of 2D field theories, or, equivalently, coordinates for a space of string backgrounds.