Abstract:Timelike Liouville theory admits the sphere S 2 as a real saddle point, about which quantum fluctuations can occur. An issue occurs when computing the expectation values of specific types of quantities, like the distance between points. The problem being that the gauge redundancy of the path integral over metrics is not completely fixed even after fixing to conformal gauge by imposing g µν = e 2 bφ g µν , where φ is the Liouville field and g µν is a reference metric. The physical metric g µν , and therefore th… Show more
“…Defining the timelike Liouville theory from the action (3) has been the goal of ongoing efforts for 25 years [12,[27][28][29][30][31][32][65][66][67][68][69][70][71]. Some earlier references to the timelike theory can be found in [21,72]).…”
Section: A Brief History Of the Timelike Structure Constantmentioning
A proper definition of the path integral of quantum gravity has been a long-standing puzzle because the Weyl factor of the Euclidean metric has a wrong-sign kinetic term. We propose a definition of two-dimensional Liouville quantum gravity with cosmological constant using conformal bootstrap for the timelike Liouville theory coupled to supercritical matter. We prove a no-ghost theorem for the states in the BRST cohomology. We show that the four-point function constructed by gluing the timelike Liouville three-point functions is well defined and crossing symmetric for external Liouville energies corresponding to all physical states in the BRST cohomology with the choice of the Ribault-Santachiara contour for the internal energy.
“…Defining the timelike Liouville theory from the action (3) has been the goal of ongoing efforts for 25 years [12,[27][28][29][30][31][32][65][66][67][68][69][70][71]. Some earlier references to the timelike theory can be found in [21,72]).…”
Section: A Brief History Of the Timelike Structure Constantmentioning
A proper definition of the path integral of quantum gravity has been a long-standing puzzle because the Weyl factor of the Euclidean metric has a wrong-sign kinetic term. We propose a definition of two-dimensional Liouville quantum gravity with cosmological constant using conformal bootstrap for the timelike Liouville theory coupled to supercritical matter. We prove a no-ghost theorem for the states in the BRST cohomology. We show that the four-point function constructed by gluing the timelike Liouville three-point functions is well defined and crossing symmetric for external Liouville energies corresponding to all physical states in the BRST cohomology with the choice of the Ribault-Santachiara contour for the internal energy.
“…5 The reason this is a good gauge fixing procedure, as delineated in appendix C, is that the variation of δϕ under the three non-compact generators of PSL(2, C) is precisely equal to the l = 1 modes. A different gauge fixing procedure is discussed in [25]. This condition will remain unchanged under the action of the SO(3) subgroup of PSL(2, C), fixing three of the six parameters of PSL(2, C).…”
We study the Euclidean path integral of two-dimensional quantum gravity with positive cosmological constant coupled to conformal matter with large and positive central charge. The problem is considered in a semiclassical expansion about a round two-sphere saddle. We work in the Weyl gauge whereby the computation reduces to that for a (timelike) Liouville theory. We present results up to two-loops, including a discussion of contributions stemming from the gauge fixing procedure. We exhibit cancelations of ultraviolet divergences and provide a path integral computation of the central charge for timelike Liouville theory. Combining our analysis with insights from the DOZZ formula we are led to a proposal for an all orders result for the two-dimensional gravitational partition function on the two-sphere.
“…Here the metric is put into conformal gauge [66,107,113,115] g ab = e φc η ab and e φc = 3 Λ cos 2 η . Via the Liouville equation of motion we have…”
Section: The 1 + 1 Dimensional Action In Liouville Gravitymentioning
confidence: 99%
“…The decay to Hats, Friedmann-Roberston-Walker (FRW) bubbles of vacua in the supermoduli space, Λ = 0, provides an opportunity to define a rigorous framework for dS. This conjectured framework known as FRW/CFT [34,55,56,[61][62][63][64][65][66][67] [68].…”
The aim of this work is to provide the details of a calculation summarized in the recent paper by Maltz and Susskind which conjectured a potentially rigorous framework where the status of de Sitter space is the same as that of a resonance in a scattering process. The conjecture is that transition amplitudes between certain states with asymptotically supersymmetric flat vacua contain resonant poles characteristic metastable intermediate states. A calculation employing constrained instantons is presented that illustrates this idea.
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