Week 04.12.2022 – 10.12.2022

A new approach to strong turbulence based on ideas of dynamical geometry and topological conservation laws is developed. In terms of the quantum field theory, this is another example of the duality between a fluctuating vector field and fluctuating geometry. Some new exact solutions of Navier-Stokes and Euler equations are found. The loop equation suggested in the early 90-ties is investigated in detail. The loop equation plays the same role in our theory as the Boltzmann kinetic equation in statistical physics. It has the form of the Schrödinger equation with a complex Hamiltonian in loop space. The viscosity plays the role of Planck's constant. Strong turbulence corresponds to the WKB limit of the loop equation. In this limit, we find a fixed point of the loop equation we call Kelvinon. Kelvinon has a conserved circulation for a fixed loop in space, generalizing Kelvin's theorem. Clebsch field of this solution has nontrivial topology with two winding numbers. These topological conservation laws allow us to compute the PDF of circulation in a WKB limit (large circulation in the viscosity units). This PDF perfectly matches the results of numerical simulations of the conventional forced Navier-Stokes equation.

In this talk, I will discuss the construction of 2d (0,2) supersymmetric gauge theories corresponding to the 18 smooth Fano 3-folds and the families of Y^(p,k)(CP1xCP1) and Y^(p,k)(CP2) Sasaki-Einstein 7-manifolds. These 2d (0,2) gauge theories can be considered as the worldvolume theories of D1-branes probing toric Calabi-Yau 4-folds. The talk will illustrate how the map between gauge theory and the corresponding geometry is considerably simplified by a Type IIA brane configuration called brane brick models.

The problem of time is often discussed as an obstacle in canonical quantisation of gravity: general covariance means there is no preferred time parameter with respect to which evolution could be defined. We can instead characterise dynamics in relational terms by defining one degree of freedom to play the role of an internal clock for the other variables; this leads to a "multiple choice problem". I will review recent results obtained in a quantum cosmological model with three dynamical degrees of freedom: a volume or scale factor variable for the geometry, a massless scalar matter field, and a perfect fluid. Each of these variables can be used as a clock for the other two. We obtain three different theories which, if we require them to have unitary time evolution with respect to the given clock, make very different statements about the fate of the Universe. Only one resolves the classical singularity, and only one leads to a quantum recollapse of the Universe at large volume. Nonclassical behaviour arises whenever a classical solution terminates in finite time so that reflecting boundary conditions are needed to make the theory unitary.

A new approach to strong turbulence based on ideas of dynamical geometry and topological conservation laws is developed. In terms of the quantum field theory, this is another example of the duality between a fluctuating vector field and fluctuating geometry. Some new exact solutions of Navier-Stokes and Euler equations are found. The loop equation suggested in the early 90-ties is investigated in detail. The loop equation plays the same role in our theory as the Boltzmann kinetic equation in statistical physics. It has the form of the Schrödinger equation with a complex Hamiltonian in loop space. The viscosity plays the role of Planck's constant. Strong turbulence corresponds to the WKB limit of the loop equation. In this limit, we find a fixed point of the loop equation we call Kelvinon. Kelvinon has a conserved circulation for a fixed loop in space, generalizing Kelvin's theorem. Clebsch field of this solution has nontrivial topology with two winding numbers. These topological conservation laws allow us to compute the PDF of circulation in a WKB limit (large circulation in the viscosity units). This PDF perfectly matches the results of numerical simulations of the conventional forced Navier-Stokes equation.

The worldsheet theory of string backgrounds is a CFT with zero central charge. This is the definition of on-shell string theory. In off-shell string theory, on the other hand, conformal invariance on the worldsheet is explicitly broken, and the worldsheet theory is therefore a QFT rather than a CFT, with a UV cutoff. In the first part of the talk, I will explain Tseytlin’s prescriptions for constructing classical (tree-level) off-shell effective actions and provide a general proof, using conformal perturbation theory, that it gives the correct equations of motion, to all orders in perturbation theory and α′. I will also show how Tseytlin's prescriptions are equivalent to quotienting out by the gauge orbits of a regulated moduli space with "n" operator insertions. In the second part of the talk, I will explain the underlying conceptual structure of the Susskind and Uglum black hole entropy argument. There I will show how the classical (tree-level) effective action and entropy S = A/4G_N can be calculated from the sphere diagrams. Time permitting, I will also discuss ongoing work for deriving the holographic entanglement entropy (the RT formula) in AdS3/CFT2. I will end with mentioning some important insights into how the ER=EPR hypothesis can be implemented using tachyon condensation on orbifolds in string theory.