arXiv subjects
Publications and source records attributed to Daniel Friedan.
We derive a formula for the curvature tensor of the natural Riemannian metric on the space of two-dimensional conformal field theories and also a formula for the curvature tensor of the space of boundary conformal field theories.
The gradient flow of the Yang-Mills action acts pointwise on closed loops of gauge fields. We construct a topologically nontrivial loop of SU(2) gauge fields on S4 that is locally stable under the flow. The stable loop is written explicitly as a path between two gauge fields equivalent under a topologically nontrivial SU(2) gauge transformation. Local stability is demonstrated by calculating the flow equations to leading order in perturbations of the loop. The stable loop might play a role in physics as a classical winding mode of the lambda model, a 2-d quantum field theory that was proposed as a mechanism for generating spacetime quantum field theory. We also present evidence for 2-manifolds of SU(2) and SU(3) gauge fields that are stable under the Yang-Mills flow. These might provide 2-d instanton corrections in the lambda model. For Isidore M. Singer in celebration of his eighty-fifth birthday.
We give a non-perturbative proof of a gradient formula for beta functions of two-dimensional quantum field theories. The gradient formula has the form \partial_{i}c = - (g_{ij}+Δg_{ij} +b_{ij})β^{j} where β^{j} are the beta functions, c and g_{ij} are the Zamolodchikov c-function and metric, b_{ij} is an antisymmetric tensor introduced by H. Osborn and Δg_{ij} is a certain metric correction. The formula is derived under the assumption of stress-energy conservation and certain conditions on the infrared behaviour the most significant of which is the condition that the large distance limit of the field theory does not exhibit spontaneously broken global conformal symmetry. Being specialized to non-linear sigma models this formula implies a one-to-one correspondence between renormalization group fixed points and critical points of c.
We consider the general supersymmetric one-dimensional quantum system with boundary, critical in the bulk but not at the boundary. The renormalization group flow on the space of boundary conditions is generated by the boundary beta functions β^{a}(λ) for the boundary coupling constants λ^{a}. We prove a gradient formula \partial\ln z/\partialλ^{a} =-g_{ab}^{S}β^{b} where z(λ) is the boundary partition function at given temperature T=1/β, and g_{ab}^{S}(λ) is a certain positive-definite metric on the space of supersymmetric boundary conditions. The proof depends on canonical ultraviolet behavior at the boundary. Any system whose short distance behavior is governed by a fixed point satisfies this requirement. The gradient formula implies that the boundary energy, -\partial\ln z/\partialβ= -Tβ^{a}\partial_{a}\ln z, is nonnegative. Equivalently, the quantity \ln z(λ) decreases under the renormalization group flow.
We present some partial results on the general infrared behavior of bulk-critical 1-d quantum systems with boundary. We investigate whether the boundary entropy, s(T), is always bounded below as the temperature T decreases towards 0, and whether the boundary always becomes critical in the IR limit. We show that failure of these properties is equivalent to certain seemingly pathological behaviors far from the boundary. One of our approaches uses real time methods, in which locality at the boundary is expressed by analyticity in the frequency. As a preliminary, we use real time methods to prove again that the boundary beta-function is the gradient of the boundary entropy, which implies that s(T) decreases with T. The metric on the space of boundary couplings is interpreted as the renormalized susceptibility matrix of the boundary, made finite by a natural subtraction.
The boundary beta-function generates the renormalization group acting on the universality classes of one-dimensional quantum systems with boundary which are critical in the bulk but not critical at the boundary. We prove a gradient formula for the boundary beta-function, expressing it as the gradient of the boundary entropy s at fixed non-zero temperature. The gradient formula implies that s decreases under renormalization except at critical points (where it stays constant). At a critical point, the number exp(s) is the ``ground-state degeneracy,'' g, of Affleck and Ludwig, so we have proved their long-standing conjecture that g decreases under renormalization, from critical point to critical point. The gradient formula also implies that s decreases with temperature except at critical points, where it is independent of temperature. The boundary thermodynamic energy u then also decreases with temperature. It remains open whether the boundary entropy of a 1-d quantum system is always bounded below. If s is bounded below, then u is also bounded below.
These talks present an overview of a tentative theory of large distance physics. For each large distance L (in dimensionless units), the theory gives two complementary descriptions of spacetime physics: quantum field theory at distances larger than L, string scattering amplitudes at distances smaller than L. The mechanism of the theory is a certain 2d nonlinear model, the lambda-model, whose target manifold is the manifold of general nonlinear models of the worldsurface, the background spacetimes for string scattering. So far, the theory has only been formulated and its basic working described, in general terms. The theory's only claims to interest at present are matters of general principle. It is a self-contained nonperturbative theory of large distance physics, operating entirely at large distance. The lambda-model constructs an actual QFT at large distance, a functional integral over spacetime fields. It constructs an effective background spacetime for string scattering at relatively small distances. It is background independent, dynamically. Nothing is adjustable in its formulation. It is a mechanical theory, not an S-matrix theory. String scattering takes place at small distances within a mechanical large distance environment. The lambda-model constructs QFT in a way that offers possibilities of novel physical phenomena at large distances. The task now is to perform concrete calculations in the lambda-model, to find out if it produces a physically useful QFT.
A theoretical mechanism is devised to determine the large distance physics of spacetime. It is a two dimensional nonlinear model, the lambda model, set to govern the string worldsurface to remedy the failure of string theory. The lambda model is formulated to cancel the infrared divergent effects of handles at short distance on the worldsurface. The target manifold is the manifold of background spacetimes. The coupling strength is the spacetime coupling constant. The lambda model operates at 2d distance $Λ^{-1}$, very much shorter than the 2d distance $μ^{-1}$ where the worldsurface is seen. A large characteristic spacetime distance $L$ is given by $L^2=\ln(Λ/μ)$. Spacetime fields of wave number up to 1/L are the local coordinates for the manifold of spacetimes. The distribution of fluctuations at 2d distances shorter than $Λ^{-1}$ gives the {\it a priori} measure on the target manifold, the manifold of spacetimes. If this measure concentrates at a macroscopic spacetime, then, nearby, it is a measure on the spacetime fields. The lambda model thereby constructs a spacetime quantum field theory, cutoff at ultraviolet distance $L$, describing physics at distances larger than $L$. The lambda model also constructs an effective string theory with infrared cutoff $L$, describing physics at distances smaller than $L$. The lambda model evolves outward from zero 2d distance, $Λ^{-1} = 0$, building spacetime physics starting from $L=\infty$ and proceeding downward in $L$. $L$ can be taken smaller than any distance practical for experiments, so the lambda model, if right, gives all actually observable physics. The harmonic surfaces in the manifold of spacetimes are expected to have novel nonperturbative effects at large distances.
This article is a collection of letters solicited by the editors of the Bulletin in response to a previous article by Jaffe and Quinn [math.HO/9307227]. The authors discuss the role of rigor in mathematics and the relation between mathematics and theoretical physics.