New entry critics of string theory to collect the references on controversies. I think they are often rambling and vague, not technically useful s the main references we want to collect under string theory and books in string theory. I have changed the sentence in string theory about mathematical definition of parts to somewhat more precise

But every now and then some aspect of string theory, some mathematical gadget or consequence found there is isolated and redefined independently and mathematically rigorously, retaining many features originally predicted.

The point is that most often one does not make rigorous the way some thing is defined via string theory, but one isolates an invariant of manifolds for example and defines a similar one via completely different foundations. The typical example is quantum cohomology which is defined in geometric terms and not in terms of field theory any more.

I have one disagreement with the entry: it says that topological quantum field theory has been discovered as part of string theory research, This is not true, TQFTs were found in 1977, 1978, 1980 articles of Albert Schwartz which had nothing to do with string theory. Only much later Atiyahâs formulation is influenced by string theory.

]]>Stub for topological string with redirect topological string theory.

]]>A very short note and a question.

Given a space $X$ with an action of a group $G$ on it, an $U(1)$-cocycle on the quotient groupoid $X// G$, i.e., a functor $X//G \to \mathbf{B} U(1)$, can be explicitly described as a function $\lambda: G\times X \to U(1)$ such that

$\lambda(h g,x)=\lambda(h, g x)\lambda(g,x)$An extremely nice and nontrivial example of this is the *Liouville cocycle*. Fix a Riemann surface $\Sigma$ and take as $X$ the space of Riemannian metrics on $\Sigma$, and as $G$ the additive group of smooth real-valued fucnctions on $\Sigma$, acting on metrics by conformal rescaling: $(f,g_{i j})\mapsto e^f g_{i j}$. Then the Liouville cocycle is the function

defined by

$\lambda(f,g)=exp(\frac{i}{2} \int_\Sigma(d f\wedge *_g d f +2 f R_g d \mu_g)),$where $*_g$ is the Hodge star operator defined by the Riemannian metric $g$, $R_g$ is the scalar curvarure and $d \mu_g$ is the volume form. It would be interesting if under suitable hypothesis (e.g., compatibility with glueing of Riemann surfaces in view of CFT), the Lioville cocycle would be the only possibility (up to a scalar factor: *central charge*). Maybe Zoran knows the answer to this.