definitions of primary fields - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T13:30:29Z http://mathoverflow.net/feeds/question/105489 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/105489/definitions-of-primary-fields definitions of primary fields Daniel 2012-08-26T00:56:03Z 2012-11-28T21:37:21Z <p>I have come across two similar definitions of <em>primary fields</em> in conformal field theory. Depending on what I am doing each definition has its own usefulness. I expect both definitions to be compatible but I can't seem to be able to show it. By compatible I mean definition 1 $\iff$ definition 2. I will write both definitions in the two-dimensional case and restricting to holomorphic transformations.</p> <p>Def #1 from D'Francesco et al's CFT: A field $f(z)$ if it transforms as $f(z) \rightarrow g(\omega)=\left( \frac{d\omega}{dz}\right)^{-h}f(z), h\in\mathbb{R}$ under an infinitesimal conformal transformation $z \rightarrow \omega(z)$.</p> <p>Def #2 from Blumenhagen et al's Intro to CFT: A field $f(z)$ is primary if it transforms as $f(z) \rightarrow g(z)=\left( \frac{d\omega}{dz}\right)^{h}f(\omega), h\in\mathbb{R}$ under an infinitesimal conformal transformation $z \rightarrow \omega(z)$.</p> http://mathoverflow.net/questions/105489/definitions-of-primary-fields/114813#114813 Answer by orbifold for definitions of primary fields orbifold 2012-11-28T21:37:21Z 2012-11-28T21:37:21Z <p>Primary fields are "operator valued" sections of the canonical bundle $K_S$ of your surface $S$ raised to some power $h$. The transformation rule simply expresses that fact. This should be a comment, but I do not have enough reputation to comment.</p>