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  state-operator correspondence breaks down

+ 3 like - 0 dislike

Is there a simple example where the state-operator correspondence breaks down or a general reason why this happens?

I'm reading this paper http://arxiv.org/pdf/hep-th/0208104.pdf where for the B model it is shown that the space of boundary vertex operators between two holomorphic branes $\mathcal{E}$ and $\mathcal{F}$ supported on the same submanifold $M$ is computed by the sheaf cohomology

$H^p(M,\mathcal{E}\otimes\mathcal{F}^*\otimes \wedge^qNM)$,

while the massless Ramond states are computed by

$Ext^{p+q}_X(i_* \mathcal{E}, i_*\mathcal{F})$,

where $i:M\to X$ is the inclusion of the submanifold into the ambient (Calabi-Yau) $X$.

There is a spectral sequence with $E_2^{p,q}$ equal to the former and converging to the latter. However, in the presence of a $B$-field, ie. curvature for the brane gauge fields, this spectral sequence has non-trivial differentials and so we have a breakdown of the state-operator correspondence.

Even simpler, if we are considering states stretching between the same brane, then the spectral sequence is trivial iff the tangent bundle of $X$ restricted to $M$ splits holomorphically as $TM \oplus NM$.

We always have a map from the $E^2$ page of vertex operators to the $E^\infty$ page of states. This probably coincides with the ordinary map in the state-operator correspondence where we put an operator at the tip of a (in this case half-)cigar and look at the state at the end. For some reason the usual argument about this being an isomorphism breaks down. Can we tell which situations this map fails to be injective or surjective?

asked Jun 19, 2014 in Theoretical Physics by Ryan Thorngren (1,925 points) [ revision history ]
edited Jun 19, 2014 by Ryan Thorngren

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