On the central charge of spacetime current algebras and correlators in string theory on AdS 3

Jihun Kim,Massimo Porrati

doi:10.1007/jhep05(2015)076

Jihun Kim, Massimo Porrati

Open Access

https://doi.org/10.1007/jhep05(2015)076

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Abstract

Spacetime Virasoro and affine Lie algebras for strings propagating in AdS3 are known to all orders in $\alpha'$. The central extension of such algebras is a string vertex, whose expectation value can depend on the number of long strings present in the background but should be otherwise state-independent. In hep-th/0106004, on the other hand, a state-dependent expectation value was found. Another puzzling feature of the theory is lack of cluster decomposition property in certain connected correlators. This note shows that both problems can be removed by defining the free energy of the spacetime boundary conformal field theory as the Legendre transform of the formula proposed in the literature. This corresponds to pass from a canonical ensemble, where the number of fundamental strings that create the background can fluctuate, to a microcanonical one, where it is fixed.

Highlights

Spacetime Virasoro and affine Lie algebras for strings propagating in AdS3 are known to all orders in α′
The central extension of such algebras is a string vertex, whose expectation value can depend on the number of long strings present in the background but should be otherwise state-independent
This note shows that both problems can be removed by defining the free energy of the spacetime boundary conformal field theory as the Legendre transform of the formula proposed in the literature

Summary

The spacetime central charge is proportional to the vertex

We recognize in W the generator of connected correlators for the spacetime CFT, that is the free energy. Δ δA(x, x) δǫJ I (x, x) δJ δ (x, x) This equation is wrong because the spacetime current algebra contains a central term, which is reflected in the vertex identity [6]. An anomalous term in the Ward identity would be Gǫ′W = ∆(ǫ), with ∆(ǫ) a local functional of the background gauge field Axonly, which obeys the standard Wess-Zumino consistency conditions [12]. Such term is canceled by adding to W a term linear in λ. The free energy W obeys another identity: thanks to eqs. (16), (17), we have δW δλ(x, x)

The solution to this linear equation is

Now the Ward identity on Γ has the correct form