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Geometries from field theories

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arxiv 1505.00131 v4 pith:FPD7ZTNT submitted 2015-05-01 hep-th hep-lat

classification hep-thhep-lat
keywords dimensionalfieldmetricdefineflowinducedlargelimit
verification ladder T0 review T1 audit T2 compute T3 formal
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abstract

We propose a method to define a $d+1$ dimensional geometry from a $d$ dimensional quantum field theory in the $1/N$ expansion. We first construct a $d+1$ dimensional field theory from the $d$ dimensional one via the gradient flow equation, whose flow time $t$ represents the energy scale of the system such that $t\rightarrow 0$ corresponds to the ultra-violet (UV) while $t\rightarrow\infty$ to the infra-red (IR). We then define the induced metric from $d+1$ dimensional field operators. We show that the metric defined in this way becomes classical in the large $N$ limit, in a sense that quantum fluctuations of the metric are suppressed as $1/N$ due to the large $N$ factorization property. As a concrete example, we apply our method to the O(N) non-linear $\sigma$ model in two dimensions. We calculate the three dimensional induced metric, which is shown to describe an AdS space in the massless limit. We finally discuss several open issues in future studies.

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  1. Derivation of the GKP-Witten relation by symmetry without Lagrangian

    hep-th 2024-11 conditional novelty 6.0 of 10

    For arbitrary spin and to all orders in an external source, the boundary limit of conformally smeared bulk operators equals the CFT generating functional, so the GKP-Witten relation follows from symmetry and OPE alone.

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