Virasoro Conformal Blocks and Thermality from Classical Background Fields (1501.05315v3)

Abstract: We show that in 2d CFTs at large central charge, the coupling of the stress tensor to heavy operators can be re-absorbed by placing the CFT in a non-trivial background metric. This leads to a more precise computation of the Virasoro conformal blocks between heavy and light operators, which are shown to be equivalent to global conformal blocks evaluated in the new background. We also generalize to the case where the operators carry U(1) charges. The refined Virasoro blocks can be used as the seed for a new Virasoro block recursion relation expanded in the heavy-light limit. We comment on the implications of our results for the universality of black hole thermality in $AdS_3$, or equivalently, the eigenstate thermalization hypothesis for $CFT_2$ at large central charge.

• The paper demonstrates that transforming CFT2 to a non-trivial background field simplifies the computation of heavy-light Virasoro conformal blocks.
• The authors show an equivalence between heavy-light Virasoro blocks and global blocks, bridging classical conformal geometry and quantum gravitational insights.
• The study introduces new recursion relations and extends analysis to include U(1) charges, enhancing methods for exploring black hole thermality in holographic theories.

Overview of Virasoro Conformal Blocks and Classical Background Fields in CFT$_2$

The paper by Fitzpatrick, Kaplan, and Walters explores the dynamics of heavy-light correlation functions in two-dimensional conformal field theories (CFT$_2$) with a large central charge. The intricate relationship between Virasoro conformal blocks and thermal effects in such CFTs is thoroughly examined through a novel approach involving classical background fields. The authors present a detailed investigation of how the coupling of stress tensors to heavy operators can be characterized within a transformed conformal geometry, elucidating implications for black hole thermality in AdS$_3$/CFT$_2$ correspondence.

Main Results

The authors focus on the computation of Virasoro conformal blocks in CFT$_2$ with particular attention to the case where heavy operators distort the geometry by inducing a classical background field. They demonstrate that for CFT$_2$ at large central charge, the intrinsic properties of Virasoro conformal blocks in the heavy-light limit can be effectively resolved using a non-trivial background metric. The primary findings can be summarized as follows:

• Transformation to a Non-trivial Background: The authors show that heavy operators can be handled elegantly by considering a conformal transformation that reabsorbs the contribution of the stress tensor into a modified metric background. This insight simplifies the computation of correlation functions by reducing the need for explicit evaluation of stress tensor-related terms.
• Virasoro Blocks & Global Blocks Equivalence: In the transformed coordinates, the refined heavy-light Virasoro conformal blocks resemble their global counterparts, calculated in the altered background, bridging a gap between global conformal symmetry effects and quantum gravitational dynamics.
• New Recursion Relations: The paper introduces recursion relations derived from the refined block computation, enhancing the efficiency of evaluating conformal blocks in the heavy-light approximation. This marks a significant advancement over prior methods limited to partial block contributions.
• Inclusion of $U(1)$ Charges: The research generalizes existing models by incorporating operators with $U(1)$ symmetry, leading to a more comprehensive understanding of how internal symmetries interact with gravitational fields in the CFT description.

Implications

The theoretical implications of this paper extend into several domains of interest:

• Universality in AdS/CFT: The classical background fields computed here could generally represent universal features in three-dimensional gravity theories, thus crucially supporting the eigenstate thermalization hypothesis. The methodology underpins a rigorous, quantifiable approach to visualizing AdS$_3$ black holes' thermal properties through heavy-light OPEs.
• Entropy and Thermodynamics: By validating the equivalence between periodicity in Euclidean time for CFT correlators and classical thermodynamic identities, the work offers new perspectives on computing entanglement entropy and related thermodynamic quantities in holographic settings.
• Extensions to Higher Dimensions: Although detailed in two dimensions, the methods proposed might inspire similar analyses in higher dimensions, potentially expanding our understanding of AdS/CFT beyond the two-dimensional boundary theories considered here.

Future Directions

Further research inspired by this work could explore connections between these classical manipulations in CFT$_2$ and analogous phenomena in higher-dimensional CFTs or in theories with varying symmetry and charge structures. Additionally, understanding the limit of their results in the presence of finite c corrections, or within contexts diverging from holography, remains an engaging challenge. Investigating methods to integrate more intricate background geometries or rich operator dynamics could also enhance the comprehensiveness of spacetime-operator duality examinations.

In conclusion, the paper by Fitzpatrick et al. enriches the theoretical landscape by offering novel computational techniques for challenging aspects of CFT correlation functions influenced by massive operator insertions, with broad implications for quantum gravity and holographic theories.