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The Geometric Dual of a–Maximisation for Toric Sasaki–Einstein Manifolds

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Abstract

We show that the Reeb vector, and hence in particular the volume, of a Sasaki–Einstein metric on the base of a toric Calabi–Yau cone of complex dimension n may be computed by minimising a function Z on \(\mathbb {R}^{n}\) which depends only on the toric data that defines the singularity. In this way one can extract certain geometric information for a toric Sasaki–Einstein manifold without finding the metric explicitly. For complex dimension n  =  3 the Reeb vector and the volume correspond to the R–symmetry and the a central charge of the AdS/CFT dual superconformal field theory, respectively. We therefore interpret this extremal problem as the geometric dual of a–maximisation. We illustrate our results with some examples, including the Y p,q singularities and the complex cone over the second del Pezzo surface.

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Authors and Affiliations

  1. Department of Physics, CERN Theory Division, 1211, Geneva 23, Switzerland

    Dario Martelli

  2. Department of Mathematics, Harvard University, One Oxford Street, Cambridge, MA, 02138, U.S.A

    James Sparks & Shing-Tung Yau

  3. Jefferson Physical Laboratory, Harvard University, Cambridge, MA, 02138, U.S.A

    James Sparks

Authors
  1. Dario Martelli
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  2. James Sparks
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  3. Shing-Tung Yau
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Correspondence to James Sparks.

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Communicated by G.W. Gibbons

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Martelli, D., Sparks, J. & Yau, ST. The Geometric Dual of a–Maximisation for Toric Sasaki–Einstein Manifolds. Commun. Math. Phys. 268, 39–65 (2006). https://doi.org/10.1007/s00220-006-0087-0

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  • DOI: https://doi.org/10.1007/s00220-006-0087-0

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