Sphere Partition Functions and the Zamolodchikov Metric

Abstract

We study the finite part of the sphere partition function of d-dimensional Conformal Field Theories (CFTs) as a function of exactly marginal couplings. In odd dimensions, this quantity is physical and independent of the exactly marginal couplings. In even dimensions, this object is generally regularization scheme dependent and thus unphysical. However, in the presence of additional symmetries, the partition function of even-dimensional CFTs can become physical. For two-dimensional N=(2,2) supersymmetric CFTs, the continuum partition function exists and computes the Kahler potential on the chiral and twisted chiral superconformal manifolds. We provide a new elementary proof of this result using Ward identities on the sphere. The Kahler transformation ambiguity is identified with a local term in the corresponding N=(2,2) supergravity theory. We derive an analogous, new, result in the case of four-dimensional N=2 supersymmetric CFTs: the S^4 partition function computes the Kahler potential on the superconformal manifold. Finally, we show that N=1 supersymmetry in four dimensions and N=(1,1) supersymmetry in two dimensions are not sufficient to make the corresponding sphere partition functions well-defined functions of the exactly marginal parameters.

arXiv comment32 pages; added references and minor corrections

References in corpus (27)

• Towards a derivation of holographic entanglement entropy
• Localization of gauge theory on a four-sphere and supersymmetric Wilson loops
• On Renormalization Group Flows in Four Dimensions
• Exactly Marginal Operators and Duality in Four Dimensional N=1 Supersymmetric Gauge Theory
• Towards the F-Theorem: N=2 Field Theories on the Three-Sphere
• Rigid Supersymmetric Theories in Curved Superspace
• Geometric Classification of Conformal Anomalies in Arbitrary Dimensions
• On the RG running of the entanglement entropy of a circle
• Partition functions of N=(2,2) gauge theories on S^2 and vortices
• Exact Results in D=2 Supersymmetric Gauge Theories
• The Constraints of Conformal Symmetry on RG Flows
• Contact Terms, Unitarity, and F-Maximization in Three-Dimensional Superconformal Theories
• Comments on Chern-Simons Contact Terms in Three Dimensions
• Exactly Marginal Deformations and Global Symmetries
• Exact Kahler Potential from Gauge Theory and Mirror Symmetry
• New supersymmetric higher-derivative couplings: Full N=2 superspace does not count!
• New higher-derivative invariants in N=2 supergravity and the Gauss-Bonnet term
• On the instantons and the hypermultiplet mass of N=2* super Yang-Mills on S^4
• Topological Anti-Topological Fusion in Four-Dimensional Superconformal Field Theories
• Comments on N=(2,2) Supersymmetry on Two-Manifolds
• Vortex partition functions, wall crossing and equivariant Gromov-Witten invariants
• Landscaping with fluxes and the E8 Yukawa Point in F-theory
• Gauge Theory Dynamics and Kahler Potential for Calabi-Yau Complex Moduli
• On metric geometry of conformal moduli spaces of four-dimensional superconformal theories
• Two-Sphere Partition Functions and Gromov-Witten Invariants
• The Seiberg-Witten Kahler Potential as a Two-Sphere Partition Function
• 2d, N=2 and N=4 supergravity and the Liouville theory in superspace

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