Sasaki-Einstein Manifolds

This article is an overview of some of the remarkable progress that has been made in Sasaki-Einstein geometry over the last decade, which includes a number of new methods of constructing Sasaki-Einstein manifolds and obstructions.Comment: 58 pages. Invited contribution to Surveys in Differential Geometry. v2: references and discussion adde

Global Worldsheet Anomalies from M-Theory

We present an M-theory proof of the anomaly of Freed and Witten which in general shifts the quantisation law for the U(1) gauge field on a D6-brane. The derivation requires an understanding of how fields on the D6-brane lift to M-theory, together with a localisation formula which we prove using a U(1)-index theorem. We also show how the anomaly is related to the K-theory classification of Ramond-Ramond fields. In addition we discuss the M-theory origin of the D6-brane effective action, and illustrate the general arguments with a concrete example.Comment: 25 pages; v2: minor alterations, references added; v3: various arguments clarified, mathematical background added. Published versio

Symmetry-breaking vacua and baryon condensates in AdS/CFT

We study the gravity duals of symmetry-breaking deformations of superconformal field theories, AdS/CFT dual to Type IIB string theory on AdS_5 x Y where Y is a Sasaki-Einstein manifold. In these vacua both conformal invariance and baryonic symmetries are spontaneously broken. We present a detailed discussion of the supergravity moduli space, which involves flat form fields on asymptotically conical Calabi-Yau manifolds, and match this to the gauge theory vacuum moduli space. We discuss certain linearised fluctuations of the moduli, identifying the Goldstone bosons associated with spontaneous breaking of non-anomalous baryonic symmetries. The remaining moduli fields are related to spontaneous breaking of anomalous baryonic symmetries. We also elaborate on the proposal that computing condensates of baryon operators is equivalent to computing the partition function of a non-compact Euclidean D3-brane in the background supergravity solution, with fixed boundary conditions at infinity.Comment: 121 pages; v2: references adde

D2-brane Chern-Simons theories: F-maximization = a-maximization

We study a system of N D2-branes probing a generic Calabi-Yau three-fold singularity in the presence of a non-zero quantized Romans mass n. We argue that the low-energy effective N = 2 Chern-Simons quiver gauge theory flows to a superconformal fixed point in the IR, and construct the dual AdS_4 solution in massive IIA supergravity. We compute the free energy F of the gauge theory on S^3 using localization. In the large N limit we find F = c(nN)^{1/3}a^{2/3}, where c is a universal constant and a is the a-function of the "parent" four-dimensional N = 1 theory on N D3-branes probing the same Calabi-Yau singularity. It follows that maximizing F over the space of admissible R-symmetries is equivalent to maximizing a for this class of theories. Moreover, we show that the gauge theory result precisely matches the holographic free energy of the supergravity solution, and provide a similar matching of the VEV of a BPS Wilson loop operator.Comment: 19 pages; v2: minor correction

The character of the supersymmetric Casimir energy

We study the supersymmetric Casimir energy $E_\mathrm{susy}$ of $\mathcal{N}=1$ field theories with an R-symmetry, defined on rigid supersymmetric backgrounds $S^1\times M_3$, using a Hamiltonian formalism. These backgrounds admit an ambi-Hermitian geometry, and we show that the net contributions to $E_\mathrm{susy}$ arise from certain twisted holomorphic modes on $\mathbb{R}\times M_3$, with respect to both complex structures. The supersymmetric Casimir energy may then be identified as a limit of an index-character that counts these modes. In particular this explains a recent observation relating $E_\mathrm{susy}$ on $S^1\times S^3$ to the anomaly polynomial. As further applications we compute $E_\mathrm{susy}$ for certain secondary Hopf surfaces, and discuss how the index-character may also be used to compute generalized supersymmetric indices.Comment: 47 pages; v2: footnote 6 added, formula (5.29) changed, Section 6 moved to Appendix

Baryonic branches and resolutions of Ricci-flat Kahler cones

We consider deformations of N=1 superconformal field theories that are AdS/CFT dual to Type IIB string theory on Sasaki-Einstein manifolds, characterised by non-zero vacuum expectation values for certain baryonic operators. Such baryonic branches are constructed from (partially) resolved, asymptotically conical Ricci-flat Kahler manifolds, together with a choice of point where the stack of D3-branes is placed. The complete solution then describes a renormalisation group flow between two AdS fixed points. We discuss the use of probe Euclidean D3-branes in these backgrounds as a means to compute expectation values of baryonic operators. The Y^{p,q} theories are used as illustrative examples throughout the paper. In particular, we present supergravity solutions describing flows from the Y^{p,q} theories to various different orbifold field theories in the infra-red, and successfully match this to an explicit field theory analysis.Comment: 51 pages, v2: reference added and minor changes; v3: minor changes, published versio

The large N limit of quiver matrix models and Sasaki-Einstein manifolds

We study the matrix models that result from localization of the partition functions of N=2 Chern-Simons-matter theories on the three-sphere. A large class of such theories are conjectured to be holographically dual to M-theory on Sasaki-Einstein seven-manifolds. We study the M-theory limit (large N and fixed Chern-Simons levels) of these matrix models for various examples, and show that in this limit the free energy reproduces the expected AdS/CFT result of N^{3/2}/Vol(Y)^{1/2}, where Vol(Y) is the volume of the corresponding Sasaki-Einstein metric. More generally we conjecture a relation between the large N limit of the partition function, interpreted as a function of trial R-charges, and the volumes of Sasakian metrics on links of Calabi-Yau four-fold singularities. We verify this conjecture for a family of U(N)^2 Chern-Simons quivers based on M2 branes at hypersurface singularities, and for a U(N)^3 theory based on M2 branes at a toric singularity.Comment: 38 pages, 4 figures; v2: minor changes, typos and factor of 2 in eq. (5.19) fixed, references and 2 figures added; v3: new section 4.5 added; v4: more typos fixed, range of validity of (4.19) clarifie

Toric Geometry, Sasaki-Einstein Manifolds and a New Infinite Class of AdS/CFT Duals

Recently an infinite family of explicit Sasaki-Einstein metrics Y^{p,q} on S^2 x S^3 has been discovered, where p and q are two coprime positive integers, with q<p. These give rise to a corresponding family of Calabi-Yau cones, which moreover are toric. Aided by several recent results in toric geometry, we show that these are Kahler quotients C^4//U(1), namely the vacua of gauged linear sigma models with charges (p,p,-p+q,-p-q), thereby generalising the conifold, which is p=1,q=0. We present the corresponding toric diagrams and show that these may be embedded in the toric diagram for the orbifold C^3/Z_{p+1}xZ_{p+1} for all q<p with fixed p. We hence find that the Y^{p,q} manifolds are AdS/CFT dual to an infinite class of N=1 superconformal field theories arising as IR fixed points of toric quiver gauge theories with gauge group SU(N)^{2p}. As a non-trivial example, we show that Y^{2,1} is an explicit irregular Sasaki-Einstein metric on the horizon of the complex cone over the first del Pezzo surface. The dual quiver gauge theory has already been constructed for this case and hence we can predict the exact central charge of this theory at its IR fixed point using the AdS/CFT correspondence. The value we obtain is a quadratic irrational number and, remarkably, agrees with a recent purely field theoretic calculation using a-maximisation.Comment: 54 pages, 5 figures; minor changes; further minor changes, ref [8] added - published version; eqns 1.3, 1.4 remove

The supersymmetric NUTs and bolts of holography

We show that a given conformal boundary can have a rich and intricate space of supersymmetric supergravity solutions filling it, focusing on the case where this conformal boundary is a biaxially squashed Lens space. Generically we find that the biaxially squashed Lens space S^3/Z_p admits Taub-NUT-AdS fillings, with topology R^4/Z_p, as well as smooth Taub-Bolt-AdS fillings with non-trivial topology. We show that the Taub-NUT-AdS solutions always lift to solutions of M-theory, and correspondingly that the gravitational free energy then agrees with the large N limit of the dual field theory free energy, obtained from the localized partition function of a class of N=2 Chern-Simons-matter theories. However, the solutions of Taub-Bolt-AdS type only lift to M-theory for appropriate classes of internal manifold, meaning that these solutions exist only for corresponding classes of three-dimensional N=2 field theories.Comment: 54 pages plus 7 appendices; v2: discussion of global properties of gauge field A in eleven dimensions improved and some parts of the paper rewritten to reflect this, new subsections 6.4.1, 6.4.2 and appendix F added, references added; v3: final version published in Nuclear Physics