Superconformal Manifolds Research Articles

On the Kähler-Hodge structure of superconformal manifolds

We show that conformal manifolds in d ≥ 3 conformal field theories with at least 4 supercharges are Kähler-Hodge, thus extending to 3d mathcal{N} = 2 and 4d mathcal{N} = 1 similar results previously derived for 4d mathcal{N} = 2 and mathcal{N} = 4 and various types of 2d SCFTs. Conformal manifolds in SCFTs are equipped with a holomorphic line bundle ℒ, which encodes the operator mixing of supercharges under marginal deformations. Using conformal perturbation theory and superconformal Ward identities, we compute the curvature of ℒ at a generic point on the conformal manifold. We show that the Kähler form of the Zamolodchikov metric is proportional to the first Chern class of ℒ, with a constant of proportionality given by the two-point function coefficient of the stress tensor, CT. In cases where certain additional conditions about the nature of singular points on the conformal manifold hold, this implies a quantization condition for the total volume of the conformal manifold.

Weyl anomalies on conformal manifolds and moduli spaces

A Weyl (conformal) anomaly signals a subtle quantum breaking of classical conformal invariance in conformal field theory. Over the years, Weyl anomalies have been used to characterize nonperturbative properties of conformal field theory. Anomalies associated with the energy–momentum tensor, like the coefficients [Formula: see text] and [Formula: see text] in four space–time dimensions, are generic and have been studied extensively. More generally, in even dimensions, there are also conformal anomalies associated with any primary operator that has integer scaling dimension. Some of the most interesting features of Weyl anomalies have to do with their behavior under continuous deformations or in vacua with spontaneously broken conformal symmetry. In this review, we summarize the defining properties of conformal anomalies, their classification into A- and B-type, and their implications on the structure of correlation functions. We point out that type-B anomalies can exhibit complicated dynamics and review recent progress in the study of this dynamics with special focus on four-dimensional [Formula: see text] superconformal field theories. We emphasize two applications of type-B anomalies in this context: potential constraints on the holonomy of superconformal manifolds and the deconstruction of anomalies in higher dimensions from anomalies in broken phases of lower-dimensional conformal field theories.

Geometry of Higgs-branch superconformal primary bundles

It is known that the two- and three-point functions of Higgs-branch superconformal primaries in 4d N=2 superconformal field theories obey non-renormalization theorems on N=2 superconformal manifolds. In this paper we prove a stronger statement, that the bundles of Higgs-branch superconformal primaries over N=2 superconformal manifolds are endowed with a flat connection, or equivalently that Higgs-branch superconformal primaries have vanishing Berry phases under N=2 exactly marginal deformations. This statement fits well with the proposed correspondence between the rigid structures of 2d chiral algebras and the sector of Schur operators in 4d N=2 theories. We also discuss the general interplay between non-renormalization theorems and the curvature of bundles of protected operators and provide a new simpler proof of the vanishing curvature of 1/2-BPS operators in 4d N=4 SYM theory that does not require the use of the 4d tt* equations.

Boundary Weyl anomaly of mathcal{N} = (2, 2) superconformal models

We calculate the trace and axial anomalies of mathcal{N} = (2, 2) superconformal theories with exactly marginal deformations, on a surface with boundary. Extending recent work by Gomis et al, we derive the boundary contribution that captures the anomalous scale dependence of the one-point functions of exactly marginal operators. Integration of the bulk super-Weyl anomaly shows that the sphere partition function computes the Kähler potential Kleft(lambda, overline{lambda}right) on the superconformal manifold. Likewise, our results confirm the conjecture that the partition function on the supersymmetric hemisphere computes the holomorphic central charge, cΩ(λ), associated with the boundary condition Ω. The boundary entropy, given by a ratio of hemispheres and sphere, is therefore fully determined by anomalies.

On exact correlation functions in SU(N) N = 2 $$ \mathcal{N}=2 $$ superconformal QCD

We consider the exact coupling constant dependence of extremal correlation functions of ${\cal N} = 2$ chiral primary operators in 4d ${\cal N} = 2$ superconformal gauge theories with gauge group SU(N) and N_f=2N massless fundamental hypermultiplets. The 2- and 3-point functions, viewed as functions of the exactly marginal coupling constant and theta angle, obey the tt* equations. In the case at hand, the tt* equations form a set of complicated non-linear coupled matrix equations. We point out that there is an ad hoc self-consistent ansatz that reduces this set of partial differential equations to a sequence of decoupled semi-infinite Toda chains, similar to the one encountered previously in the special case of SU(2) gauge group. This ansatz requires a surprising new non-renormalization theorem in ${\cal N} = 2$ superconformal field theories. We derive a general 3-loop perturbative formula for 2- and 3-point functions in the ${\cal N} = 2$ chiral ring of the SU(N) theory, and in all explicitly computed examples we find agreement with the tt* equations, as well as the above-mentioned ansatz. This is suggestive evidence for an interesting non-perturbative conjecture about the structure of the ${\cal N} = 2$ chiral ring in this class of theories. We discuss several implications of this conjecture. For example, it implies that the holonomy of the vector bundles of chiral primaries over the superconformal manifold is reducible. It also implies that a specific subset of extremal correlation functions can be computed in the SU(N) theory using information solely from the S^4 partition function of the theory obtained by supersymmetric localization.

Sphere partition functions and the Zamolodchikov metric

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.

THE DEFINITION OF NEVEU–SCHWARZ SUPERCONFORMAL FIELDS AND UNCHARGED SUPERCONFORMAL TRANSFORMATIONS

The construction of Neveu–Schwarz superconformal field theories for any N is given via a superfield formalism. We also review some results and definitions of superconformal manifolds and generalise contour integration and Taylor expansion to superconformal spaces. For arbitrary N we define (uncharged) primary fields and give their infinitesimal change under superconformal transformations. This leads us to the operator product expansion of the stress-energy tensor with itself and with primary fields. In this way we derive the well-known commutation relations of the Neveu–Schwarz superconformal algebras K N . In this context we observe that the central extension term disappears for N≥4 for the Neveu–Schwarz theories. Finally, we give the global transformation rules of primary fields under theaction of the algebra generators.

The space of super light rays for complex conformal spacetimes

After defining a superconformal structure on a 4|4 N supermanifold, its space of super light rays is constructed and shown to have a natural supercontact structure. We next construct, for a 5|2 N dimensional supercontact manifold, its space of normal quadrics. This is shown to be a 4|4 N superconformal manifold. Every four dimensional conformal manifold is then proved to have an extension to a superconformal manifold of dimension 4|4 N for N < 4. After the equivalence of the N = 3 Supersymmetric Yang-Mills equations and integrability along super light rays is shown, a one to one correspondence between solutions to the N = 3 Supersymmetric Yang-Mills equations and certain vector bundles over the N = 3 space of super light rays is established.

SUPERCONFORMAL GEOMETRY FROM THE GRASSMANN AND HARMONIC ANALYTICITIES I: THE N=2 SUPERCONFORMAL CASE

The geometry of the N=2 superconformal manifolds is studied. It is shown that the underlying intrinsic operators of these manifolds form an N=2ℤ2-graded algebra. We also show that the N=2 superconformal structure is equivalent to the Grassmann analyticity. Superconformal chiral tensors and integral measures are analyzed and the invariant superpotential, built of the primary chiral superfields, is given. A new realization of the N=2 supercurrent is obtained.

Superconformal geometry and string theory

We give a formula for the determinant of the super Laplace operator in a holomorphic hermitian line bundle over a superconformal manifold. This is then used to obtain an expression for the fermion string measure.

Geometry of superconformal manifolds

The main facts about complex curves are generalized to superconformal manifolds. The results thus obtained are relevant to the fermion string theory and, in particular, they are useful for computation of determinants of super laplacians which enter the string partition function.

Geometry of two-dimensional superconformal field theories

The concept of a superconformal manifold is analyzed. The space of equivalence classes of superconformal manifolds is investigated. A connection with the theory of a fermionic string in the critical dimension is pointed out.