Abstract:We propose a construction for the quantum-corrected Coulomb branch of a general 3d gauge theory with N = 4 supersymmetry, in terms of local coordinates associated with an abelianized theory. In a fixed complex structure, the holomorphic functions on the Coulomb branch are given by expectation values of chiral monopole operators. We construct the chiral ring of such operators, using equivariant integration over BPS moduli spaces. We also quantize the chiral ring, which corresponds to placing the 3d theory in a 2d Omega background. Then, by unifying all complex structures in a twistor space, we encode the full hyperkähler metric on the Coulomb branch. We verify our proposals in a multitude of examples, including SQCD and linear quiver gauge theories, whose Coulomb branches have alternative descriptions as solutions to Bogomolnyi and/or Nahm equations.

Abstract:We introduce several families of N = (2, 2) UV boundary conditions in 3d N = 4 gauge theories and study their IR images in sigma-models to the Higgs and Coulomb branches. In the presence of Omega deformations, a UV boundary condition defines a pair of modules for quantized algebras of chiral Higgs-and Coulomb-branch operators, respectively, whose structure we derive. In the case of abelian theories, we use the formalism of hyperplane arrangements to make our constructions very explicit, and construct a half-BPS interface that implements the action of 3d mirror symmetry on gauge theories and boundary conditions. Finally, by studying two-dimensional compactifications of 3d N = 4 gauge theories and their boundary conditions, we propose a physical origin for symplectic duality -an equivalence of categories of modules associated to families of Higgs and Coulomb branches that has recently appeared in the mathematics literature, and generalizes classic results on Koszul duality in geometric representation theory. We make several predictions about the structure of symplectic duality, and identify Koszul duality as a special case of wall crossing.

We conjecture that for superconformal field theories in even dimensions, the supersymmetric Casimir energy on a space with topology S 1 × S D−1 is equal to an equivariant integral of the anomaly polynomial. The equivariant integration is defined with respect to the Cartan subalgebra of the global symmetry algebra that commutes with a given supercharge. We test our proposal extensively by computing the supersymmetric Casimir energy for large classes of superconformal field theories, with and without known Lagrangian descriptions, in two, four and six dimensions.

Abstract:We study the Nekrasov partition function of the five dimensional U(N ) gauge theory with maximal supersymmetry on R 4 ×S 1 in the presence of codimension two defects. The codimension two defects can be described either as monodromy defects, or by coupling to a certain class of three dimensional quiver gauge theories on R 2 × S 1 . We explain how these computations are connected with both classical and quantum integrable systems. We check, as an expansion in the instanton number, that the aforementioned partition functions are eigenfunctions of an elliptic integrable many-body system, which quantizes the Seiberg-Witten geometry of the five-dimensional gauge theory.

In three-dimensional gauge theories, monopole operators create and destroy vortices. We explore this idea in the context of 3d N = 4 gauge theories in the presence of an Ω-background. In this case, monopole operators generate a non-commutative algebra that quantizes the Coulomb-branch chiral ring. The monopole operators act naturally on a Hilbert space, which is realized concretely as the equivariant cohomology of a moduli space of vortices.The action furnishes the space with the structure of a Verma module for the Coulomb-branch algebra. This leads to a new mathematical definition of the Coulomb-branch algebra itself, related to that of Braverman-Finkelberg-Nakajima. By introducing additional boundary conditions, we find a construction of vortex partition functions of 2d N = (2, 2) theories as overlaps of coherent states (Whittaker vectors) for Coulomb-branch algebras, generalizing work of Braverman-Feigin-Finkelberg-Rybnikov on a finite version of the AGT correspondence. In the case of 3d linear quiver gauge theories, we use brane constructions to exhibit vortex moduli spaces as handsaw quiver varieties, and realize monopole operators as interfaces between handsaw-quiver quantum mechanics, generalizing work of Nakajima. 4.4 Recovering the Coulomb-branch algebra 50 4.5 Verma modules 53 5 Boundary conditions and overlaps 55 5.1 Boundaries and modules 56 5.2 Local operators on a Neumann b.c. 57 5.3 Whittaker states 60 5.4 Overlaps and vortex partition functions 63 5.5 Example: SQCD 70 -i -6 Vortex quantum mechanics 71 6.1 SQCD 71 6.2 Triangular quiver 79 6.3 Equivalence to vortex moduli space 84 7 Case study: abelian quiver 86 7.1 Vortex quantum mechanics 90

This article reviews the recent progress in twistor approaches to Wilson loops, amplitudes and their duality for N = 4 super Yang-Mills. Wilson loops and amplitudes are derived from first principles using the twistor action for maximally supersymmetric Yang-Mills theory. We start by deriving the MHV rules for gauge theory amplitudes from the twistor action in an axial gauge in twistor space, and show that this gives rise to the original momentum space version given by Cachazo, Svrček and Witten. We then go on to obtain from these the construction of the momentum twistor space loop integrand using (planar) MHV rules and show how it arises as the expectation value of a holomorphic Wilson loop in twistor space. We explain the connection between the holomorphic Wilson loop and certain light-cone limits of correlation functions. We give a brief review of other ideas in connection with amplitudes in twistor space: twistor-strings, recursion in twistor space, the Grassmannian residue formula for leading singularities and amplitudes as polytopes. This article is an invited review for a special issue of Journal of Physics A devoted to 'Scattering Amplitudes in Gauge Theories'.

We derive a systematic procedure for obtaining explicit, -loop leading singularities of planar N = 4 super Yang-Mills scattering amplitudes in twistor space directly from their momentum space channel diagram. The expressions are given as integrals over the moduli of connected, nodal curves in twistor space whose degree and genus matches expectations from twistor-string theory. We propose that a twistor-string theory for pure N = 4 super Yang-Mills -if it exists -is determined by the condition that these leading singularity formulae arise as residues when an unphysical contour for the path integral is used, by analogy with the momentum space leading singularity conjecture. We go on to show that the genus g twistor-string moduli space for g-loop N k−2 MHV amplitudes may be mapped into the Grassmannian G(k, n). For a leading singularity, the image of this map is a 2(n − 2)-dimensional subcycle of G(k, n) and, when 'primitive', it is of exactly the type found from the Grassmannian residue formula of Arkani-Hamed, Cachazo, Cheung & Kaplan. Based on this correspondence and the Grassmannian conjecture, we deduce restrictions on the possible leading singularities of multi-loop N p MHV amplitudes. In particular, we argue that no new leading singularities can arise beyond 3p loops. arXiv:0912.0539v3 [hep-th] 30 Dec 20091 With Penrose conventions for twistor space, MHV amplitudes -those whose 'pure glue' sector involves two negative and arbitrarily many positive helicity gluons -are supported on holomorphic lines in dual twistor space PT * . We abuse notation by taking PT * to be variously a copy of CP 3 , CP 3|4 or the neighbourhood of a line in either of these spaces, according to context. We will often describe PT * in terms of its homogeneous coordinates W α = (λ A , µ A ) and χ a in the supersymmetric case. This space was called twistor space by Witten in [1].

Abstract:We compute the supersymmetric partition function of the six-dimensional (2, 0) theory of type A N −1 on S 1 × S 5 in the presence of both codimension two and codimension four defects. We concentrate on a limit of the partition function depending on a single parameter. From the allowed supersymmetric configurations of defects we find a precise match with the characters of irreducible modules of W N algebras and affine Lie algebras of type A N −1 .