arXiv subjects

Mathew Bullimore

Publications and source records attributed to Mathew Bullimore.

At least 37 records · Page 2Linked to original sources

arXiv2016

Boundaries, Mirror Symmetry, and Symplectic Duality in 3d $\mathcal{N}=4$ Gauge Theory

We introduce several families of $\mathcal{N}=(2,2)$ UV boundary conditions in 3d $\mathcal N=4$ 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 $\mathcal{N}=4$ 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.

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arXiv2016

The Superconformal Index of the (2,0) Theory with Defects

We compute the supersymmetric partition function of the six-dimensional $(2,0)$ theory of type $A_{N-1}$ on $S^1 \times S^5$ 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 $A_{N-1}$.

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arXiv2016

Supersymmetric Casimir Energy and the Anomaly Polynomial

We conjecture that for superconformal field theories in even dimensions, the supersymmetric Casimir energy on a space with topology $S^1\times S^{D-1}$ 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.

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arXiv2015

The Coulomb Branch of 3d $\mathcal{N}=4$ Theories

We propose a construction of the quantum-corrected Coulomb branch of a general 3d gauge theory with $\mathcal{N}=4$ 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 the Bogomolnyi and/or Nahm equations.

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arXiv2015

Defects and Quantum Seiberg-Witten Geometry

We study the Nekrasov partition function of the five dimensional U(N) gauge theory with maximal supersymmetry on R^4 x 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 x 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.

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arXiv2014

The superconformal index and an elliptic algebra of surface defects

In this paper we continue the study of the superconformal index of four-dimensional $\mathcal{N}=2$ theories of class $\mathcal{S}$ in the presence of surface defects. Our main result is the construction of an algebra of difference operators, whose elements are labeled by irreducible representations of $A_{N-1}$. For the fully antisymmetric tensor representations these difference operators are the Hamiltonians of the elliptic Ruijsenaars-Schneider system. The structure constants of the algebra are elliptic generalizations of the Littlewood-Richardson coefficients. In the Macdonald limit, we identify the difference operators with local operators in the two-dimensional TQFT interpretation of the superconformal index. We also study the dimensional reduction to difference operators acting on the three-sphere partition function, where they characterize supersymmetric defects supported on a circle, and show that they are transformed to supersymmetric Wilson loops under mirror symmetry. Finally, we compare to the difference operators that create 't Hooft loops in the four-dimensional $\mathcal{N}=2^*$ theory on a four-sphere by embedding the three-dimensional theory as an S-duality domain wall.

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arXiv2013

Defect Networks and Supersymmetric Loop Operators

We consider topological defect networks with junctions in $A_{N-1}$ Toda CFT and the connection to supersymmetric loop operators in $\mathcal{N} = 2$ theories of class S on a four-sphere. Correlation functions in the presence of topological defect networks are computed by exploiting the monodromy of conformal blocks, generalising the notion of a Verlinde operator. Concentrating on a class of topological defects in $A_2$ Toda theory, we find that the Verlinde operators generate an algebra whose structure is determined by a set of generalised skein relations. These relations encode the representation theory of a quantum group. In the second half of the paper, we explore the dictionary between topological defect networks and supersymmetric loop operators in the $\mathcal{N}=2^*$ star theory by comparing to exact localisation computations. In this context, the the generalised skein relations are related to the operator product expansion of loop operators.

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arXiv2013

Surface defects, the superconformal index and q-deformed Yang-Mills

Recently a prescription to compute the four-dimensional N = 2 superconformal index in the presence of certain BPS surface defects has been given. These surface defects are labelled by symmetric representations of SU(N). In the present paper we give a prescription to compute the superconformal index in the presence of surface defects labelled by arbitrary representations of SU(N). Furthermore, we extend the dictionary between the N = 2 superconformal Schur-index and correlators of q-deformed Yang-Mills to incorporate such surface defects.

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arXiv2013

On S-duality of the Superconformal Index on Lens Spaces and 2d TQFT

We consider the 4d superconformal index for ${\cal N}=2$ gauge theories on $S^1 \times L(r,1)$, where $L(r,1)$ is a Lens space. We focus on a one-parameter slice of the three-dimensional fugacity space and in that sector we show S-duality. We do so by rewriting the index in a way that resembles a correlation function of a 2d TFT, which however, we do not identify.

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arXiv2012

New Formulae for Gravity Amplitudes: Parity Invariance and Soft Limits

Cachazo and Skinner have recently conjectured a new formula for the complete tree-level S-matrix of N = 8 supergravity. The formula is an integral over the moduli space of rational curves in supertwistor space, and remarkably, manifests the full permutation symmetry of graviton scattering amplitudes. We check that the formula is parity invariance and reproduces the correct universal soft behaviour of graviton amplitudes.

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arXiv2011

Descent equations for superamplitudes

At loop level in planar N=4 super Yang-Mills, the dual superconformal symmetry of tree amplitudes is lost. This is true even if one uses a supersymmetry preserving regulator, and even for finite quantities that remain dual conformally invariant. We examine this breaking from the dual point of view of the super Wilson Loop, tracing it to the difference between supersymmetries of the self-dual and of the full theories. We show that the anomaly is controlled by a descent equation that determines the derivative of an L-loop amplitude in terms of a single non-trivial integral of an (L-1)-loop amplitude. We propose that this equation can be used recursively to construct multi-loop amplitudes in a way that makes their transcendentality manifest.

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arXiv2011

A Proof of the Supersymmetric Correlation Function / Wilson Loop Correspondence

We prove that in the limit when its insertion points become pairwise null-separated, the ratio of certain n-point correlation functions in N=4 SYM is equal to a supersymmetric Wilson loop on twistor space, acting in the adjoint representation. In the planar limit, each of these objects reduces to the square of the complete n-particle planar superamplitude. Our proof is at the level of the integrand.

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arXiv2011

Scattering Amplitudes and Wilson Loops in Twistor Space

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, Svrcek 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'.

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arXiv2011

Holomorphic Linking, Loop Equations and Scattering Amplitudes in Twistor Space

We study a complex analogue of a Wilson Loop, defined over a complex curve, in non-Abelian holomorphic Chern-Simons theory. We obtain a version of the Makeenko-Migdal loop equation describing how the expectation value of these Wilson Loops varies as one moves around in a holomorphic family of curves. We use this to prove (at the level of the integrand) the duality between the twistor Wilson Loop and the all-loop planar S-matrix of N=4 super Yang-Mills by showing that, for a particular family of curves corresponding to piecewise null polygons in space-time, the loop equation reduce to the all-loop extension of the BCFW recursion relations. The scattering amplitude may be interpreted in terms of holomorphic linking of the curve in twistor space, while the BCFW relations themselves are revealed as a holomorphic analogue of skein relations.

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arXiv2010

MHV Diagrams from an All-Line Recursion Relation

We consider the recursion relation for loop integrands in planar N = 4 SYM generated by an all-line shift of momentum twistors. We examine the behaviour of the rational loop integrands when the shift parameter becomes large, and find that a valid recursion relation may be obtained in all cases. The recursion relation is then formulated both in region momentum space and in momentum twistor space, and solved in detail for some one and two-loop examples. Finally, we show that the general iterative solution of the recursion relation generates the MHV vertex expansion for all loop integrands, providing a proof of the MHV diagram formalism for all loop amplitudes in planar N = 4 SYM.

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arXiv2010

Inverse Soft Factors and Grassmannian Residues

The action of inverse soft factors on scattering amplitudes in N=4 SYM is shown to take a remarkably simple form in momentum twistor space. This is used to identify individual residues of the grassmannian with primitive leading singularities at NMHV and N^2MHV and to derive explicit expressions in terms of momentum twistors.

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arXiv2010

Kinetic mixing of U(1)s for local string models

We study kinetic mixing between massless U(1)s in toroidal orbifolds with D3-branes at orbifold singularities. We focus in particular on C^3/Z_4 singularities but also study C^3/Z_6 and C^3/Z'_6 singularities. We find kinetic mixing can be present and describe the conditions for it to occur. Kinetic mixing comes from winding modes in the N=2 sector of the orbifold. If kinetic mixing is present its size depends only on the complex structure modulus of the torus and is independent of the Kähler moduli. We also study gauge threshold corrections for local Z_M x Z_N orbifold models finding that, consistent with previous studies, gauge couplings run from the bulk winding scale rather than the string scale.

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arXiv2010

MHV Diagrams in Momentum Twistor Space

We show that there are remarkable simplifications when the MHV diagram formalism for N=4 super Yang-Mills is reformulated in momentum twistor space. The vertices are replaced by unity while each propagator becomes a dual superconformal `R-invariant' whose arguments may be read off from the diagram. The momentum twistor MHV rules generate a formula for the full, all-loop planar integrand of the super Yang-Mills S-matrix that is manifestly dual superconformally invariant up to the choice of a reference twistor. We give a general proof of this reformulation and illustrate its use by computing the momentum twistor NMHV and NNMHV tree amplitudes and the integrands of the MHV and NMHV 1-loop and the MHV 2-loop planar amplitudes.

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