arXiv subjects

Nick Dorey

Publications and source records attributed to Nick Dorey.

At least 19 recordsLinked to original sources

arXiv2026

An M2/M5 Duality from the Giant Graviton Expansion

We conjecture a precise relation between the superconformal indices of two theories defined in different spacetime dimensions. The first is the three-dimensional ABJM theory describing the worldvolume of parallel M2-branes in M-theory on $\mathbb{R}^{10,1}$. The second is the $\mathcal{N}=(2,0)$ theory in six dimensions which describes the worldvolume of parallel M5-branes in the same background. As we review, the existence of such a duality is closely related to Imamura's proposal for the giant graviton expansion of the three-dimensional index. We check our conjecture against various results for the two indices available in the literature. Using an existing proposal of Hristov for the ABJM superconformal index, we verify our conjecture to the first three orders in an expansion around the six-dimensional Cardy limit.

hep-th↗

arXiv2025

Giant Gravitons, Fermionic Forms and Vertex Algebras

We investigate the mathematical and physical content of the giant graviton expansion of three-dimensional $\mathcal{N}=4$ superconformal field theories in a simplifying limit. We uncover an interesting relation between the coefficients in this expansion, the Hilbert series of certain quiver varieties and the representation theory of vertex algebras. In particular, for the worldvolume theory of $N$ M2-branes at the tip of a toric hyper-K\"{a}hler four-fold cone: $X_{4}=\mathbb{C}^2 /{\mathbb{Z}_L} \times \mathbb{C}^2/{\mathbb{Z}_K}$, we derive an explicit expression for the coefficients in terms of affine fermionic forms and show that they coincide with characters of a direct sum of parafermionic W-algebras.

hep-th↗

arXiv2024

Giant Gravitons and Volume Minimisation

We establish a precise correspondence between the giant graviton expansion of the superconformal index of field theories in $D\leq 4$, and the master volume formalism of Gauntlett, Martelli and Sparks (GMS) which determines the near horizon geometries of certain BPS black holes and black strings in supergravity. We focus on 4d $\mathcal{N}=1$ superconformal field theories arising on the world volume of $N$ D3 branes placed at the tip of a cone over a toric Sasaki-Einstein manifold SE$_{5}$, the simplest example of which is $S^5$, corresponding to $\mathcal{N}=4$ super-Yang-Mills. The giant graviton expansion realises the superconformal index as the sum of contributions from wrapped D3 branes in the dual AdS$_{5}\times \text{SE}_{5}$. We argue that, for large wrapping numbers, the asymptotics of each such contribution is governed by the master volume of a particular metric deformation of $\text{SE}_5$ (suitably fibred over $S^{3}$). In particular, the wrapping numbers of a generic giant graviton configuration are identified with K\"{a}hler moduli of the corresponding metric. We further show that at large $N$ the entropy function of the relevant AdS$_5\times \text{SE}_5$ BPS rotating black hole is recovered by extremising over these moduli. Our results suggest that the complex Euclidean geometries corresponding to rotating BPS black holes in AdS$_{5}$ are determined by a close analogue of GMS volume minimisation, and that conversely, the off-shell geometries considered in such minimisation procedures should be understood as the near-horizon geometries of back-reacted giant gravitons. We present analogous results for 3d $\mathcal{N}=2$ theories holographically dual to M-theory on AdS$_4\times \text{SE}_7$.

hep-th↗

arXiv2023

Conformal Quantum Mechanics, Holomorphic Factorisation, and Ultra-Spinning Black Holes

We study a limit in which a relativistic CFT reduces to conformal quantum mechanics, and relate the partition functions of the two theories. When the initial CFT is holographic, our limit coincides with an ultra-spinning limit in the gravity dual. We therefore propose that ultra-spinning black holes are dual to an appropriate ensemble in finite-dimensional conformal quantum mechanics. The limit is studied in detail for SCFTs in four and six dimensions. These theories have a superconformal index which can be computed by gluing together two or more blocks. Applying our limit to the index effectively isolates a single such block. Our results therefore suggest that ultra-spinning black holes play the role of blocks in the gravitational dual of holomorphic factorisation.

hep-th↗

arXiv2022

Superconformal Quantum Mechanics and Growth of Sheaf Cohomology

We give a geometric interpretation for superconformal quantum mechanics defined on a hyper-Kahler cone which has an equivariant symplectic resolution. BPS states are identified with certain twisted Dolbeault cohomology classes on the resolved space and their index degeneracies can also be related to the Euler characteristic computed in equivariant sheaf cohomology. In the special case of the Hilbert scheme of K points on C2, we obtain a rigorous estimate for the exponential growth of the index degeneracies of BPS states as K goes to infinity. This growth serves as a toy model for our recently proposed duality between a seven dimensional black hole and superconformal quantum mechanics.

hep-th↗

arXiv2022

Black Hole Entropy from Quantum Mechanics

We provide evidence for a holographic duality between superconformal quantum mechanics on the moduli space of Yang-Mills instantons and M-theory in certain asymptotically $AdS_{7}\times S^{4}$ backgrounds with a plane-wave boundary metric. We show that the gravitational background admits a supersymmetric black hole solution whose entropy is precisely reproduced by the superconformal index of the dual quantum mechanics.

hep-th↗

arXiv2021

Blocks and Vortices in the 3d ADHM Quiver Gauge Theory

We study the hemisphere partition function of a three-dimensional $\mathcal{N}=4$ supersymmetric $U(N)$ gauge theory with one adjoint and one fundamental hypermultiplet -- the ADHM quiver theory. In particular, we propose a distinguished set of UV boundary conditions which yield Verma modules of the quantised chiral rings of the Higgs and Coulomb branches. In line with a recent proposal by two of the authors in collaboration with M. Bullimore, we show explicitly that the hemisphere partition functions recover the characters of these modules in two limits, and realise blocks gluing exactly to the partition functions of the theory on closed three-manifolds. We study the geometry of the vortex moduli space and investigate the interpretation of the vortex partition functions as equivariant indices of quasimaps to the Hilbert scheme of points in $\mathbb{C}^2$. We also investigate half indices of the ADHM quiver gauge theory in the presence of a line operator and discuss their geometric interpretation. Along the way we find interesting relations between our hemisphere blocks and related quantities in topological string theory and equivariant quantum K-theory.

hep-th↗

arXiv2020

Factorisation of 3d $\mathcal{N}=4$ Twisted Indices and the Geometry of Vortex Moduli Space

We study the twisted indices of $\mathcal{N}=4$ supersymmetric gauge theories in three dimensions on spatial $S^{2}$ with an angular momentum refinement. We demonstrate factorisation of the index into holomorphic blocks for the $T[SU(N)]$ theory in the presence of generic fluxes and fugacities. We also investigate the relation between the twisted index, Hilbert series and the moduli space of vortices. In particular, we show that each holomorphic block coincides with a generating function for the $χ_{t}$ genera of the moduli spaces of "local" vortices. The twisted index itself coincides with a corresponding generating function for the $χ_{t}$ genera of moduli spaces of "global" vortices in agreement with a proposal of Bullimore et. al. We generalise this geometric interpretation of the twisted index to include fluxes and Chern-Simons levels. For the $T[SU(N)]$ theory, the relevant moduli spaces are the local and global versions of Laumon space respectively and we demonstrate the proposed agreements explicitly using results from the mathematical literature. Finally, we exhibit a precise relation between the Coulomb branch Hilbert series and the Poincaré polynomials of the corresponding vortex moduli spaces.

hep-th↗

arXiv2020

Superconformal Quantum Mechanics on Kähler Cones

We consider supersymmetric quantum mechanics on a Kähler cone, regulated via a suitable resolution of the conical singularity. The unresolved space has a $\mathfrak{u}(1,1|2)$ superconformal symmetry and we propose the existence of an associated quantum mechanical theory with a discrete spectrum consisting of unitary, lowest weight representations of this algebra. We define a corresponding superconformal index and compute it for a wide range of examples.

hep-th↗

arXiv2018

An Index for Superconformal Quantum Mechanics

We study quantum mechanical systems with $\mathfrak{osp}(4^{*}|4)$ superconformal symmetry. We classify unitary lowest-weight representations of this superconformal algebra and define an index which receives contributions from short and semi-short multiplets only. We consider the example of a quantum mechanical $σ$-model with hyper-Kähler target $\mathcal{M}$ equipped with a triholomorphic homothety. The superconformal index coincides with the Witten index of a novel form of supersymmetric quantum mechanics for a particle moving on $\mathcal{M}$ in a background magnetic field in which an unbroken $\mathfrak{su}(1|2)$ subalgebra of the superconformal algebra is linearly realised as a global symmetry.

hep-th↗

arXiv2018

A Superconformal Index for HyperKähler Cones

We define an index for $\mathfrak{osp}(4^{*}|4)$ superconformal quantum mechanics on a hyperKähler cone. The index is defined on an equivariant symplectic resolution of the cone, which acts as a regulator. We present evidence that the index does not depend on the choice of resolution parameters and encodes information about the spectrum of (semi-) short representations of the superconformal algebra of the unresolved space. In particular, there are two types of multiplet which can be counted exactly using the index. These correspond to holomorphic functions on the cone and to the generators of the Borel-Moore homology on the resolved space respectively. We calculate the resulting index by localisation for a large class of examples.

hep-th↗

arXiv2017

ADHM and the 4d Quantum Hall Effect

Yang-Mills instantons are solitonic particles in d=4+1 dimensional gauge theories. We construct and analyse the quantum Hall states that arise when these particles are restricted to the lowest Landau level. We describe the ground state wavefunctions for both Abelian and non-Abelian quantum Hall states. Although our model is purely bosonic, we show that the excitations of this 4d quantum Hall state are governed by the Nekrasov partition function of a certain five dimensional supersymmetric gauge theory with Chern-Simons term. The partition function can also be interpreted as a variant of the Hilbert series of the instanton moduli space, counting holomorphic sections rather than holomorphic functions. It is known that the Hilbert series of the instanton moduli space can be rewritten using mirror symmetry of 3d gauge theories in terms of Coulomb branch variables. We generalise this approach to include the effect of a five dimensional Chern-Simons term. We demonstrate that the resulting Coulomb branch formula coincides with the corresponding Higgs branch Molien integral which, in turn, reproduces the standard formula for the Nekrasov partition function.

hep-th↗

arXiv2016

Solution of quantum integrable systems from quiver gauge theories

We construct new integrable systems describing particles with internal spin from four-dimensional $\mathcal{N}=2$ quiver gauge theories. The models can be quantized and solved exactly using the quantum inverse scattering method and also using the Bethe/Gauge correspondence.

hep-th↗

arXiv2016

A Matrix Model for WZW

We study a U(N) gauged matrix quantum mechanics which, in the large N limit, is closely related to the chiral WZW conformal field theory. This manifests itself in two ways. First, we construct the left-moving Kac-Moody algebra from matrix degrees of freedom. Secondly, we compute the partition function of the matrix model in terms of Schur and Kostka polynomials and show that, in the large $N$ limit, it coincides with the partition function of the WZW model. This same matrix model was recently shown to describe non-Abelian quantum Hall states and the relationship to the WZW model can be understood in this framework.

hep-th↗

arXiv2016

A Matrix Model for Non-Abelian Quantum Hall States

We propose a matrix quantum mechanics for a class of non-Abelian quantum Hall states. The model describes electrons which carry an internal SU(p) spin. The ground states of the matrix model include spin-singlet generalisations of the Moore-Read and Read-Rezayi states and, in general, lie in a class previously introduced by Blok and Wen. The effective action for these states is a U(p) Chern-Simons theory. We show how the matrix model can be derived from quantisation of the vortices in this Chern-Simons theory and how the matrix model ground states can be reconstructed as correlation functions in the boundary WZW model.

cond-mat.str-el↗

arXiv2014

Instantons, Integrability and Discrete Light-Cone Quantisation

We study supersymmetric quantum mechanics on the moduli space of Yang-Mills instantons on R^2 x T^2 and its application to the discrete light-cone quantisation (DLCQ) of N=4 SUSY Yang-Mills. In the presence of a target space magnetic field, the model has a discrete spectrum with the wavefunctions of generic energy eigenstates supported away from the singular points of the moduli space. The corresponding Hamiltonian is part of an osp(1,1|4) superalgebra which enlarges to su(1,1|4) superconformal invariance in the sector corresponding to the N=4 theory. The Hamiltonian is isospectral to the light-cone dilatation operator of the N=4 theory in this sector. The model also has an interesting scaling limit where it becomes integrable. We determine the semiclassical spectrum in this limit. We discuss a possible approach to constructing the dilatation operator of N=4 supersymmetric Yang-Mills theory in DLCQ.

hep-th↗

arXiv2014

Superconformal Quantum Mechanics and the Discrete Light-Cone Quantisation of N=4 SUSY Yang-Mills

We study the quantum mechanical sigma model arising in the discrete light-cone quantisation of N=4 supersymmetric Yang-Mills theory. The target space is a certain torus fibration over a scale-invariant special Kahler manifold. We show that the expected SU(1,1|4) light-cone superconformal invariance of the N=4 theory emerges in a limit where the volume of the fibre goes to zero and give an explicit construction of the generators. The construction given here yields a large new family of superconformal quantum mechanical models with SU(1,1|4) invariance.

hep-th↗

arXiv2012

On the BPS Spectrum at the Root of the Higgs Branch

We study the BPS spectrum and walls of marginal stability of the $\mathcal{N}=2$ supersymmetric theory in four dimensions with gauge group SU(n) and $n\le N_{f}<2n$ fundamental flavours at the root of the Higgs branch. The strong-coupling spectrum of this theory was conjectured in hep-th/9902134 to coincide with that of the two-dimensional supersymmetric $\mathbb{CP}^{2n-N_{f}-1}$ sigma model. Using the Kontsevich--Soibelman wall-crossing formula, we start with the conjectured strong-coupling spectrum and extrapolate it to all other regions of the moduli space. In the weak-coupling regime, our results precisely agree with the semiclassical analysis of hep-th/9902134: in addition to the usual dyons, quarks, and $W$ bosons, if the complex masses obey a particular inequality, the resulting weak-coupling spectrum includes a tower of bound states consisting of a dyon and one or more quarks. In the special case of $\mathbb{Z}_{n}$-symmetric masses, there are bound states with one quark for odd $n$ and no bound states for even $n$.

hep-th↗