arXiv subjects

Andrew Neitzke

Publications and source records attributed to Andrew Neitzke.

At least 37 records · Page 2Linked to original sources

arXiv2017

From $S^1$-fixed points to $\mathcal{W}$-algebra representations

We study a set $\mathcal{M}_{K,N}$ parameterizing filtered $SL(K)$-Higgs bundles over $\mathbb{CP}^1$ with an irregular singularity at $z = \infty$, such that the eigenvalues of the Higgs field grow like $\lvert λ\rvert \sim \lvert z ^{N/K} \mathrm{d} z \rvert$, where $K$ and $N$ are coprime. $\mathcal{M}_{K,N}$ carries a $\mathbb{C}^\times$-action analogous to the famous $\mathbb{C}^\times$-action introduced by Hitchin on the moduli spaces of Higgs bundles over compact curves. The construction of this $\mathbb{C}^\times$-action on $\mathcal{M}_{K,N}$ involves the rotation automorphism of the base $\mathbb{CP}^1$. We classify the fixed points of this $\mathbb{C}^\times$-action, and exhibit a curious $1$-$1$ correspondence between these fixed points and certain representations of the vertex algebra $\mathcal{W}_K$; in particular we have the relation $μ= \frac{1}{12} \left(K - 1 - c_{\mathrm{eff}} \right)$, where $μ$ is a regulated version of the $L^2$ norm of the Higgs field, and $c_{\mathrm{eff}}$ is the effective Virasoro central charge of the corresponding $W$-algebra representation. We also discuss a Bialynicki-Birula-type stratification of $\mathcal{M}_{K,N}$, where the strata are labeled by isomorphism classes of the underlying filtered vector bundles.

math.DG↗

arXiv2017

Integral iterations for harmonic maps

We study minimal harmonic maps $g: {\mathbb{C}} \to SO(3) \backslash SL(3,{\mathbb{R}})$, parameterized by polynomial cubic differentials $P$ in the plane. The asymptotic structure of such a $g$ is determined by a convex polygon $Y(P)$ in ${\mathbb{RP}^2}$. We give a conjectural method for determining $Y(P)$ by solving a fixed-point problem for a certain integral operator. The technology of spectral networks and BPS state counts is a key input to the formulation of this fixed-point problem. We work out two families of examples in detail.

math.DG↗

arXiv2016

Opers versus nonabelian Hodge

For a complex simple simply connected Lie group $G$, and a compact Riemann surface $C$, we consider two sorts of families of flat $G$-connections over $C$. Each family is determined by a point ${\mathbf u}$ of the base of Hitchin's integrable system for $(G,C)$. One family $\nabla_{\hbar,{\mathbf u}}$ consists of $G$-opers, and depends on $\hbar \in {\mathbb C}^\times$. The other family $\nabla_{R,ζ,{\mathbf u}}$ is built from solutions of Hitchin's equations, and depends on $ζ\in {\mathbb C}^\times, R \in {\mathbb R}^+$. We show that in the scaling limit $R \to 0$, $ζ= \hbar R$, we have $\nabla_{R,ζ,{\mathbf u}} \to \nabla_{\hbar,{\mathbf u}}$. This establishes and generalizes a conjecture formulated by Gaiotto.

math.DG↗

arXiv2016

BPS states in the Minahan-Nemeschansky E6 theory

We use the method of spectral networks to compute BPS state degeneracies in the Minahan-Nemeschansky $E_6$ theory, on its Coulomb branch, without turning on a mass deformation. The BPS multiplicities come out in representations of the $E_6$ flavor symmetry. For example, along the simplest ray in electromagnetic charge space, we give the first $14$ numerical degeneracies, and the first $7$ degeneracies as representations of $E_6$. We find a complicated spectrum, exhibiting exponential growth of multiplicities as a function of the electromagnetic charge. There is one unexpected outcome: the spectrum is consistent (in a nontrivial way) with the hypothesis of "spin purity," that if a BPS state in this theory has electromagnetic charge equal to $n$ times a primitive charge, then it appears in a spin-$\frac{n}{2}$ multiplet.

hep-th↗

arXiv2016

Spectral networks and Fenchel-Nielsen coordinates

We explain that spectral networks are a unifying framework that incorporates both shear (Fock-Goncharov) and length-twist (Fenchel-Nielsen) coordinate systems on moduli spaces of flat SL(2,C) connections, in the following sense. Given a spectral network W on a punctured Riemann surface C, we explain the process of "abelianization" which relates flat SL(2)-connections (with an additional structure called "W-framing") to flat C*-connections on a covering. For any W, abelianization gives a construction of a local Darboux coordinate system on the moduli space of W-framed flat connections. There are two special types of spectral network, combinatorially dual to ideal triangulations and pants decompositions; these two types of network lead to Fock-Goncharov and Fenchel-Nielsen coordinates respectively.

math.GT↗

arXiv2015

$R^3$ index for four-dimensional $N=2$ field theories

In theories with $N=2$ supersymmetry on $R^{3,1}$, BPS bound states can decay across walls of marginal stability in the space of Coulomb branch parameters, leading to discontinuities in the BPS indices $Ω(γ,u)$. We consider a supersymmetric index $I$ which receives contributions from 1/2-BPS states, generalizing the familiar Witten index $Tr (-1)^F e^{-βH}$. We expect $I$ to be smooth away from loci where massless particles appear, thanks to contributions from the continuum of multi-particle states. Taking inspiration from a similar phenomenon in the hypermultiplet moduli space of $N=2$ string vacua, we conjecture a formula expressing $I$ in terms of the BPS indices $Ω(γ,u)$, which is continuous across the walls and exhibits the expected contributions from single particle states at large $β$. This gives a universal prediction for the contributions of multi-particle states to the index $I$. This index is naturally a function on the moduli space after reduction on a circle, closely related to the canonical hyperkähler metric and hyperholomorphic connection on this space.

hep-th↗

arXiv2014

Hitchin systems in N=2 field theory

This is the second article in the collection of reviews "Exact results on N=2 supersymmetric gauge theories", ed. J. Teschner. The space of vacua of class $\cal S$ theories on $R^3 \times S^1$ can be identified as the moduli space of solutions to the self-duality equations in two dimensions on Riemann surfaces studied by Hitchin. This fact plays a fundamental role for recent studies of the spectrum of BPS states in class $\cal S$ theories, and it is related to the integrable structure underlying Seiberg-Witten theory of theories of class $\cal S$. This article reviews important aspects of the role of the Hitchin system for the infrared physics of class $\cal S$ theories.

hep-th↗

arXiv2014

Twistorial Topological Strings and a tt* Geometry for N=2 Theories in 4d

We define twistorial topological strings by considering tt* geometry of the 4d N=2 supersymmetric theories on the Nekrasov-Shatashvili half-Omega background, which leads to quantization of the associated hyperKahler geometries. We show that in one limit it reduces to the refined topological string amplitude. In another limit it is a solution to a quantum Riemann-Hilbert problem involving quantum Kontsevich-Soibelman operators. In a further limit it encodes the hyperKahler integrable systems studied by GMN. In the context of AGT conjecture, this perspective leads to a twistorial extension of Toda. The 2d index of the half-Omega theory leads to the recently introduced index for N=2 theories in 4d. The twistorial topological string can alternatively be viewed, using the work of Nekrasov-Witten, as studying the vacuum geometry of 4d N=2 supersymmetric theories on T^2x I where I is an interval with specific boundary conditions at the two ends.

hep-th↗

arXiv2014

Line Defects, Tropicalization, and Multi-Centered Quiver Quantum Mechanics

We study BPS line defects in N=2 supersymmetric four-dimensional field theories. We focus on theories of "quiver type," those for which the BPS particle spectrum can be computed using quiver quantum mechanics. For a wide class of models, the renormalization group flow between defects defined in the ultraviolet and in the infrared is bijective. Using this fact, we propose a way to compute the BPS Hilbert space of a defect defined in the ultraviolet, using only infrared data. In some cases our proposal reduces to studying representations of a "framed" quiver, with one extra node representing the defect. In general, though, it is different. As applications, we derive a formula for the discontinuities in the defect renormalization group map under variations of moduli, and show that the operator product algebra of line defects contains distinguished subalgebras with universal multiplication rules. We illustrate our results in several explicit examples.

hep-th↗

arXiv2014

Hyperkahler Sigma Model and Field Theory on Gibbons-Hawking Spaces

We describe a novel deformation of the 3-dimensional sigma model with hyperkähler target, which arises naturally from the compactification of a 4-dimensional $\mathcal{N}=2$ theory on a hyperkähler circle bundle (Gibbons-Hawking space). We derive the condition for which the deformed sigma model preserves 4 out of the 8 supercharges. We also study the contribution from a NUT center to the sigma model path integral, and find that supersymmetry implies it is a holomorphic section of a certain holomorphic line bundle over the hyperkähler target. We study explicitly the case where the original 4-dimensional theory is pure $U(1)$ super Yang-Mills, and show that the contribution from a NUT center in this case is simply the Jacobi theta function.

hep-th↗

arXiv2013

Spectral networks

We introduce new geometric objects called spectral networks. Spectral networks are networks of trajectories on Riemann surfaces obeying certain local rules. Spectral networks arise naturally in four-dimensional N=2 theories coupled to surface defects, particularly the theories of class S. In these theories spectral networks provide a useful tool for the computation of BPS degeneracies: the network directly determines the degeneracies of solitons living on the surface defect, which in turn determine the degeneracies for particles living in the 4d bulk. Spectral networks also lead to a new map between flat GL(K,C) connections on a two-dimensional surface C and flat abelian connections on an appropriate branched cover Sigma of C. This construction produces natural coordinate systems on moduli spaces of flat GL(K,C) connections on C, which we conjecture are cluster coordinate systems.

hep-th↗

arXiv2013

Notes on a new construction of hyperkahler metrics

We review a construction of hyperkahler metrics proposed in joint work of Davide Gaiotto, Greg Moore and the author. A key ingredient in this construction is a collection of integer "DT invariants" obeying the wall-crossing formula of Kontsevich-Soibelman.

math.DG↗

arXiv2013

Wild Wall Crossing and BPS Giants

We show that the BPS spectrum of pure SU(3) four-dimensional super Yang-Mills with N=2 supersymmetry exhibits a surprising phenomenon: there are regions of the Coulomb branch where the growth of the BPS degeneracies with the charge is exponential. We show this using spectral networks and independently using wall-crossing formulae and quiver methods. The computations using spectral networks provide a very nontrivial example of how these networks determine the four-dimensional BPS spectrum. We comment on some physical implications of the wild spectrum: for example, exponentially many field-theoretic BPS states with large charge are gigantic. Finally, we exhibit some surprising, thus far unexplained, regularities of the BPS spectrum.

hep-th↗

arXiv2012

Framed BPS States

We consider a class of line operators in d=4, N=2 supersymmetric field theories which leave four supersymmetries unbroken. Such line operators support a new class of BPS states which we call "framed BPS states." These include halo bound states similar to those of d=4, N=2 supergravity, where (ordinary) BPS particles are loosely bound to the line operator. Using this construction, we give a new proof of the Kontsevich-Soibelman wall-crossing formula for the ordinary BPS particles, by reducing it to the semiprimitive wall-crossing formula. After reducing on S1, the expansion of the vevs of the line operators in the IR provides a new physical interpretation of the "Darboux coordinates" on the moduli space M of the theory. Moreover, we introduce a "protected spin character" which keeps track of the spin degrees of freedom of the framed BPS states. We show that the generating functions of protected spin characters admit a multiplication which defines a deformation of the algebra of functions on M. As an illustration of these ideas, we consider the six-dimensional (2,0) field theory of A1 type compactified on a Riemann surface C. Here we show (extending previous results) that line operators are classified by certain laminations on a suitably decorated version of C, and we compute the spectrum of framed BPS states in several explicit examples. Finally we indicate some interesting connections to the theory of cluster algebras.

hep-th↗

arXiv2012

Spectral Networks and Snakes

We apply and illustrate the techniques of spectral networks in a large collection of A_{K-1} theories of class S, which we call "lifted A_1 theories." Our construction makes contact with Fock and Goncharov's work on higher Teichmuller theory. In particular we show that the Darboux coordinates on moduli spaces of flat connections which come from certain special spectral networks coincide with the Fock-Goncharov coordinates. We show, moreover, how these techniques can be used to study the BPS spectra of lifted A_1 theories. In particular, we determine the spectrum generators for all the lifts of a simple superconformal field theory.

hep-th↗

arXiv2011

On a hyperholomorphic line bundle over the Coulomb branch

Given an N=2 supersymmetric field theory in four dimensions, its dimensional reduction on S^1 is a sigma model with hyperkahler target space M. We describe a canonical line bundle V on M, equipped with a hyperholomorphic connection. The construction of this connection is similar to the known construction of the metric on M itself: one begins with a simple "semiflat" connection and then improves it by including contributions weighed by the degeneracies of BPS particles going around S^1. We conjecture that V describes the physics of the theory dimensionally reduced on NUT space.

hep-th↗

arXiv2011

Wall-crossing, Hitchin Systems, and the WKB Approximation

We consider BPS states in a large class of d=4, N=2 field theories, obtained by reducing six-dimensional (2,0) superconformal field theories on Riemann surfaces, with defect operators inserted at points of the Riemann surface. Further dimensional reduction on S^1 yields sigma models, whose target spaces are moduli spaces of Higgs bundles on Riemann surfaces with ramification. In the case where the Higgs bundles have rank 2, we construct canonical Darboux coordinate systems on their moduli spaces. These coordinate systems are related to one another by Poisson transformations associated to BPS states, and have well-controlled asymptotic behavior, obtained from the WKB approximation. The existence of these coordinates implies the Kontsevich-Soibelman wall-crossing formula for the BPS spectrum. This construction provides a concrete realization of a general physical explanation of the wall-crossing formula which was proposed in 0807.4723. It also yields a new method for computing the spectrum using the combinatorics of triangulations of the Riemann surface.

hep-th↗

arXiv2011

Wall-Crossing in Coupled 2d-4d Systems

We introduce a new wall-crossing formula which combines and generalizes the Cecotti-Vafa and Kontsevich-Soibelman formulas for supersymmetric 2d and 4d systems respectively. This 2d-4d wall-crossing formula governs the wall-crossing of BPS states in an N=2 supersymmetric 4d gauge theory coupled to a supersymmetric surface defect. When the theory and defect are compactified on a circle, we get a 3d theory with a supersymmetric line operator, corresponding to a hyperholomorphic connection on a vector bundle over a hyperkahler space. The 2d-4d wall-crossing formula can be interpreted as a smoothness condition for this hyperholomorphic connection. We explain how the 2d-4d BPS spectrum can be determined for 4d theories of class S, that is, for those theories obtained by compactifying the six-dimensional (0,2) theory with a partial topological twist on a punctured Riemann surface C. For such theories there are canonical surface defects. We illustrate with several examples in the case of A_1 theories of class S. Finally, we indicate how our results can be used to produce solutions to the A_1 Hitchin equations on the Riemann surface C.

hep-th↗