Instanton Moduli Space Research Articles

Higgs branch RG flows via decay and fission

Magnetic quivers have been an instrumental technique for advancing our understanding of Higgs branches of supersymmetric theories with eight supercharges. In this work, we present the “decay and fission” algorithm for unitary magnetic quivers. It enables the derivation of the complete phase (Hasse) diagram and is characterized by the following key attributes: First and foremost, the algorithm is inherently simple, just relying on convex linear algebra. Second, any magnetic quiver can only undergo decay or fission processes; these reflect the possible Higgs branch RG flows (Higgsings), and the quivers thereby generated are the magnetic quivers of the new RG fixed points. Third, the geometry of the decay or fission transition (i.e., the transverse slice) is simply read off. As a consequence, the algorithm does not rely on a complete list of minimal transitions, but rather outputs the transverse slice geometry automatically. As a proof of concept, its efficacy is showcased across various scenarios, encompassing superconformal field theories from dimensions 3 to 6, instanton moduli spaces, and little string theories. Published by the American Physical Society 2024

Moduli spaces of instantons in flag manifold sigma models. Vortices in quiver gauge theories

In this paper, we discuss lumps (sigma model instantons) in flag manifold sigma models. In particular, we focus on the moduli space of BPS lumps in general Kähler flag manifold sigma models. Such a Kähler flag manifold, which takes the form Un1+⋯+nL+1Un1×⋯×UnL+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\frac{\ extrm{U}\\left({n}_1+\\cdots +{n}_{L+1}\\right)}{\ extrm{U}\\left({n}_1\\right)\ imes \\cdots \ imes \ extrm{U}\\left({n}_{L+1}\\right)} $$\\end{document}, can be realized as a vacuum moduli space of a U(N1) × ··· × U(NL) quiver gauged linear sigma model. When the gauge coupling constants are finite, the gauged linear sigma model admits BPS vortex configurations, which reduce to BPS lumps in the low energy effective sigma model in the large gauge coupling limit. We derive an ADHM-like quotient construction of the moduli space of BPS vortices and lumps by generalizing the quotient construction in U(N) gauge theories by Hanany and Tong. As an application, we check the dualities of the 2d models by computing the vortex partition functions using the quotient construction.

Saw varieties

We introduce a new class of quiver varieties, called saw (quiver) varieties. These can be seen as a generalization of the three existing theories: the quiver gauge theory of monopoles, Kac's representation theory of Dynkin types, and Donaldson's theory on the based maps of P1. E.g., in our terminology, the handsaw (quiver) varieties and chainsaw varieties are finite and affine A type saw varieties respectively with the maximal symmetry breaking. These varieties are the monopole spaces and the caloron spaces in 3d gauge theory, respectively. We expand the theory over to the DE type quiver gauge theory.The saw varieties are constructed via a Hamiltonian formalism, so that the moment maps defining these varieties naturally arise. The main assertion of the paper is that the moment map μ defining a saw variety is a flat morphism for arbitrary dimension vector, if and only if the underlying graph is of finite ADE types. Furthermore under the assumption of maximal symmetry breaking or all the higher frame dimension vectors, the affine ADE types also attain the flatness property of μ. We give an algebro-geometric description of the momentum-zero scheme μ−1(0) and the saw variety defined as the Hamiltonian reduction.Various partition functions given rise to by the above mentioned theories are the byproducts of the geometric structure of the corresponding saw varieties.

Canonical orientations for moduli spaces of $G_2$-instantons with gauge group $\mathrm{SU}(m)$ or $\mathrm{U}(m)$

Suppose $(X, g)$ is a compact, spin Riemannian $7$-manifold, with Dirac operator ${D\mspace{-12mu}/\mspace{4mu}}^g : C^\infty (X, {S\mspace{-10mu}/\mspace{2mu}}) \to C^\infty (X, {S\mspace{-10mu}/\mspace{2mu}})$. Let $G$ be $\mathrm{SU}(m)$ or $\mathrm{U}(m)$, and $E \to X$ be a rank $m$ complex bundle with $G$-structure. Write $\mathcal{B}_E$ for the infinite-dimensional moduli space of connections on $E$, modulo gauge. There is a natural principal $\mathbb{Z}_2$-bundle $O^{{D\mspace{-12mu}/\mspace{4mu}}^g}_E \to \mathcal{B}_E$ parametrizing orientations of $\operatorname{det} {D\mspace{-12mu}/\mspace{4mu}}^g_{\mathrm{Ad} \, A}$ for twisted elliptic operators ${D\mspace{-12mu}/\mspace{4mu}}^g_{\mathrm{Ad} \, A}$ at each $[A]$ in $\mathcal{B}_E$. A theorem of Walpuski [33] shows $O^{{D\mspace{-12mu}/\mspace{4mu}}^g}_E$ is trivializable. We prove that if we choose an orientation for $\operatorname{det} {D\mspace{-12mu}/\mspace{4mu}}^g$ , and a flag structure on $X$ in the sense of [17], then we can define canonical trivializations of $O^{{D\mspace{-12mu}/\mspace{4mu}}^g}_E$ for all such bundles $E \to X$, satisfying natural compatibilities. Now let $(X, \varphi, g)$ be a compact $G_2$-manifold, with $d (\ast \varphi) = 0$. Then we can consider moduli spaces $\mathcal{M}^{G_2}_E$ of $G_2$-instantons on $E \to X$, which are smooth manifolds under suitable transversality conditions, and derived manifolds in general, with $\mathcal{M}^{G_2}_E \subset \mathcal{B}_E$. The restriction of $O^{{D\mspace{-12mu}/\mspace{4mu}}^g}_E$ to $\mathcal{M}^{G_2}_E$ is the $\mathbb{Z}_2$-bundle of orientations on $\mathcal{M}^{G_2}_E$. Thus, our theorem induces canonical orientations on all such $G_2$-instanton moduli spaces $\mathcal{M}^{G_2}_E$. This contributes to the Donaldson–Segal programme [11], which proposes defining enumerative invariants of $G_2$-manifolds $(X, \varphi, g)$ by counting moduli spaces $\mathcal{M}^{G_2}_E$, with signs depending on a choice of orientation.

Multi-instanton calculus in c = 1 string theory

We formulate a strategy for computing the complete set of non-perturbative corrections to closed string scattering in c = 1 string theory from the worldsheet perspective. This requires taking into account the effect of multiple ZZ-instantons, including higher instantons constructed from ZZ boundary conditions of type (m, 1), with a careful treatment of the measure and contour in the integration over the instanton moduli space. The only a priori ambiguity in our prescription is a normalization constant mathcal{N} m that appears in the integration measure for the (m, 1)-type ZZ instanton, at each positive integer m. We investigate leading corrections to the closed string reflection amplitude at the n-instanton level, i.e. of order {e}^{-n/{g}_s} , and find striking agreement with our recent proposal on the non-perturbative completion of the dual matrix quantum mechanics, which in turn fixes mathcal{N} m for all m.

Anti-self-dual connections over the 5D Heisenberg group and the twistor method

By introducing notions of an α-plane in the 5D complex Heisenberg group and the twistor space as the moduli space of all α-planes, we can define an anti-self-dual (ASD) connection as a connection flat over all α-planes. This geometric approach allows us to establish the Penrose-Ward correspondence between ASD connections over the 5D complex Heisenberg group and a class of holomorphic vector bundles on the twistor space. By Atiyah-Ward ansätz, we also construct ASD connections on the 5D complex Heisenberg group. When restricted to the 5D real Heisenberg group, the flat model of 5D contact manifolds, an ASD connection satisfies the horizontal part of the contact instanton equation.

Universal Structures in C-Linear Enumerative Invariant Theories

An enumerative invariant theory in Algebraic Geometry, Differential Geometry, or Representation Theory, is the study of invariants which 'count' $\tau$-(semi)stable objects $E$ with fixed topological invariants $[E]=\alpha$ in some geometric problem, using a virtual class $[{\cal M}_\alpha^{\rm ss}(\tau)]_{\rm virt}$ in some homology theory for the moduli spaces ${\cal M}_\alpha^{\rm st}(\tau)\subseteq{\cal M}_\alpha^{\rm ss}(\tau)$ of $\tau$-(semi)stable objects. Examples include Mochizuki's invariants counting coherent sheaves on surfaces, Donaldson-Thomas type invariants counting coherent sheaves on Calabi-Yau 3- and 4-folds and Fano 3-folds, and Donaldson invariants of 4-manifolds. We make conjectures on new universal structures common to many enumerative invariant theories. Such theories have two moduli spaces ${\cal M},{\cal M}^{\rm pl}$, where the second author gives $H_*({\cal M})$ the structure of a graded vertex algebra, and $H_*({\cal M}^{\rm pl})$ a graded Lie algebra, closely related to $H_*({\cal M})$. The virtual classes $[{\cal M}_\alpha^{\rm ss}(\tau)]_{\rm virt}$ take values in $H_*({\cal M}^{\rm pl})$. Defining $[{\cal M}_\alpha^{\rm ss}(\tau)]_{\rm virt}$ when ${\cal M}_\alpha^{\rm st}(\tau)\ne{\cal M}_\alpha^{\rm ss}(\tau)$ (in gauge theory, when the moduli space contains reducibles) is a difficult problem. We conjecture that there is a natural way to define $[{\cal M}_\alpha^{\rm ss}(\tau)]_{\rm virt}$ in homology over $\mathbb Q$, and that the resulting classes satisfy a universal wall-crossing formula under change of stability condition $\tau$, written using the Lie bracket on $H_*({\cal M}^{\rm pl})$. We prove our conjectures for moduli spaces of representations of quivers without oriented cycles. Our conjectures in Algebraic Geometry using Behrend-Fantechi virtual classes are proved in the sequel arXiv:2111.04694.

The Hasse diagram of the moduli space of instantons

Hasse diagrams (or phase diagrams) for moduli spaces of supersymmetric field theories have been intensively studied in recent years, and many tools to compute them have been developed. The moduli space of instantons, despite being well studied, has proven difficult to deal with. In this note we explore the Hasse diagram of this moduli space from several perspectives — using the partial Higgs mechanism, using brane systems and using quiver subtraction — having to refine previously developed techniques. In particular we introduce the new concept of decorated quiver, which allows to deal with a large class of unitary quivers, including those with adjoint matter.

Hermitian Yang–Mills equations and their generalizations

This is a review of the results on Hermitian Yang–Mills equation and their generalization called the deformed Hermitian Yang–Mills equation. Solutions of Hermitian Yang–Mills equation in complex dimension 2 are given by antiself-dual connections so this equation may be considered as a multidimensional generalization of the instanton equations. Deformed Hermitian Yang–Mills equation reduce to the Hermitian Yang–Mills equation in the large volume limit.

Virtual Segre and Verlinde numbers of projective surfaces

Recently, Marian–Oprea–Pandharipande established (a generalization of) Lehn's conjecture for Segre numbers associated to Hilbert schemes of points on surfaces. Extending the work of Johnson, they provided a conjectural correspondence between Segre and Verlinde numbers. For surfaces with holomorphic 2-form, we propose conjectural generalizations of their results to moduli spaces of stable sheaves of any rank. Using Mochizuki's formula, we derive a universal function which expresses virtual Segre and Verlinde numbers of surfaces with holomorphic 2-form in terms of Seiberg–Witten invariants and intersection numbers on products of Hilbert schemes of points. We prove that certain canonical virtual Segre and Verlinde numbers of general type surfaces are topological invariants and we verify our conjectures in examples. The power series in our conjectures are algebraic functions, for which we find expressions in several cases and which are permuted under certain Galois actions. Our conjectures imply an algebraic analog of the Mariño–Moore conjecture for higher rank Donaldson invariants. For ranks 3 and 4, we obtain explicit expressions for Donaldson invariants in terms of Seiberg–Witten invariants.

Design and Parametric Analysis of a Supersonic Turbine for Rotating Detonation Engine Applications

Pressure gain combustion is a promising alternative to conventional gas turbine technologies and within this class the Rotating Detonation Engine has the greatest potential. The Fickett–Jacobs cycle can theoretically increase the efficiency by 15% for medium pressure ratios, but the combustion chamber delivers a strongly non-uniform flow; in these conditions, conventionally designed turbines are inadequate with an efficiency below 30%. In this paper, an original mean-line code was developed to perform an advanced preliminary design of a supersonic turbine; self-starting capability of the supersonic channel has been verified through Kantrowitz and Donaldson theory; the design of the supersonic profile was carried out employing the Method of Characteristics; an accurate evaluation of the aerodynamic losses has been achieved by considering shock waves, profile, and mixing losses. Afterwards, an automated Computational Fluid Dynamics (CFD) based optimization process was developed to find the optimal loading condition that minimizes losses while delivering a sufficiently uniform flow at outlet. Finally, a novel parametric analysis was performed considering the effect of inlet angle, Mach number, reaction degree, peripheral velocity, and blade height ratio on the turbine stage performance. This analysis has revealed for the first time, in authors knowledge, that this type of machines can achieve efficiencies over 70%.

A consistent two-skyrmion configuration space from instantons

To study a nuclear system in the Skyrme model one must first construct a space of low energy Skyrme configurations. However, there is no mathematical definition of this configuration space and there is not even consensus on its fundamental properties, such as its dimension. Here, we propose that the full instanton moduli space can be used to construct a consistent skyrmion configuration space, provided that the Skyrme model is coupled to a vector meson which we identify with the ρ-meson. Each instanton generates a unique skyrmion and we reinterpret the 8N instanton moduli as physical degrees of freedom in the Skyrme model. In this picture a single skyrmion has six zero modes and two non-zero modes: one controls the overall scale of the solution and one the energy of the ρ-meson field. We study the N = 1 and N = 2 systems in detail. Two interacting skyrmions can excite the ρ through scattering, suggesting that the ρ and Skyrme fields are intrinsically linked. Our proposal is the first consistent manifold description of the two-skyrmion configuration space. The method can also be generalised to higher N and thus provides a general framework to study any skyrmion configuration space.

Mocking the u-plane integral

The u-plane integral is the contribution of the Coulomb branch to correlation functions of {mathcal {N}}=2 gauge theory on a compact four-manifold. We consider the u-plane integral for correlators of point and surface observables of topologically twisted theories with gauge group mathrm{SU}(2), for an arbitrary four-manifold with (b_1,b_2^+)=(0,1). The u-plane contribution equals the full correlator in the absence of Seiberg–Witten contributions at strong coupling, and coincides with the mathematically defined Donaldson invariants in such cases. We demonstrate that the u-plane correlators are efficiently determined using mock modular forms for point observables, and Appell–Lerch sums for surface observables. We use these results to discuss the asymptotic behavior of correlators as function of the number of observables. Our findings suggest that the vev of exponentiated point and surface observables is an entire function of the fugacities.

Gauge theories on compact toric manifolds

We compute the mathcal{N}=2 supersymmetric partition function of a gauge theory on a four-dimensional compact toric manifold via equivariant localization. The result is given by a piecewise constant function of the Kähler form with jumps along the walls where the gauge symmetry gets enhanced. The partition function on such manifolds is written as a sum over the residues of a product of partition functions on mathbb {C}^2. The evaluation of these residues is greatly simplified by using an “abstruse duality” that relates the residues at the poles of the one-loop and instanton parts of the mathbb {C}^2 partition function. As particular cases, our formulae compute the SU(2) and SU(3) equivariant Donaldson invariants of mathbb {P}^2 and mathbb {F}_n and in the non-equivariant limit reproduce the results obtained via wall-crossing and blow up methods in the SU(2) case. Finally, we show that the U(1) self-dual connections induce an anomalous dependence on the gauge coupling, which turns out to satisfy a mathcal {N}=2 analog of the mathcal {N}=4 holomorphic anomaly equations.

Correspondence between the twisted N=2 super-Yang-Mills and conformal Baulieu-Singer theories

We characterize the correspondence between the twisted $N=2$ super-Yang-Mills theory and the Baulieu-Singer topological theory quantized in the self-dual Landau gauges. While the first is based on an on-shell supersymmetry, the second is based on an off-shell Becchi-Rouet-Stora-Tyutin symmetry. Because of the equivariant cohomology, the twisted $N=2$ in the ultraviolet regime and Baulieu-Singer theories share the same observables, the Donaldson invariants for 4-manifolds. The triviality of the Gribov copies in the Baulieu-Singer theory in these gauges shows that working in the instanton moduli space on the twisted $N=2$ side is equivalent to working in the self-dual gauges on the Baulieu-Singer one. After proving the vanishing of the $\beta$ function in the Baulieu-Singer theory, we conclude that the twisted $N=2$ in the ultraviolet regime, in any Riemannian manifold, is correspondent to the Baulieu-Singer theory in the self-dual Landau gauges -- a conformal gauge theory defined in Euclidean flat space.

Argyres-Douglas theories, S-duality and AGT correspondence

We propose a Nekrasov-type formula for the instanton partition functions of four-dimensional mathcal{N} = 2 U(2) gauge theories coupled to (A1, D2n) Argyres-Douglas theories. This is carried out by extending the generalized AGT correspondence to the case of U(2) gauge group, which requires us to define irregular states of the direct sum of Virasoro and Heisenberg algebras. Using our formula, one can evaluate the contribution of the (A1, D2n) theory at each fixed point on the U(2) instanton moduli space. As an application, we evaluate the instanton partition function of the (A3, A3) theory to find it in a peculiar relation to that of SU(2) gauge theory with four fundamental flavors. From this relation, we read off how the S-duality group acts on the UV gauge coupling of the (A3, A3) theory.

On the D(\u20131)/D7-brane systems

We study non-perturbative effects in supersymmetric U(N) gauge theories in eight dimensions realized by means of D(–1)/D7-brane systems with non-trivial world-volume fluxes turned on. Using an explicit string construction in terms of vertex operators, we derive the action for the open strings ending on the D(–1)-branes and exhibit its BRST structure. The space of vacua for these open strings is shown to be in correspondence with the moduli space of generalized ADHM gauge connections which trigger the non-perturbative corrections in the eight-dimensional theory. These corrections are computed via localization and turn out to depend on the curved background used to localize the integrals on the instanton moduli space, and vanish in flat space. Finally, we show that for specific choices of the background the instanton partition functions reduce to weighted sums of the solid partitions of the integers.

Topological strings on toric geometries in the presence of Lagrangian branes

We propose expressions for refined open topological string partition function on certain non-compact Calabi Yau 3-folds with topological branes wrapped on the special lagrangian submanifolds. The corresponding web diagrams are partially compact and a lagrangian brane is inserted on one of the external legs. Partial compactification introduces a mass deformation in the corresponding gauge theory. We propose conjectures that equate these open topological string partition functions with the generating function of equivaraint indices on certain quiver moduli spaces. To obtain these conjectures we use the identification of topological string partition functions with equivariant indices on the instanton moduli spaces.

Verlinde formulae on complex surfaces: K-theoretic invariants

Abstract We conjecture a Verlinde type formula for the moduli space of Higgs sheaves on a surface with a holomorphic 2-form. The conjecture specializes to a Verlinde formula for the moduli space of sheaves. Our formula interpolates between K-theoretic Donaldson invariants studied by Göttsche and Nakajima-Yoshioka and K-theoretic Vafa-Witten invariants introduced by Thomas and also studied by Göttsche and Kool. We verify our conjectures in many examples (for example, on K3 surfaces).

Supersymmetric soliton \u03c3-models from non-Lorentzian field theories

Recently, a class of non-Lorentzian supersymmetric Lagrangian field theories was considered, in some cases describing M-theory brane configurations, but more generally found as fixed points of non-Lorentzian RG flows induced upon Lorentzian theories. In this paper, we demonstrate how the dynamics of such theories can be reduced to motion on the supersymmetric moduli space of BPS solitons of the parent theory. We focus first on the mathcal{N} = (1, 1) σ-model in (1 + 1)-dimensions with potential, where we produce a su persymmetric extension to the standard geodesic approximation for slow kink motion. We then revisit the (4 + 1)-dimensional Yang-Mills-like theory with 24 supercharges describing a null compactification of M5-branes. We show that the theory reduces to a σ-model on instanton moduli space, extended by couplings to additional fields from the parent theory, and possessing (8+8) super(conformal) symmetries along with 8 further fermionic shift symmetries. We derive this model explicitly for the single SU(2) instanton.