We propose a simple formula for the 4d-2d partition function of half-BPS surface defects in $d=4,\ \mathcal{N}=2$ gauge theories: $Z^{\text{4d-2d}}=\langle Z^{\text{2d}} \rangle_{\text{4d}}$. Our results are applicable for any surface defect obtained by gauging a 2d flavour symmetry using a 4d gauge group. For defects obtained via the Higgsing procedure, our formula reproduces the recent calculation by Pan and Peelaers. For Gukov-Witten defects our results reproduce the orbifold calculation by Kanno and Tachikawa. We emphasize the role of "negative vortices" which are realized as negative D0 branes.
We analyze intersecting surface defects inserted in interacting four-dimensional N = 2 supersymmetric quantum field theories. We employ the realization of a class of such systems as the infrared fixed points of renormalization group flows from larger theories, triggered by perturbed Seiberg-Witten monopole-like configurations, to compute their partition functions. These results are cast into the form of a partition function of 4d/2d/0d coupled systems. Our computations provide concrete expressions for the instanton partition function in the presence of intersecting defects and we study the cor…
We derive the exact vortex partition function in 2d $\mathcal{N}$ = (2,2) gauge theory on the Omega-background, applying the localization scheme in the Higgs phase. We show that the partition function at a finite Omega-deformation parameter $ε$ satisfies a system of differential equations, which can be interpreted as a quantized version of the twisted F-term equations characterizing the SUSY vacua. Using the differential equations derived in this paper, we show the correspondence between the partition function of the two-dimensional vortex string worldsheet theory and the Nekrasov partition fu…
Abelian and center gauges are considered in continuum Yang-Mills theory in order to detect the magnetic monopole and center vortex content of gauge field configurations. Specifically we examine the Laplacian Abelian and center gauges, which are free of Gribov copies, as well as the center gauge analog of the (Abelian) Polyakov gauge. In particular, we study meron, instanton and instanton-anti-instanton field configurations in these gauges and determine their monopole and vortex content. While a single instanton does not give rise to a center vortex, we find center vortices for merons. Furtherm…
We consider certain line defect operators in five-dimensional SUSY gauge theories, whose interaction with the self-dual instantons is described by 1d ADHM-like gauged quantum mechanics constructed by Tong and Wong. The partition function in the presence of these operators is known to be a generating function of BPS Wilson loops in skew symmetric tensor representations of the gauge group. We calculate the partition function and explicitly prove that it is a finite polynomial of the defect mass parameter $x$, which is an essential property of the defect operator and the Wilson loop generating fu…
We study the superconformal index of five-dimensional SCFTs and the sphere partition function of four-dimensional gauge theories with eight supercharges in the presence of co-dimension two half-BPS defects. We derive a prescription which is valid for defects which can be given a "vortex construction", i.e. can be defined by RG flow from vortex configurations in a larger theory. We test the prescription against known results and expected dualities. We employ our prescription to develop a general computational strategy for defects defined by coupling the bulk degrees of freedom to a Gauged Linea…
We consider the Yang-Mills instanton equations on the four-dimensional manifold S^2xSigma, where Sigma is a compact Riemann surface of genus g>1 or its covering space H^2=SU(1,1)/U(1). Introducing a natural ansatz for the gauge potential, we reduce the instanton equations on S^2xSigma to vortex-type equations on the sphere S^2. It is shown that when the scalar curvature of the manifold S^2xSigma vanishes, the vortex-type equations are integrable, i.e. can be obtained as compatibility conditions of two linear equations (Lax pair) which are written down explicitly. Thus, the standard methods of …
There seems to be close relationship between the moduli space of vortices and the moduli space of instantons, which is not yet clearly understood from a standpoint of the field theory. We clarify the reasons why many similarities are found in the methods for constructing the moduli of instanton and vortex, viewed in the light of the notion of the self-duality. We show that the non-Abelian vortex is nothing but the instanton in $R^{2} \times Z_{2}$ from a viewpoint of the noncommutative differential geometry and the gauge theory in discrete space. The action for pure Yang-Mills theory in $R^{2}…
We study codimension-two BPS defects in 2d N=(2,2) supersymmetric gauge theories, focusing especially on those characterized by vortex-like singularities in the dynamical or non-dynamical gauge field. We classify possible SUSY-preserving boundary conditions on charged matter fields around the vortex defects, and derive a formula for defect correlators on the squashed sphere. We also prove an equivalence relation between vortex defects and 0d-2d coupled systems. Our defect correlators are shown to be consistent with the mirror symmetry duality between Abelian gauged linear sigma models and Land…
We consider a complex vector bundle E endowed with a connection A over the eight-dimensional manifold R^2 x G/H, where G/H = SU(3)/U(1)xU(1) is a homogeneous space provided with a never integrable almost complex structure and a family of SU(3)-structures. We establish an equivalence between G-invariant solutions A of the Spin(7)-instanton equations on R^2 x G/H and general solutions of non-Abelian coupled vortex equations on R^2. These vortices are BPS solitons in a d=4 gauge theory obtained from N=1 supersymmetric Yang-Mills theory in ten dimensions compactified on the coset space G/H with an…
By using the self-dual vortex background on extra two-dimensional Riemann surfaces in 5+1 dimensions, the localization mechanism of bulk fermions on a string-like defect with the exponentially decreasing warp-factor is obtained. We give the conditions under which localized spin 1/2 and 3/2 fermions can be obtained.
Field theory with instantons can be partially regularized by adding degrees of freedom at some scale. These extra degrees of freedom lead to the appearence of the new topological defects. These defects which we call freckles have some characteristic size depending on the scale at which the extra degrees of freedom revive. The examples of two dimensional sigma model, four dimensional gauge theory are studied. The compactification of the four dimensional supersymmetric gauge theory down to two dimensions is also considered and the new phenomena are found.
We investigate the relation between instantons and monopoles in the Laplacian Abelian Gauge using analytical methods in the continuum. Our starting point is the fact that the 't Hooft instanton with its high symmetry leads to a pointlike defect with Hopf invariant one. In order to generalise this result we partly break the symmetry by a local perturbation. We find that for generic configurations near the 't Hooft instanton the defects become loops. The analytical results show explicitly that these defects are magnetic monopoles with unit charge. In addition, the monopoles are twisted to accoun…
Generalizing self-duality on R^2 x S^2 to higher dimensions, we consider the Donaldson-Uhlenbeck-Yau equations on R^{2n} x S^2 and their noncommutative deformation for the gauge group U(2). Imposing SO(3) invariance (up to gauge transformations) reduces these equations to vortex-type equations for an abelian gauge field and a complex scalar on R^{2n}_θ. For a special S^2-radius R depending on the noncommutativity θwe find explicit solutions in terms of shift operators. These vortex-like configurations on R^{2n}_θdetermine SO(3)-invariant multi-instantons on R^{2n}_θx S^2_R for R=R(θ). The latt…
It is known that monopoles can be confined by vortex-strings in d=3+1 while vortices can be confined by domain-lines in d=2+1. Here, as a higher dimensional generalization of these, we show that Yang-Mills instantons can be confined by monopole-strings in d=4+1. We achieve this by putting the system into the Higgs phase in which the configuration can be constructed inside a non-Abelian vortex sheet.
The Abelian Higgs model on a compact Riemann surface Σof genus g is considered. We show that for g > 1 the Bogomolny equations for multi-vortices at critical coupling can be obtained as compatibility conditions of two linear equations (Lax pair) which are written down explicitly. These vortices correspond precisely to SO(3)-symmetric Yang-Mills instantons on the (conformal) gravitational instanton Σ\times S^2 with a scalar-flat Kahler metric. Thus, the standard methods of constructing solutions and studying their properties by using Lax pairs (twistor approach, dressing method etc.) can be app…
A supersymmetric extension of the Skyrme model was obtained recently, which consists of only the Skyrme term in the Nambu-Goldstone (pion) sector complemented by the same number of quasi-Nambu-Goldstone bosons. Scherk-Schwarz dimensional reduction yields a kinetic term in three or lower dimensions and a potential term in two dimensions, preserving supersymmetry. Euclidean solitons (instantons) are constructed in the supersymmetric Skyrme model. In four dimensions, the soliton is an instanton first found by Speight. Scherk-Schwarz dimensional reduction is then performed once to get a 3-dimensio…
We clarify relations among topological solitons in various dimensions: a domain wall, non-Abelian vortex, magnetic monopole and Yang-Mills instanton, together with a (non-Abelian) sine-Gordon soliton, baby Skyrmion (lump) and Skyrmion. We construct a composite configuration consisting of a domain wall, vortex, magnetic monopole and Yang-Mills instanton (wall-vortex-monopole-instanton) using the effective theory technique or moduli approximation. Removing some solitons from such a composite, we obtain all possible composite solitons in the form of solitons within a soliton, including all the pr…
A topological defect in the form of the Abrikosov-Nielsen-Olesen vortex in a space of arbitrary dimension is considered as a gauge-flux-carrying tube that is impenetrable for quantum matter. Charged scalar matter field is quantized in the vortex background with the perfectly rigid (Neumann) boundary condition imposed at the side surface of the vortex. We show that a current circulating around the vortex is induced in the vacuum, if the Compton wavelength of the matter field exceeds the transverse size of the vortex considerably. The vacuum current is periodic in the value of the gauge flux of …
The relation between defects of Abelian gauges and instantons is discussed for explicit examples in the Laplacian Abelian gauge. The defect coming from an instanton is pointlike and becomes a monopole loop with twist upon perturbation. The interplay between magnetic charge, twist and instanton number - encoded as a Hopf invariant - is investigated with the help of a new method, an auxiliary Abelian fibre bundle.
We study the Abelian Higgs vortex solutions to the sinh-Gordon equation and the elliptic Tzitzeica equation. Starting from these particular vortices, we construct solutions to the Taubes equation with higher vortex number, on surfaces with conical singularities. We then, analyse more general properties of vortices on such singular surfaces and propose a method to obtain vortices on conifolds from vortices on surfaces of revolution. We apply our method to construct explicit vortex solutions on the Poincaré disk with a conical singularity in the centre, to which we refer as the "hyperbolic cone"…
Maxwell-Chern-Simons models in the presence of an instanton anti-instanton background are studied. The saddle-point configuration corresponds to the creation and annihilation of a vortex localized around the Dirac string needed to support the nontrivial background. This configuration is generalized to the case in which a nonlocal Maxwell term is allowed in order to fulfill the finite action requirement. Following 't Hooft procedure, we compute the vortex correlation functions and we study the possibility of obtaining spin 1/2 excitations. A possible connection with the bosonization of interact…
Gauge theories in the presence of codimension two vortex defects are known to be related to the theories on orbifolds. By using this relation we study the localized path integrals of 2D N=(2,2) SUSY gauge theories with point-like vortex defects. We present a formula for the correlation functions of vortex defects inserted at the north and the south poles of squashed spheres. For Abelian gauge theories the correlators are locally constant as functions of the parameters of the defect, but exhibit discontinuity at some threshold values determined by the deficit angle at the poles and the R-charge…
We show that when a topological defect with extended world-volume annihilates with an anti-defect, there arise topological defects with dimensions less than those of the original defects by one. Domain wall annihilations create vortices while monopole-string annihilations result in instantons. We find that twisted domain wall rings are vortices, whereas twisted monopole rings are instantons.
We consider Yang-Mills theories with general gauge groups $G$ and twists on the four torus. We find consistent boundary conditions for gauge fields in all instanton sectors. An extended Abelian projection with respect to the Polyakov loop operator is presented, where $A_0$ is independent of time and in the Cartan subalgebra. Fundamental domains for the gauge fixed $A_0$ are constructed for arbitrary gauge groups. In the sectors with non-vanishing instanton number such gauge fixings are necessarily singular. The singularities can be restricted to Dirac strings joining magnetically charged defec…
Gauge invariance plays an important role in forming topological defects. In this work, from the AdS/CFT correspondence, we realize the clusters of equal-sign vortices during the course of critical dynamics of a strongly coupled superconductor. This is the first time to achieve the equal-sign vortex clusters in strongly coupled systems, considering that vortices with equal sign repel. The appearance of clusters of equal-sign vortices is a typical character of flux trapping mechanism, distinct from Kibble-Zurek mechanism which merely presents vortex-antivortex pair distributions resulting from g…
We consider U(n+1) Yang-Mills instantons on the space Σ\times S^2, where Σis a compact Riemann surface of genus g. Using an SU(2)-equivariant dimensional reduction, we show that the U(n+1) instanton equations on Σ\times S^2 are equivalent to non-Abelian vortex equations on Σ. Solutions to these equations are given by pairs (A,ϕ), where A is a gauge potential of the group U(n) and ϕis a Higgs field in the fundamental representation of the group U(n). We briefly compare this model with other non-Abelian Higgs models considered recently. Afterwards we show that for g>1, when Σ\times S^2 becomes a…
The phenomenon of the finite-temperature induced quantum numbers in fermionic systems with topological defects is analyzed. We consider an ideal gas of twodimensional relativistic massive electrons in the background of a defect in the form of a pointlike magnetic vortex with arbitrary flux. This system is found to acquire, in addition to fermion number, also orbital angular momentum, spin, and induced magnetic flux, and we determine the functional dependence of the appropriate thermal averages and correlations on the temperature, the vortex flux, and the continuous parameter of the boundary co…
We define a class of supersymmetric defect loop operators in N = 2 gauge theories in 2+1 dimensions. We give a prescription for computing the expectation value of such operators in a generic N = 2 theory on the three-sphere using localization. We elucidate the role of defect loop operators in IR dualities of supersymmetric gauge theories, and write down their transformation properties under the SL(2, Z) action on conformal theories with abelian global symmetries.
We introduce and study a surface defect in four dimensional gauge theories supporting nested instantons with respect to the parabolic reduction of the gauge group at the defect. This is engineered from a D3/D7-branes system on a non compact Calabi-Yau threefold $X$. For $X=T^2\times T^*{\mathcal C}_{g,k}$, the product of a two torus $T^2$ times the cotangent bundle over a Riemann surface ${\mathcal C}_{g,k}$ with marked points, we propose an effective theory in the limit of small volume of ${\mathcal C}_{g,k}$ given as a comet shaped quiver gauge theory on $T^2$, the tail of the comet being ma…
We present a detailed study of a single vortex in a holographic symmetry breaking phase. At low energies the system flows to an nontrivial conformal fixed point. Novel vortex physics arises from the interaction of these gapless degrees of freedom with the vortex: at low energies the vortex may be understood as a conformal defect in this low energy theory. Defect conformal symmetry allows the construction of a simple infrared geometry describing a new kind of extremal horizon: a Poincare horizon with a small bubble of magnetic Reissner-Nordstrom horizon inside it that carries a single unit of m…
Employing the fracton-elastic duality, we develop a low-energy effective theory of a zero-temperature vortex crystal in a two-dimensional bosonic superfluid which naturally incorporates crystalline topological defects. We extract static interactions between these defects and investigate several continuous quantum transitions triggered by the Higgs condensation of vortex vacancies/interstitials and dislocations. We propose that the quantum melting of the vortex crystal towards the hexatic or smectic phase may occur via a pair of continuous transitions separated by an intermediate vortex superso…
In this paper we discuss new flavored space-like defects in confined QCD which can be considered as the Euclidean extended instantons carrying the topologically quantized currents. We focus on the simplest 1d space-like defect - the S-Skyrmion solution extended in one space coordinate and localized in Euclidean time. It can be identified both in the holographic QCD and in the Chiral Perturbation Theory(ChPT). The Skyrmion charges get transformed into the corresponding currents for S-Skyrmion. The analogy with the Thouless pump and the quantum phase slip phenomena is mentioned.
The partition function of $\mathcal{N}=2$ super Yang-Mills theories with arbitrary simple gauge group coupled to a self-dual $Ω$-background is shown to be fully determined by studying the renormalization group equations relevant to the surface operators generating its one-form symmetries. The corresponding system of equations results in a ${\it non-autonomous}$ Toda chain on the root system of the Langlands dual, the evolution parameter being the RG scale. A systematic algorithm computing the full multi-instanton corrections is derived in terms of recursion relations whose gauge theoretical so…
Our understanding of the mechanism by which topological defects are formed in symmetry breaking phase transitions has recently changed. We examine the non-equilibrium dynamics of defect formation for weakly-coupled global O(N) theories possessing vortices (strings) and monopoles. It is seen that, as domains form and grow, defects are swept along on their boundaries at a density of about one defect per coherence area (strings) or per coherence volume (monopoles).
In this work, we derive a recently proposed Abelian model to describe the interaction of correlated monopoles, center vortices, and dual fields in three dimensional SU(2) Yang-Mills theory. Following recent polymer techniques, special care is taken to obtain the end-to-end probability for a single interacting center vortex, which constitutes a key ingredient to represent the ensemble integration.
A topological defect in the form of the Abrikosov-Nielsen-Olesen vortex is considered as a gauge-flux-carrying tube that is impenetrable for quantum matter. The relativistic spinor matter field is quantized in the vortex background in $2+1$-dimensional conical space-time which is a section orthogonal to the vortex axis; the most general set of boundary conditions ensuring the impenetrability of the vortex core is employed. We find the induced vacuum current circulating around the vortex and the induced vacuum magnetic field strength pointing along the vortex axis. The requirement of finiteness…
We calculate the energetic structure of defect cores and propose controlled methods to imprint a nontrivially entangled vortex pair that undergoes non-Abelian vortex reconnection in a biaxial nematic spin-2 condensate. For a singular vortex, we find three superfluid cores in addition to depletion of the condensate density. These exhibit order parameter symmetries that are different from the discrete symmetry of the biaxial nematic phase, forming an interface between the defect and the bulk superfluid. We provide a detailed analysis of phase mixing in the resulting vortex cores and find an inst…
We consider 5d supersymmetric gauge theories with unitary groups in the $Ω$-background and study codim-2/4 BPS defects supported on orthogonal planes intersecting at the origin along a circle. The intersecting defects arise upon implementing the most generic Higgsing (geometric transition) to the parent higher dimensional theory, and they are described by pairs of 3d supersymmetric gauge theories with unitary groups interacting through 1d matter at the intersection. We explore the relations between instanton and generalized vortex calculus, pointing out a duality between intersecting defects s…
Gieseker-Nakajima moduli spaces $M_{k}(n)$ parametrize the charge $k$ noncommutative $U(n)$ instantons on ${\bf R}^{4}$ and framed rank $n$ torsion free sheaves $\mathcal{E}$ on ${\bf C\bf P}^{2}$ with ${\rm ch}_{2}({\mathcal{E}}) = k$. They also serve as local models of the moduli spaces of instantons on general four-manifolds. We study the generalization of gauge theory in which the four dimensional spacetime is a stratified space $X$ immersed into a Calabi-Yau fourfold $Z$. The local model ${\bf M}_{k}({\vec n})$ of the corresponding instanton moduli space is the moduli space of charge $k$ …
We successfully exhaust the complete set of exact solutions of non-Abelian vortices in a quiver gauge theory, that is, the S[U(N) x U(N)] gauge theory with a bi-fudamental scalar field on a hyperbolic plane with a certain curvature, from SO(3)-invariant SU(2N) Yang-Mills instanton solutions. This work provides, for the first time, exact non-Abelian vortex solutions. We establish the ADHM construction for non-Abelian vortices and identify all the moduli parameters and the complete moduli space.
By making use of the complete decomposition of SO(3) spin connection, the topological defect in 3-dimensional Euclidean gravity is studied in detail. The topological structure of disclination is given as the combination of a monopole structure and a vortex structure. Furthermore, the Kac-Moody algebra generated by the monopole and vortex is discussed in three dimensional Chern-Simons theory.
We show that the degrees of freedom associated with magnetic monopole- and vortexlike gluonic configurations make a strong contribution to the anomaly of the energy-momentum tensor of Yang-Mills theory in the deconfinement phase immediately above the critical temperature. As is well known in zero-temperature Yang-Mills theory, the monopoles and vortices are constituents of a generic gluonic object in which the two neighbor monopoles are connected together by a segment of vortex string. Our results provide evidence that the monopole-vortex chains in SU(2) gauge theory and their SU(3) counterpar…
