Making use of ansatzs for the form fields in the 10d type IIA supergravity version of the ABJM model, we come with a solution in the Euclidean signature recognized as a monopole instanton-like object. Indeed we will see that we can have a (anti) self-dual solution at a special limit. While as a topological object, its back-reaction on the original background should be ignorable, we show the energy-momentum tensors vanish exactly. On the field theory side, the best counterpart is an U(1) gauge field of a gauge transformation. To adjust with bulk, the gauge field must prompt to a dynamic one without adding any kinetic term for this dual photon except a marginal, abelian AB-type Chern-Simons term on the boundary. We will see how both side solutions match next to another confirmation from some earlier works of this vortex-particle duality.
The coefficients of the membrane instantons in the ABJM theory are known to be quadratic polynomials of the chemical potential. We show that, after deforming the ABJM theory into more general superconformal Chern-Simons theories labelled by (q,p) where the original ABJM theory corresponds to (q,p)=(1,1), we can decompose the membrane instanton into three types of non-perturbative effects with constant coefficients independent of the chemical potential. We find that, although these constants contain poles at certain values of q and p including the ABJM case, all of the poles cancel among themse…
We study the partition function of the orbifold ABJM theory on $S^3$, which is the $\mathcal{N}=4$ necklace quiver Chern-Simons-matter theory with alternating levels, in the Fermi gas formalism. We find that the grand potential of the orbifold ABJM theory is expressed explicitly in terms of that of the ABJM theory. As shown previously, the ABJM grand potential consists of the naive but primary non-oscillatory term and the subsidiary infinitely-replicated oscillatory terms. We find that the subsidiary oscillatory terms of the ABJM theory actually give a non-oscillatory primary term of the orbif…
We investigate another supersymmetric Chern-Simons theory called the orientifold ABJM theory, which replaces the unitary supergroup structure of the ABJM theory with an orthosymplectic one. Its non-perturbative structure is completely clarified by considering the duplication of the quiver.
We study instanton corrections to the vacuum expectation value (VEV) of 1/6 BPS Wilson loops in ABJM theory from the Fermi gas approach. We mainly consider Wilson loops in the fundamental representation and winding Wilson loops, but we also initiate the study of Wilson loops with two boundaries. We find that the membrane instanton corrections to the Wilson loop VEV are determined by the refined topological string in the Nekrasov-Shatashvili limit, and the pole cancellation mechanism between membrane instantons and worldsheet instantons works also in the Wilson loop VEVs as in the case of the p…
We review recent progress in determining the partition function of the ABJM theory in the large N expansion, including all of the perturbative and non-perturbative corrections. Especially, we will focus on how these exact expansions are obtained from various beautiful relations to Fermi gas system, topological string theory, integrable model and supergroup.
In the presence of a background supergravity flux, N M2-branes will expand via the Myers effect into M5-branes wrapped on a fuzzy three-sphere. In previous work the fluctuations of the M2-branes were shown to be described by the five-dimensional Yang-Mills gauge theory associated to D4-branes. We show that the ABJM prescription for eleven-dimensional momentum in terms of magnetic flux lifts to an instanton flux of the effective five-dimensional Yang-Mills theory on the sphere, giving an M-theory interpretation for these instantons.
We extend the hermitian three-algebra formulation of ABJM theory to include $U(1)$ factors. With attention payed to extra $U(1)$ factors, we refine the classification of $\N=6$ ABJM theories. We argue that essentially the only allowed gauge groups are $SU(N)\times SU(N)$, $U(N)\times U(M)$ and $Sp(N)\times U(1)$ and that we have only one independent Chern-Simons level in all these cases. Our argument is based on integrality of the $U(1)$ Chern-Simons levels and supersymmetry. A relation between monopole operators and Wilson lines in Chern-Simons theory suggests certain gauge representations of…
We study the partition function of the orientifold ABJM theory, which is a superconformal Chern-Simons theory associated with the orthosymplectic supergroup. We find that the partition function associated with any orthosymplectic supergroup can be realized as the partition function of a Fermi gas system whose density matrix is identical to that associated with the corresponding unitary supergroup with a projection to the even or odd chirality. Furthermore we propose an identity which gives directly all of the Gopakumar-Vafa invariants for the worldsheet instanton effects in the chirally projec…
We study the instanton effects of the ABJM partition function using the Fermi gas formalism. We compute the exact values of the partition function at the Chern-Simons levels k=1,2,3,4,6 up to N=44,20,18,16,14 respectively, and extract non-perturbative corrections from these exact results. Fitting the resulting non-perturbative corrections by their expected forms from the Fermi gas, we determine unknown parameters in them. After separating the oscillating behavior of the grand potential, which originates in the periodicity of the grand partition function, and the worldsheet instanton contributi…
The partition function of the ABJM theory receives non-perturbative corrections due to instanton effects. We study these non-perturbative corrections, including bound states of worldsheet instantons and membrane instantons, in the Fermi-gas approach. We require that the total non-perturbative correction should be always finite for arbitrary Chern-Simons level. This finiteness is realized quite non-trivially because each bound state contribution naively diverges at some levels. The poles of each contribution should be canceled out in total. We use this pole cancellation mechanism to find unknow…
We study the BPS spectrum of monopole operators in ABJM theory. First we work out the complete spectrum of the chiral ring by using a semiclassical analysis of the field theory compactified on a two sphere. By properly taking into account the full quantization condition of monopole charges, we show that the moduli space of ABJM theory with Chern-Simons level k, -k is a particular Z_k cover of the symmetric product of C^4/Z_k. We also work out in detail the spectrum of fluctuations around half-BPS monopole configurations and we find candidate states for a dual BPS configuration to the giant tor…
We consider the quantum spectral problem appearing the Fermi gas formulation of the ABJM (Aharony-Bergman-Jafferis-Maldacena) matrix model. This is known to related to the refined topological string on local P^1*P^1 Calabi-Yau geometry. In the ABJM setting the problem is formulated by an integral equation, and is somewhat different from the one formulated directly in terms of the Calabi-Yau geometry and studied in our earlier paper. We use the similar method in our earlier paper to determine the non-perturbative contributions to the quantum phase volume in the ABJM case from the Bohr-Sommerfel…
We construct monopole operators for 3d Yang-Mills-matter theories and Chern-Simons-matter theories in canonical formalism. In this framework, monopole operators, although as the disorder operators, could be written in terms of the fundamental fields of the theory, and thus could be treated in the same way as the ordinary operators. We study the properties of the constructed monopole operators. In Chern-Simons-matter theories, monopole operators transform as the local operators with the classical conformal dimension $ 0 $ under the action of the dilation and are also covariantly constant. In su…
Using the matrix model which calculates the exact free energy of ABJM theory on S^3 we study non-perturbative effects in the large N expansion of this model, i.e., in the genus expansion of type IIA string theory on AdS4xCP^3. We propose a general prescription to extract spacetime instanton actions from general matrix models, in terms of period integrals of the spectral curve, and we use it to determine them explicitly in the ABJM matrix model, as exact functions of the 't Hooft coupling. We confirm numerically that these instantons control the asymptotic growth of the genus expansion. Further…
We study the large N expansion of the partition function of the quiver superconformal Chern-Simons theories deformed by two continuous parameters which correspond to the general R-charge assignment to the matter fields. Though the deformation breaks the conformal symmetry, we find that the partition function shares various structures with the superconformal cases, such as the Airy function expression of the perturbative expansion in 1/N with the overall constant A(k) related to the constant map in the ABJM case through a simple rescaling of k. We also identify five kinds of the non-perturbativ…
We study vacuum expectation values (VEVs) of circular half BPS Wilson loops in arbitrary representations in ABJM theory. We find that those in hook representations are reduced to elementary integrations thanks to the Fermi gas formalism, which are accessible from the numerical studies similar to the partition function in the previous studies. For non-hook representations, we show that the VEVs in the grand canonical formalism can be exactly expressed as determinants of those in the hook representations. Using these facts, we can study the instanton effects of the VEVs in various representation…
We study the supersymmetry enhancement of ABJM theory. Starting from a ${\cal N}=2$ supersymmetric Chern-Simons matter theory with gauge group U(2)$\times$U(2) which is a truncated version of the ABJM theory, we find by using the monopole operator that there is additional ${\cal N}=2$ supersymmetry related to the gauge group. We show this additional supersymmetry can combine with ${\cal N}=6$ supersymmetry of the original ABJM theory to an enhanced ${\cal N}=8$ SUSY with gauge group U(2)$\times$U(2) in the case $k=1,2$. We also discuss the supersymmetry enhancement of the ABJM theory with U($N…
It was known that the ABJM matrix model is dual to the topological string theory on a Calabi-Yau manifold. Using this relation it was possible to write down the exact instanton expansion of the partition function of the ABJM matrix model. The expression consists of a universal function constructed from the free energy of the refined topological string theory with an overall topological invariant characterizing the Calabi-Yau manifold. In this paper we explore two other superconformal Chern-Simons theories of the circular quiver type. We find that the partition function of one theory enjoys the…
We describe a method which allows one to study hidden symmetries in a large class of strongly coupled supersymmetric gauge theories in three dimensions. We apply this method to the ABJM theory and to the infrared limit of N=4 SQCD with adjoint and fundamental matter. We show that the U(N) ABJM model with Chern-Simons level k=1 or k=2 has hidden N=8 supersymmetry. Hidden supersymmetry is also shown to occur in N=4 d=3 SQCD with one fundamental and one adjoint hypermultiplet. The latter theory, as well as the U(N) ABJM theory at k=1, are shown to have a decoupled free sector. This provides evide…
Budapest University of Technology and Economics（布达佩斯技术与经济大学）；HUN-REN Wigner Research Centre for Physics（匈牙利研究与教育网络温格物理研究中心）；Institut de Physique Théorique, Université Paris Saclay, CNRS（巴黎萨克雷大学CNRS理论物理研究所）
The ABJM theory refers to superconformal Chern-Simons-matter theory with product gauge group U(L)xU(R) and level +k, -k, respectively. The theory is a candidate for worldvolume dynamics of M2-branes sitting at C(4)/Z(k)k. By utilizing monopole operators, we prove that ABJM theory gets enhanced N=8 supersymmetry and SO(8) R-symmetry at Chern-Simons levels k=1,2. We first show that the ABJM Lagrangian can be written in a manifestly SO(8) invariant form up to certain extra terms. We then show that upon integrating out Chern-Simons gauge fields these extra terms vanish precisely at levels k=1,2. U…
We show that baryon-like operators exist in the N=6 superconformal Chern-Simons-matter theory constructed by Aharony, Bergman, Jafferis and Maldacena (ABJM). This involves the introduction of Wilson(or 't Hooft) lines ending on baryon-like operators and we show that the presence of such lines cannot be detected by the fields in the theory. The same construction can be used to make magnetic monopoles charged under the diagonal U(1) gauge group with arbitrary non-integral charge. If we do not include such monopole configurations in the path integral, the moduli space is known to be given by the …
Correlation between instantons and QCD-monopoles is studied in the abelian-gauge-fixed QCD. From a simple topological consideration, instantons are expected to appear only around the QCD-monopole trajectory in the abelian-dominating system. The QCD-monopole in the multi-instanton solution is studied in the Polyakov-like gauge, where $A_4(x)$ is diagonalized. The world line of the QCD-monopole is found to be penetrate the center of each instanton. For the single-instanton solution, the QCD-monopole trajectory becomes a simple straight line. On the other hand, in the multi-instanton system, the …
We study monopole-instantons in M2-brane theories, focussing on the ABJM class of Chern-Simons gauge theories coupled to matter. We calculate calculate explicitly the 8-fermion term in the effective action induced by these monopole-instantons, and discuss their role in resolving a classical singularity in the moduli space. The results are compared with monopole-instantons in N=8 3d SYM and D-brane theories, as well the dual supergravity description as a membrane scattering process.
A relation between the total instanton number and the quantum-numbers of magnetic monopoles that arise in general Abelian gauges in SU(2) Yang-Mills theory is established. The instanton number is expressed as the sum of the `twists' of all monopoles, where the twist is related to a generalized Hopf invariant. The origin of a stronger relation between instantons and monopoles in the Polyakov gauge is discussed.
We show that the BPS configurations of uniform field strength can be interpreted as those for sheets of infinite number of BPS magnetic monopoles, and found that the number of normalizable zero modes per each magnetic monopole charge is four. We identify monopole sheets as the intersecting planes of D3 branes. Similar analysis on self-dual instanton configurations is worked out and the number of zero modes per each instanton number is found to match that of single isolated instanton.
We review the physics of topological objects in QCD. Topics include: solitons, vortices, magnetic monopoles, instantons, (effective theories of) confinement.
We show that in certain non-Abelian Born-Infeld-Higgs theories there are monopole solutions obeying Bogomol'nyi-like equations in the Prasad-Sommerfield limit. We also discuss the existence of instanton solution in an SU(2) Born-Infeld theory.
We construct families of SO(3)-symmetric charge 1 instantons and calorons on the space H^3 x R. We show how the calorons include instantons and hyperbolic monopoles as limiting cases. We show how Euclidean calorons are the flat space limit of this family.
We discuss three classes of solitonic solutions in string theory: instantons, monopoles and string-like solitons. Instantons may provide a nonperturbative understanding of the vacuum structure of string theory, while monopoles may appear in string predictions for grand unification. The particular monopole solution shown has finite action as the result of the cancellation of gauge and gravitational singularities, a feature, which if it survives quantization, may yield insight into the structure of string theory as a finite theory of quantum gravity. In $D=10$, both monopoles and instantons poss…
We extend the relation between instanton and monopole solutions of the selfduality equations in SU(2) gauge theory to noncommutative space-times. Using this approach and starting from a noncommutative multi-instanton solution we construct a U(2) monopole configuration which lives in 3 dimensional ordinary space. This configuration resembles the Wu-Yang monopole and satisfies the selfduality (Bogomol'nyi) equations for a U(2) Yang-Mills-Higgs system.
We calculate the non-abelian R-charges of BPS monopole operators in three-dimensional gauge theories with N=3 supersymmetry. This class of models includes ABJM theory, the proposed gauge theory dual of M-theory on AdS_4 x S^7/Z_k, as a special case. In the UV limit of the N=3 theories the Yang-Mills coupling becomes weak and the monopole operators are described by classical backgrounds. This allows us to find their SU(2)_R charges in a one-loop computation which by virtue of the non-renormalization of non-abelian R-charges yields the exact result for any value of the coupling. The spectrum of …
Motivated by the recent D-brane constructions of world-volume monopoles and instantons, we study the supersymmetric SU(N) Yang-Mills theory on $S^1 \times R^{3+1}$, spontaneously broken by a Wilson loop. In addition to the usual N-1 fundamental monopoles, the N-th BPS monopole appears from the Kaluza-Klein sector. When all N monopoles are present, net magnetic charge vanishes and the solution can be reinterpreted as a Wilson-loop instanton of unit Pontryagin number. The instanton/multi-monopole moduli space is explicitly constructed, and seen to be identical to a Coulomb phase moduli space of …
We study how instantons arise in the low energy effective theory of the SU(2) Yang-Mills theory in the context of the non-linear sigma model recently propose by Faddeev and Niemi. We find a simple relation between the instanton number $ν$ and the charge m of the monopole that appears in the effective theory. It is given by $ν= m Φ/(2π)$, where $Φ$ is the quantized flux associated with a U(1) gauge field passing through the loop formed by the singularity of the monopole.
The physics of the bare Seiberg-Witten action, without supersymmetric partners, is considered in the framework of standard Quantum Field Theory. The topological analysis related to the solutions of the Seiberg-Witten equations is performed and the phase structure of the model is analysed.
In recent work, several classes of solitonic solutions of string theory with higher-membrane structure have been obtained. These solutions can be classified according to the symmetry possessed by the solitons in the subspace of the spacetime transverse to the membrane. Solitons with four-dimensional spherical symmetry represent instanton solutions in string theory, while those with three-dimensional spherical symmetry represent magnetic monopole-type solutions. For both of these classes, we discuss bosonic as well as heterotic solutions.
We consider states with large angular momentum to facilitate the study of the M-theory regime of the AdS_4/CFT_3 correspondence. We study the duality between M-theory in AdS_4xS^7/Z_k and the ABJM N=6 Chern-Simons-matter theory with gauge group U(N)xU(N) and level k, taking N large and k of order 1. In this regime the lack of an explicit formulation of M-theory in AdS_4xS^7/Z_k makes the gravity side difficult, while the CFT is strongly coupled and the planar approximation is not applicable. To overcome these difficulties, we focus on states on the gravity side with large angular momentum J>>1…
We calculate instanton corrections to three dimensional gauge theories with N=4 and N=8 supersymmetry and SU(n) gauge groups. The N=4 results give new information about the moduli space of n BPS SU(2) monopoles, including the leading order non-pairwise interaction terms. A few comments are made on the relationship of the N=8 results to membrane scattering in matrix theory.
We calculate fermionic response in domain wall backgrounds of four-dimensional gauged supergravity interpolating between distinct stable AdS vacua. The backgrounds, found by Bobev et al., are holographically dual to zero-temperature states of ABJM theory at finite density for monopole charge and are similar to zero-temperature limits of holographic superconductors, but with a symmetry-breaking source as well. The condensed scalar mixes charged and neutral fields dual to composite fermionic operators in the Dirac equations. Both gapped and gapless bands of stable quasiparticles are found.
The Kaluza-Klein monopole is a well known object in both gravity and string theory, related by T-duality to a "smeared" NS5-brane which retains the isometry around the duality circle. As the true NS5-brane solution is localized at a point on the circle, duality implies that the Kaluza-Klein monopole should show some corresponding behavior. In this paper, we express the Kaluza-Klein monopole as a gauged linear sigma model in two dimensions and show that worldsheet instantons give corrections to its geometry. These corrections can be understood as a localization in "winding space" which could be…
We consider the superposition of infinitely many instantons on a circle in R^4. The construction yields a self-dual solution of the Yang-Mills equations with action density concentrated on the ring. We show that this configuration is reducible in which case magnetic charge can be defined in a gauge invariant way. Indeed, we find a unit charge monopole (worldline) on the ring. This is an analytic example of the correlation between monopoles and action/topological density, however with infinite action. We show that both the Maximal Abelian Gauge and the Laplacian Abelian Gauge detect the monopol…
We search for an abelian description of the Yang-Mills instantons on certain eight dimensional manifolds with the special holonomies $Spin(7)$ and SU(4). By mimicing the Seiberg-Witten theory in four dimensions, we propose a set of monopole-like equations governing the 8-dimensional U(1) connections and spinors, which are supposed to be the dual theory of the nonabelian instantons. We also give a naive test of the generalized $S$-duality in the abelian sector of 8-dimensional Yang-Mills theory. Some problems in this approach are pointed out.
We consider Yang-Mills theory on manifolds ${\mathbb R}\times X$ with a $d$-dimensional Riemannian manifold $X$ of special holonomy admitting gauge instanton equations. Instantons are considered as particle-like solutions in $d+1$ dimensions whose static configurations are concentrated on $X$. We study how they evolve in time when considered as solutions of the Yang-Millsequations on ${\mathbb R}\times X$ with moduli depending on time $t\in{\mathbb R}$. It is shown that in the adiabatic limit, when the metric in the $X$ direction is scaled down, the classical dynamics of slowly moving instanto…
We discuss non-perturbative effects in the ABJM model due to monopole instantons. We begin by constructing the instanton solutions in the $U(2)\times U(2)$ model, explicitly, and computing the Euclidean action. The Wick-rotated Lagrangian is complex and its BPS monopole instantons are found to be a delicate version of the usual 't Hooft-Polyakov monopole solutions. They are generically 1/3 BPS but become 1/2 BPS at special locus in the moduli space of two M2-branes, yet each instanton carries eight fermionic zero modes, regardless of the vacuum choice. The low energy effective action induced b…
In this paper, the metric on the moduli space of the k=1 SU(n) periodic instanton -or caloron- with arbitrary gauge holonomy at spatial infinity is explicitly constructed. The metric is toric hyperKaehler and of the form conjectured by Lee and Yi. The torus coordinates describe the residual U(1)^{n-1} gauge invariance and the temporal position of the caloron and can also be viewed as the phases of n monopoles that constitute the caloron. The (1,1,..,1) monopole is obtained as a limit of the caloron. The calculation is performed on the space of Nahm data, which is justified by proving the isome…
In our previous work arXiv:1704.08675, we pointed out that various multi-cut solutions exist in the Chern-Simons (CS) matrix models at large-$N$ due to a curious structure of the saddle point equations. In the ABJM matrix model, these multi-cut solutions might be regarded as the condensations of the D2-brane instantons. However many of these multi-cut solutions including the ones corresponding to the condensations of the D2-brane instantons were obtained numerically only. In the current work, we propose an ansatz for the multi-cut solutions which may allow us to derive the analytic expressions…
We introduce two ansatzs for the 3-form potential of Euclidean 11d supergravity on skew-whiffed AdS_4 X S^7 background which results in two scalar modes with m^2=-2 on AdS_4. Being conformally coupled with a quartic interaction it is possible to find the exact solutions of the scalar equation on this background. These modes turn out to be invariant under SU(4) subgroup of SO(8) isometry group, whereas there are no corresponding SU(4) singlet BPS operators of dimensions one or two on the boundary ABJM theory. Noticing the interchange of 8_s and 8_c representations under skew-whiffing in the bul…
Various string theory realizations of three-dimensional gauge theories relate them to gravitational instantons, Nahm equations and monopoles. We use this correspondence to model self-dual gravitational instantons of $D_k$-type as moduli spaces of singular monopoles, find their twistor spaces and metrics.
We show that the instantons induce the abelian monopoles in the Maximal Abelian Projection of $SU(2)$ gluodynamics. As an example we consider the case of one instanton and the case of a set of instantons arranged along a straight line. The abelian monopoles which are induced by instantons may play some role in the confinement scenario.
In our previous paper (Fukui et al., 2010), we have shown that the Jackiw-Nohl-Rebbi two-instanton generates a circular loop of magnetic monopole in the four-dimensional Euclidean SU(2) Yang-Mills theory. On the other hand, it is claimed in Brower, Orginos and Tan (1997) that the 't Hooft two-instanton does not generate magnetic monopole loop. It seems that two results are inconsistent with each other, since the JNR two-instanton converges to the 't Hooft two-instanton in a certain limit. In this paper, we clarify that two results are compatible with each other by demonstrating how the magneti…
We construct a number of explicit examples of hyperbolic monopoles, with various charges and often with some platonic symmetry. The fields are obtained from instanton data in four-dimensional Euclidean space that are invariant under a circle action, and in most cases the monopole charge is equal to the instanton charge. A key ingredient is the identification of a new set of constraints on ADHM instanton data that are sufficient to ensure the circle invariance. Unlike for Euclidean monopoles, the formulae for the squared Higgs field magnitude in the examples we construct are rational functions …
Three dimensional Euclidean gravity in the dreibein-spin connection formalism is investigated. We use the monopole-instanton ansatz for the dreibein and the spin connection. The equations of motion are solved. We point out a two dimensional solution with a vanishing action.
We investigate 't Hooft-Mandelstam monopoles in QCD in the presence of a single classical instanton configuration. The solution to the Maximal Abelian projection is found to be a circular monopole trajectory with radius $R$ centered on the instanton. At zero loop radius, there is a marginally stable (or flat) direction for loop formation to $O(R^4 logR)$. We argue that loops will form, in the semi-classical limit, due to small perturbations such as the dipole interaction between instanton anti-instanton pairs. As the instanton gas becomes a liquid, the percolation of the monopole loops may the…
Monopoles and instantons are sheets (membranes) and strings in d=5+1, respectively, and instanton strings can terminate on monopole sheets. We consider a pair of monopole and anti-monopole sheets which is unstable to decay and results in a creation of closed instanton strings. We show that when an instanton string is stretched between the monopole sheets, there remains a new topological soliton of codimension five after the pair annihilation, i.e., a twisted closed instanton string or a knotted instanton.
It is well known that there are no static non-Abelian monopole solutions in pure Yang-Mills theory on Minkowski space R^{3,1}. We show that such solutions exist in SU(N) gauge theory on the spaces R^2\times S^2 and R^1\times S^1\times S^2 with Minkowski signature (-+++). In the temporal gauge they are solutions of pure Yang-Mills theory on T^1\times S^2, where T^1 is R^1 or S^1. Namely, imposing SO(3)-invariance and some reality conditions, we consistently reduce the Yang-Mills model on the above spaces to a non-Abelian analog of the ϕ^4 kink model whose static solutions give SU(N) monopole (-…
