Sergey Cherkis

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• Cherkis, S. A., Braden, H. W., & Quinones, J. M. (2023). Construction of exact solutions to Nahm’s equations for the multimonopole. Journal of Mathematical Physics, 64(1), 011701. doi:10.1063/5.0098288
• Bielawski, R., Borchard, Y., & Cherkis, S. A. (2022). Deformations of instanton metrics.
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  We discuss a class of bow varieties which can be viewed as Taub-NUTdeformations of moduli spaces of instantons on noncommutative $\mathbb{R}^4$.Via the Generalized Legendre Transform we find the K\"ahler potential on eachof these spaces.[Journal_ref: ]
• Braden, H. W., Cherkis, S. A., & Quinones, J. M. (2022). Construction of Exact Solutions to Nahm's Equations for the Multimonopole.
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  We construct high rank solutions to Nahm's equations for boundary conditionsthat correspond to the Dirac multimonopole. Here, the spectral curve isexplicitly known and we achieve the integration by constructing a basis ofpolynomial tuples that forms a frame for the flow of the eigenline bundle overthe curve.[Journal_ref: ]
• Cherkis, S. (2021). Instantons on multi-Taub-NUT Spaces II: Bow Construction. submitted to Journal of Differential Geometry.
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  Unitary anti-self-dual connections on Asymptotically Locally Flat (ALF)hyperk\"ahler spaces are constructed in terms of data organized in a bow. Bowsgeneralize quivers, and the relevant bow gives rise to the underlying ALF spaceas the moduli space of its particular representation -- the smallrepresentation. Any other representation of that bow gives rise toanti-self-dual connections on that ALF space. We prove that each resulting connection has finite action, i.e. it is aninstanton. Moreover, we derive the asymptotic form of such a connection andcompute its topological class.[Journal_ref: ]
• Cherkis, S. (2021). Magnetic topology in coupled binaries, spin-orbital resonances, and flares. The Astrophysical Journal, 923(1), 13. doi:https://doi.org/10.3847/1538-4357/ac29b8
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  We consider topological configurations of the magnetically coupled spinningstellar binaries (e.g., merging neutron stars or interacting star-planetsystems). We discuss conditions when the stellar spins and the orbital motionnearly `compensate' each other, leading to very {\it slow} overall winding ofthe coupled magnetic fields; slowly winding configurations allow gradualaccumulation of magnetic energy, that is eventually released in a flare whenthe instability threshold is reached. We find that this slow winding can beglobal and/or local. We describe the topology of the relevant space$\mathbb{F}=T^1S^2$ as the unit tangent bundle of the two-sphere and findconditions for slowly winding configurations in terms of magnetic moments,spins and orbital momentum. These conditions become ambiguous near thetopological bifurcation points; in certain cases they also depend on therelative phases of the spin and orbital motions. In the case of mergingmagnetized neutron stars, if one of the stars is a millisecond pulsar, spinningat $\sim$ 10 msec, the global resonance $\omega_1+\omega_2= 2 \Omega$(spin-plus beat is two times the orbital period) occurs approximately a secondbefore the merger; the total energy of the flare can be as large as $10\%$ ofthe total magnetic energy, producing bursts of luminosity $\sim 10^{44}$ ergs$^{-1}$. Higher order local resonances may have similar powers, since theamount of involved magnetic flux tubes may be comparable to the total connectedflux.[Journal_ref: ]
• Cherkis, S. A., Larrain-Hubach, A., & Stern, M. (2021).

  Instantons on multi-Taub-NUT Spaces I: Asymptotic Form and Index Theorem

  . J. Differential Geom., 119(1), 1-72. doi:10.4310/jdg/1631124166
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  We study finite action anti-self-dual Yang-Mills connections on the multi-Taub-NUT space. We establish the curvature and the harmonic spinors decay rates and compute the index of the associated Dirac operator. This is the first in a series of papers proving the completeness of the bow construction of instantons on multi-Taub-NUT spaces and exploring it in detail.
• Cherkis, S., Stern, M., & Larrain-Hubach, A. (2021). Instantons on multi-Taub-NUT Spaces I: Asymptotic Form and Index Theorem. Journal of Differential Geometry, 119(1), 1-72. doi:10.4310/jdg/1631124166
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  We study finite action anti-self-dual Yang-Mills connections on themulti-Taub-NUT space. We establish the curvature and the harmonic spinors decayrates and compute the index of the associated Dirac operator. This is the first in a series of papers proving the completeness of the bowconstruction of instantons on multi-Taub-NUT spaces and exploring it in detail.[Journal_ref: ]
• Cherkis, S. (2020). INSTANTONS AND BOWS FOR THE CLASSICAL GROUPS. The Quarterly Journal Of Mathematics, Quart. J. Math. 00(2020), 1–48; doi:10.1093/qmath/haaa034, 1–48. doi:doi:10.1093/qmath/haaa034
• Cherkis, S. (2021). Instantons and Bows for the Classical Groups. The Quarterly Journal of Mathematics, 72(1-2), 339–386. doi:10.1093/qmath/haaa034
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  The construction of Atiyah, Drinfeld, Hitchin, and Manin [ADHM78] providedcomplete description of all instantons on Euclidean four-space. It was extendedby Kronheimer and Nakajima to instantons on ALE spaces, resolutions oforbifolds $\mathbb{R}^4/\Gamma$ by a finite subgroup $\Gamma\subset SU(2).$ Weconsider a similar classification, in the holomorphic context, of instantons onsome of the next spaces in the hierarchy, the ALF multi-Taub-NUT manifolds,showing how they tie in to the bow solutions to Nahm's equations [Che09] viathe Nahm correspondence. Recently in [Nak18a] and [NT17], based on [Nak03],Nakajima and Takayama constructed the Coulomb branch of the moduli space ofvacua of a quiver gauge theory, tying them to the same space of bow solutions.One can view our construction as describing the same manifold as the Higgsbranch of the mirror gauge theory [COS11]. Our construction also yields themonad construction of holomorphic instanton bundles on the multi-Taub-NUT spacefor any classical compact Lie structure group.[Journal_ref: ]
• Hurtubise, J., & Cherkis, S. A. (2020). Instantons and Bows for the Classical Groups. Quarterly Journal of Mathematics. doi:10.1093/qmath/haaa034
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  The construction of Atiyah, Drinfeld, Hitchin, and Manin [ADHM78] provided complete description of all instantons on Euclidean four-space. It was extended by Kronheimer and Nakajima to instantons on ALE spaces, resolutions of orbifolds $\mathbb{R}^4/\Gamma$ by a finite subgroup $\Gamma\subset SU(2).$ We consider a similar classification, in the holomorphic context, of instantons on some of the next spaces in the hierarchy, the ALF multi-Taub-NUT manifolds, showing how they tie in to the bow solutions to Nahm's equations [Che09] via the Nahm correspondence. Recently in [Nak18a] and [NT17], based on [Nak03], Nakajima and Takayama constructed the Coulomb branch of the moduli space of vacua of a quiver gauge theory, tying them to the same space of bow solutions. One can view our construction as describing the same manifold as the Higgs branch of the mirror gauge theory [COS11]. Our construction also yields the monad construction of holomorphic instanton bundles on the multi-Taub-NUT space for any classical compact Lie structure group.
• , S. C., & , R. C. (2019). Crystal Volumes and Monopole Dynamics.
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  The low velocity dynamic of a doubly periodic monopole, also called amonopole wall or monowall for short, is described by geodesic motion on itsmoduli space. This moduli space is hyperkaehler and non-compact. We establish arelation between the Kaehler potential of this moduli space and the volume of aregion in Euclidean three-space cut out by a plane arrangement associated witheach monowall.[Journal_ref: ]
• Cherkis, S. (2019). Monads for Instantons and Bows. Advances in Theoretical and Mathematical Physics, 23(1).
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  Instantons on the Taub-NUT space are related to `bow solutions' via ageneralization of the ADHM-Nahm transform. Both are related to complexgeometry, either via the twistor transform or via the Kobayashi-Hitchincorrespondence. We explore various aspects of this complex geometry, exhibitingequivalences. For both the instanton and the bow solution we produce two monadsencoding each of them respectively. Identifying these monads we establish theone-to-one correspondence between the instanton and the bow solution.[Journal_ref: ]
• Cherkis, S. A., & Cross, R. (2019). Crystal Volumes and Monopole Dynamics. (arXiv:1906.04454v2 [hep-th] UPDATED). arXiv High Energy Physics - Theory.
• Cherkis, S. A., & Cross, R. (2019). Crystal Volumes and Monopole Dynamics. Communications in Mathematical Physics.
• Cherkis, S. A., & Hurtubise, J. (2019). Monads for instantons and bows. Advances in Theoretical and Mathematical Physics, 23(1), 167-251. doi:10.4310/atmp.2019.v23.n1.a4
• Cherkis, S., Zaitsev, D., & O'Hara, C. (2016). A Compact Expression for Periodic Instantons. Journal of Geometry and Physics, 110, 382-392.
• Cherkis, S. A., & Hurtubise, J. (2017). Monads for Instantons and Bows. arXiv: Differential Geometry.
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  Instantons on the Taub-NUT space are related to `bow solutions' via a generalization of the ADHM-Nahm transform. Both are related to complex geometry, either via the twistor transform or via the Kobayashi-Hitchin correspondence. We explore various aspects of this complex geometry, exhibiting equivalences. For both the instanton and the bow solution we produce two monads encoding each of them respectively. Identifying these monads we establish the one-to-one correspondence between the instanton and the bow solution.
• Cherkis, S. (2015). Octonions, Monopoles, and Knots. Letters in Mathematical Physics, 105, 641–659. doi:10.1007/s11005-015-0755-0
• Cherkis, S. A. (2015). Octonions, Monopoles, and Knots. Letters in Mathematical Physics, 105(5), 641-659. doi:10.1007/s11005-015-0755-0
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  Witten’s approach to Khovanov homology of knots is based on the ve-dimensional system of partial dierential equations, which we call Haydys-Witten equations. We argue for a one-to-one correspondence between its solutions and solutions of the seven-dimensional system of equations. The latter can be formulated on any G2 holonomy manifold and is a close cousin of the monopole equation of Bogomolny. Octonions play the central role in our view, in which both the sevendimensional equations and the Haydys-Witten equations appear as reductions of the eight-dimensional Spin(7) instanton equation.
• Cherkis, S. (2014). Phases of Five-dimensional Theories, Monopole Walls, and Melting Crystals. Journal of High Energy Physics, 27.
• Cherkis, S. A., & Ward, R. S. (2012). Moduli of monopole walls and amoebas. Journal of High Energy Physics, 2012(5).
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  Abstract: We study doubly-periodic monopoles, also called monopole walls, determining their spectral data and computing the dimensions of their moduli spaces. Using spectral data we identify the moduli, and compare our results with a perturbative analysis. We also identify an SL(2, ℤ) action on monopole walls, in which the S transformation corresponds to the Nahm transform. © SISSA 2012.
• Blair, C. D., & Cherkis, S. A. (2011). Singular monopoles from Cheshire bows. Nuclear Physics B, 845(1), 140-164.
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  Abstract: Singular monopoles are nonabelian monopoles with prescribed Dirac-type singularities. All of them are delivered by Nahm's construction. In practice, however, the effectiveness of the latter is limited to the cases of one or two singularities. We present an alternative construction of singular monopoles formulated in terms of Cheshire bows. To illustrate the advantages of our bow construction we obtain an explicit expression for one U(2) gauge group monopole with any given number of singularities of Dirac type. © 2010 Elsevier B.V.
• Cherkis, S. A. (2011). Instantons on Gravitons. Communications in Mathematical Physics, 306(2), 449-483.
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  Abstract: Yang-Mills instantons on ALE gravitational instantons were constructed by Kronheimer and Nakajima in terms of matrices satisfying algebraic equations. These were conveniently organized into a quiver. We construct generic Yang-Mills instantons on ALF gravitational instantons. Our data are formulated in terms of matrix-valued functions of a single variable, that are in turn organized into a bow. We introduce the general notion of a bow, its representation, its associated data and moduli space of solutions. For a judiciously chosen bow the Nahm transform maps any bow solution to an instanton on an ALF space. We demonstrate that this map respects all complex structures on the moduli spaces, so it is likely to be an isometry, and use this fact to study the asymptotics of the moduli spaces of instantons on ALF spaces. © 2011 Springer-Verlag.
• Cherkis, S. A., O'Hara, C., & Sämann, C. (2011). Super Yang-Mills theory with impurity walls and instanton moduli spaces. Physical Review D - Particles, Fields, Gravitation and Cosmology, 83(12).
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  Abstract: We explore maximally supersymmetric Yang-Mills theory with walls of impurities respecting half of the supersymmetries. The walls carry fundamental or bifundamental matter multiplets. We employ three-dimensional N=2 superspace language to identify the Higgs branch of this theory. We find that the vacuum conditions determining the Higgs branch are exactly the bow equations yielding Yang-Mills instantons on a multi-Taub-NUT space. Under electric-magnetic duality, the super Yang-Mills theory describing the bulk is mapped to itself, while the fundamental- and bifundamental-carrying impurity walls are interchanged. We perform a one-loop computation on the Coulomb branch of the dual theory to find the asymptotic metric on the original Higgs branch. © 2011 American Physical Society.
• Blair, C. D., & Cherkis, S. A. (2010). One monopole with k singularities. Journal of High Energy Physics, 2010(11).
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  Abstract: We present all charge one monopole solutions of the Bogomolny equation with k prescribed Dirac singularities for the gauge groups U(2), SO(3), or SU(2). We analyze these solutions comparing them to the previously known expressions for the cases of one or two singularities.
• Cherkis, S. A. (2010).

  Instantons on Gravitons

  . Communications in Mathematical Physics, 306, 449–483. doi:10.1007/s00220-011-1293-y
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  Yang-Mills instantons on ALE gravitational instantons were constructed by Kronheimer and Nakajima in terms of matrices satisfying algebraic equations. These were conveniently organized into a quiver. We construct generic Yang-Mills instantons on ALF gravitational instantons. Our data are formulated in terms of matrix-valued functions of a single variable, that are in turn organized into a bow. We introduce the general notion of a bow, its representation, its associated data and moduli space of solutions. For a judiciously chosen bow the Nahm transform maps any bow solution to an instanton on an ALF space. We demonstrate that this map respects all complex structures on the moduli spaces, so it is likely to be an isometry, and use this fact to study the asymptotics of the moduli spaces of instantons on ALF spaces.
• Cherkis, S. A. (2010). Instantons on the Taub-NUT space. Advances in Theoretical and Mathematical Physics, 14(2), 609-641.
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  Abstract: We present a construction of self-dual Yang-Mills connections on the Taub-NUT space. We illustrate it by finding explicit expressions for all SU(2) instantons of instanton number one and generic monodromy at infinity. © 2010 International Press.
• Cherkis, S. A. (2009).

  Instantons on the Taub-NUT Space

  . Adv. Theor. Math. Phys. 14 (2) 609 - 642.
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  We present a construction of self-dual Yang-Mills connections on the Taub-NUT space. We illustrate it by finding explicit expressions for all SU(2) instantons of instanton number one and generic monodromy at infinity.
• Cherkis, S. A. (2009). Moduli spaces of instantons on the Taub-NUT space. Communications in Mathematical Physics, 290(2), 719-736.
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  Abstract: We present ADHM-Nahm data for instantons on the Taub-NUT space and encode these data in terms of Bow Diagrams. We study the moduli spaces of the instantons and present these spaces as finite hyperkähler quotients. As an example, we find an explicit expression for the metric on the moduli space of one SU(2) instanton. We motivate our construction by identifying a corresponding string theory brane configuration. By following string theory dualities we are led to supersymmetric gauge theories with impurities. © 2009 Springer-Verlag.
• Cherkis, S. A., & Durcan, B. (2009). The 't Hooft-Polyakov monopole in the presence of a 't Hooft operator. Physics Letters, Section B: Nuclear, Elementary Particle and High-Energy Physics, 671(1), 123-127.
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  Abstract: We present explicit BPS field configurations representing one non-Abelian monopole with one minimal weight 't Hooft operator insertion. We explore the SO (3) and SU (2) gauge groups. In the case of SU (2) gauge group the minimal 't Hooft operator can be completely screened by the monopole. If the gauge group is SO (3), however, such screening is impossible. In the latter case we observe a different effect of the gauge symmetry enhancement in the vicinity of the 't Hooft operator. © 2008 Elsevier B.V. All rights reserved.
• Cherkis, S. A., Dotsenko, V., & Sämann, C. (2009). Superspace actions for multiple M2-branes, metric 3-algebras, and their classification. Physical Review D - Particles, Fields, Gravitation and Cosmology, 79(8).
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  Abstract: We classify canonical metric 3-algebra structures on matrix algebras and find novel three-dimensional conformally invariant actions in N=4 projective superspace based on them. These can be viewed as Chern-Simons theories with special matter content and couplings. We explore the relations of these actions with the N=2 actions based on generalized 3-Lie algebras found earlier and relate them to the original Bagger-Lambert-Gustavsson action. © 2009 The American Physical Society.
• Cherkis, S. A., & Durcan, B. (2008). Singular monopoles via the Nahm transform. Journal of High Energy Physics, 2008(4).
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  Abstract: We present explicit expressions for the fields of a charge one BPS monopole with two Dirac singularities. These are solutions of the nonlinear Bogomolny equations with the gauge group U(2) or SO(3). We derive these expressions by applying the technique of the Nahm transform. By exploring various limits we find a number of other solutions.
• Cherkis, S., & Sämann, C. (2008). Multiple M2-branes and generalized 3-Lie algebras. Physical Review D - Particles, Fields, Gravitation and Cosmology, 78(6).
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  Abstract: We propose a generalization of the Bagger-Lambert-Gustavsson action as a candidate for the description of an arbitrary number of M2-branes. The action is formulated in terms of N=2 superfields in three dimensions and corresponds to an extension of the usual superfield formulation of Chern-Simons matter theories. Demanding gauge invariance of the resulting theory does not imply the total antisymmetry of the underlying 3-Lie algebra structure constants. We relax this condition and propose a class of examples for these generalized 3-Lie algebras. We also discuss various associated ordinary Lie algebras. © 2008 The American Physical Society.
• Cherkis, S. A. (2007). A journey between two curves. Symmetry, Integrability and Geometry: Methods and Applications (SIGMA), 3.
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  Abstract: A typical solution of an integrable system is described in terms of a holomorphic curve and a line bundle over it. The curve provides the action variables while the time evolution is a linear flow on the curve's Jacobian. Even though the system of Nahm equations is closely related to the Hitchin system, the curves appearing in these two cases have very different nature. The former can be described in terms of some classical scattering problem while the latter provides a solution to some Seiberg-Witten gauge theory. This note identifies the setup in which one can formulate the question of relating the two curves.
• Cherkis, S. A., & Hitchin, N. J. (2005). Gravitational instantons of type D_k. Communications in Mathematical Physics, 260(2), 299-317.
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  Abstract: We use two different methods to obtain Asymptotically Locally Flat hyperkähler metrics of type D k . © Springer-Verlag 2005.
• Berenstein, D., & Cherkis, S. A. (2004). Deformations of N = 4 SYM and integrable spin chain models. Nuclear Physics B, 702(1-2), 49-85.
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  Abstract: Beginning with the planar limit of N = 4 SYM theory, we study planar diagrams for field theory deformations of N = 4 which are marginal at the free field theory level. We show that the requirement of integrability of the full one-loop dilatation operator in the scalar sector, places very strong constraints on the field theory, so that the only soluble models correspond essentially to orbifolds of N = 4 SYM. For these, the associated spin chain model gets twisted boundary conditions that depend on the length of the chain, but which are still integrable. We also show that theories with integrable subsectors appear quite generically, and it is possible to engineer integrable subsectors to have some specific symmetry, however these do not generally lead to full integrability. We also try to construct a theory whose spin chain has quantum group symmetry SOq (6) as a deformation of the SO (6) R-symmetry structure of N = 4 SYM. We show that it is not possible to obtain a spin chain with that symmetry from deformations of the scalar potential of N = 4 SYM. We also show that the natural context for these questions can be better phrased in terms of multi-matrix quantum mechanics rather than in four-dimensional field theories. © 2004 Published by Elsevier B.V.
• Berkovits, N., & Cherkis, S. A. (2004).

  Pure Spinors are Higher-Dimensional Twistors

  . JHEP 12 (2004) 049, 049. doi:10.1088/1126-6708/2004/12/049
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  In any even (Euclidean) dimension d=2n, projective pure spinors parameterize the coset space SO(2n)/U(n), which is the space of all complex structures on R^{2n}. For d=4 and d=6, these spaces are CP^1 and CP^3, and the corresponding pure spinors have been interpreted as four and six-dimensional twistor variables. In this paper, we argue that the identification of pure spinors and twistors holds in any even dimension, and we use pure spinors to construct massless solutions in higher dimensions.
• Berkovits, N., & Cherkis, S. A. (2004). Higher-dimensional twistor transforms using pure spinors. Journal of High Energy Physics, 8(12), 1103-1117.
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  Abstract: Hughston has shown that projective pure spinors can be used to construct massless solutions in higher dimensions, generalizing the four-dimensional twistor transform of Penrose. In any even (euclidean) dimension d = 2n, projective pure spinors parameterize the coset space SO(2n)/U(n), which is the space of all complex structures on ℝ2n. For d = 4 and d = 6, these spaces are ℂℙ1 and ℂℙ3 and the appropriate twistor transforms can easily be constructed. In this paper, we show how to construct the twistor transform for d > 6 when the pure spinor satisfies nonlinear constraints, and present explicit formulas for solutions of the massless field equations. © SISSA/ISAS 2005.
• Cherkis, S. A., & Kapustin, A. (2003). Periodic monopoles with singularities and N = 2 super-QCD. Communications in Mathematical Physics, 234(1), 1-35.
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  Abstract: We study solutions of the Bogomolny equation on ℝ2 × S1 with prescribed singularities. We show that the Nahm transform establishes a one-to-one correspondence between such solutions and solutions of the Hitchin equations on a punctured cylinder with the eigenvalues of the Higgs field growing at infinity in a particular manner. The moduli spaces of solutions have natural hyperkähler metrics of a novel kind. We show that these metrics describe the quantum Coulomb branch of certain N = 2 d = 4 supersymmetric gauge theories on ℝ3 × S1. The Coulomb branches of the corresponding uncompactified theories have been previously determined by E. Witten using the M-theory fivebrane. We show that the Seiberg-Witten curves of these theories are identical to the spectral curves associated to solutions of the Bogomolny equation on ℝ2 × S1. In particular, this allows us to rederive Witten's results without recourse to the M-theory fivebrane.
• Cherkis, S. A., & Kapustin, A. (2002). Hyper-Kähler metrics from periodic monopoles. Physical Review D - Particles, Fields, Gravitation and Cosmology, 65(8 A), 840151-8401510.
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  Abstract: Relative moduli spaces of periodic monopoles provide novel examples of asymptotically locally flat hyper-Kähler manifolds. By considering the interactions between well-separated periodic monopoles, we infer the asymptotic behavior of their metrics. When the monopole moduli space is four dimensional, this construction yields interesting examples of metrics with a self-dual curvature (gravitational instantons). We discuss their topology and complex geometry. An alternative construction of these gravitational instantons using moduli spaces of Hitchin equations is also described.
• Cherkis, S., & Hashimoto, A. (2002). Supergravity solution of intersecting branes and AdS/CFT with flavor. Journal of High Energy Physics, 6(11), 731-746.
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  Abstract: We construct the supergravity solution for fully localized D2/D6 intersection. The near horizon limit of this solution is the supergravity dual of supersymmetric Yang-Mills theory in 2+1 dimensions with flavor. We use this solution to formulate mirror symmetry of 2+1 dimensional gauge theories in the language of AdS/CFT correspondence. We also construct the supergravity dual of a non-commutative gauge theory with fundamental matter. © SISSA/ISAS 2002.
• Cherkis, S. A. (2001).

  PERIODIC MONOPOLES IN STRING THEORY

  . International Journal of Modern Physics. doi:10.1142/s0217751x01008631
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  Periodic solutions of Bogomolny equation have novel Gravitational Instantons as their moduli spaces. String theory allows to identify these moduli spaces with moduli spaces of Hitchin systems as well as with Coulomb branches of Seiberg-Witten gauge theories on a space with one compact direction. We classify these Gravitatial Instantons. We also perform Nahm transform of Periodic Monopoles establishing the correspondence with Hitchin systems predicted by String Theory.
• Cherkis, S. A. (2001). Periodic monopoles in string theory. International Journal of Modern Physics A, 16(SUPPL. 1C), 970-974.
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  Abstract: Periodic solutions of Bogomolny equation have novel Gravitational Instantons as their moduli spaces. String theory allows to identify these moduli spaces with moduli spaces of Hitchin systems as well as with Coulomb branches of Seiberg-Witten gauge theories on a space with one compact direction. We classify these Gravitational Instantons. We also perform Nahm transform of Periodic Monopoles establishing the correspondence with Hitchin systems predicted by String Theory.
• Cherkis, S., & Kapustin, A. (2001). Nahm transform for periodic monopoles and N = 2 super Yang-Mills theory. Communications in Mathematical Physics, 218(2), 333-371.
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  Abstract: We study Bogomolny equations on ℝ2 × double-struck S sign11. Although they do not admit nontrivial finite-energy solutions, we show that there are interesting infinite-energy solutions with Higgs field growing logarithmically at infinity. We call these solutions periodic monopoles. Using the Nahm transform, we show that periodic monopoles are in one-to-one correspondence with solutions of Hitchin equations on a cylinder with Higgs field growing exponentially at infinity. The moduli spaces of periodic monopoles belong to a novel class of hyperkähler manifolds and have applications to quantum gauge theory and string theory. For example, we show that the moduli space of k periodic monopoles provides the exact solution of N = 2 super Yang-Mills theory with gauge group SU(k) compactified on a circle of arbitrary radius.
• Cherkis, S. A., & Kapustin, A. (2000).

  Periodic Monopoles with Singularities and 𝒩=2 Super-QCD

  . Communications in Mathematical Physics. doi:10.1007/s00220-002-0786-0
• Cherkis, S. A. (1999).

  Supergravity Solution for M5-brane Intersection

  . arXiv: High Energy Physics - Theory.
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  Supergravity solution describing two intersecting M5-branes is presented. The branes are fixed in the relative transverse directions and are delocalized along the overall transverse ones. The intersection can be smoothed, so that the M5-branes present one holomorphic cycle. We also obtain a solution corresponding to an M5-brane on a holomorphic cycle of multi-Taub-NUT space. All these solutions preserve 1/4 of supersymmetry.
• Cherkis, S. A., & Kapustin, A. (1999).

  D_k Gravitational Instantons and Nahm Equations

  . Adv.Theor.Math.Phys., 2, 1287-1306. doi:10.4310/ATMP.1998.v2.n6.a3
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  We construct D_k asymptotically locally flat gravitational instantons as moduli spaces of solutions of Nahm equations. This allows us to find their twistor spaces and Kahler potentials.
• Cherkis, S. A., & Kapustin, A. (1999).

  Singular Monopoles and Gravitational Instantons

  . Communications in Mathematical Physics, 203, 713–728. doi:10.1007/s002200050632
• Cherkis, S. A., & Kapustin, A. (1999). Singular monopoles and gravitational instantons. Communications in Mathematical Physics, 203(3), 713-728.
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  Abstract: We model Ak and Dk asymptotically locally flat gravitational instantons on the moduli spaces of solutions of U (2) Bogomolny equations with prescribed singularities. We study these moduli spaces using Ward correspondence and find their twistor description. This enables us to write down the Kähler potential for Ak and Dk gravitational instantons in a relatively explicit form.
• Cherkis, S. A., & Kapustin, A. (1998). D_k gravitational instantons and nahm equations. Advances in Theoretical and Mathematical Physics, 2(6), 1287-1306.
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  Abstract: We construct Dk Asymptotically Locally Flat gravitational instantons as moduli spaces of solutions of Nahm equations. This allows us to find their twistor spaces and Kähler potentials. © 1999 International Press.
• Cherkis, S. A., & Kapustin, A. (1998). Singular monopoles and supersymmetric gauge theories in three dimensions. Nuclear Physics B, 525(1-2), 215-234.
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  Abstract: According to the proposal of Hanany and Witten, Coulomb branches of N = 4 SU(n) gauge theories in three dimensions are isometric to moduli spaces of BPS monopoles. We generalize this proposal to gauge theories with matter, which allows us to describe the metrics on their spaces of vacua by means of the hyper-Kähler quotient construction. To check the identification of moduli spaces a comparison is made with field theory predictions. For SU(2) theory with k fundamental hypermultiplets the Coulomb branch is expected to be the Dk ALF gravitational instanton, so our results lead to a construction of such spaces. In the special case of SU(2) theory with four or fewer fundamental hypermultiplets we calculate the complex structures on the moduli spaces and compare them with field-theoretical results. We also discuss some puzzles with brane realizations of three-dimensional N = 4 theories. © 1998 Elsevier Science B.V.
• Cherkis, S., & Schwarz, J. H. (1997). Wrapping the M theory five-brane on K3. Physics Letters, Section B: Nuclear, Elementary Particle and High-Energy Physics, 403(3-4), 225-232.
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  Abstract: Using a recently constructed M5-brane world-volume action, we demonstrate that wrapping the M5-brane on K3 gives the heterotic string in seven dimensions. To facilitate the comparison, a new version of the world-sheet action for the Narain-compactified heterotic string, with manifest T duality invariance, is formulated. © 1997 Elsevier Science B.V.

Proceedings Publications

• Cherkis, S. A. (2012).

  Doubly-periodic monopoles and their moduli spaces

  . In Advances in hyperkähler and holomorphic symplectic geometry, BIRS, Banff, Canada.
• Cherkis, S. A. (1999).

  Supersymmetric Gauge Theories and Gravitational Instantons

  . In Proceedings, 6th International Symposium on Particles, strings and cosmology (PASCOS 1998) : Boston, USA, March 22-29, 1998, 763-766.
  More info

  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.
• Cherkis, S. A. (1998).

  Supersymmetric Gauge Theories and Gravitational Instantons

  . In Proceedings, 6th International Symposium on Particles, strings and cosmology (PASCOS 1998) : Boston, USA, March 22-29, 1998, 763-766.
  More info

  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.

Presentations

• Cherkis, S. (2023, 01). Octonions, Monopoles,and Knots. RTG Seminar, UArizona.
• Cherkis, S. (2023, 02).

  Up and Down

  the Bow Construction

  . RIMS Conference in Kyoto. Kyoto, Japan: RIMS.
• Cherkis, S. (2023, 03). Up and Downthe Bow Construction. MIT Geometry and Topology Seminar. MIT, Cambridge, MA: MIT.
• Cherkis, S. (2023, 04). Exploded Geometry of Monopole Moduli Spaces. Berkeley String-Math Seminar. UC Berkeley: CTP Berkeley.
• Cherkis, S. (2023, 09). Tesserons from Monopoles. Topological Solitons Conference. virtual conference in Nagoya, Japan: Nagoya University.
• Cherkis, S. (2022, July). ALG metrics from string theory. Geometry and Physics of ALX Metrics in Gauge Theory Workshop. American Institute of Mathematics, San Jose, CA: American Institute of Mathematics.
• Cherkis, S. (2022, May). Topology of Magnetic Flux Tubes in Coupled Binaries. Relativistic Plasma Astrophysics Conference. Purdue University: Purdue University.
• Cherkis, S. (2022, October). All Gravitational Instantons from Monopole Moduli Spaces. New Four-Dimensional Gauge Theories Workshop. SLMath (MSRI), Berkeley: SLMath (former MSRI).
• Cherkis, S. (2021, 10/2021). The Geometry of Instantons on multi-Taub-NUT Space. Budapest-Vienna Algebraic Geometry Seminar. Budapest-Vienna (via Zoom).
• Cherkis, S. (2021, June). Bows to Instantons and Back. Oberwolfach Workshop Analysis, Geometry and Topology of Singular PDE. Oberwolfach, Germany.
• Cherkis, S. (2021, November). Bow Construction of Yang-Mills Instantons on Taub-NUT space. York Geometry Seminar. University of York, UK.
• Cherkis, S. (2020, February). Compactification of Monopole Moduli Spaces. Seminar at Waterloo University. Waterloo, Canada: University of Waterloo.
• Cherkis, S. (2020, October). Doubly Periodic Monopoles and Exploded Geometry. Kansas State University Geometry Seminar. KSU via Zoom.
• Cherkis, S. (2019, December). Moduli paces of Monopoles and Quantum Field Theories. Geometric Representation Theory and Quantum Field Theories. Sanya, Hainan, China: International Mathematics Forum, Sanya.
• Cherkis, S. (2019, February). Hyperfields and Compactification of Monowall Moduli Spaces. Hoppin Miniconference 2019. Department of Mathematics, University of Arizona: Department of Mathematics, University of Arizona.
• Cherkis, S. (2019, July). ALH Gravitational Instantons as Classical Gauge-theoretic Moduli Spaces. Geometric and analytic aspects of moduli spaces. Leibniz University, Hannover: Leibniz University, Hannover.
• Cherkis, S. (2019, June). Monopole Moduli Space Compactification and Exploding Manifolds. Jussieu Geometry Seminar. Jussieu, Paris: Jussieu, Paris.
• Cherkis, S. (2019, March). Affine Lie Algebras and Instantons Constructions. Martin B. Halpern Memorial Conference, Berkeley. Berkeley University: UC Berkeley.
• Cherkis, S. (2019, March). Compactification of Doubly Periodic Monopole Moduli Spaces. A Workshop on Challenges at the Interface of Hitchin Systems and String Theory. SCGP, Stony Brooke, NY: SCGP.
• Cherkis, S. (2019, March). Moduli Spaces of Monowalls and their Compactification. Geometry Seminar. ICMS, Edinburgh: Maxwell Institute.
• Cherkis, S. (2019, November). Moduli Spaces of Monopole and Their Compactification. Field Institute, Geometric Structures Laboratory. Geometric Structures Laboratory: Fields Institute.
• Cherkis, S. (2016, May). On Compactification of Monopole Wall Moduli Spaces. Microlocal Methods in Analysis and Geometry. CIRM, Luminy, France: CIRM.
• Cherkis, S. (2018, January). Octonionic Monopoles and another look at the Twistor Transform. Gauge Theory and Special Holonomy, Imperial College, January 8-12, 2018. Imperial College, London: Imperial College.
• Cherkis, S. (2017, Aug. 23, 2017). Yang-Mills Instantons on Gravitational Instantons: Index and Monads. CRM conference honoring Jacques Hurtubise. CRM, University of Montreal.
• Cherkis, S. (2017, Aug. 30, 2017). Instantons, Bows, and Gauge Theory Mirror Symmetry. The Analysis of Gauge-Theoretic Moduli Spaces. BIRS, Banff, Canada.
• Cherkis, S. (2017, Feb. 13, 2017). Generalizing Quivers: Bows, Slings, Monowalls. Hitchin Systems in Mathematics and Physics. Perimeter Institute.
• Cherkis, S. (2017, June 23, 2017). Yang-Mills Instantons on Gravitational Instantons: Index and Monads. The Sen Conjecture and Beyond. University College London.
• Cherkis, S. (2017, March 10, 2017). Monad for Instantons on multi-Taub-NUT and Bows. Gauge Theory and Categorification. IPAM, UCLA.
• Cherkis, S. (2016, July). From the Bow Integrable System to the Kähler Potential on the Moduli Spaces of G-monopoles and instantons. London Mathematical Society Durham Symposium. Durham, UK: London Mathematical Society - EPSRC.
  More info

  We study monopoles in three-space and instantons on a multi-Taub-NUT space with some gauge group G. The index bundle of the associated family of Dirac operators defines a bow representation as well as an integrable system associated with it. Solutions of this integrable system are in one-to-one correspondence with the gauge equivalence classes of these monopoles or instantons. This view leads to a natural Kaehler potential on their moduli space. The first part of this talk is based on the recent work with Mark Stern and Andres Larrain-Hubach
• Cherkis, S. (2016, June). Isometry of Bow and Instanton Moduli Spaces. Moduli, Integrability and Dynamics. Institut Mittag-Leffler: Institut Mittag-Leffler.
• Cherkis, S. (2016, November). The Slings and Arrows: Instantons on Gravitons. SYMPOSIUM TO CELEBRATE THE 75TH BIRTHDAY OF JOHN SCHWARZ. CALTECH: Burke Institute.
• Cherkis, S. (2016, October). Analysis of Yang-Mills Instantons on multi-Taub-NUT and Bows.. Recent Developments in Gauge Theory Simons Center. Simons Center for Geometry and Physics, Stony Brook, NY.
• Cherkis, S. (2015, 04). Monopole Wall Dynamics. The Ninth IMACS International Conference on Nonlinear Evolution Equations and Wave Phenomena: Computation and Theory. Georgia Center for Continuing Education University of Georgia, Athens, GA, USA.
  More info

  Invited Talk
• Cherkis, S. (2015, 07). Instanton and Bow Moduli Spaces. Metric and Analytic Aspects of Moduli Spaces. Isaac Newton Institute for Mathematical Sciences, Cambridge.
• Cherkis, S. (2014, June). Phases of 5D Quantum Theories, Monopole Walls, and Secondary Polyhedra. Institute of Theoretical Physics Seminar. IPhT, Saclay, France: IPhT.
• Cherkis, S. (2014, March). Monopole Walls, Quantum Field Theories in Five Dimensions, and Melting Crystals. Mitchell Institute Seminar. Texas A&M University: Texas A&M University.
• Cherkis, S. (2014, March). Octonions, Monopoles, and Knots. Stanford Geometry Seminar. Stanford, CA: Stanford University.
• Cherkis, S. (2014, May). Monopole Walls, 5D Field Theories, and Amoebas. Edinburgh Geometry Seminar. International Centre for Mathematical Sciences, Edinburgh, UK: International Centre for Mathematical Sciences, Edinburgh, UK.
• Cherkis, S. (2014, November). Hyperkahler Spaces and Monopole Walls. Geometry Seminar. Kansas State University: Kansas State University.
• Cherkis, S. (2014, November). Octonions Monopoles, and Knots. Geometry Seminar. Duke University, NC: Duke University Mathematics Department.
• Cherkis, S. (2014, October). Hyperkaehler Spaces from Monopole Walls. Geometry Seminar. University of Texas, Austin: UT Austin.