Seiberg-Witten Theory Research Articles

Framed BPS states, moduli dynamics, and wall-crossing

We formulate supersymmetric low energy dynamics for BPS dyons in strongly-coupled N=2 Seiberg-Witten theories, and derive wall-crossing formulae thereof. For BPS states made up of a heavy core state and n probe (halo) dyons around it, we derive a reliable supersymmetric moduli dynamics with 3n bosonic coordinates and 4n fermionic superpartners. Attractive interactions are captured via a set of supersymmetric potential terms, whose detail depends only on the charges and the special Kaehler data of the underlying N=2 theories. The small parameters that control the approximation are not electric couplings but the mass ratio between the core and the probe, as well as the distance to the marginal stability wall where the central charges of the probe and of the core align. Quantizing the dynamics, we construct BPS bound states and derive the primitive and the semi-primitive wall-crossing formulae from the first principle. We speculate on applications to line operators and Darboux coordinates, and also about extension to supergravity setting.

Deformed prepotential, quantum integrable system and Liouville field theory

We study the dual descriptions recently discovered for the Seiberg–Witten theory in the presence of surface operators. The Nekrasov partition function for a four-dimensional N = 2 gauge theory with a surface operator is believed equal to the wave-function of the corresponding integrable system, or the Hitchin system, and is identified with the conformal block with a degenerate field via the AGT relation. We verify the conjecture by showing that the null state condition leads to the Schrödinger equations of the integrable systems. Furthermore, we show that the deformed prepotential emerging from the period integrals of the principal function corresponds to monodromy operation of the conformal block. We also give the instanton partition functions for the asymptotically free SU ( 2 ) gauge theories in the presence of the surface operator via the AGT relation. We find that these partition functions involve the counting of two- and four-dimensional instantons.

Seiberg-Witten theory, matrix model and AGT relation

We discuss the Penner-type matrix model which has been proposed to explain the AGT relation between the 2-dimensional Liouville theory and 4-dimensional $$ \mathcal{N} = 2 $$ superconformal gauge theories. In our previous communication we have obtained the spectral curve of the matrix model and showed that it agrees with that derived from M-theory. We have also discussed the decoupling limit of massive flavors and proposed new matrix models which describe Seiberg-Witten theory with flavors N f = 2, 3. In this article we explicitly evaluate the free energy of these matrix models and show that they in fact reproduce the amplitudes of Seiberg-Witten theory.

Holomorphicity and modularity in Seiberg-Witten theories with matter

We calculate the gravitational corrections to the effective action of N=2 SU(2) Seiberg-Witten theory with matter using modularity, the holomorphic anomaly equation and expected behavior at the boundaries of the moduli space. As in pure gauge theory we show that the gap condition at the dyon singularities completely fixes the gravitational corrections. We discuss the behavior of the gravitational corrections at the conformal points. We compare our results with the recursive solution of the loop equation in the matrix model approach, which provides in addition open amplitudes.

Four-Dimensional Wall-Crossing via Three-Dimensional Field Theory

We give a physical explanation of the Kontsevich-Soibelman wall-crossing formula for the BPS spectrum in Seiberg-Witten theories. In the process we give an exact description of the BPS instanton corrections to the hyperkahler metric of the moduli space of the theory on R^3 x S^1. The wall-crossing formula reduces to the statement that this metric is continuous. Our construction of the metric uses a four-dimensional analogue of the two-dimensional tt* equations.

Genus-one correction to asymptotically free Seiberg-Witten prepotential from Dijkgraaf-Vafa matrix model

We find perfect agreements on the genus-one correction to the prepotential of SU(2) Seiberg-Witten theory with N_f=2, 3 between field theoretical and Dijkgraaf-Vafa-Penner type matrix model results.

A study of Wall-Crossing: flavored kinks in D = 2 QED

We study spectrum of D=2 N=(2,2) QED with N+1 massive charged chiral multiplets, with care given to precise supermultiplet countings. In the infrared the theory flows to CP^N model with twisted masses, where we construct generic flavored kink solitons for the large mass regime, and study their quantum degeneracies. These kinks are qualitatively different and far more numerous than those of small mass regime, with features reminiscent of multi-pronged (p,q) string web, complete with the wall-crossing behavior. It has been also conjectured that spectrum of this theory is equivalent to the hypermultiplet spectrum of a certain D=4 Seiberg-Witten theory. We find that the correspondence actually extends beyond hypermultiplets in D=4, and that many of the relevant indices match. However, a D=2 BPS state is typically mapped to several different kind of dyons whose individual supermultiplets are rather complicated; the match of index comes about only after summing over indices of these different dyons. We note general wall-crossing behavior of flavored BPS kink states, and compare it to those of D=4 dyons.

Matrix model conjecture for exact BS periods and Nekrasov functions

We give a concise summary of the impressive recent development unifying a number of different fundamental subjects. The quiver Nekrasov functions (generalized hypergeometric series) form a full basis for all conformal blocks of the Virasoro algebra and are sufficient to provide the same for some (special) conformal blocks of W-algebras. They can be described in terms of Seiberg-Witten theory, with the SW differential given by the 1-point resolvent in the DV phase of the quiver (discrete or conformal) matrix model (\beta-ensemble), dS = ydz + O(\epsilon^2) = \sum_p \epsilon^{2p} \rho_\beta^{(p|1)}(z), where \epsilon and \beta are related to the LNS parameters \epsilon_1 and \epsilon_2. This provides explicit formulas for conformal blocks in terms of analytically continued contour integrals and resolves the old puzzle of the free-field description of generic conformal blocks through the Dotsenko-Fateev integrals. Most important, this completes the GKMMM description of SW theory in terms of integrability theory with the help of exact BS integrals, and provides an extended manifestation of the basic principle which states that the effective actions are the tau-functions of integrable hierarchies.

Conformality or confinement (II): one-flavor CFTs and mixed-representation QCD

We study QCD-like four dimensional theories in the theoretically controlled framework of deformation theory and/or twisted partition function on S*1 x R*3. By using duality, we show that a class of one-flavor theories exhibit new physical phenomena: discrete chiral symmetry breaking induced by the condensation of topological disorder operators, and confinement and the generation of mass gap due to new non-selfdual topological excitations. In the R*4 limit, we argue that the mass gap disappears, the chiral symmetry breaking vacua are of runaway type, and the theory flows to a CFT. We also study mixed-representation theories and find abelian chiral symmetry breaking by topological operators charged under abelian chiral symmetries. These are reminiscent to, but distinct, from Seiberg-Witten theory with matter, where 4d monopoles have non-abelian chiral charge. This examination also helps us refine our recent bounds on the conformal window. In an Addendum, we also discuss mixed vectorlike/chiral representation theories, obtain bounds on their conformal windows, and compare with the all-order beta function results of arXiv:0911.0931.

Taubes’s proof of the Weinstein conjecture in dimension three

Does every smooth vector field on a closed three-manifold, for example the three-sphere, have a closed orbit? No, according to counterexamples by K. Kuperberg and others. On the other hand there is a special class of vector fields, called Reeb vector fields, which are associated to contact forms. The three-dimensional case of the Weinstein conjecture asserts that every Reeb vector field on a closed oriented three-manifold has a closed orbit. This conjecture was recently proved by Taubes using Seiberg-Witten theory. We give an introduction to the Weinstein conjecture, the main ideas in Taubes’s proof, and the bigger picture into which it fits. Taubes’s proof of the Weinstein conjecture is the culmination of a large body of work, both by Taubes and by others. In an attempt to make this story accessible to nonspecialists, much of the present article is devoted to background and context, and Taubes’s proof itself is only partially explained. Hopefully this article will help prepare the reader to learn the full story from Taubes’s paper [62]. More exposition of this subject (which was invaluable in the preparation of this article) can be found in the online video archive from the June 2008 MSRI hot topics workshop [44], and in the article by Auroux [5]. Below, in §1–§3 we introduce the statement of the Weinstein conjecture and discuss some examples. In §4–§6 we discuss a natural strategy for approaching the Weinstein conjecture, which proves it in many but not all cases, and provides background for Taubes’s work. In §7 we give an overview of the big picture surrounding Taubes’s proof of the Weinstein conjecture. Readers who already have some familiarity with the Weinstein conjecture may wish to start here. In §8–§9 we recall necessary material from Seiberg-Witten theory. In §10 we give an outline of Taubes’s proof, and in §11 we explain some more details of it. To conclude, in §12 we discuss some further results and open problems related to the Weinstein conjecture. 1. Statement of the Weinstein conjecture The Weinstein conjecture asserts that certain vector fields must have closed orbits. Before stating the conjecture at the end of this section, we first outline its origins. This is discussion is only semi-historical, because only a sample of the relevant works will be cited, and not always in chronological order. 1.1. Closed orbits of vector fields. Let Y be a closed manifold (in this article all manifolds and all objects defined on them are smooth unless otherwise stated), 2000 Mathematics Subject Classification. 57R17,57R57,53D40. Partially supported by NSF grant DMS-0806037.

Multiflavor QCD∗ on [formula omitted]: Studying transition from Abelian to non-Abelian confinement

The center-stabilized multiflavor QCD∗ theories formulated on R3×S1 exhibit both Abelian and non-Abelian confinement as a function of the S1 radius, similar to the Seiberg–Witten theory as a function of the mass deformation parameter. For sufficiently small number of flavors and small r(S1), we show occurrence of a mass gap in gauge fluctuations, and linear confinement. This is a regime of confinement without continuous chiral symmetry breaking (χSB). Unlike one-flavor theories where there is no phase transition in r(S1), the multiflavor theories possess a single phase transition associated with breaking of the continuous χS. We conjecture that the scale of the χSB is parametrically tied up with the scale of Abelian to non-Abelian confinement transition.

Exotic prepotentials from D(-1)D7 dynamics

We compute the partition functions of D(-1)D7 systems describing the multi-instanton dynamics of SO(N) gauge theories in eight dimensions. This is the simplest instance of the so called exotic instantons. In analogy with the Seiberg-Witten theory in four space-time dimensions, the prepotential and correlators in the chiral ring are derived via localization formulas and found to satisfy relations of the Matone type. Exotic prepotentials of SO(N) gauge theories with N=2 supersymmetries in four-dimensions are also discussed.

Surface operators in 𝒩 = 2 abelian gauge theory

We generalise the analysis in [arXiv:0904.1744] to superspace, and explicitly prove that for any embedding of surface operators in a general, twisted N=2 pure abelian theory on an arbitrary four-manifold, the parameters transform naturally under the SL(2,Z) duality of the theory. However, for nontrivially-embedded surface operators, exact S-duality holds if and only if the quantum parameter effectively vanishes, while the overall SL(2,Z) duality holds up to a c-number at most, regardless. Nevertheless, this observation sets the stage for a physical proof of a remarkable mathematical result by Kronheimer and Mrowka--that expresses a ramified analog of the Donaldson invariants solely in terms of the ordinary Donaldson invariants--which, will appear, among other things, in forthcoming work. As a prelude to that, the effective interaction on the corresponding u-plane will be computed. In addition, the dependence on second Stiefel-Whitney classes and the appearance of a Spin^c structure in the associated low-energy Seiberg-Witten theory with surface operators, will also be demonstrated. In the process, we will stumble upon an interesting phase factor that is otherwise absent in the unramified case.

Extended supersymmetric moduli space and a SUSY/non-SUSY duality

We study N = 1 supersymmetric U ( N ) gauge theories coupled to an adjoint chiral field with superpotential. We consider the full supersymmetric moduli space of these theories obtained by adding all allowed chiral operators. These include higher-dimensional operators that introduce a field-dependence for the gauge coupling. We show how Feynman diagram/matrix model/string theoretic techniques can all be used to compute the IR glueball superpotential. Moreover, in the limit of turning off the superpotential, this leads to a deformation of N = 2 Seiberg–Witten theory. In the case where the superpotential drives the squared gauge coupling to a negative value, we find that supersymmetry is spontaneously broken, which can be viewed as a novel mechanism for breaking supersymmetry. We propose a new duality between a class of N = 1 supersymmetric U ( N ) gauge theories with field-dependent gauge couplings and a class of U ( N ) gauge theories where supersymmetry is softly broken by nonzero expectation values for auxiliary components of spurion superfields.

ORIENTABILITY AND REAL SEIBERG–WITTEN INVARIANTS

We investigate the Seiberg–Witten theory in the presence of real structures. Certain conditions are obtained so that integer-valued real Seiberg–Witten invariants can be defined. In general, we study properties of the real Seiberg–Witten projection map from the point of view of Fredholm map degrees.

Seiberg–Witten theory and matrix models

We derive a family of matrix models which encode solutions to the Seiberg–Witten theory in 4 and 5 dimensions. Partition functions of these matrix models are equal to the corresponding Nekrasov partition functions, and their spectral curves are the Seiberg–Witten curves of the corresponding theories. In consequence of the geometric engineering, the 5-dimensional case provides a novel matrix model formulation of the topological string theory on a wide class of non-compact toric Calabi–Yau manifolds. This approach also unifies and generalizes other matrix models, such as the Eguchi–Yang matrix model, matrix models for bundles over P 1 , and Chern–Simons matrix models for lens spaces, which arise as various limits of our general result.

Surgery, Yamabe invariant, and Seiberg–Witten theory

By using the gluing formula of the Seiberg–Witten invariant, we compute the Yamabe invariant Y ( X ) of 4-manifolds X obtained by performing surgeries along points, circles or tori on compact Kähler surfaces. For instance, if M is a compact Kähler surface of nonnegative Kodaira dimension, and N is a smooth closed oriented 4-manifold with b 2 + ( N ) = 0 and Y ( N ) ≥ 0 , then we show that Y ( M # N ) = Y ( M ) .

Extended MQCD and SUSY/non-SUSY duality

We study the SUSY/non-SUSY duality proposed by Aganagic et al. from Type IIA string and M-theory perspectives. We find that our brane configuration generalizes the so-called $extended$ Seiberg-Witten theory on the one hand, and provides a way to realize non-SUSY vacua by intersecting NS5-branes on the other hand. We also argue how the partial SUSY breaking from $\Ncal=2$ down to $\Ncal=1$ can be clearly visualized through the brane picture.

Extended 5d Seiberg–Witten theory and melting crystal

We study an extension of the Seiberg–Witten theory of 5d N = 1 supersymmetric Yang–Mills on R 4 × S 1 . We investigate correlation functions among loop operators. These are the operators analogous to the Wilson loops encircling the fifth-dimensional circle and give rise to physical observables of topologically-twisted 5d N = 1 supersymmetric Yang–Mills in the Ω background. The correlation functions are computed by using the localization technique. Generating function of the correlation functions of U ( 1 ) theory is expressed as a statistical sum over partitions and reproduces the partition function of the melting crystal model with external potentials. The generating function becomes a τ function of 1-Toda hierarchy, where the coupling constants of the loop operators are interpreted as time variables of 1-Toda hierarchy. The thermodynamic limit of the partition function of this model is studied. We solve a Riemann–Hilbert problem that determines the limit shape of the main diagonal slice of random plane partitions in the presence of external potentials, and identify a relevant complex curve and the associated Seiberg–Witten differential.

Product formulae for Ozsváth–Szabó 4–manifold invariants

We give formulae for the Ozsvath-Szabo invariants of 4-manifolds X obtained by fiber sum of two manifolds M_1, M_2 along surfaces S_1, S_2 having trivial normal bundle and genus g>0. The formulae follow from a general theorem on the Ozsvath-Szabo invariants of the result of gluing two 4-manifolds along a common boundary, which is phrased in terms of relative invariants of the pieces. These relative invariants take values in a version of Heegaard Floer homology with coefficients in modules over certain Novikov rings; the fiber sum formula follows from the theorem that this "perturbed" version of Heegaard Floer theory recovers the usual Ozsvath-Szabo invariants, when the 4-manifold in question has b^+>1. The construction allows an extension of the definition of the Ozsvath-Szabo invariants to 4-manifolds having b^+ = 1 depending on certain choices, in close analogy with Seiberg-Witten theory. The product formulae lead quickly to calculations of the Ozsvath-Szabo invariants of various 4-manifolds; in all cases the results are in accord with the conjectured equivalence between the Ozsvath-Szabo and Seiberg-Witten invariants.