The main goal of the paper is the development of a method for the determination of the spectrum of the so-called BPS (Bogomol’nyi-Prasad-Sommerfield) states in a class of \(\mathcal{N}=2\) theories in four dimensions. These states are, by definition, eigenstates of the quantum Hamiltonian in short representations of the \(\mathcal{N}=2\) supersymmetry algebra. The set of such states depends on a) the parameters of the Lagrangian of the theory, such as various couplings (including masses) and b) the chosen vacuum of the theory, i.e., the expectation values of the adjoint scalar fields of the vector multiplet (a point on the “Coulomb branch” of the theory).
The theories, for which the construction of the paper works, are of a particular class. They may be obtained as compactifications of the \(\mathrm{SU}(2)\) (2,0) six-dimensional superconformal theory with codimension-two defect operators on a Riemann surface. In fact, in the paper all examples concern the case when the Riemann surface is a sphere with punctures. These low-energy theories are described by asymptotically free or conformal linear quivers (those linear quivers, for which the beta functions of all gauge groups are non-positive). The matter fields of the theories are either bi-fundamental or fundamental. The number of gauge groups in the quiver of the four-dimensional theory is related to the number of punctures of the sphere. Fully analyzed is the case of a single \(\mathrm{SU}(2)\) gauge group with four fundamental hyper-multiplets (Section 10), which corresponds to a sphere with four punctures (and various degenerations of this case). In principle, an algorithm for the determination of the BPS states in such theories was developed in [A. Klemm et al., Nucl. Phys., B 477, No. 3, 746–764 (1996;Zbl 0925.81196)] – it involved the determination of geodesics in a particular metric on the Riemann surface, constructed using the Seiberg-Witten differential. In the present paper a more algebraic approach is proposed, which is based on the so-called “spectrum generator” (see below).
The key ingredient in the description of the low-energy effective action of an \(\mathcal{N}=2\) theory on its Coulomb branch is the so-called Seiberg-Witten curve, equipped with a Seiberg-Witten meromorphic differential. For the construction of the Seiberg-Witten curve of the theories at hand the authors recall an interpretation of these theories as of low-energy theories describing oscillations of certain configurations of intersecting D4-D6-NS5 branes in type IIA string theory (This description was established in the paper [E. Witten, Nucl. Phys., B 500, No. 1–3, 3–42 (1997;Zbl 0934.81066)]). The approach to \(\mathcal{N}=2\) theories based on intersecting branes has the advantage that the Seiberg-Witten curve is in fact “built into” the string-theoretic geometry – in particular, if one lifts the corresponding brane configurations to 11D, one obtains (on the Coulomb branch) a single M5-brane wrapping a complex curve (the fact that it is a complex curve follows from the required amount of supersymmetry), and this complex curve is identified with the Seiberg-Witten curve. In the conformal limit (i.e., when all expectation values are zero), the curve becomes reducible, and as a result the previously “single” M5-brane splits into a union of several branes, reproducing the picture of brane intersections.
Technically, the construction of the paper relies on the description of the moduli space of solutions to a Hitchin system of a particular sort, defined on the sphere with punctures. As we will see, the relation to the Hitchin system also opens an alternative view at the construction of the corresponding Seiberg-Witten theory. This Hitchin system arises as follows. First, one considers the compactification of the \(\mathcal{N}=2\) four-dimensional theory on a circle of radius \(R\) down to three dimensions. The \(\mathcal{N}=2\) supersymmetry of the 4D theory then translates into \(\mathcal{N}=4\) supersymmetry of the 3D theory. As a result, the 3D low-energy effective theory is described by a sigma-model with hyper-Kähler target space, which is a toric fibration over the target space of the original 4D theory (the coordinates on the latter are the scalar fields from the vector multiplet). The “doubling” of the degrees of freedom upon dimensional reduction is achieved as follows. The bosonic part of a vector multiplet in 4D comprises a complex scalar field and a vector field. The reduction of the complex scalar field remains a complex scalar field, whereas the reduction of the abelian gauge field produces two scalar fields: one from the 4-th component of the vector field, and one from the Hodge-dualized vector field in 3D. A more careful analysis shows that these two scalars are in fact periodic, and therefore may be thought of as living in a torus. One of the by-products of the straightforward dimensional reduction is the explicit knowledge of a metric on this hyper-Kähler manifold. The form of the metric implies that the above-mentioned torus is fibered over a base space, which is the target space of the low-energy 4D theory. The metric obtained in this way provides the correct low-energy description only when the radius of the circle is sufficiently large, since for \({1\over R}\gtrsim m\) (where \(m\) is the mass of a particle in the 4D theory) one needs to take into account the massive particles of the 4D theory in the 3D low-energy approximation as well (Löscher-type corrections). This is how the massive BPS states of the 4D theory enter the picture. To summarize, the target space of the 3D sigma-model is rather highly constrained: it is a hyper-Kähler manifold, endowed with a fibration by complex tori (the tori are complex in one of the complex structures of this hyper-Kähler manifold). (The dimensional reduction down to 3D of the low-energy effective theories on the Coulomb branch of N=2 theories was first elaborated in the work [N. Seiberg andE. Witten, Adv. Ser. Math. Phys. 24, 333–366 (1997;Zbl 1058.81717)]. Such manifolds are usually referred to as phase spaces of algebraic integrable systems. In fact, the integrable system at hand is a Hitchin system, as suggested by the following argument.
So far we have described how the 4D \(\mathcal{N}=2\) theory is compactified down to 3D. Since the 4D theory itself was obtained as a compactification of a 6D theory, one can interchange the orders of compactification, first compactifying from 6D to 5D on a circle, and then to 3D on a Riemann surface (a punctured sphere, as above). The 5D theory is then the maximally supersymmetric Yang-Mills theory. If one wishes to define the theory on a curved manifold (i.e., flat space \(R^{1,3} \times\) Riemann surface), one has to twist the theory in such a way as to preserve \(\mathcal{N}=2\) supersymmetry in four dimensions, and the BPS equations of the resulting theory are the Hitchin equations (For a detailed discussion of how this is done see [K. Yonekura, “Supersymmetric gauge theory, (2,0) theory and twisted 5d Super-Yang-Mills”, J. High Energy Phys. 2014, Paper No. 142 (2014;doi:10.1007/JHEP01(2014)142)]. Therefore a natural guess is that the target space of the 3D sigma-model is the moduli space of solutions to these equations. This moduli space is a hyper-Kähler manifold, on general grounds. Moreover, one of the crucial properties of Hitchin moduli spaces is that they are fibered by complex tori (and as such are very special examples of hyper-Kähler manifolds, representing algebraic integrable systems). As a hyper-Kähler manifold, the Hitchin moduli space has a twistor sphere worth of complex structures. For every point “u” in the twistor sphere, one can pick a complex structure and a respective symplectic form, which is holomorphic in this complex structure. The twistor theory of hyper-Kähler manifolds assures that the knowledge of Darboux coordinates for this holomorphic symplectic form, as functions of the twistor sphere parameter “u”, grants full knowledge of the differential geometry of these manifolds: in particular, it allows to derive the hyper-Kähler metrics on them.
Before one can speak of the moduli space of solutions to Hitchin’s equations, however, one has to supplement the latter by an additional piece of data – the behavior of the fields at the punctures. This is done most invariantly by specifying the behavior of the (gauge-invariant) quadratic differential on the considered Riemann surface, which is holomorphic away from the punctures and whose existence follows directly from the \(\mathrm{SU}(2)\) Hitchin system. (The theory of quadratic differentials is presented in great detail in the book [K. Strebel, Quadratic differentials. Berlin etc.: Springer-Verlag (1984;Zbl 0547.30001)]). This quadratic differential defines a ramified double covering of this Riemann surface (The zeros of the differential are the branch points); this double cover is identified with the Seiberg-Witten curve (By this construction it is the spectral curve of the Hitchin system). The square root of the quadratic differential, which is well-defined on the double cover, is identified with the Seiberg-Witten differential. In the case of conformal linear quivers, the quadratic differential generally has double poles at the punctures. The locations of the poles themselves are related to the gauge couplings and topological angles (recall that the number of poles is related to the number of gauge groups in a quiver theory). The residues at the poles (by definition, these are the coefficients of the quadratic singularity) are related to the masses of the fundamental hyper-multiplets. The remaining parameters of the quadratic differential are in one-to-one correspondence with the expectation values of the adjoint scalars of the theory, which is compatible with the fact that its square root is the Seiberg-Witten differential. Certain limits of the conformal linear quiver theories are analyzed in the paper as well. One such limit is when the masses of the fundamental hyper-multiplets are sent to infinity – as a result, one obtains an asymptotically-free theory (Section 10 of the paper). In this way one can consider, for instance, pure \(\mathcal{N}=2\) \(\mathrm{SU}(2)\) gauge theory as in the original work ofN. Seiberg andE. Witten [Nucl. Phys., B 426, No. 1, 19–52 (1994;Zbl 0996.81510)]. However, as a result of taking the limit, one obtains irregular singularities of the quadratic differential. Another limit considered in detail in Section 9 of the paper concerns the description of a patch of the moduli space where some of the BPS particles become nearly massless. From the point of view of the spectral curve, this corresponds to the situation when some of the zeros of the quadratic differential (i.e., some of the branch points) coalesce. The scaling limit corresponds to “zooming in” this part of the spectral curve and forgetting (formally sending to infinity) the rest of it.
The paper mainly studies the properties of the holomorphic Darboux coordinates on the moduli spaces of solutions to Hitchin’s equations, with the defects described above. The coordinates themselves are constructed in the large-\(R\) limit, in which case they can be deduced directly from the Seiberg-Witten solution. The explicit construction of the coordinates for finite \(R\) would presumably follow the route explained by the same authors in their other paper [Commun. Math. Phys. 299, No. 1, 163–224 (2010;Zbl 1225.81135)]. The moduli space of solutions to the Hitchin equations may be viewed as the moduli space of flat \(\mathrm{sl}(2,\mathbb{C})\)-connections. On the other hand, there is an elegant construction of Fock-Goncharov for the description of moduli spaces of flat connections on Riemann surfaces with punctures [V. Fock andA. Goncharov, Publ. Math., Inst. Hautes Étud. Sci. 103, 1–211 (2006;Zbl 1099.14025)]. The singular behavior of the gauge field at the punctures gives rise to “defects”. The existence of the Fock-Goncharov coordinates in this situation is the practical reason, why the defects are useful for the analysis of the Hitchin system. The Fock-Goncharov coordinates are defined using a triangulation of the Riemann surface with vertices at the punctures. Given a vertex of a triangulation (i.e., a puncture) and viewing the flat connection as being a connection on a rank-2 vector bundle over the Riemann surface, one can choose a covariantly constant section of this bundle that is an eigenvector of the monodromy matrix around this puncture (Such a choice for all vertices is called a “decoration” of the triangulation). The Fock-Goncharov coordinates are certain cross-ratios of the Wronskians of these sections, labeled by edges of the triangulation (the precise definition involves the structure of the triangulation). One can check that these coordinates (or more exactly their logarithms) are Darboux coordinates for the holomorphic symplectic structure on the moduli space. These coordinates depend on the choice of triangulation – when a triangulation changes, they undergo symplectomorphisms, which are identified with the Kontsevich-Soibelman transformations [M. Kontsevich andY. Soibelman, “Stability structures, motivic Donaldson-Thomas invariants and cluster transformations”, Preprint,arXiv:0811.2435]. The next question that is considered is the choice of triangulation. It turns out that, when the Riemann surface is equipped with a quadratic differential, there is a canonical family of triangulations given by the so-called “WKB trajectories”, or “\(\theta\)-trajectories” in the terminology of Strebel’s book. By definition, the phase of the quadratic differential (\(\theta\)) is constant along such trajectory. One can show on general grounds, that such “WKB-curves” foliate the Riemann surface, moreover they provide a cell decomposition for it. When the zeros of the quadratic differential are simple, this cell decomposition is a triangulation (which has to do with the fact that there are exactly three WKB trajectories emanating from a simple zero). By construction, the edges of the WKB triangulation are “generic WKB trajectories”, interpolating between two punctures (i.e., poles of the quadratic differential). There is a more invariant way of labeling the edges of a WKB triangulation – to every edge defined above one can assign a certain homologically non-trivial cycle on the Riemann surface (with punctures). Since to every edge corresponds one Fock-Goncharov coordinate, in the end one has a set of Fock-Goncharov coordinates labeled by homology cycles on the Riemann surface and a choice of the angle \(\theta\) (using which WKB-trajectories were defined).
The main result of the paper is the proposal of how to compute the spectrum of BPS particles, given a Seiberg-Witten curve (i.e., when the masses, gauge couplings, and expectation values of the scalar fields are fixed). First, one fixes the angle \(\theta\) and computes the corresponding triangulation of the Riemann surface. If one were to evolve the triangulation continuously with \(\theta\), one would find that it would “jump” at those values of \(\theta\), which correspond to the appearance of BPS hyper- and vector-multiplets. This way one would essentially arrive at the algorithm of [A. Klemm et al., Nucl. Phys., B 477, No. 3, 746–764 (1996;Zbl 0925.81196)]. There is, however, a different way, which lies in the realization that the triangulations corresponding to angles \(\theta\) and \(\theta+\pi\) are essentially the same. The only thing that changes is the “decoration”, i.e., a choice of a particular covariantly constant eigenvector of the monodromy matrix near every puncture. When one switches from \(\theta\) to \(\theta+\pi\) the eigenvector at each puncture changes to the opposite one. It turns out that the corresponding rational transformation of the Fock-Goncharov coordinates may be computed exactly, once the triangulation is given. This transformation is called the “spectrum generator”. It is then explained that this transformation may be uniquely factorized into a product of elementary transformations, corresponding to the BPS vector- and hyper-multiplets of various electric/magnetic charges – this is the same product that one would get, if one were continuously evolving the triangulation from \(\theta\) to \(\theta+\pi\). Therefore the spectrum of BPS particles may be read off this decomposition. Applications of this algorithm to \(\mathrm{SU}(2)\) theories with fundamental hyper-multiplets are considered.

Reviewer: Dmitri Bykov (Moskva)

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