University of Amsterdam

University of Amsterdam MSc Physics Theoretical Physics Master Thesis Dualities in Gauge Theories and their Geometric Realization by S.P.G. van Leuven 5756561 May 2014 60 ECTS February 2013-May 2014 Supervisor: Second Supervisor: Prof.dr. E.P. Verlinde Prof.dr. J. de Boer ITFA Abstract In this thesis, exact results in N = 2 super Yang-Mills theories are discussed. Starting from the exact solution of pure SYM provided by Seiberg and Witten through an elliptic curve construction, generalizations are discussed such as the inclusion of hypermultiplets. Special emphasis is put on the selfdual Nf = 4 theory, which will play a central role in remainder. Furthermore, a brief summary is given of the M-theory construction of Witten to determine the Seiberg-Witten curves of gauge theories with arbitrary unitary product gauge groups, also dubbed quiver gauge theories. This provides the necessary introduction to understand the recent developments made by Gaiotto. Superconformal quiver theories are associated in a very precise manner to punctured Riemann surfaces Cn;g. Gaiotto's conjecture that the UV moduli space of the quiver theories equals the moduli space of the associated Riemann surface is explained in detail. Furthermore, explicit checks of his proposal are performed by comparing the boundaries of the moduli spaces for Tn;g[A1], g = 0; 1. We briefly re-examine the M-theory construction in this new light, essentially explaining the reduction of the elusive 6d (2; 0) theory on Cn;g. At last, the extension to A2 and AN−1 quiver gauge theories is discussed. Contents 1 Introduction 1 2 Supersymmetry 3 2.1 Superspace and Superfields................................3 2.2 Super Yang-Mills Theory.................................5 2.3 Coupling SYM to Matter.................................8 2.4 R-Symmetry and an Anomaly...............................8 2.5 Exact Renormalization of the Prepotential........................ 11 3 Seiberg-Witten Theory 15 3.1 The Classical Moduli Space................................ 15 3.2 Duality of the Super Yang-Mills Action......................... 18 3.3 Central Charge and Dual Theories............................ 20 3.4 Monodromies and the Quantum Moduli Space..................... 22 3.5 Solution of the Model................................... 27 3.6 Discussion.......................................... 33 3.7 Confinement of Electric Charge.............................. 34 3.8 Adding Matter....................................... 36 3.8.1 Global Symmetries of Nf ≤ 4........................... 37 3.8.2 Moduli Spaces of Nf ≤ 3............................. 39 3.8.3 Nf = 4 Theory................................... 41 3.9 Seiberg-Witten Curves from M-Theory.......................... 45 4 Gaiotto Dualities 49 4.1 Quiver Gauge Theories................................... 50 4.2 A Geometric Realization of Duality............................ 56 4.3 Mathematical Description................................. 61 4.3.1 Massless T4;0[A1].................................. 62 4.3.2 Massive T4;0[A1].................................. 64 4.3.3 Massless Tm;0[A1]................................. 67 4.3.4 Massive Tm;0[A1].................................. 67 4.3.5 Tm;1[A1]....................................... 69 4.4 Summary: A Six Dimensional Construction....................... 73 4.5 Generalizations....................................... 75 4.5.1 T(f1;f3);g[A2]..................................... 76 4.5.2 T(fa);g[AN−1].................................... 80 5 Summary and Outlook 84 A Electric-magnetic Duality 86 i CONTENTS ii B Supersymmetry Multiplets and Superfields Expansions 88 B.1 Vector and Hypermultiplet................................ 88 B.2 Expansion of Superfields.................................. 88 C Half Hypers, Bifundamentals and Trifundamentals 90 D Explicit Expression Quadratic Differential 94 Chapter 1 Introduction Yang-Mills theories with N = 2 supersymmetry present an intriguing playground to help understand the non-perturbative dynamics of non-abelian gauge theories, which arise inescapably in the IR of UV free theories. Starting with the developments of Seiberg and Witten in [59] and [60], it was realized that the low energy effective actions of a large class of spontaneously broken su- persymmetric Yang-Mills theories, possibly coupled to hypermultiplets, allow a purely geometrical description in terms of (hyper)elliptic curves. The basic property of N = 2 gauge theories which underlies the appearance of elliptic curves is the complex moduli space of vacua on which the effective action depends holomorphically. A beautiful interplay between physical assumptions and the mathematical rigidity of complex analysis fixes the effective action completely. An infinite series of non-perturbative corrections was exactly computed, standing in great contrast to increasingly difficult explicit instanton calculations.1 Whereas Seiberg and Witten originally studied the low energy effective behaviour of SU(2) theories coupled to Nf ≤ 4 flavours, the analysis was subsequently generalized to SU(N); SO(N); Sp(N) gauge groups with and without the inclusion of hypermultiplets[2][9][11][48]. The fact that the appearance of geometry seems a generic property of N = 2 gauge theories leads to the question whether the elliptic curves are purely auxiliary or if they carry an actual physical interpretation. A natural place to search for an answer to this question is in the realm of string- or M- theory, since these theories try to encode our four dimensional world through geometric constructions in extra dimensions. And indeed, it was found by Witten in [72] that Seiberg-Witten curves can be constructed in Type IIA or M-theory using certain brane configurations. Whereas four-dimensional spacetime is wrapped by the branes, they additionally extend in extra dimensions to form the Seiberg-Witten curves. This insight extended the class of gauge theories described by complex curves enormously, primarily opening the door to arbitrary product gauge theories.2 But not only does the M-theory construction give a straightforward recipe to construct Seiberg-Witten curves, it also naturally knows about rather non-trivial aspects of gauge charges in the theories[31][40]. By natural it is meant that the natural geometry of M-theory inadvertently explains these fundamental questions. This demonstrates the power of M-theory in its deep capability of explaining complicated four-dimensional physics in terms of only two basic objects: the M2 and M5 brane. The M-theory construction of Witten was further scrutinized by Gaiotto, with success[29]. Gaiotto studied N = 2 superconformal field theories and was able to determine elementary building blocks, in particular for SU(2) and SU(3) gauge theories but also the road towards SU(N), to construct product gauge theories in the form of generalized quiver diagrams. Generalized quiver 1In 2002, Nekrasov devised a method to calculate instanton corrections in a more direct manner[45]. Although this method provides a check on the Seiberg-Witten solution, it will not be discussed in this thesis. 2Witten found the brane constructions for products of unitary gauge groups. See [1] for products of symplectic and orthogonal gauge groups. 1 2 diagrams are trivalent graphs which provide a convenient depiction of both gauge and flavour symmetries and the corresponding matter representations. This realization was subsequently understood to allow for a natural identification of the quiver theories with genus g Riemann surfaces with a variety of punctures, corresponding to possible flavour symmetries. A degeneration of the Riemann surface is interpreted as a weak coupling limit of the gauge theory; different degenerations of the Riemann surface correspond to different decoupling limits of the gauge theory. This suggests the conjecture made by Gaiotto: the UV moduli space of the gauge theory coincides with moduli space of these Riemann surfaces. In particular, the boundaries of the moduli spaces are matched, mathematically checked through the construction of the Seiberg-Witten curves, and surprising dual descriptions for certain quiver gauge theories are found. Theories underlying all dual descriptions are denoted by Tf1;:::;fN ;g[AN ], making a clear two quivers are dual whenever the number and type of punctures and genus coincides. Although in the SU(2) case the possible dual descriptions of a single theory are relatively mild, the analysis extends to higher rank gauge theories. For general SU(N) quiver gauge theories, a rich zoo of building blocks exist to produce SCFTs[17]. For these higher rank theories, dual descriptions of a seemingly ordinary SU(N) theory generically include non-Lagrangian isolated interacting SCFTs whose flavour symmetries are partly gauged and coupled to the rest of the quiver. This is much in the spirit of Argyres-Seiberg duality [10] and indeed, Argyres-Seiberg duality is recovered as a special case of this much larger web of dualities. Apart from the fact the Riemann surfaces provide a useful tool to understand dualities in gauge theories, they also provide an explicit UV-IR correspondence. The distinction between the exactly marginal gauge coupling, parametrizing the UV, and the physical gauge coupling as understood from the Seiberg-Witten curve arises naturally and explains the discrepancy between the original assumptions of Seiberg-Witten of no renormalization and the explicit calculations in [44] in a beautiful geometrical way. From the M-theory construction, it becomes apparent the Seiberg-Witten curves

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