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Higher-Dimensional Instantons on Cones Over Sasaki-Einstein Spaces and Coulomb Branches for 3-Dimensional N = 4 Gauge Theories

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Higher-Dimensional Instantons on Cones Over Sasaki-Einstein Spaces and Coulomb Branches for 3-Dimensional N = 4 Gauge Theories Two aspects of gauge theories higher-dimensional instantons on cones over Sasaki-Einstein spaces and Coulomb branches for 3-dimensional N = 4 gauge theories Von der Fakultät für Mathematik und Physik der Gottfried Wilhelm Leibniz Universität Hannover zur Erlangung des Grades Doktor der Naturwissenschaften – Dr. rer. nat. – genehmigte Dissertation von Dipl.-Phys. Marcus Sperling, geboren am 11. 10. 1987 in Wismar 2016 Eingereicht am 23.05.2016 Referent: Prof. Olaf Lechtenfeld Korreferent: Prof. Roger Bielawski Korreferent: Prof. Amihay Hanany Tag der Promotion: 19.07.2016 ii Abstract Solitons and instantons are crucial in modern field theory, which includes high energy physics and string theory, but also condensed matter physics and optics. This thesis is concerned with two appearances of solitonic objects: higher-dimensional instantons arising as supersymme- try condition in heterotic (flux-)compactifications, and monopole operators that describe the Coulomb branch of gauge theories in 2+1 dimensions with 8 supercharges. In PartI we analyse the generalised instanton equations on conical extensions of Sasaki- Einstein manifolds. Due to a certain equivariant ansatz, the instanton equations are reduced to a set of coupled, non-linear, ordinary first order differential equations for matrix-valued functions. For the metric Calabi-Yau cone, the instanton equations are the Hermitian Yang-Mills equations and we exploit their geometric structure to gain insights in the structure of the matrix equations. The presented analysis relies strongly on methods used in the context of Nahm equations. For non-Kähler conical extensions, focusing on the string theoretically interesting 6-dimensional case, we first of all construct the relevant SU(3)-structures on the conical extensions and subsequently derive the corresponding matrix equations. For the Kähler-torsion sine-cone the problem is reduced to the Calabi-Yau cone by means of conformal invariance. For the nearly Kähler and half-flat 6-manifolds we derive new explicit solutions. The extension of a quiver bundle construction, defined via equivariant dimensional reduction over product manifolds M d×G/H with Kähler cosets, to the case of the Sasaki-Einstein coset S5 = SU(3)/SU(2) (and A-type orbifolds thereof) is considered in Part II. The construction of the quiver bundle via SU(3)-equivariant dimensional reduction over M d×S5 is described in detail and turns out to be an extension of SU(3)-equivariant dimensional reduction over d 2 M ×CP . The extension is manifest in vertex edge loops, associated to the Hopf fibration 5 2 U(1) ,→ S → CP , and is exemplified with specific examples. As the arising quiver gauge theories are conceptually different from the conventional quiver gauge theories found in the physics literature, we aim to provide a first step in the comparison between these two. For that, we associate two different instanton moduli spaces on the Calabi-Yau orbifold C(S5/Γ): on the one hand, equivariant instantons, wherein the underlying structure of the gauge theory is fixed by the Sasakian quiver associated to S5, and, on the other hand, translationally invariant instantons, which exhibit an appearance as moduli spaces of world volume gauge theories on D-branes located at orbifold singularities. We contrast the resulting quiver gauge theories, their relations, and construct the formal Kähler structure on the moduli space. Recently, the Coulomb branch MC of 3-dimensional N = 4 gauge theories has experienced an algebraic description in terms of the monopole formula which is the Hilbert series associated to the chiral ring C[MC ]. In Part III we analyse the monopole formula and provide a geome- tric derivation of the set of chiral ring generators. The matter content divides the dominant Weyl chamber of the GNO-dual of the gauge group into a fan, via the weight vectors of the matter field representation. Upon intersection of the fan with the weight lattice, the monopole formula turns into a sum over affine semi-groups twisted by the Casimir invariance. This new understanding provides unprecedented insights into the combinatorics of the Hilbert series: we can explicitly sum the Hilbert series (in principle for any gauge group), we identify a sufficient set of bare monopole operators, we derive the number and degree of the corresponding dressed monopole operators, and we provide a clean approximation of the plethystic logarithm. This novel interpretation is tested against a comprehensive list of rank two examples and one rank three scenario. key words: higher-dimensional instantons, monopole operators, gauge theories iii Kurzdarstellung Solitonen und Instantonen sind fundamentale Bestandteile moderner Feldtheorien, nicht nur in Hochenergiephysik oder Stringtheorie, sondern auch in der Theorie kondensierter Materie und Optik. In der vorliegenden Arbeit werden zwei Beispiele solcher solitonischen Objekte betrachtet: höher-dimensionale Yang–Mills-Instantonen, welche als Supersymmetriebedingungen in heterotischen (Fluss-)Kompaktifizierungen erscheinen, und Monopoloperatoren, welche für die Beschreibung des Coulomb-Zweiges von Eichtheorien in 2+1 Dimensionen mit 8 Superladungen notwendig sind. DerI. Teil der Arbeit ist der Analyse der verallgemeinerten Instantongleichungen auf kegel- förmigen Erweiterungen von Sasaki–Einstein-Mannigfaltigkeiten gewidmet. Die Instantonglei- chungen können durch einen äquivarianten Ansatz auf ein System gekoppelter, nichtlinearer, gewöhnlicher Differentialgleichungen erster Ordnung für matrixwertige Funktionen reduziert werden. Für den Fall des metrischen Calabi–Yau-Kegels sind die Instantongleichungen identisch zu den hermiteschen Yang–Mills-Gleichungen, sodass wir deren geometrische Strukturen nutzen können, um die Geometrie der Matrixgleichungen zu untersuchen. Die in dieser Arbeit präsen- tierte Analyse basiert auf Methoden, die ihre Anwendung bereits bei den Nahm-Gleichungen gefunden haben. Im Falle von Kegelerweiterungen die nicht Kähler sind, wobei wir uns auf die stringtheoretisch interessanten 6-dimensionalen Fälle beschränken, konstruieren wir zuerst die relevanten SU(3)-Strukturen und leiten anschließend die korrespondierenden Matrixgleichungen her. Der dabei auftretende Fall Kähler-Torsion ist mittles konformer Invarianz äquivalent zum Calabi–Yau-Kegel. Für die nearly Kähler und halb-flachen 6-Mannigfaltigkeiten präsentieren wir neue explizite Lösungen. Im Mittelpunkt desII. Teils steht die Übertragung einer Konstruktion von Köcherbündeln, welche ursprünglich durch die äquivariante dimensionale Reduktion auf Produkten M d×G/H mit einer Kähler-Faktorgruppe G/H definiert ist, auf den Fall der Sasaki-Einstein-Faktorgruppe S5 = SU(3)/SU(2) und Orbifaltigkeiten derer. Die Konstruktion des Köcherbündels mittels SU(3)-äquivarianter dimensionaler Reduktion über M d×S5 wird detailliert beschrieben und ent- d 2 puppt sich als Erweiterung der SU(3)-äquivarianten dimensionalen Reduktion über M ×CP . 5 2 Markenzeichen der Erweiterung sind Vertexschleifen, welche zur Faserung U(1) ,→ S → CP assoziiert sind, und wir verdeutlichen diesen Umstand anhand von Beispielen. Da diese so erzeug- ten Köchereichtheorien konzeptionell verschieden von den gewöhnlichen Köchereichtheorien der Hochenergiephysik sind, beabsichtigen wir einen ersten Vergleich zwischen diesen zu ziehen. Dar- um betrachten wir zwei verschiedene Instantonlösungsräume auf der Calabi–Yau-Orbifaltigkeit C(S5/Γ): zum einen den der äquivarianten Instantonen, deren zugrunde-liegende Eichtheorien strukturell durch die Sasaki-Köcher bestimmt sind, und zum anderen den der translationsin- varianten Instantonen, deren Lösungsraum eine Realisierung als Weltvolumeneichtheorien auf D-Branen findet. Für diese beiden Fälle stellen wir die resultierenden Köchereichtheorien sowie deren Relationen gegenüber und konstruieren die formale Kählerstruktur auf den Lösungsräu- men. Der Coulomb-Zweig MC von 3-dimensionalen N =4 Eichtheorien wurde kürzlich algebraisch durch die Monopolformel beschrieben, welche der Hilbertreihe des chiralen Rings C[MC ] ent- spricht. Im III. Teil analysieren wir die Monopolformel und liefern eine geometrische Herleitung einer hinreichenden Menge von Generatoren des chiralen Rings. Der Materiegehalt der Theo- rie, genauer gesagt die Gewichtsvektoren der Darstellung in welcher die Materie transformiert, zerlegt die fundamentale Weylkammer des GNO-Dualen der Eichgruppe in einen Fächer. Nach Schnitt des Fächers mit dem Gewichtsgitter kann die Monopolformel als Summe über affine Halbgruppen verstanden werden, welche noch zusätzlich mit der Casimir-Invarianz nicht-trivial verflochten wird. Dieses neuartige Verständnis erlaubt weitere Einblicke in die Kombinatorik der Hilbertreihe: Man kann die Hilbertreihe explizit summieren (im Prinzip für einen beliebige v Eichtheorie), einen hinreichenden Satz von nackten Monopoloperatoren identifizieren, sowie die Anzahl und die Dimensionen der bekleideten Monopoloperatoren herleiten. Außerdem erlaubt dieses Wissen eine saubere Näherung des plethystischen Logarithmus. Die entwickelte neue Interpretation wird mittels einer umfassende Liste von Beispielen mit Eichgruppen vom Rang zwei und einer Gruppe vom Rang drei verifiziert sowie verdeutlicht. Schlagworte: verallgemeinerte Instantonen, Monopoloperatoren, Eichtheorien vi Contents overview 1 Introduction1 1.1 Motivation......................................1 1.2 Thesis overview....................................4 I Instantons on cones over Sasaki-Einstein manifolds7 2 Introduction and motivation 11 2.1 Compactifications of the heterotic
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