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On Generalisations of the AGT Correspondence for Non-Lagrangian Theories of Class S

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On Generalisations of the AGT Correspondence for Non-Lagrangian Theories of Class S On generalisations of the AGT correspondence for non-Lagrangian theories of class S Dissertation zur Erlangung des Doktorgrades an der Fakultat¨ fur¨ Mathematik, Informatik und Naturwissenschaften Fachbereich Physik der Universitat¨ Hamburg vorgelegt von Ioana Coman-Lohi Hamburg 2018 Gutachter/innen der Dissertation: Prof. Dr. Joerg Teschner Prof. Dr. Volker Schomerus Zusammensetzung der Prfungskommission: Prof. Dr. Joerg Teschner Prof. Dr. Volker Schomerus Prof. Dr. Gleb Arutyunov Prof. Dr. Gunter¨ H. W. Sigl Prof. Dr. Dieter Horns Vorsitzende/r der Prfungskommission: Prof. Dr. Dieter Horns Datum der Disputation: 18.04.2018 Vorsitzender Fach-Promotionsausschusses PHYSIK: Prof. Dr. Wolfgang Hansen Leiter des Fachbereichs PHYSIK: Prof. Dr. Michael Potthoff Dekan der Fakultat¨ MIN: Prof. Dr. Heinrich Graener To Karolina, Nikolas, Ana, Doru and Cristina Abstract Non-perturbative aspects of N = 2 supersymmetric field theories of class S are deeply encoded in the algebra of functions on moduli spaces Mflat of flat SL(N)-connections on Riemann surfaces. Expectation values of Wilson and ’t Hooft supersymmetric line operators are related to holonomies of flat connections, while expectation values of line operators in the low energy effective theory are related to Fock-Goncharov coordinates on Mflat. Through the decomposition of UV line operators into IR line operators, we determine their noncommutative algebra from the quantisation of Fock-Goncharov Laurent polynomials, and find that it coincides with the skein algebra studied in the context of Chern-Simons theory through quantum group theory methods. We also investigate the relations between the quantisation of moduli spaces of flat connections, slN Toda conformal field theories and quantum group theory, and find that another realisation of the skein algebra is generated by Verlinde network oper- ators on Toda conformal blocks. To arrive here we use the free field representation of W-algebras to define natural bases for spaces of conformal blocks in Toda field theory. For sl3 conformal blocks on a three-punctured sphere with generic representa- tions associated to the three punctures we construct operator valued monodromies of degenerate fields which can be used to describe the quantisation of the moduli spaces of flat SL(3)-connections. The matrix elements of these quantum monodromy ma- trices can be expressed as Laurent polynomials of more elementary operators, which have a simple definition in the free field representation and are identified as quantised counterparts of natural higher rank Fenchel-Nielsen type coordinates. Arguments from quantum group theory then lead to the identification of the quantum skein algebra of functions on moduli spaces of flat connections and the algebra of Verlinde network op- erators. Comparing the spectra of these two realisations provides non-trivial support for their equivalence. Our results can be viewed as evidence for the generalisation of the AGT correspondence to higher-rank class S theories, in particular those with no known weakly-coupled Lagrangian description. We investigate possible applications to the study of non-Lagrangian SUSY field theo- ries through their relation to topological string theory. Beginning with the topological string partition functions corresponding to the TN trinion theories of class S, we ex- plore the relation between the geometric data of the local Calabi-Yau manifold used in the geometric engineering of these theories and certain parameters which enter the free field construction of slN Toda conformal blocks. A meaningful way to take the geometric engineering limit is discussed in the most simple case. Zusammenfassung Nichtperturbative Aspekte N = 2 supersymmetrischer Feldtheorien der Klasse S sind in der Algebra der Funktionen auf Modulraumen¨ Mflat flacher SL(N)-Zusammenhange¨ auf Riemannschen Flachen¨ verschlusselt.¨ Erwartungswerte supersymmetrischer Wilson und ’t Hooft Linienoperatoren stehen in Verbindung mit Holonomien flacher Zusammenhange,¨ wahrend¨ Erwartungswerte von Linienoperatoren in der effektiven Niedrigenergie-Theorie mit Fock-Goncharov Koordinaten auf Mflat verbunden sind. Uber¨ die Zerlegung von UV- Linienoperatoren in IR-Linienoperatoren bestimmen wir ihre nichtkommutative Algebra aus der Quantisierung der Fock-Goncharov Laurent-Polynome und finden mithilfe quantengrup- pentheoretischer Methoden, dass sie mit der Skein-Algebra ubereinstimmt,¨ welche im Kontext von Chern-Simons Theorie behandelt wird. Daruber¨ hinaus untersuchen wir Verbindungen zwischen der Quantisierung von Modulraumen¨ flacher Zusammenhange,¨ slN Toda konformen Feldtheorien und Quantengruppentheorie. Wir finden, dass eine weitere Realisierung der Skein-Algebra durch Verlinde-Netzwerkoperatoren auf konformen Blocken¨ von Toda konformen Feldtheorien erzeugt wird. Um zu diesem Ergeb- nis zu gelangen, nutzen wir die freie Felddarstellung von W-Algebren, um eine naturliche¨ Basis fur¨ Raume¨ konformer Blocke¨ in Toda konformen Feldtheorien zu definieren. Fur¨ sl3 konforme Blocke¨ auf einer 3-punktierten Sphare¨ mit beliebigen zu den drei Punkten assozi- ierten Darstellungen konstruieren wir operatorwertige Monodromien entarteter Felder, welche zur Beschreibung der Quantisierung von Modulraumen¨ flacher SL(3)-Zusammenhange¨ genutzt werden konnen.¨ Die Matrixelemente dieser Quanten-Monodromiematrizen konnen¨ als Laurent- Polynome elementarerer Operatoren ausgedruckt¨ werden, welche eine einfache Definition in der freien Felddarstellung haben und die quantisierten Gegenstucke¨ naturlicher¨ Fenchel-Nielsen Koordinaten hoheren¨ Ranges sind. Argumente aus der Quantengruppentheorie fuhren¨ dann zur Identifizierung der Quanten-Skein-Algebra der Funktionen auf Modulraumen¨ flacher Zusam- menhange¨ und der Algebra der Verlinde-Netzwerkoperatoren. Der Vergleich der Spektren dieser beiden Realisierungen liefert nicht-triviale Evidenz fur¨ ihre Aquivalenz.¨ Unsere Resul- tate konnen¨ als Untermauerung der Verallgemeinerung der AGT Korrespondenz auf Theorien der Klasse S hoheren¨ Rangs, insbesondere fur¨ jene ohne bekannte Lagrange-Beschreibung im schwach gekoppelten Regime, verstanden werden. Wir untersuchen mogliche¨ Anwendungen in der Erforschung supersymmetrischer Theorien ohne Lagrange-Beschreibung durch ihre Verbindung zur topologischen Stringtheorie. Begin- nend mit der Zustandssumme des topologischen Strings, welche dem TN Trinion in Theorien der Klasse S entspricht, erkunden wir die Verbindung zwischen den geometrischen Daten der lokalen Calabi-Yau Mannigfaltigkeit, welche im Kontext des Geometric Engineering dieser Theorien benutzt werden, und bestimmten Parametern, welche in der freien Feldkonstruktion von slN Toda konformen Blocken¨ auftauchen. Fur¨ den einfachsten Fall diskutieren wir einen sinnvollen Weg, den Grenzwert fur¨ das Geometric Engineering zu bestimmen. This thesis is based on the publications: • I. Coman, M. Gabella, J. Teschner, Line operators in theories of class S, quan- tized moduli space of flat connections, and Toda field theory, JHEP 1510 (2015) 143, [arXiv:1505.05898]; • I. Coman, E. Pomoni, J. Teschner, Toda conformal blocks, quantum groups, and flat con- nections, (2017), [arXiv:1712.10225]; • I. Coman, E. Pomoni, J. Teschner, Liouville conformal blocks from topological strings, to appear; • I. Coman, E. Pomoni, J. Teschner, Toda conformal blocks from topological strings, in progress. Other publications by the author: • I. Coman, E. Pomoni, F. Yagi, M. Taki, Spectral curves of N=1 theories of class Sk, JHEP 1706, (2017) 136, [arXiv:1512.06079]; • C.I. Lazaroiu, E.M. Babalic, I.A. Coman, The geometric algebra of supersymmetric back- grounds, Proceedings of Symposia in Pure Mathematics, Vol. 90 (2015) 227-237; • C.I. Lazaroiu, E.M. Babalic, I.A. Coman, The geometric algebra of Fierz identities in ar- bitrary dimensions and signatures, JHEP 09 (2013) 156, [arXiv:1304.4403]; • C.I. Lazaroiu, E.M. Babalic, I.A. Coman, A unified approach to Fierz identities, AIP Conf. Proc. 1564, 57 (2013), [arXiv:1303.1575]; • E.M. Babalic, I.A. Coman, C. Condeescu, C.I. Lazaroiu, A. Micu, On N=2 compactifica- tions of M-theory to AdS3 using geometric algebra techniques, AIP Conf. Proc. 1564, 63 (2013); • C.I. Lazaroiu, E.M. Babalic, I.A. Coman, Geometric algebra techniques in flux compacti- fications, Advances of High Energy Physics (2016), [arXiv:1212.6766v1]. Contents I Introduction 1 1 Introduction 1 1.1 Recent advances . .2 1.2 Correspondence to two dimensional conformal field theory . .6 1.3 Relation to topological string theory . .8 1.4 Parallel developments and an alternative route towards AGT . .9 1.5 Overview . 13 II Quantisation of moduli spaces of flat connections on punctured Riemann surfaces 17 2 Algebra of loop and network operators 21 2.1 Moduli space of flat connections . 21 2.2 Trace functions . 22 2.3 Poisson structure . 24 2.4 Classical skein algebra . 25 2.5 Loop and network functions . 26 2.6 Commuting Hamiltonians . 29 2.7 Tinkertoys . 30 2.8 Skein quantisation . 31 3 Quantisation of tinkertoys 37 3.1 Pants networks . 37 3.2 One-punctured torus . 45 3.3 Four-punctured sphere . 50 4 Fock-Goncharov holonomies 53 CONTENTS 4.1 Fock-Goncharov coordinates . 53 4.2 Holonomies . 54 4.3 Quantisation . 56 4.4 Pants networks . 58 4.5 One-punctured torus . 65 4.6 Four-punctured sphere . 72 5 First conclusions 75 Appendices 77 A Fock-Goncharov coordinates 79 A.1 Coordinates associated with N-triangulations . 79 A.2 Snakes and projective bases . 81 III Toda conformal blocks, moduli spaces of flat connections and quantum groups 87 6 Relation to conformal field
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