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Localization, supersymmetric gauge theories and toric geometry
Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Theoretical Physics.
2017 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Gauge theories is one of the most pervasive and important subject of modern theoretical physics, and there are still many things about them we do not understand. In particular dealing with strongly coupled theories where normal perturbative techniques do not apply is a fundamental open problem. In this thesis, we study a particular class of toy-models that have supersymmetry, which makes them much easier to deal with. We employ the mathematical technique of localization, which for supersymmetric theories lets us evaluate certain path integrals exactly and for any value of the coupling. This is used to study the 5d N=1 theories placed on toric Sasaki-Einstein manifolds and compute their partition functions, finding that they factorize into a product of contributions from each closed Reeb orbit of the manifold. This computation leads us to define two new hierarchies of special functions associated to these manifolds, and we study their properties. Finally, we use the 5d N=1 theories to construct new 4d N=2 theories on a large class of curved backgrounds. These theories have some interesting features, such as supporting both instantons and anti-instantons, and having a position-dependent complexified coupling constant.

Place, publisher, year, edition, pages
Uppsala: Acta Universitatis Upsaliensis, 2017. , 114 p.
Series
Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology, ISSN 1651-6214 ; 1500
Keyword [en]
Supersymmetry, gauge theory, localization, toric geometry
National Category
Subatomic Physics
Research subject
Theoretical Physics
Identifiers
URN: urn:nbn:se:uu:diva-318668ISBN: 978-91-554-9882-5 (print)OAI: oai:DiVA.org:uu-318668DiVA: diva2:1086479
Public defence
2017-05-26, Häggsalen, Ångströmslaboratoriet, Lägerhyddsvägen 1, Uppsala, 10:00 (English)
Opponent
Supervisors
Available from: 2017-05-02 Created: 2017-04-03 Last updated: 2017-05-05
List of papers
1. Gluing Nekrasov partition functions
Open this publication in new window or tab >>Gluing Nekrasov partition functions
2015 (English)In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 337, no 2, 785-816 p.Article in journal (Refereed) Published
Abstract [en]

In this paper we summarise the localisation calculation of 5D superYang-Mills on simply connected toric Sasaki-Einstein (SE) manifolds. We showhow various aspects of the computation, including the equivariant index, theasymptotic behaviour and the factorisation property are governed by thecombinatorial data of the toric geometry. We prove that the full perturbativepartition function on a simply connected SE manifold corresponding to an n-gontoric diagram factorises to n copies of perturbative Nekrasov partitionfunction. This leads us to conjecture the full partition function as gluing ncopies of full Nekrasov partition function. This work is a generalisation ofsome earlier computation carried out on $Y^{p,q}$ manifolds, whose moment mapcone has a quadrangle and our result is valid for manifolds whose moment mapcones have pentagon base, hexagon base, etc. The algorithm we used for dealingwith general cones may also be of independent interest.

National Category
Subatomic Physics
Research subject
Theoretical Physics
Identifiers
urn:nbn:se:uu:diva-252090 (URN)10.1007/s00220-015-2351-7 (DOI)000353505300008 ()
Note

Preprint number:  UUITP-02/14

Available from: 2015-04-29 Created: 2015-04-29 Last updated: 2017-04-03Bibliographically approved
2. Modular properties of full 5D SYM partition function
Open this publication in new window or tab >>Modular properties of full 5D SYM partition function
2016 (English)In: Journal of High Energy Physics (JHEP), ISSN 1126-6708, E-ISSN 1029-8479, Vol. 16, no 03, 193Article in journal (Refereed) Published
Abstract [en]

We study properties of the full partition function for the U(1) 5D N=2∗ gauge theory with adjoint hypermultiplet of mass M. This theory is ultimately related to abelian 6D (2,0) theory. We construct the full non-perturbative partition function on toric Sasaki-Einstein manifolds by gluing flat copies of the Nekrasov partition function and we express the full partition function in terms of the generalized double elliptic gamma function GC2 associated with a certain moment map cone C. The answer exhibits a curious SL(4,ℤ) modular property. Finally, we propose a set of rules to construct the partition function that resembles the calculation of 5D supersymmetric partition function with the insertion of defects of various co-dimensions.

National Category
Subatomic Physics
Research subject
Theoretical Physics
Identifiers
urn:nbn:se:uu:diva-287828 (URN)10.1007/JHEP03(2016)193 (DOI)000373391900002 ()
Funder
Swedish Research Council, 2014-5517The Swedish Foundation for International Cooperation in Research and Higher Education (STINT)Knut and Alice Wallenberg Foundation
Available from: 2016-04-26 Created: 2016-04-26 Last updated: 2017-04-03Bibliographically approved
3. Multiple elliptic gamma functions associated to cones
Open this publication in new window or tab >>Multiple elliptic gamma functions associated to cones
(English)Manuscript (preprint) (Other academic)
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
urn:nbn:se:uu:diva-318666 (URN)
Available from: 2017-03-27 Created: 2017-03-27 Last updated: 2017-04-03
4. N=2 supersymmetric gauge theory on connected sums of S2xS2
Open this publication in new window or tab >>N=2 supersymmetric gauge theory on connected sums of S2xS2
2017 (English)In: Journal of High Energy Physics (JHEP), ISSN 1126-6708, E-ISSN 1029-8479, Vol. 03, 026Article in journal (Refereed) Published
Abstract [en]

We construct 4D N = 2 theories on an infinite family of 4D toric manifolds with the topology of connected sums of S-2 x S-2. These theories are constructed through the dimensional reduction along a non -trivial U(1) -fiber of 5D theories on toric Sasaki Einstein manifolds. We discuss the conditions under which such reductions can be carried out and give a partial classification result of the resulting 4D manifolds. We calculate the partition functions of these 4D theories and they involve both instanton and anti-instanton contributions, thus generalizing Pestun's famous result on S-4.

National Category
Subatomic Physics
Research subject
Theoretical Physics
Identifiers
urn:nbn:se:uu:diva-318662 (URN)10.1007/JHEP03(2017)026 (DOI)000397669900008 ()
Available from: 2017-03-27 Created: 2017-03-27 Last updated: 2017-04-19Bibliographically approved

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