Zoran Skoda diplomski higher gauge theories

This page offers a description of a possible mentorship for a 5th year student of theoretical physics (University of Zagreb) who is interested to write a diploma thesis with the topic “Higher gauge theories” (only basics should be covered; approximate focused study and writing should fit, if possible, within about 3 months from starting acquainting with the topic). A version of the project can be alternatively taken by a senior student of the mathematics department as well.

In nlab one can find a lot of material related to that topic; though the corresponding material in $n$lab is far more mathematically oriented, we expect a diploma thesis treatment from possibly far more physical viewpoint and in more physicists-style conventions and notation; the exact balance will be found in the process. Available literature is wide and varies from very physically minded all up to the works of pure mathematics, and the author will find with his mentor what is the best suited option, according to his/her interests and background. Interested to possibly undertake this diploma work should talk to Zoran ?koda?.

Tema (Topic): Vi?e ba?darske teorije (Higher gauge theories)

Sa?etak: Ba?darski potencijali imaju geometrijsku interpretaciju kao koneksije u diferencijalnoj geometriji i samim time one su primjeri diferencijalnih 1-formi s koeficijentima u Liejevoj algebri ba?darske grupe. ?estica koja se giba u polju ba?darskog potencijala, dobiva fazu koja je direktno povezana s tzv. holonomijom ba?darskog polja uzdu? putanje. Diracova kvantizacija elektri?nog naboja je zapravo uvjet dobre globalne definiranosti holonomije ba?darskog polja. Danas postoji niz modela u kojima ulogu analognu 1-formi koneksija imaju neke 2-forme ili ?ak vi?i analogoni. Osnovni primjer su Kalb-Ramondova polja (?B-field?) u teoriji superstruna i D-brana. Holonomija uzdu? putanja zamijenjena je sli?nim veli?inama koje odgovaraju paralelnom prijenosu uzdu? povr?ina, radije nego linija. Cilj ovog diplomskog rada je obraditi zorno geometrijsko porijeklo ovih veli?ina kao diferencijalnih formi koneksija na vi?im analogonima sve?enjeva ili alternativno kao paralelnog prijenosa uzdu? vi?edimenzionalnih povr?ina. To je vrlo aktualna tematika u teorijskoj fizici, a vode?i autori su Daniel Freed, Greg Moore, John Baez i na? suradnik Urs Schreiber.

Abstract: Gauge potentials have a geometric interpretation as connections in differential geometry and they are hence examples of differential 1-forms with coefficients in the Lie algebra of the gauge group. A particle moving in a gauge field is acquiring a phase directly related to the parallel transport of the gauge field along a path in the base space. Dirac quantization of electric field is in fact a condition of that the holonomy/parallel transport is globally well defined. There is a number of models today where instead of connection 1-forms some differential $n$-forms appear. The basic example is the Kalb-Ramond field (?B-field?) in superstring theory with D-branes. The parallel transport along paths is is replaced by similar operators which denote the transport along higher dimensional surfaces, rather than line paths. The diploma work should cover the connection differential forms on higher analogues of bundles, and the parallel transport along multidimensional surfaces with emphasis on their clear geometric origins. This is a contemporary problematics in physics, and the leading authors are Daniel Freed, Greg Moore, John Baez and our collaborator Urs Schreiber.

Osnovna literatura (Basic references):

• John C. Baez, John Huerta, An invitation to higher gauge theory, arxiv/1003.4485
• John C. Baez, Urs Schreiber, Higher gauge theory, in: Categories in algebra, geometry and mathematical physics, 7–30, Contemp. Math. 431, Amer. Math. Soc. 2007, math.DG/0511710, Higher gauge theory II: 2-connections on 2-bundles, hep-th/0412325

Osnove diferencijalne geometrije za fizi?are (Basic background of differential geometry for physicists):

• Bernard F. Schutz, Geometrical methods of mathematical physics, amazon, google

Physical background of gauge theories in geometrical treatment is also in

• Albert Schwartz, Quantum field theory and topology, Grundlehren der Math. Wissen. 307, Springer 1993. (translated from Russian original)

Neki prijedlozi mogu?e dopunske literature (Possible additional references and background):

• nlab: gauge theory, holonomy, connection on a bundle
• Johan Louis Dupont, Fibre bundles and Chern-Weil theory, Lecture Notes Series 69, Dept. of Math., University of Aarhus, Aarhus, 2003, 115 pp. pdf
• Rafael Rani, On parallel transport and curvature, pdf
• J. Baez, A. Crans, Higher dimensional algebra VI: Lie 2-algebras, Theory and Applications of Categories 12 (2004), 492?538. math.QA/0307263

Naprednija literatura (Selected advanced references):

• Charles Nash, Differential topology and quantum field theory, Acad. Press 1991.

• Daniel S. Freed, Dirac charge quantization and generalized differential cohomology, Surveys in differential geometry, 129–194, Surv. Differ. Geom., VII, Int. Press, Somerville, MA, 2000.

• Urs Schreiber, Zoran ?koda, Categorified Symmetries, arxiv/1004.2472

• R. Bott, L. W. Tu, Differential forms in algebraic topology, Graduate Texts in Mathematics 82, Springer 1982. xiv+331 pp.

• Yvonne Choquet-Bruhat, Cecile Dewitt-Morette, Analysis, manifolds and physics, 1982 and 2001

• differential cohomology in an (∞,1)-topos -- survey

• Hisham Sati, Geometric and topological structures related to M-branes, arxiv/1001.5020

• Anton Kapustin, Topological field theory, higher categories, and their applications, Proc. ICM 2010, arxiv/1004.2307

• Dan Freed, Mike Hopkins, Jacob Lurie, Constantin Teleman, Topological Quantum Field Theories from Compact Lie Groups (arXiv)

• Tom Leinster, Higher categories, higher operads, math.CT/0305049

• Thomas Nikolaus, Higher categorical structures in geometry, general theory and applications to quantum field theory, Ph.D. thesis, Hamburg 2011, pdf and slides