# Marcel Bischoff

## Recent News

## Education

Ph.D. in Mathematics at the University of Rome “Tor Vergata,” 2012. Ph.D. thesis “Construction of Models in low-dimensional Quantum Field Theory using Operator Algebraic Methods” supervised by Prof. Roberto Longo.

Diploma in Physics at the Georg-August-Universität Göttingen, 2009. Diploma thesis: “On the Pole Structure of Higher Correlation Functions in Globally Conformal Invariant Quantum Field Theory” (in german) supervised by Prof. Karl-Henning Rehren.

## Research Interests

- Operator Algebras, in particular Subfactors
- Algebraic Quantum Field Theory, in particular Conformal Field Theory
- Tensor Categories

## NSF Grant

Bischoff is the PI on an NSF grant titled Quantum Symmetries and Conformal Nets.

**Abstract**: Symmetries, which can be mathematically described by groups, play an important role in science. In quantum physics, the fundamental theory of physics at small scales, operator algebras provide a mathematical framework to study quantum systems and their symmetries. To describe quantum particles and matter in low dimensions (for example, on a thin layer) one needs a generalization of symmetry that goes beyond groups, often referred to as quantum symmetries. A main focus of this project is to find models realizing such symmetries in quantum field theory, which combines the principles of quantum physics and the theory of special relativity. The goal is to use the rich interplay between operator algebras and quantum field theory via conformal nets and to better understand possible quantum symmetries in mathematics and low-dimensional physics. One potential application is topological quantum computing, where the goal is to use quantum operations coming from non-trivial quantum symmetries to perform computations.

Conformal nets give rise to interesting quantum symmetries in terms of subfactors and unitary modular tensor categories. The project will focus on three main directions. First, the principal investigator will extend the understanding of boundaries or defects between conformal nets, generalize the abstract description of defects in terms of braided subfactors, and explore the relation to Jones' planar algebras. The second project focus is to provide new structural results and examples of rational conformal nets realizing the quantum doubles subfactors. Lastly, the principal investigator will develop methods for using operator-algebraic techniques to construct conformal nets from their quantum symmetries and give relations between the operator algebraic approach of conformal nets and the purely algebraic approach of vertex operator algebras.

## Former Positions

2014/08 – 2017/08 Assistant Professor (non-tenure track, Postdoc) of Mathematics, Vanderbilt University, Nashville, TN. Supervisor: Vaughan F. R. Jones.

2012/10 – 2014/08 Postdoctoral researcher in the DFG (German Research Foundation) Research Training Group 1493 “Mathematical Structures in Modern Quantum Physics”, University of Göttingen, Germany.

## Scientific/Academic Honors and Grants

2015-2016 Summer research support through NSF Grant DMS-1362138, principal investigator Vaughan F.R. Jones.

2014-2017 Vanderbilt University, College of Arts and Science annual research fund and Postdoc appointment with reduced (1-1) teaching load.

2009-2012 PhD scholarship in Mathematics at the University of Rome “Tor Vergata”.

## Publications

Marcel Bischoff, Alexei Davydov: Hopf algebra actions in tensor categories, accepted in Transformation Groups. [arXiv:1811.10528 [math.QA]]

Marcel Bischoff, Corey Jones, Yuan-Ming Lu, David Penneys: Spontaneous symmetry breaking from anyon condensation. Journal of High Energy Physics (2019) 2019:62. DOI: 10.1007/JHEP02(2019)062M [Journal Link (open access)] [arXiv:1811.00434 [math.QA]]

Marcel Bischoff: Generalized Orbifold Construction for Conformal Nets. Reviews in Mathematical Physics Vol. 29, No. 1 (2017) 1750002 (53 pages). DOI: 10.1142/S0129055X17500027 [Journal link] [arXiv:1608.00253 [math-ph]] [MathSciNet:MR3595480] [zbMATH:ZBl 06680086]

Marcel Bischoff: A Remark on CFT Realization of Quantum Doubles of Subfactors: Case Index < 4, Letters in Mathematical Physics, March 2016, Volume 106, Issue 3, pp 341-363. DOI: 10.1007/s11005-016-0816-z [Journal Link] [arXiv:1506.02606 [math-ph]] [MathSciNet:MR3462031] [zbMATH:ZBl 1333.81364]

Marcel Bischoff, Yasuyuki Kawahigashi, Roberto Longo, Karl-Henning Rehren: Phase boundaries in algebraic conformal QFT, Communications in Mathematical Physics, February 2016, Volume 342, Issue 1, pp 1-45. DOI: 10.1007/s00220-015-2560-0 [Journal link] [arXiv:1405.7863 [math-ph]] [MathSciNet:MR3455144] [zbMATH:ZBl 1355.81101]

Marcel Bischoff, Yasuyuki Kawahigashi, Roberto Longo: Characterization of 2D rational local conformal nets and its boundary conditions: the maximal case, Documenta Math. 20 (2015) 1137--1184 [Journal Link (open access)] [arXiv:1410.8848 [math-ph]] [MathSciNet:MR3424476] [zbMATH:ZBl 1337.81103]

Marcel Bischoff, Yasuyuki Kawahigashi, Roberto Longo, Karl-Henning Rehren: Tensor categories of endomorphisms of von Neumann algebras (with applications to Quantum Field Theory), to appear as a book in Springer Briefs in Mathematical Physics, Vol 3. ISBN 978-3-319-14300-2, 2015. [Journal Link] [arXiv:1407.4793 [math.OA]] [MathSciNet:MR3308880] [zbMATH:ZBl 06398995]

Marcel Bischoff, Yoh Tanimoto: Integrable QFT and Longo-Witten endomorphisms, Annales Henri Poincaré. February 2015, Volume 16, Issue 2, pp 569-608. DOI: 10.1007/s00023-014-0337-1 [Journal link (open access)] [Erratum (open access)] [arXiv:1305.2171 [math-ph]] [MathSciNet:MR3302606] [zbMATH:ZBl 1317.81192]

Marcel Bischoff, Yoh Tanimoto: Construction of wedge-local nets of observables through Longo-Witten endomorphisms. II Commun. Math. Phys. 317 (3), 2013 p. 667-695, DOI: 10.1007/s00220-012-1593-x [Journal link (open access)] [arXiv:1111.1671 [math-ph]] [MathSciNet:MR3009721] [zbMATH:ZBl 1260:81215]

Marcel Bischoff: Models in Boundary Quantum Field Theory Associated with Lattices and Loop Group Models, Commun. Math. Phys. 315 (3), 2012 p. 827-858, DOI: 10.1007/s00220-012-1511-2 [Journal link] [arXiv:1108.4889 [math-ph]] [MathSciNet:MR2981815] [zbMATH:ZBl 1256.81078]

Marcel Bischoff, Daniel Meise, Karl-Henning Rehren, Ingo Wagner: Conformal quantum field theory in various dimensions Bulg. J. Phys. 36 (2009) 170-185. [Journal Link (open access)] [arXiv:0908.3391 [math-ph]] [LQP] [MathSciNet:MR2640829] [zbMATH:ZBl 1202.81189]

### Preprints

Marcel Bischoff, A remark about the anomalies of cyclic holomorphic permutation orbifolds.

[arXiv:1812.11910 [math-ph]]

Marcel Bischoff: The rank of G-crossed braided extensions of modular tensor categories.

[arxiv:1807.06131 [math.QA]]

Marcel Bischoff: Conformal Net Realizability of Tambara-Yamagami Categories and Generalized Metaplectic Modular Categories.

[arXiv:1803.04949 [math.QA]]

### Proceedings

Marcel Bischoff, Karl-Henning Rehren: The Hypergroupoid Of Boundary Conditions for Local Quantum Observables, Adv. Stud. Pure Math. 80 (2019) 23-42.

[Journal Link] [arXiv:1612.02972 [math-ph]]

Marcel Bischoff: The Relation between Subfactors arising from Conformal Nets and the Realization of Quantum Doubles, Proc. Centre Math. Appl., Proceedings of the 2014 Maui and 2015 Qinhuangdao Conferences in Honour of Vaughan F.R. Jones’ 60th Birthday. Scott Morrison and David Pennys, eds. Proceedings of the Centre for Mathematics and its Applications, v. 46. (Canberra AUS: Centre for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University, 2017), 15 - 24 2017.

[Journal Link (open access)] [arXiv:1511.08931 [math-ph]]

Marcel Bischoff (joint with Yasuyuki Kawahigashi, Roberto Longo, Karl-Henning Rehren): An Algebraic Conformal Quantum Field Theory Approach to Defects, p. 906-908. In: Oberwolfach Reports, Subfactors and Conformal Field Theory, Organised by Dietmar Bisch, Terry J. Gannon, Vaughan F. R. Jones and Yasuyuki Kawahigashi, Volume 12, Issue 2, 2015, pp. 849–926. DOI: 10.4171/OWR/2015/16.

[Journal Link]

### Theses

PhD Thesis (Mathematics): Construction Of Models In Low-Dimensional Quantum Field Theory Using Operator Algebraic Methods, University of Rome Tor Vergata, September 2012

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Diploma Thesis (Physics): Über die Polstruktur höherer Korrelationsfunktionen in global konform-invarianter Quantenfeldtheorie (in German), University of Göttingen 2009

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