Publications of João Faria Martins

Anyons moving in 2D space
Surface holonomy along the surface Σ
      Stasheff's polytope

Preprints (comments are very welcome):

- Joao Faria Martins, Frank W Nijhoff, Daniel Riccombeni: The Darboux-KP system as an integrable Chern-Simons multiform theory in infinite dimensional space. arXiv:2305.03182 [math-ph]

- Faria Martins J, Porter T : A categorification of Quinn's finite total homotopy TQFT with application to TQFTs and once-extended TQFTs derived from strict omega-groupoids. arXiv:2301.02491 [math.CT]. (235 pages, preliminary version.)

- Koppen V, Faria Martins J, Martin P: Exactly solvable models for 2+1D topological phases derived from crossed modules of semisimple Hopf algebras. arXiv:2104.02766 [math-ph]

Publications (comments are very welcome as well):

- Motion groupoids and mapping class groupoids. arXiv:2103.10377 [math-ph], with Fiona Torzewska and Paul Martin. To appear in Commmunications in Mathematical Physics.

-On a canonical lift of Artin's representation to loop braid groups. With Celeste Damiani and Paul Martin. Journal of Pure and Applied Algebra Volume 225, Issue 12, December 2021, 106760. Preliminary version.

-Representations of the Loop Braid Group and Aharonov-Bohm like effects in discrete (3+1)-dimensional higher gauge theory. With Alex Bullivant and Paul Martin. Advances in Theoretical and Mathematical Physics.(To appear.) Preliminary version.

Higher lattices, discrete two-dimensional holonomy and topological phases in (3+1) D with higher gauge symmetry. With Alex Bullivant, Marcos Calçada, Zoltán Kádár and Paul Martin. Reviews in Mathematical Physics, Vol. 32, No. 04, 2050011 (2020)| Research Paper. Postprint version. Earlier version (arXiv:1702.00868 [math-ph]).

-Topological phases from higher gauge symmetry in 3+1D. With Alex Bullivant, Marcos Calçada, Zoltán Kádár and Paul Martin. PHYSICAL REVIEW B 95, 155118 (2017). Preliminary version.

-Categorifying the $sl(2,C)$ Knizhnik-Zamolodchikov Connection via an Infinitesimal 2-Yang-Baxter Operator in the String Lie-2-Algebra, with Lucio Simone Cirio. Advances in Theoretical and Mathematical Physics. Volume 21 (2017) Number 1, Pages: 147 - 229. Preliminary version.

-Crossed modules of Hopf algebras and of associative algebras and two-dimensional holonomy. Journal of Geometry and Physics. Volume 99, January 2016, Pages 68-110 Preliminary version.

-Two-Fold Homotopy of 2-Crossed Module Maps of Commutative Algebras. With İ.İlker Akça and Kadir Emir. Communications in Algebra 47 (2019), no. 1, 289–311. Preliminary version.

-Pointed Homotopy of Maps Between 2-Crossed Modules of Commutative Algebras, with İ.İlker Akça and Kadir Emir. Homology, Homotopy and Applications18 (2016), no. 1, 99–128. Preliminary version.

-Link invariants from finite categorical groups, with Roger Picken. Homology, Homotopy & Applications. 2015, Vol. 17 Issue 2, p205-233. 29p. Preliminary version. This is a very reduced version of: Link invariants from finite categorical groups and braided crossed modules.

-Infinitesimal 2-braidings and differential crossed modules, with Lucio Cirio. Advances in Mathematics, Volume 277, 4 June 2015, Pages 426–491. Preliminary version.

-Pointed homotopy and pointed lax homotopy of 2-crossed module maps, with Björn Gohla, Advances in Mathematics Volume 248, 25 November 2013, Pages 986–1049. Preliminary version. Very preliminary version.

-Spin Foam State Sums and Chern-Simons Theory,  with  Aleksandar Miković . Chern-Simons gauge theory: 20 years after, 277–284, AMS/IP Stud. Adv. Math., 50, Amer. Math. Soc., Providence, RI, 2011. Preliminary version.

-Categorifying the Knizhnik–Zamolodchikov connection, with Lucio Simone Cirio, Differential Geometry and its Applications, Volume 30, Issue 3, June 2012, Pages 238–261. Preliminary version.

-Four-Dimensional Spin Foam Perturbation Theory,  with  Aleksandar Miković . Symmetry, Integrability and Geometry: Methods and Applications (SIGMA). 7 (2011), 094, 22 pages.

-The fundamental 2-crossed complex of a reduced CW-complex. Homology, Homotopy and Applications, Vol. 13 (2011), No. 2, pp.129-157.

-Lie crossed modules and gauge-invariant actions for 2-BF theories,  with  Aleksandar Miković. Advances in Theoretical and Mathematical Physics, Volume 15, Number 4 (August 2011) p.1059-1084. Preliminary version

-Surface Holonomy for Non-Abelian 2-Bundles via Double Groupoids, with Roger Picken. Advances in Mathematics Volume 226, Issue 4, 1 March 2011, Pages 3309-3366. Preliminary version. Very preliminary version.

-The fundamental Gray 3-groupoid of a smooth manifold and local 3-dimensional holonomy based on a 2-crossed module, with Roger Picken. Differential Geometry and its Applications Volume 29, Issue 2, March 2011, Pages 179-206. Preliminary version. Very preliminary version.

-On Two-Dimensional Holonomy, with Roger Picken. Transactions of the American Mathematical Society, 362 (2010), 5657-5695.

-Spin Foam Perturbation Theory for Three-Dimensional Quantum Gravity,  with  Aleksandar Miković . Communications in Mathematical Physics. Volume 288, Number 2 / June, 2009.

-The Fundamental Crossed Module of the Complement of a Knotted Surface.  Transactions of the American Mathematical Society.   361 (2009), 4593-4630.

-Invariants of Welded Virtual Knots Via Crossed Module Invariants of Knotted Surfaces, with Louis H. Kauffman.   Compositio Mathematica. Volume 144, Issue 04, July 2008, pp 1046-1080.

-Invariants of Spin Networks Embedded  in Three-Manifolds,  with  Aleksandar Miković.   Communications in Mathematical Physics, Volume  279, Number 2/ Abril 2008, pages 381-399.

-Observables in the Turaev-Viro and Crane-Yetter models
, with John W. Barrett and J. Manuel Garcia Islas.   Journal of Mathematical Physics, 48, 093508 (2007) (18 pages).

-On Yetter's Invariant and an Extension of the Dijkgraaf-Witten Invariant to Categorical Groups, with Timothy Porter.   Theory and Application of Categories, Vol. 18, 2007, No. 4, pp 118-150.

-On the Homotopy Type and the Fundamental Crossed Complex of the Skeletal Filtration of a CW-Complex. Homology Homotopy and Applications, Vol. 9 (2007), No. 1, pp.295-329.

-Categorical Groups, Knots and Knotted Surfaces, J. Knot Theory Ramifications 16 (2007), no 9, 1181-1217.

Other work:

-On 2-Dimensional Homotopy Invariants of Complements of Knotted Surfaces.
Note: This is an archaic version of "The Fundamental Crossed Module of the Complement of a Knotted Surface, Trans AMS. 361 (2009), 4593-4630", using movies of knotted surfaces with bands and spanners rather than hyperbolic splittings of knotted surfaces for constructing a representation of the algebraic homotopy 2-type of the complement of a knotted surface. May still be of interest.    

PhD Thesis:

-Quantum Topology and the Lorentz Group.

Thesis related publications

-Knot Theory with the Lorentz Group FUNDAMENTA MATHEMATICAE 188 (2005), 59-93. (Special volume based on papers submitted by the participants of the Second International Conference “Knots in Poland 2003”).  

-On the Analytic Properties of the $z$-coloured Jones Polynomial, J. Knot Theory Ramifications  14  (2005),  no. 4, 435--466.