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Super Wilson loops and holonomy on supermanifolds. (English) Zbl 1316.58004
In this interesting paper the author makes precise mathematical sense of super Wilson loops and the notion of holonomy for supermanifolds. The motivation is clearly rooted in super Yang-Mills theory. He does this by trying to follow more-or-less the classical constructions by employing the functor of points. In doing so the author presents the notion of a S-path (also see A. J. Bruce [Arch. Math., Brno 50, No. 2, 115–130 (2014; Zbl 1340.58002)]) using internal Homs objects in the category of supermanifolds. Loosely, S-paths are paths that are parametrised by external Grassmann odd parameters.
From there S-loops, connections, parallel transport and the holonomy of an S-point are developed in the ‘categorical language’. This leads to the notion of the holonomy group functor, which turns out not to be representable in general; we have a generalised supermanifold and not a genuine supermanifold. An Ambrose–Singer theorem is also established in this context.
Finally the relation with Galaev’s notion of holonomy for supermanifolds [A. S. Galaev, Abh. Math. Semin. Univ. Hamb. 79, No. 1, 47–78 (2009; Zbl 1182.58004)] is explored.

##### MSC:
 58A50 Supermanifolds and graded manifolds 53C29 Issues of holonomy in differential geometry 18F05 Local categories and functors
##### Keywords:
supermanifolds; holonomy; group functor
Full Text:
##### References:
 [1] Alday, L., Maldacena, J.: Gluon scattering amplitudes at strong coupling. JHEP, 2007, 06, 2007, · Zbl 1245.81256 [2] Alday, L., Roiban, R.: Scattering amplitudes, Wilson loops and the string/gauge theory correspondence. Phys. Reports, 468, 5, 2008, 153-211, [3] Ballmann, W.: Vector bundles and connections. 2002, Lecture notes, Universität Bonn, [4] Bär, C.: Gauge theory. 2009, Lecture notes, Universität Potsdam, [5] Belitsky, A.: Conformal anomaly of super Wilson loop. Nucl. Phys. B, 862, 2012, 430-449, · Zbl 1246.81105 [6] Belitsky, A., Korchemsky, G., Sokatchev, E.: Are scattering amplitudes dual to super Wilson loops?. Nucl. Phys. B, 855, 2012, 333-360, · Zbl 1229.81176 [7] Brandhuber, A., Heslop, P., Travaglini, G.: MHV amplitudes in $$N=4$$ super Yang-Mills and Wilson loops. Nucl. Phys. B, 794, 2008, 231-243, · Zbl 1273.81201 [8] Carmeli, C., Caston, L., Fioresi, and R.: Mathematical Foundations of Supersymmetry. 2011, European Mathematical Society, · Zbl 1226.58003 [9] Caron-Huot, S.: Notes on the scattering amplitude / Wilson loop duality. JHEP, 2011, 07, 2011, · Zbl 1298.81357 [10] Deligne, P., Freed, D.: Supersolutions. Quantum Fields and Strings: A Course for Mathematicians, 1999, American Mathematical Society, · Zbl 1170.81431 [11] Drummond, J., Korchemsky, G., Sokatchev, E.: Conformal properties of four-gluon planar amplitudes and Wilson loops. Nucl. Phys. B, 795, 2008, 385-408, · Zbl 1219.81227 [12] Galaev, A.: Holonomy of supermanifolds. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 79, 2009, 47-78, · Zbl 1182.58004 [13] Gorbatsevich, V., Onishchik, A., Vinberg, E.: Foundations of Lie Theory and Lie Transformation Groups. 1997, Springer, · Zbl 0999.17500 [14] Groeger, J.: Holomorphic supercurves and supersymmetric sigma models. J. Math. Phys., 52, 12, 2011, · Zbl 1273.81148 [15] Groeger, J.: Vertex operators of super Wilson loops. Phys. Rev. D, 86, 10, 2012, [16] Hanisch, F.: Variational problems on supermanifolds. 2012, Dissertation, Universität Potsdam, [17] Hélein, F.: A representation formula for maps on supermanifolds. J. Math. Phys., 49, 2, 2008, · Zbl 1153.81374 [18] Hélein, F.: An introduction to supermanifolds and supersymmetry. Systèmes intégrables et théorie des champs quantiques, 2009, 103-157, Hermann, [19] Khemar, I.: Supersymmetric harmonic maps into symmetric spaces. Journal of Geometry and Physics, 57, 8, 2007, 1601-1630, · Zbl 1120.53039 [20] Leites, D.: Introduction to the theory of supermanifolds. Russian Math. Surveys, 35, 1, 1980, · Zbl 0462.58002 [21] Mason, L., Skinner, D.: The complete planar $$S$$-matrix of $$N=4$$ SYM as a Wilson loop in twistor space. JHEP, 2010, 12, 2010, · Zbl 1294.81122 [22] Molotkov, V.: Infinite-dimensional $$\mathbb{Z}^k_2$$-supermanifolds. 1984, ICTP Preprints, IC/84/183, [23] Monterde, J., Sánchez-Valenzuela, O.: Existence and uniqueness of solutions to superdifferential equations. Journal of Geometry and Physics, 10, 4, 1993, 315-343, · Zbl 0772.58007 [24] Sachse, C.: A categorical formulation of superalgebra and supergeometry. 2008, Preprint, Max Planck Institute for Mathematics in the Sciences, [25] Sachse, C., Wockel, C.: The diffeomorphism supergroup of a finite-dimensional supermanifold. Adv. Theor. Math. Phys., 15, 2, 2011, 285-323, · Zbl 1268.58009 [26] Tennison, B.: Sheaf Theory. 1975, Cambridge University Press, · Zbl 0313.18010 [27] Varadarajan, V.: Supersymmetry for Mathematicians: An Introduction. 2004, American Mathematical Society, · Zbl 1142.58009 [28] Yamabe, H.: On an arcwise connected subgroup of a Lie group. Osaka Math. J., 2, 1, 1950, 13-14, · Zbl 0039.02101
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