| Title:
|
Super Wilson Loops and Holonomy on Supermanifolds (English) |
| Author:
|
Groeger, Josua |
| Language:
|
English |
| Journal:
|
Communications in Mathematics |
| ISSN:
|
1804-1388 |
| Volume:
|
22 |
| Issue:
|
2 |
| Year:
|
2014 |
| Pages:
|
185-211 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
The classical Wilson loop is the gauge-invariant trace of the parallel transport around a closed path with respect to a connection on a vector bundle over a smooth manifold. We build a precise mathematical model of the super Wilson loop, an extension introduced by Mason-Skinner and Caron-Huot, by endowing the objects occurring with auxiliary Graßmann generators coming from $S$-points. A key feature of our model is a supergeometric parallel transport, which allows for a natural notion of holonomy on a supermanifold as a Lie group valued functor. Our main results for that theory comprise an Ambrose-Singer theorem as well as a natural analogon of the holonomy principle. Finally, we compare our holonomy functor with the holonomy supergroup introduced by Galaev in the common situation of a topological point. It turns out that both theories are different, yet related in a sense made precise. (English) |
| Keyword:
|
supermanifolds |
| Keyword:
|
holonomy |
| Keyword:
|
group functor |
| MSC:
|
18F05 |
| MSC:
|
53C29 |
| MSC:
|
58A50 |
| idZBL:
|
Zbl 1316.58004 |
| idMR:
|
MR3303138 |
| . |
| Date available:
|
2015-01-27T09:44:34Z |
| Last updated:
|
2020-01-05 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144130 |
| . |
| Reference:
|
[1] Alday, L., Maldacena, J.: Gluon scattering amplitudes at strong coupling.JHEP, 2007, 06, 2007, MR 2326587 |
| Reference:
|
[2] Alday, L., Roiban, R.: Scattering amplitudes, Wilson loops and the string/gauge theory correspondence.Phys. Reports, 468, 5, 2008, 153-211, MR 2475059, 10.1016/j.physrep.2008.08.002 |
| Reference:
|
[3] Ballmann, W.: Vector bundles and connections.2002, Lecture notes, Universität Bonn, |
| Reference:
|
[4] Bär, C.: Gauge theory.2009, Lecture notes, Universität Potsdam, |
| Reference:
|
[5] Belitsky, A.: Conformal anomaly of super Wilson loop.Nucl. Phys. B, 862, 2012, 430-449, Zbl 1246.81105, MR 2924694, 10.1016/j.nuclphysb.2012.04.022 |
| Reference:
|
[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, MR 2854542, 10.1016/j.nuclphysb.2011.10.014 |
| Reference:
|
[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, MR 2386887, 10.1016/j.nuclphysb.2007.11.002 |
| Reference:
|
[8] Carmeli, C., Caston, L., Fioresi, and R.: Mathematical Foundations of Supersymmetry.2011, European Mathematical Society, MR 2840967 |
| Reference:
|
[9] Caron-Huot, S.: Notes on the scattering amplitude / Wilson loop duality.JHEP, 2011, 07, 2011, Zbl 1298.81357, MR 2875966 |
| Reference:
|
[10] Deligne, P., Freed, D.: Supersolutions.Quantum Fields and Strings: A Course for Mathematicians, 1999, American Mathematical Society, Zbl 1170.81431, MR 1701600 |
| Reference:
|
[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, MR 2392215, 10.1016/j.nuclphysb.2007.11.041 |
| Reference:
|
[12] Galaev, A.: Holonomy of supermanifolds.Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 79, 2009, 47-78, Zbl 1182.58004, MR 2541343 |
| Reference:
|
[13] Gorbatsevich, V., Onishchik, A., Vinberg, E.: Foundations of Lie Theory and Lie Transformation Groups.1997, Springer, Zbl 0999.17500, MR 1631937 |
| Reference:
|
[14] Groeger, J.: Holomorphic supercurves and supersymmetric sigma models.J. Math. Phys., 52, 12, 2011, Zbl 1273.81148, MR 2907649, 10.1063/1.3665710 |
| Reference:
|
[15] Groeger, J.: Vertex operators of super Wilson loops.Phys. Rev. D, 86, 10, 2012, 10.1103/PhysRevD.86.105039 |
| Reference:
|
[16] Hanisch, F.: Variational problems on supermanifolds.2012, Dissertation, Universität Potsdam, |
| Reference:
|
[17] Hélein, F.: A representation formula for maps on supermanifolds.J. Math. Phys., 49, 2, 2008, Zbl 1153.81374, MR 2392866, 10.1063/1.2840464 |
| Reference:
|
[18] Hélein, F.: An introduction to supermanifolds and supersymmetry.Systèmes intégrables et théorie des champs quantiques, 2009, 103-157, Hermann, |
| Reference:
|
[19] Khemar, I.: Supersymmetric harmonic maps into symmetric spaces.Journal of Geometry and Physics, 57, 8, 2007, 1601-1630, Zbl 1120.53039, MR 2316716 |
| Reference:
|
[20] Leites, D.: Introduction to the theory of supermanifolds.Russian Math. Surveys, 35, 1, 1980, Zbl 0462.58002, MR 0565567, 10.1070/RM1980v035n01ABEH001545 |
| Reference:
|
[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, MR 2818481 |
| Reference:
|
[22] Molotkov, V.: Infinite-dimensional $\mathbb{Z}^k_2$-supermanifolds.1984, ICTP Preprints, IC/84/183, |
| Reference:
|
[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, MR 1218555, 10.1016/0393-0440(93)90003-W |
| Reference:
|
[24] Sachse, C.: A categorical formulation of superalgebra and supergeometry.2008, Preprint, Max Planck Institute for Mathematics in the Sciences, |
| Reference:
|
[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, MR 2924232, 10.4310/ATMP.2011.v15.n2.a2 |
| Reference:
|
[26] Tennison, B.: Sheaf Theory.1975, Cambridge University Press, Zbl 0313.18010, MR 0404390 |
| Reference:
|
[27] Varadarajan, V.: Supersymmetry for Mathematicians: An Introduction.2004, American Mathematical Society, Zbl 1142.58009, MR 2069561 |
| Reference:
|
[28] Yamabe, H.: On an arcwise connected subgroup of a Lie group.Osaka Math. J., 2, 1, 1950, 13-14, Zbl 0039.02101, MR 0036766 |
| . |
