Abstract
We study 5d fermionic CS theory with a fermionic 2-form gauge potential. This theory can be obtained from 5d maximally supersymmetric YM theory by performing the maximal topological twist. We put the theory on a five-manifold and compute the partition function. We find that it is a topological quantity, which involves the Ray-Singer torsion of the five-manifold. For abelian gauge group we consider the uplift to the 6d theory and find a mismatch between the 5d partition function and the 6d index, due to the nontrivial dimensional reduction of a selfdual two-form gauge field on a circle. We also discuss an application of the 5d theory to generalized knots made of 2d sheets embedded in 5d.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
E. Witten, Quantum Field Theory and the Jones Polynomial, Commun. Math. Phys. 121 (1989) 351 [INSPIRE].
C. Beasley and E. Witten, Non-Abelian localization for Chern-Simons theory, J. Diff. Geom. 70 (2005) 183 [hep-th/0503126] [INSPIRE].
M. Blau and G. Thompson, Chern-Simons theory on S1-bundles: Abelianisation and q-deformed Yang-Mills theory, JHEP 05 (2006) 003 [hep-th/0601068] [INSPIRE].
J. Kallen, Cohomological localization of Chern-Simons theory, JHEP 08 (2011) 008 [arXiv:1104.5353] [INSPIRE].
E. Witten, Two-dimensional gauge theories revisited, J. Geom. Phys. 9 (1992) 303 [hep-th/9204083] [INSPIRE].
V. Pestun, Localization of gauge theory on a four-sphere and supersymmetric Wilson loops, Commun. Math. Phys. 313 (2012) 71 [arXiv:0712.2824] [INSPIRE].
D.S. Freed and R.E. Gompf, Computer calculation of Witten’s three manifold invariant, Commun. Math. Phys. 141 (1991) 79 [INSPIRE].
L. Rozansky, A large k asymptotics of Witten’s invariant of Seifert manifolds, Commun. Math. Phys. 171 (1995) 279 [hep-th/9303099] [INSPIRE].
D.H. Adams and S. Sen, Partition function of a quadratic functional and semiclassical approximation for Witten’s three manifold invariant, hep-th/9503095 [INSPIRE].
D.H. Adams and S. Sen, Phase and scaling properties of determinants arising in topological field theories, Phys. Lett. B 353 (1995) 495 [hep-th/9506079] [INSPIRE].
D.H. Adams, A Note on the Faddeev-Popov determinant and Chern-Simons perturbation theory, Lett. Math. Phys. 42 (1997) 205 [hep-th/9704159] [INSPIRE].
D.H. Adams, The semiclassical approximation for the Chern-Simons partition function, Phys. Lett. B 417 (1998) 53 [hep-th/9709147] [INSPIRE].
M. Blau and G. Thompson, Topological Gauge Theories of Antisymmetric Tensor Fields, Annals Phys. 205 (1991) 130 [INSPIRE].
D. Bak and A. Gustavsson, The geometric Langlands twist in five and six dimensions, JHEP 07 (2015) 013 [arXiv:1504.00099] [INSPIRE].
J-H. Park and N. Nekrasov, private notes.
A.S. Schwarz, The Partition Function of Degenerate Quadratic Functional and Ray-Singer Invariants, Lett. Math. Phys. 2 (1978) 247 [INSPIRE].
M. Blau and G. Thompson, Do metric independent classical actions lead to topological field theories?, Phys. Lett. B 255 (1991) 535 [INSPIRE].
D.B Ray, Reidemeister torsion and the laplacian on lens spaces, Adv. Math. 4 (1970) 109.
C. Nash and D.J. O’Connor, Determinants of Laplacians, the Ray-Singer torsion on lens spaces and the Riemann zeta function, J. Math. Phys. 36 (1995) 1462 [Erratum ibid. 36 (1995) 4549] [hep-th/9212022] [INSPIRE].
T. Friedmann and E. Witten, Unification scale, proton decay and manifolds of G 2 holonomy, Adv. Theor. Math. Phys. 7 (2003) 577 [hep-th/0211269] [INSPIRE].
U. Bunke, Lectures on analytic torsion, http://www.uni-regensburg.de/Fakultaeten/nat Fak I/Bunke/sixtorsion.pdf.
P. Mnev, Lecture notes on torsions, arXiv:1406.3705 [INSPIRE].
Y. Imamura, H. Matsuno and D. Yokoyama, Factorization of the \( {S}^3/{\mathrm{\mathbb{Z}}}_n \) partition function, Phys. Rev. D 89 (2014) 085003 [arXiv:1311.2371] [INSPIRE].
P.A. Kirk and E.P. Klassen, Chern-Simons invariants of 3-manifolds and representation spaces of knot groups, Math. Ann. 287 (1990) 343, http://eudml.org/doc/164690.
E. Witten, On quantum gauge theories in two-dimensions, Commun. Math. Phys. 141 (1991) 153 [INSPIRE].
M. Blau and G. Thompson, Derivation of the Verlinde formula from Chern-Simons theory and the G/G model, Nucl. Phys. B 408 (1993) 345 [hep-th/9305010] [INSPIRE].
M. Mariño, Lectures on localization and matrix models in supersymmetric Chern-Simons-matter theories, J. Phys. A 44 (2011) 463001 [arXiv:1104.0783] [INSPIRE].
J. Källén and M. Zabzine, Twisted supersymmetric 5D Yang-Mills theory and contact geometry, JHEP 05 (2012) 125 [arXiv:1202.1956] [INSPIRE].
E. Guadagnini and F. Thuillier, Path-integral invariants in abelian Chern-Simons theory, Nucl. Phys. B 882 (2014) 450 [arXiv:1402.3140] [INSPIRE].
W. Siegel, Hidden Ghosts, Phys. Lett. B 93 (1980) 170.
T. Kimura, Quantum Theory of Antisymmetric Higher Rank Tensor Gauge Field in Higher Dimensional Space-time, Prog. Theor. Phys. 65 (1981) 338 [INSPIRE].
M.R. Douglas, On D = 5 super Yang-Mills theory and (2, 0) theory, JHEP 02 (2011) 011 [arXiv:1012.2880] [INSPIRE].
N. Lambert, C. Papageorgakis and M. Schmidt-Sommerfeld, M5-Branes, D4-branes and Quantum 5D super-Yang-Mills, JHEP 01 (2011) 083 [arXiv:1012.2882] [INSPIRE].
D. Bak and A. Gustavsson, Witten indices of abelian M5 brane on \( \mathrm{\mathbb{R}}\times {S}^5 \), JHEP 11 (2016) 177 [arXiv:1610.06255] [INSPIRE].
S. Rosenberg, The Laplacian on a Riemannian Manifold: An Introduction to Analysis on Manifolds, Cambridge University Press, Cambridge U.K. (1997).
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1710.02841
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Bak, D., Gustavsson, A. Five-dimensional fermionic Chern-Simons theory. J. High Energ. Phys. 2018, 37 (2018). https://doi.org/10.1007/JHEP02(2018)037
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP02(2018)037