Physics Letters B 789 (2019) 114–118
Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
Topological actions via gauge variations of higher structures
Hisham Sati
a,∗
, Matthew Wheeler
b
a
Division of Science and Mathematics, New York University, Abu Dhabi, United Arab Emirates
b
Department of Mathematics, The University of Arizona, Tucson, AZ 85721, USA
a r t i c l e i n f o a b s t r a c t
Article history:
Received
27 October 2018
Received
in revised form 26 November 2018
Accepted
5 December 2018
Available
online 10 December 2018
Editor:
N. Lambert
In this note we provide a new perspective on the topological parts of several action functionals in string
and M-theory. We show that rationally these can be viewed as large gauge transformations corresponding
to variations of higher structures, such as String, Fivebrane, and Ninebrane structures.
© 2018 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP
3
.
1. Introduction
Dirac’s formulation of spinors establishes that the tangent space
with its rotation group (in the Euclidean setting) or Lorentz group
(in the Minkowski case) is not enough to account for their behav-
ior
and properties. The proper formulation is via lifting to the Spin
group, which is the double cover of the orthogonal group. From
a topological point of view, one requires that the spacetime mani-
fold M
n
admit a Spin structure, whose existence can be established
at the level of classifying space BSpin(n) and requires the vanish-
ing
of the second Stiefel–Whitney class w
2
of M
n
. Had the Spin
group not already been known, one could have further defined the
Spin group to be an appropriate loop space of BSpin(n).
Generalizing
from point particles to strings, one realizes that a
Spin structure is no longer able to fully account for anomalies, and
an appropriate formulation requires lifting to a String structure [1].
This still allows for using Dirac operators and to extending to other
backgrounds, such as orbifolds [2]. To account for other signatures,
including Lorentzian, semi-Riemannian analogues of String struc-
tures
are constructed in [3]. From the point of view of (higher)
groups, one has the String group as the loop space of the classi-
fying
space BString(n) of such structures. Similar, but more sub-
tle,
arguments hold in part for the generalization to the fivebrane
[4][5] and the ninebrane [6], where one can also lift to groups
(in a more general homotopy sense) that are loop spaces of the
corresponding classifying spaces BFivebrane(n) and BNinebrane(n),
respectively. The structures are related via the Pontrjagin classes
*
Corresponding author.
E-mail
address: hsati@nyu.edu (H. Sati).
SO Spin
p
Q
1
String
p
Q
2
Fivebrane
p
Q
3
Ninebrane.
The above structures are in the spirit of the higher algebraic ap-
proaches
to M-branes [7][8][9], but more on the topological side.
By uncovering such structures (see [10]for another recent illustra-
tion),
one hopes that more insight is gained into the fundamental
nature of M-theory and its relation to string theory. A priori there
are several subtleties involved in considering the above higher
structures; most notably the presence of torsion in cohomology
and the need to use higher generalization of bundles, i.e., higher
gerbes or n-bundles. While such formulations have been applied
in useful ways, e.g., to the worldvolume theories of M-branes (see
[11][12][13][14][15]for various approaches), both of these com-
plications
can be set aside by rationalizing the structures, i.e., by
taking the corresponding cohomology to be over the rational (or
real) numbers. The outcome is a resolution of both subtleties with
the same token: not only does torsion get evaded, but also the
higher structures themselves can now be described using a for-
mulation
via only the much more familiar Spin structures and
corresponding bundles, which after all is sufficient for many pur-
poses
in physics.
Indeed,
in [16]we defined and characterized these rational
Spin-String, Spin-Fivebrane, and Spin-Ninebrane structures. It is
our aim here to apply these to describe how topological action
functionals in M-theory and string theory can be interpreted as
global gauge transformations corresponding to variations of such
structures. Such a point of view on the actions via variations of
structures has been proposed in [17], where variations of framing
(essentially a parallelism, i.e. with a trivialization of the tangent
bundle) was highlighted. Our current description is a generaliza-
tion
to the case when the structure is topologically (highly) non-
trivial.
https://doi.org/10.1016/j.physletb.2018.12.012
0370-2693/
© 2018 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by
SCOAP
3
.