Squashed
extra dimensions in gauge theory and matrix models
supported by the FWF, P28590
principal
investigator: Harold
Steinacker
project
members: Marcus Sperling (postdoc), Timon Salar Gutleb (Master's
student), Clemens Kerschbaumer (Master's student)
The starting point of this
project is given by the squashed SU(3) brane solutions of SU(M) N
= 4 super-Yang-Mills deformed by a cubic potential discovered and
discussed in
- H. C. Steinacker and J. Zahn, Self-intersecting fuzzy extra
dimensions from squashed coadjoint orbits in N = 4 SYM and
matrix models, JHEP 1502, 027 (2015), 1409.1440
- H. C. Steinacker, Chiral low-energy physics from squashed
branes in deformed N = 4 SYM, JHEP 1510 (2015) 119 ,
arXiv:1411.3139.
- H. C. Steinacker, “Spinning squashed extra dimensions, chiral
gauge theory and hierarchy from N = 4 SYM,” Nucl.Phys.
B896 (2015) 212-243 [arXiv:1411.3139]
.
These solutions behave
as self-intersecting fuzzy extra dimensions, leading to a
truncated Kaluza-Klein tower of massive states, and a distinct
sector of bosonic and fermionic zero modes with rich structure.
The aim is to investigate the properties and the physical
significance of this new class of solutions both in deformed N = 4
SYM and in matrix models, and to explore possible generalizations
and applications.
Here are some pictures and computer-measurements of the
simplest of these "squashed extra dimensions", given by a
projection of fuzzy CP2:
this is a analytic plot of
the classical surface,
...
and here are some computer measurement of the fuzzy space:
These pictures in fact show only a 2-dimensional section, which
happens to be known as "Roman Surface"
(as observed by Clemens Kerschbaumer). The triple
self-intersection at the origin is very important; this is where
the 3 generations of massless fermions arise.
Here is a recent related talk on
this topic.
These and similar solutions
arise both within Yang-Mills gauge theories with large rank, as well as
within the (presumably) more fundamental and far-reaching matrix models,
notably the IKKT matrix model, provided a simple cubic term is
added to the potential (this may ultimately be replaced by a mass term.
This is still under investiation).
We have recently pulished a
long and detailed paper where these ideas are carried out and
pushed further:
- M. Sperling and H. C.
Steinacker, “Intersecting branes, Higgs sector, and
chirality from N = 4 SYM with soft SUSY breaking,” JHEP 1804 (2018) 116 [arXiv:1803.07323].
In this work, we have established the basic hypothesis of this
project, namely that there are indeed non-trivial "Higgs"
condensates which link the various sheets of the squashed branes at
their intersection loci. In fact, we have established a whole class
of new exact solutions of the underlying models (Yang-Mills or
matrix models) consisting of branes plus Higgs. The latter turn out
to form similar geometrical structures as the squashed branes
themselves. Moreover, these new solutions are exact minima of the
potential under certain conditions.
Furthermore, we showed how such solutions can provide the basis for
obtaining a low-energy effective gauge theory which is in the rough
ball-park of the standard model. This vindicates the idea underlying
the project, but as usual more work is required, which is
non-trivial. This is a good project for further work.
More general backgrounds with
a non-trivial geometric interpretation in large-N Yang-Mills gauge
theories and matrix models have been found also as extra
dimensions which are an intrinsic part of 4-dimensional covariant
quantum spaces, as discussed below.
Recently, this project has
developed more towards „covariant quantum spaces“, which
allow to reconcile the notions of covariance and quantum
space(time), and provide a promising basis for a higher spin
extension of quantum gravity. This was sparked by the paper
- H. C. Steinacker, “Emergent gravity on
covariant quantum spaces in the IKKT model,” JHEP 1612, 156 (2016) [arXiv:1606.00769]
In this paper, the
basic observation is made that the fluctuation modes on
certain "covariant" quantum spaces (=fuzzy spaces
which are equivariant bundles over some base manifold
interpreted as (space-time)) in 4 dimensions generically
lead to higher spin gauge theory, and in particular to
gravity. This promises to provide the
long-sought backgrounds of the IKKT matrix model which
leads to a (potentially) realistic emergent gravity, which
is the ultimate goal of my research efforts.
Given this
development, the project was re-focused towards studying
the physics which arises on these spaces. In fact this is
very much compatible with the ideas on
(squashed) extra dimensions as discussed above, as shown
in
- M. Sperling and H. C. Steinacker, “Covariant
4-dimensional fuzzy spheres, matrix models and
higher spin,” J. Phys. A 50, no. 37, 375202 (2017)
[arXiv:1704.02863].
Here a genrealized
class of covariant quantum spaces is studied, which has an
internal structure which is almost the same as the
squashed geometries discussed above. Hence these two lines
of work are perfectly compatible and complement each
other. It appears that the various threads are starting to
converge.
The details of the
resulting gravity theory are rather involved, as uusal,
and depend on the details of the background space-time
solution under consideration. We have developed the
underlying gauge theory systematically, starting with S^4,
- M. Sperling and H. C.
Steinacker, “Higher spin gauge theory on fuzzy
S^4_N” J. Phys. A 51, no. 7, 075201 (2018)
[arXiv:1707.00885].
moving on to the
non-compact H^4,
- M. Sperling
and H. C. Steinacker, "The fuzzy
4-hyperboloid $H^4_n$ and higher-spin in
Yang-Mills matrix models,''
arXiv:1806.05907
which is
the basis for the covariant cosmological space-time
solutions found in
- H. C. Steinacker,
“Cosmological space-times with resolved Big Bang in
Yang-Mills matrix models,'' JHEP 1802,
033 (2018) [arXiv:1709.10480 [hep-th]].
- H. C.
Steinacker, “Quantized open FRW cosmology
from Yang–Mills matrix models,'' Phys. Lett.
B 782, 176 (2018
[arXiv:1710.11495 [hep-th]].
and culminating
in a paper on the
resulting 3+1-dimensional gauge theory, which
indeed provides a very nice mechanism for
gravity:
- M.
Sperling and H. C. Steinacker,
",Covariant cosmological
quantum space-time,
higher-spin and gravity in the
IKKT matrix model''
arXiv:1901.03522
This is a
promising starting point for a quantum theory of
gravity. Stay tuned!
Here is a recent
talk on this topic.
Further work related to suqashed
extra dimension and non-commutative field theory:
- H. C. Steinacker, “String
states, loops and effective actions in noncommutative
field theory and matrix models,” Nucl.
Phys. B 910, 346 (2016)
[arXiv:1606.00646]
-
L. Schneiderbauer and H.
C. Steinacker, “Measuring finite Quantum Geometries
via Quasi-Coherent States,” J.
Phys. A 49, no. 28, 285301 (2016)
[arXiv:1601.08007]
Further
work related to supersymmetric gauge theory in
lower dimensions and the associated algebraic structures: