Urs Schreiber
2005-03-31 15:56:42 UTC
I have been thinking about 2-holonomy a lot, lately.
(http://golem.ph.utexas.edu/string/archives/000503.html). Hence of course a
paper by E. Akhmedov which appeared today
E. Akhmedov,
Towards the Theory of Non-Abelian Tensor Fields I
http://de.arxiv.org/abs/hep-th/0503234
attracted my attention with its abstract, which reads
We present a triangulation-independent area-ordering prescription which
naturally generalizes the well known path ordering one. For such a
prescription it is natural that the two--form 'connection' should carry
three 'color' indices rather than two as it is in the case of the ordinary
one-form gauge connection. To define the prescription in question we have to
define how to exponentiate a matrix with three indices. The definition uses
the fusion rule structure constants.
<<<
I have just read through this paper and I think the idea is what I am going
to summarize in the following. My presentation is a little different from E.
Akhmedov's in that I take his last remark right before the conclusions as
the starting point and motivate the construction from there.
There is a well-known way to construct 2-dimensional topological field
theories on a triangulated surface. It is a 2d version of the
Dijkgraaf-Witten model
(http://staff.science.uva.nl/~rhd/papers/group.pdf)
which I learned about this winter in John Baez's quantum gravity seminar
(http://math.ucr.edu/home/baez/qg-winter2005/)
(week 6 (http://math.ucr.edu/home/baez/qg-winter2005/w05week06.pdf) this
year)
and which is discussed in detail in
M. Fukuma, S. Hosono & H. Kawai,
Lattice Topological Field Theory in Two Dimensions
http://de.arxiv.org/abs/hep-th/9212154
The idea is simply to triangulate your manifold and associate to each
triangle a given 3-index quantity C_{ijk}, with each index associated to one
of the edges of the triangle. All edges are labelled either in-going or
out-going and if an edge is outgoing we raise the corresponding index of
C_{ijk} using a symmetric 2-index quantity g^{ij}. Then define the partition
function of this setup simply to be the contraction of all the C... by means
of g^... in the obvious way.
This partition function becomes that of a topological theory when the C and
g are such that their contraction in the above way is independent of the
triangulation of the surface. One can show that this is the case precisely
if the C_{ijk} are the structure constants of a semi-simple associative
algebra and g = C^2 is its 'Killing form'.
To my mind E. Akhmedov's central observation is that the formula for
computing the holonomy of a non-abelian connection 1-form along a line is
like a sum over n-point functions of a 1-dimensional topological field
theory with the n-th powers of the 1-form in the formula for the
path-ordered exponential playing the role of the n insertions.
Motivated by this observation, he proposes to compute nonabelian 2-holonomy
by taking the analogous sum of n-point functions in a 2d TFT of the above
type.
An insertion in the above 2d TFT corresponds to removing one of the C_{ijk}
labels from one of the triangles and replacing it with a 'vertex', which
must be a 3-index quantity, too. Guess how we call, it: B_{ijk}. Or better
yet, when this is inserted in triangle number a call it B_{ijk}(a).
Denoting by < B(a_1) B(a_2) ... B(a_n) > the n-point function of our
theory, I believe that E. Akhmedov proposes (he uses different notation) to
define the 2-holonomy
hol_B(S)
of the 3-indexed discrete 2-form B over a given closed surface S to be
hol_B(S)
=
\lim_{e -> 0}
\sum_{n=0}^oo
\frac{1}{n!}
\sum_{ {a_i}}
< B(a_1) B(a_2) \cdots B(a_n) >_S
where e is some measure for the coarseness of our triangulation.
That's it.
Plausibly, when things are set up suitably this continuum limit exists and
is well defined, i.e. independent of the triangulation.
That sounds good. In particular since the three indices carried by B suggest
themselves naturally as a source for n^3-scaling on 5-branes.
The underlying philosophy the way Akhmedov presents it is rather similar to
Thomas Larsson's ideas (http://de.arxiv.org/abs/math-ph/0205017) on 2-form
gauge theory, though different in the details.
Of course for me one big question is: Can this construction be captured
using 2-bundles with 2-connection?
In any case, one would have to think about how the above definition of
2-holonomy could be generalized to a situation where there is no gloablly
defined 2-form B. This can already be seen in the abelian case, where the
above 2-holonomy reduces to the 2-holonomy known for abelian gerbes only
when everything is defined globally.
Hmm....
[This message is also available at
http://golem.ph.utexas.edu/string/archives/000542.html, where the formulas
can be seen in pretty-printed form.]
(http://golem.ph.utexas.edu/string/archives/000503.html). Hence of course a
paper by E. Akhmedov which appeared today
E. Akhmedov,
Towards the Theory of Non-Abelian Tensor Fields I
http://de.arxiv.org/abs/hep-th/0503234
attracted my attention with its abstract, which reads
We present a triangulation-independent area-ordering prescription which
naturally generalizes the well known path ordering one. For such a
prescription it is natural that the two--form 'connection' should carry
three 'color' indices rather than two as it is in the case of the ordinary
one-form gauge connection. To define the prescription in question we have to
define how to exponentiate a matrix with three indices. The definition uses
the fusion rule structure constants.
<<<
I have just read through this paper and I think the idea is what I am going
to summarize in the following. My presentation is a little different from E.
Akhmedov's in that I take his last remark right before the conclusions as
the starting point and motivate the construction from there.
There is a well-known way to construct 2-dimensional topological field
theories on a triangulated surface. It is a 2d version of the
Dijkgraaf-Witten model
(http://staff.science.uva.nl/~rhd/papers/group.pdf)
which I learned about this winter in John Baez's quantum gravity seminar
(http://math.ucr.edu/home/baez/qg-winter2005/)
(week 6 (http://math.ucr.edu/home/baez/qg-winter2005/w05week06.pdf) this
year)
and which is discussed in detail in
M. Fukuma, S. Hosono & H. Kawai,
Lattice Topological Field Theory in Two Dimensions
http://de.arxiv.org/abs/hep-th/9212154
The idea is simply to triangulate your manifold and associate to each
triangle a given 3-index quantity C_{ijk}, with each index associated to one
of the edges of the triangle. All edges are labelled either in-going or
out-going and if an edge is outgoing we raise the corresponding index of
C_{ijk} using a symmetric 2-index quantity g^{ij}. Then define the partition
function of this setup simply to be the contraction of all the C... by means
of g^... in the obvious way.
This partition function becomes that of a topological theory when the C and
g are such that their contraction in the above way is independent of the
triangulation of the surface. One can show that this is the case precisely
if the C_{ijk} are the structure constants of a semi-simple associative
algebra and g = C^2 is its 'Killing form'.
To my mind E. Akhmedov's central observation is that the formula for
computing the holonomy of a non-abelian connection 1-form along a line is
like a sum over n-point functions of a 1-dimensional topological field
theory with the n-th powers of the 1-form in the formula for the
path-ordered exponential playing the role of the n insertions.
Motivated by this observation, he proposes to compute nonabelian 2-holonomy
by taking the analogous sum of n-point functions in a 2d TFT of the above
type.
An insertion in the above 2d TFT corresponds to removing one of the C_{ijk}
labels from one of the triangles and replacing it with a 'vertex', which
must be a 3-index quantity, too. Guess how we call, it: B_{ijk}. Or better
yet, when this is inserted in triangle number a call it B_{ijk}(a).
Denoting by < B(a_1) B(a_2) ... B(a_n) > the n-point function of our
theory, I believe that E. Akhmedov proposes (he uses different notation) to
define the 2-holonomy
hol_B(S)
of the 3-indexed discrete 2-form B over a given closed surface S to be
hol_B(S)
=
\lim_{e -> 0}
\sum_{n=0}^oo
\frac{1}{n!}
\sum_{ {a_i}}
< B(a_1) B(a_2) \cdots B(a_n) >_S
where e is some measure for the coarseness of our triangulation.
That's it.
Plausibly, when things are set up suitably this continuum limit exists and
is well defined, i.e. independent of the triangulation.
That sounds good. In particular since the three indices carried by B suggest
themselves naturally as a source for n^3-scaling on 5-branes.
The underlying philosophy the way Akhmedov presents it is rather similar to
Thomas Larsson's ideas (http://de.arxiv.org/abs/math-ph/0205017) on 2-form
gauge theory, though different in the details.
Of course for me one big question is: Can this construction be captured
using 2-bundles with 2-connection?
In any case, one would have to think about how the above definition of
2-holonomy could be generalized to a situation where there is no gloablly
defined 2-form B. This can already be seen in the abelian case, where the
above 2-holonomy reduces to the 2-holonomy known for abelian gerbes only
when everything is defined globally.
Hmm....
[This message is also available at
http://golem.ph.utexas.edu/string/archives/000542.html, where the formulas
can be seen in pretty-printed form.]