Discussion:
Akhmedov: Nonabelian 2-holonomy using TFT
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Urs Schreiber
2005-03-31 15:56:42 UTC
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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.]
Thomas Larsson
2005-04-02 11:35:33 UTC
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Post by Urs Schreiber
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.
An obvious difference is that Akhmedov uses triangles and I squares, but this
is hardly important. I prefer quads because it makes a connection with the
Yang-Baxter eqn, which was my original motivation.

More significantly, in order to be able to contract indices Akhmedov introduces
a metric kappa^ij (bottom of page 5). I avoid that by putting two four-index
quantities, and their inverses, on each plaquette. In that way I can arrange
things so that always one up and one down index are contracted, and there is no
need for a metric.
Urs Schreiber
2005-04-12 16:25:52 UTC
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Post by Thomas Larsson
Post by Urs Schreiber
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.
An obvious difference is that Akhmedov uses triangles and I squares, but this
is hardly important.
The triangles are crucial in making the setup independent of the
latticization, since using them there is a way to get a TFT using structure
constants C_ijk.
Post by Thomas Larsson
More significantly, in order to be able to contract indices Akhmedov introduces
a metric kappa^ij (bottom of page 5). I avoid that by putting two four-index
quantities, and their inverses, on each plaquette. In that way I can arrange
things so that always one up and one down index are contracted, and there is no
need for a metric.
The metric is no extra structure in Akhmedov's setup, since it follows from
the C_ijk.

To me it seems as if Akhmedov provides a way to make your idea of
associating an "internal index" with each edge consistent, i.e. to get a
well-defined continuum theory independent of the choice of discretization
used to define it.

I also believe there is a nice way to describe that continuum limit:

Let G be the space in which the internal indices take values. For instance
for the TFT defined from discrete groups, G could be the limiting Lie group.
But it might be something else, too (and one should not confuse this group
with any "gauge group" here).

Pick any connected collection of triangles, i.e. a small surface element. It
defines a tensor with n incoming and m outgoing indices. In the limit this
gives a map from pairs of smooth paths in G to the base field

T(gamma1, gamma2)

such that composition is given by "continuous index contraction"

(T o T')(gamma1, gamma2)
=
int D[gamma] T(gamma1,gamma) T(gamma,gamma2) .

(Here the functional integral is over a reparameterization gauge slice, the
choice of which is free due to the properties mentioned above.)

These have to be invertible and we can impose a "star-condition". But since
these T are nothing but integral kernels we see that the group they form is
that of unitary operators on something like L^2(PG), where PG is the path
space over G. But in fact without any loss of generality we can assume that
all paths start and end at a given point in G and have sitting instant at
that point and are parameterized by a parameter in [0,1], so we really get
OmegaG, the based loop space over G and unitary operators on L^2(Omega G).

In addition to this "vertical product" coming from composition there is a
"horizontal" product on these guys (coming from literally horizontally
composing triangles) given by

(T.T')(gamma1,gamma2)
=
T(Lgamma1,Lgamma2) T'(Rgamma1,Rgamma2)

where Lgamma is the path that traces out the first half of gamma at twice
the speed, and Rgamma similarly gives the right half and it is implicit that
T and T' vanish when their arguments are not based loops with sitting
instant at the base.

In fact, I believe it can be checked that this defines on the Ts the
structure of a weak monoidal category with all morphisms invertible. By
throwing in weak formal horizontal inverses (representing the "zig-zag
symmetry" of holonomy which is otherwise not captured) we get a weak
2-group.

I had speculated before that weak 2-groups allow for realizing n^3 scaling
behavior, and here indeed it is explicit, due to the TFT definition of the
whole thing. That's quite interesting.
Thomas Larsson
2005-04-13 14:07:33 UTC
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Post by Urs Schreiber
The triangles are crucial in making the setup independent of the
latticization, since using them there is a way to get a TFT using structure
constants C_ijk.
If you have a quadrangulation, you can always cut each square in two triangles.
Post by Urs Schreiber
Post by Thomas Larsson
More significantly, in order to be able to contract indices Akhmedov introduces
a metric kappa^ij (bottom of page 5). I avoid that by putting two four-index
quantities, and their inverses, on each plaquette. In that way I can arrange
things so that always one up and one down index are contracted, and there is no
need for a metric.
The metric is no extra structure in Akhmedov's setup, since it follows from
the C_ijk.
?????

In order to contract two lower indices, you always need something with two
upper indices, i.e. a (possibly degenerate) metric. You don't need a metric
to define holonomies and inverses in 1-gauge theory. However, you do need it
if you want you gauge group to be unitary or orthogonal.
Post by Urs Schreiber
Pick any connected collection of triangles, i.e. a small surface element. It
defines a tensor with n incoming and m outgoing indices.
E.g., pick a single triangle. n+m=3, but what are m and n separately?
In C_ijk, which of i,j,k are incoming and which are outgoing?
Post by Urs Schreiber
In addition to this "vertical product" coming from composition there is a
"horizontal" product on these guys (coming from literally horizontally
composing triangles) given by
(T.T')(gamma1,gamma2)
=
T(Lgamma1,Lgamma2) T'(Rgamma1,Rgamma2)
where Lgamma is the path that traces out the first half of gamma at twice
the speed, and Rgamma similarly gives the right half and it is implicit that
T and T' vanish when their arguments are not based loops with sitting
instant at the base.
Hm. This part I don't understand.
Post by Urs Schreiber
In fact, I believe it can be checked that this defines on the Ts the
structure of a weak monoidal category with all morphisms invertible. By
throwing in weak formal horizontal inverses (representing the "zig-zag
symmetry" of holonomy which is otherwise not captured) we get a weak
2-group.
I had speculated before that weak 2-groups allow for realizing n^3 scaling
behavior, and here indeed it is explicit, due to the TFT definition of the
whole thing. That's quite interesting.
G and dB scales as n^3, but [B,B] like n^4. So we need to contract [B,B]
with something which scales like 1/n, namely a vector parallel to the path.
Urs Schreiber
2005-04-13 15:13:38 UTC
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Post by Thomas Larsson
In order to contract two lower indices, you always need something with two
upper indices, i.e. a (possibly degenerate) metric.
These TFT's that we are talking about can be shown to be in 1-1
correspondence with semisimple algebras with structure constants C_i^j_k.
The bilinear form has to be proportional to the "Killing form" g_ij =
C_i^r_s C_j^s_r of these, as found first in hep-th/9212154. (*)
Post by Thomas Larsson
Post by Urs Schreiber
Pick any connected collection of triangles, i.e. a small surface element. It
defines a tensor with n incoming and m outgoing indices.
E.g., pick a single triangle. n+m=3, but what are m and n separately?
The edges of these are colored by an orientation which indicates if the
corresponding index is in- or outgoing.
Post by Thomas Larsson
In C_ijk, which of i,j,k are incoming and which are outgoing?
As discussed in the literature, e.g. the hep-th/9212154 and as I mentioned
in my original post (http://golem.ph.utexas.edu/string/archives/000542.html)
an upstairs index corresponds to ingoing and a downstairs index to outgoing
(or vice versa, depending on your conventions).
Post by Thomas Larsson
Post by Urs Schreiber
In addition to this "vertical product" coming from composition there is a
"horizontal" product on these guys (coming from literally horizontally
composing triangles) given by
(T.T')(gamma1,gamma2)
=
T(Lgamma1,Lgamma2) T'(Rgamma1,Rgamma2)
where Lgamma is the path that traces out the first half of gamma at twice
the speed, and Rgamma similarly gives the right half and it is implicit that
T and T' vanish when their arguments are not based loops with sitting
instant at the base.
Hm. This part I don't understand.
This is just the continuum version of the obvious horizontal composition of
finite-resolution surface elements (as I have tried to describe here:
http://golem.ph.utexas.edu/string/archives/000542.html#c002138):

For instance let your /\ ~T and /\' ~ T' be single triangles given by
tensors T^ij_k and T'^lm_n. Then we can horizonatlly compose them to get
/\./\' given by the tensor

(T.T')^ijlm_kn = T^ij_k T'^lm_n .



footnote: (*)
This applies to the ordinary TFTs. Of course Akhmedov comes along and wants
to make the C_i^j_k position dependent by replacing them (that's not how he
puts it, though) with

B_i^j_k (x) = C_i^j_k + epsilon b_i^j_j(x).

One would have to work out the conditions on b_i^j_k(x) for this to give a
well-defined continuum limit. This has not been discussed in the literature
as far as I am aware and I also don't see that Akhmedov addresses this
point.
Thomas Larsson
2005-04-14 12:28:20 UTC
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Post by Urs Schreiber
These TFT's that we are talking about can be shown to be in 1-1
correspondence with semisimple algebras with structure constants C_i^j_k.
The bilinear form has to be proportional to the "Killing form" g_ij =
C_i^r_s C_j^s_r of these, as found first in hep-th/9212154. (*)
IOW, Akhmedov assumes the existence of a Killing metric, which was exactly
my point. However, holonomies and inverses in 1-gauge theory are defined
without reference to this metric, so it does not naturally belong to the
domain of the problem. Moreover, there is no problem to define 1-gauge
theories for gauge groups which do not admit a Killing metric, although
somewhat unusual. This also works for 2-gauge theories in my sense, but
evidently not in Akhmedov's sense.
Post by Urs Schreiber
Post by Thomas Larsson
In C_ijk, which of i,j,k are incoming and which are outgoing?
As discussed in the literature, e.g. the hep-th/9212154 and as I mentioned
in my original post (http://golem.ph.utexas.edu/string/archives/000542.html)
an upstairs index corresponds to ingoing and a downstairs index to outgoing
(or vice versa, depending on your conventions).
IOW, you have several quantities for each plaquette, e.g. C_ijk, C^ijk,
C^i_jk etc. If you have a Killing metric, it is not important to distinguish
between them. Since I don't make that assumption, I need to be careful here.

The only problem I saw with the formal continuum limit is that the
different surface holonomies take values in different spaces. If the
boundary consists of n links, the lattice surface holonomy belongs to V^n
(ignoring the difference between V and its dual) and the gauge
transformations belong to G^n, and it may be problematic to take the
limit of the n-fold tensor product. However, tonight I woke up and realized
that it is not.

V^n can be identified with the space of piecewise constant V-valued
functions, with n pieces. In the limit n -> infinity, this becomes the
space of functions from the boundary to V, LV. One probably want some
continuity conditions in the limit. The gauge transformations in G^n
in G^n simply become the loop group LG, and acts on the functions in LV.

There is something definitely right about this. As you know, I am a firm
believer in locality, and this is the reason why I don't like the 2-gauge
theories of Baez, Pfeiffer and others. If a finite-dimensional G acts on a
loop, in must be some non-local process which is smeared over the whole
loop at once. However, LG can act locally, with one copy of G at each
point of the loop. From this viewpoint it is encouraging that you and
Baez now seem forced into considering loop groups.

The generalization to higher gauge theory is now obvious: the p-holonomy
associated with a p-manifold M in p-gauge theory has a gauge symmetry
given by the manifold group living on the boundary dM, i.e. G^dM.
Urs Schreiber
2005-04-14 12:53:11 UTC
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Post by Thomas Larsson
Post by Urs Schreiber
These TFT's that we are talking about can be shown to be in 1-1
correspondence with semisimple algebras with structure constants C_i^j_k.
The bilinear form has to be proportional to the "Killing form" g_ij =
C_i^r_s C_j^s_r of these, as found first in hep-th/9212154. (*)
IOW, Akhmedov assumes the existence of a Killing metric,
Akhmedov's idea is based on 2-dimensional TFTs. These are in 1-1
correspondence with semisimple associative algebras, which, by definition,
have a nondegenerate Killing form (hep-th/9212154).

The fact that these algebras have to be associative and have a nondegenerate
Killing form is an algebraic reformulation of the invariance of the
corresponding TFT under re-triangulating moves.

Associativity describes the "fusion transformation", equation 3.1 and
figure 7 of hep-th/9212154. The bilinear form gives the
"bubble transformation", equation 3.2 and figure 8 in hep-th/9212154.

These "moves" are the 2D version of the Matveev moves, which are
equivalent to the Alexander moves or the bond-flip moves.
Post by Thomas Larsson
Moreover, there is no problem to define 1-gauge
theories for gauge groups which do not admit a Killing metric, although
somewhat unusual.
The nondegenerate Killing form of the above mentioned algebras should not
be confused with any Killing form of any Lie algebra of any gauge group that might
appear once somebody manages to deforms these TFTs into something like a
surface holonomy.
Post by Thomas Larsson
IOW, you have several quantities for each plaquette, e.g. C_ijk, C^ijk,
C^i_jk etc. If you have a Killing metric, it is not important to distinguish
between them. Since I don't make that assumption, I need to be careful here.
Since you don't have that assumption you will have to do something else to
ensure that your construction is well defined (independent of details of
the latticization).


[...]
Post by Thomas Larsson
However, tonight I woke up and realized that it is not.
V^n can be identified with the space of piecewise constant V-valued
functions, with n pieces. In the limit n -> infinity, this becomes the
space of functions from the boundary to V, LV.
This is what I wrote in the second post in this thread.
Post by Thomas Larsson
One probably want some
continuity conditions in the limit. The gauge transformations in G^n
in G^n simply become the loop group LG, and acts on the functions in LV.
The concept of gauge group might require some care. For instance the
group of diffeomorphisms of the loop will play a role, too. I have
indicated in that previous post how we actually seem to get a coherent
2-group out of this (if it can indeed all be well defined).
Urs Schreiber
2005-04-20 18:34:57 UTC
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Post by Urs Schreiber
One would have to work out the conditions on b_i^j_k(x) for this to give a
well-defined continuum limit. This has not been discussed in the
literature as far as I am aware and I also don't see that Akhmedov
addresses this point.
Today he has a new preprint on this issue:

http://golem.ph.utexas.edu/string/archives/000558.html

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