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.]