- #1

MTd2

Gold Member

- 2,028

- 25

- Thread starter MTd2
- Start date

- #1

MTd2

Gold Member

- 2,028

- 25

- #2

marcus

Science Advisor

Gold Member

Dearly Missed

- 24,738

- 785

It's the beginnings of an idea. You link to the n-cat café where they discuss the draft Baez Huerta paper and where he points to page 37. To give a taste, I'll quote what he says in that passage:

==quote draft Baez Huerta==

Roberts and Schreiber go on to consider an analogous sequence of 3-groups constructed starting from a 2-group. Among these, the ‘inner automorphism 3-group’ of a 2-group plays a special role, which might make it important in understanding general relativity as a higher gauge theory.

As we have already seen in Section 4.3,** Palatini gravity in 4d spacetime** involves an so(3, 1)-valued 1-form A and a so(3, 1)-valued 2-form B = e ∧ e.

This is precisely the data we expect for a connection on a principal G-2-bundle where G is the tangent 2-group of the Lorentz group, except that the 2-form B fails to obey the equation dt(B) = F , as required by Theorem 4.5. Is there a way around this problem?

One possibility is to follow Breen and Messing [27], who, as we note, omit the condition dt(B) = F in their work on connections on nonabelian gerbes. This denies them the advantages of computing holonomies for surfaces, but they still have a coherent theory which may oﬀer some new insights into general relativity.

On the other hand, Schreiber [70] has argued that for any Lie 2-group G , the

3-group I N N (G ) allows us to deﬁne a version of parallel transport for particles, strings and 2-branes starting from an arbitrary g-valued 1-form A and h-valued 2-form B. The condition dt(B) = F is not required. So, to treat 4d Palatini gravity as a higher gauge theory, perhaps we can treat the basic ﬁelds as a 3-connection on an I N N (T SO(3, 1))-3-bundle. To entice the reader into pursuing this line of research, we optimistically dub this 3-group I N N (T (SO(3, 1)) the**gravity 3-group**.

==endquote==

draft of "Invitation to Higher Gauge Theory" is here:

http://math.ucr.edu/home/baez/invitation1.pdf

You can see that there might be some clues here as to why BF theory keeps coming up in gravity work. They point to section 4.3 where it mentions that** Palatini gravity in 4d spacetime** involves an so(3, 1)-valued 1-form A and a so(3, 1)-valued 2-form B = e ∧ e. This is tantamount to a tie-in with BF theory, since in this context the curvature of the form A is denoted F.

So maybe we should look back to section 4.3. That's on pages 32-35

==quote draft Baez Huerta==

Roberts and Schreiber go on to consider an analogous sequence of 3-groups constructed starting from a 2-group. Among these, the ‘inner automorphism 3-group’ of a 2-group plays a special role, which might make it important in understanding general relativity as a higher gauge theory.

As we have already seen in Section 4.3,

This is precisely the data we expect for a connection on a principal G-2-bundle where G is the tangent 2-group of the Lorentz group, except that the 2-form B fails to obey the equation dt(B) = F , as required by Theorem 4.5. Is there a way around this problem?

One possibility is to follow Breen and Messing [27], who, as we note, omit the condition dt(B) = F in their work on connections on nonabelian gerbes. This denies them the advantages of computing holonomies for surfaces, but they still have a coherent theory which may oﬀer some new insights into general relativity.

On the other hand, Schreiber [70] has argued that for any Lie 2-group G , the

3-group I N N (G ) allows us to deﬁne a version of parallel transport for particles, strings and 2-branes starting from an arbitrary g-valued 1-form A and h-valued 2-form B. The condition dt(B) = F is not required. So, to treat 4d Palatini gravity as a higher gauge theory, perhaps we can treat the basic ﬁelds as a 3-connection on an I N N (T SO(3, 1))-3-bundle. To entice the reader into pursuing this line of research, we optimistically dub this 3-group I N N (T (SO(3, 1)) the

==endquote==

draft of "Invitation to Higher Gauge Theory" is here:

http://math.ucr.edu/home/baez/invitation1.pdf

You can see that there might be some clues here as to why BF theory keeps coming up in gravity work. They point to section 4.3 where it mentions that

So maybe we should look back to section 4.3. That's on pages 32-35

Last edited:

- #3

MTd2

Gold Member

- 2,028

- 25

http://math.ucr.edu/home/baez/invitation.pdf

- #4

arivero

Gold Member

- 3,293

- 57

To be of some value in physics, there should be some uniqueness. The correct version of " parallel transport for particles, strings and 2-branes " would be expected to have some extra condition forcing D=11. It should relate to Evans paper on the link between division algebras and supersymmetry, and pass to topology via the link between division algebras, projective spaces and Hopf fibrations.On the other hand, Schreiber [70] has argued that for any Lie 2-group G , the

3-group I N N (G ) allows us to deﬁne a version of parallel transport for particles, strings and 2-branes starting from an arbitrary g-valued 1-form A and h-valued 2-form B. The condition dt(B) = F is not required. So, to treat 4d Palatini gravity as a higher gauge theory, perhaps we can treat the basic ﬁelds as a 3-connection on an I N N (T SO(3, 1))-3-bundle. To entice the reader into pursuing this line of research, we optimistically dub this 3-group I N N (T (SO(3, 1)) thegravity 3-group.

- #5

MTd2

Gold Member

- 2,028

- 25

http://arxiv.org/abs/1003.3436

Authors: John C. Baez, John Huerta

(Submitted on 17 Mar 2010)

Abstract: Starting from the four normed division algebras - the real numbers, complex numbers, quaternions and octonions - a systematic procedure gives a 3-cocycle on the Poincare Lie superalgebra in dimensions 3, 4, 6 and 10. A related procedure gives a 4-cocycle on the Poincare Lie superalgebra in dimensions 4, 5, 7 and 11. In general, an (n+1)-cocycle on a Lie superalgebra yields a "Lie n-superalgebra": that is, roughly speaking, an n-term chain complex equipped with a bracket satisfying the axioms of a Lie superalgebra up to chain homotopy. We thus obtain Lie 2-superalgebras extending the Poincare superalgebra in dimensions 3, 4, 6, and 10, and Lie 3-superalgebras extending the Poincare superalgebra in dimensions 4, 5, 7 and 11. As shown in Sati, Schreiber and Stasheff's work on higher gauge theory, Lie 2-superalgebra connections describe the parallel transport of strings, while Lie 3-superalgebra connections describe the parallel transport of 2-branes. Moreover, in the octonionic case, these connections concisely summarize the fields appearing in 10- and 11-dimensional supergravity.

- #6

arivero

Gold Member

- 3,293

- 57

Thanks, I had missed this one.They uploaded today to arxiv a paper about this:.