John Baez
Dec10-04, 05:02 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Some of us have spent time here talking about connections on nonabelian gerbes\nand their relation to physics. Lately Urs Schreiber and I have made a bunch\nof progress on these issues. So, some of you may be interested in a talk I\'m\ngiving in Paris next Wednesday, at a conference in honor of Larry Breen\'s 60th\nbirthday.\n\n............................... .............................................\n\nC ategorified Gauge Theory\n\nThe problem of defining "higher gauge fields" is an old one.\nThese should be generalizations of the gauge fields normally\nconsidered in physics, describing the parallel transport of\nhigher-dimensional extended objects rather than point particles.\n\nNot surprisingly, interest in this problem has been reawakened by string\ntheory, since 1-dimensional strings and even higher-dimensional membranes\nplay a crucial role here. Indeed, a higher analogue of the electromagnetic\nfield called the Kalb-Ramond field arises naturally in string theory.\nIt describes the phase acquired by a string as it moves through spacetime.\nElectromagnetism is described locally by a 1-form, which we integrate over\nthe worldline of a charged particle to compute the phase it acquires as it\nmoves. Similarly, the Kalb-Ramond field is described locally by a 2-form,\nwhich we integrate over the worldsheet of a string.\n\nHowever, mathematical physicists know that globally, the electromagnetic\nfield is described by a connection on a U(1) bundle, which would be\nnontrivial in the presence of magnetic monopoles. Similarly, the\nKalb-Ramond field is described globally by a connection on a "U(1) gerbe".\n\nIn traditional physics, forces other than electromagnetism are described\nby replacing the group U(1) by other Lie groups G and treating these\nforces as connection on G-bundles. This raises the question of whether\nwe can similarly see connections on U(1) gerbes as a special case of\nhigher gauge fields involving more general Lie groups - particularly\n*nonabelian* groups.\n\nBreen and Messing have raised the possibility that we can, by using their\ntheory of connections on "nonabelian gerbes". However, they\ndid not develop a theory of parallel transport for these connections.\n\nIn joint work with Toby Bartels, Alissa Crans, Aaron Lauda and Urs\nSchreiber, I have developed such a theory by systematically\n*categorifying* the concepts of smooth manifold, Lie group and Lie\nalgebra, and setting up a theory of bundles, connections and curvature\nin this new context.\n\nIn particular, we define a "strict Lie 2-group" to be a category C\nwhere the set of objects and the set of morphisms are Lie groups, and\nsource, target, identity and composition maps are homomorphisms of\nLie groups. This turns out to be the same as a "Lie crossed module":\na pair of Lie groups G and H with a homomorphism t: H -> G and an action\nof G on H satisfying the equations in the usual definition of crossed module.\n\nJust as a connection on a trivial G-bundle is the same as a Lie(G)-valued\n1-form, a connection on a trivial 2-bundle with C as its gauge 2-group\nturns out to be a Lie(G)-valued 1-form A together with a Lie(H)-valued\n2-form B. We show that a connection of this sort gives well-behaved\nparallel transport along both curves and surfaces if a certain quantity\ncalled the "fake curvature" vanishes. This quantity was introduced\nby Breen and Messing. It equals\n\ndA + A ^ A - dt(B)\n\nWhen this is zero, we can define nonabelian parallel transport over\nsurfaces in a way that doesn\'t depend on the parametrization of the\nsurface. This was always the sticking point in attempts at higher\ngauge theory.\n\nYou can see transparencies of my talk and also references at:\n\nhttp://math.ucr.edu/home/baez/breen/\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>Some of us have spent time here talking about connections on nonabelian gerbes
and their relation to physics. Lately Urs Schreiber and I have made a bunch
of progress on these issues. So, some of you may be interested in a talk I'm
giving in Paris next Wednesday, at a conference in honor of Larry Breen's 60th
birthday.
.................................................. ..........................
Categorified Gauge Theory
The problem of defining "higher gauge fields" is an old one.
These should be generalizations of the gauge fields normally
considered in physics, describing the parallel transport of
higher-dimensional extended objects rather than point particles.
Not surprisingly, interest in this problem has been reawakened by string
theory, since 1-dimensional strings and even higher-dimensional membranes
play a crucial role here. Indeed, a higher analogue of the electromagnetic
field called the Kalb-Ramond field arises naturally in string theory.
It describes the phase acquired by a string as it moves through spacetime.
Electromagnetism is described locally by a 1-form, which we integrate over
the worldline of a charged particle to compute the phase it acquires as it
moves. Similarly, the Kalb-Ramond field is described locally by a 2-form,
which we integrate over the worldsheet of a string.
However, mathematical physicists know that globally, the electromagnetic
field is described by a connection on a U(1) bundle, which would be
nontrivial in the presence of magnetic monopoles. Similarly, the
Kalb-Ramond field is described globally by a connection on a "U(1) gerbe".
In traditional physics, forces other than electromagnetism are described
by replacing the group U(1) by other Lie groups G and treating these
forces as connection on G-bundles. This raises the question of whether
we can similarly see connections on U(1) gerbes as a special case of
higher gauge fields involving more general Lie groups - particularly
*nonabelian* groups.
Breen and Messing have raised the possibility that we can, by using their
theory of connections on "nonabelian gerbes". However, they
did not develop a theory of parallel transport for these connections.
In joint work with Toby Bartels, Alissa Crans, Aaron Lauda and Urs
Schreiber, I have developed such a theory by systematically
*categorifying* the concepts of smooth manifold, Lie group and Lie
algebra, and setting up a theory of bundles, connections and curvature
in this new context.
In particular, we define a "strict Lie 2-group" to be a category C
where the set of objects and the set of morphisms are Lie groups, and
source, target, identity and composition maps are homomorphisms of
Lie groups. This turns out to be the same as a "Lie crossed module":
a pair of Lie groups G and H with a homomorphism t: H -> G and an action
of G on H satisfying the equations in the usual definition of crossed module.
Just as a connection on a trivial G-bundle is the same as a Lie(G)-valued
1-form, a connection on a trivial 2-bundle with C as its gauge 2-group
turns out to be a Lie(G)-valued 1-form A together with a Lie(H)-valued
2-form B. We show that a connection of this sort gives well-behaved
parallel transport along both curves and surfaces if a certain quantity
called the "fake curvature" vanishes. This quantity was introduced
by Breen and Messing. It equals
dA + A ^ A - dt(B)
When this is zero, we can define nonabelian parallel transport over
surfaces in a way that doesn't depend on the parametrization of the
surface. This was always the sticking point in attempts at higher
gauge theory.
You can see transparencies of my talk and also references at:
http://math.ucr.edu/home/baez/breen/
and their relation to physics. Lately Urs Schreiber and I have made a bunch
of progress on these issues. So, some of you may be interested in a talk I'm
giving in Paris next Wednesday, at a conference in honor of Larry Breen's 60th
birthday.
.................................................. ..........................
Categorified Gauge Theory
The problem of defining "higher gauge fields" is an old one.
These should be generalizations of the gauge fields normally
considered in physics, describing the parallel transport of
higher-dimensional extended objects rather than point particles.
Not surprisingly, interest in this problem has been reawakened by string
theory, since 1-dimensional strings and even higher-dimensional membranes
play a crucial role here. Indeed, a higher analogue of the electromagnetic
field called the Kalb-Ramond field arises naturally in string theory.
It describes the phase acquired by a string as it moves through spacetime.
Electromagnetism is described locally by a 1-form, which we integrate over
the worldline of a charged particle to compute the phase it acquires as it
moves. Similarly, the Kalb-Ramond field is described locally by a 2-form,
which we integrate over the worldsheet of a string.
However, mathematical physicists know that globally, the electromagnetic
field is described by a connection on a U(1) bundle, which would be
nontrivial in the presence of magnetic monopoles. Similarly, the
Kalb-Ramond field is described globally by a connection on a "U(1) gerbe".
In traditional physics, forces other than electromagnetism are described
by replacing the group U(1) by other Lie groups G and treating these
forces as connection on G-bundles. This raises the question of whether
we can similarly see connections on U(1) gerbes as a special case of
higher gauge fields involving more general Lie groups - particularly
*nonabelian* groups.
Breen and Messing have raised the possibility that we can, by using their
theory of connections on "nonabelian gerbes". However, they
did not develop a theory of parallel transport for these connections.
In joint work with Toby Bartels, Alissa Crans, Aaron Lauda and Urs
Schreiber, I have developed such a theory by systematically
*categorifying* the concepts of smooth manifold, Lie group and Lie
algebra, and setting up a theory of bundles, connections and curvature
in this new context.
In particular, we define a "strict Lie 2-group" to be a category C
where the set of objects and the set of morphisms are Lie groups, and
source, target, identity and composition maps are homomorphisms of
Lie groups. This turns out to be the same as a "Lie crossed module":
a pair of Lie groups G and H with a homomorphism t: H -> G and an action
of G on H satisfying the equations in the usual definition of crossed module.
Just as a connection on a trivial G-bundle is the same as a Lie(G)-valued
1-form, a connection on a trivial 2-bundle with C as its gauge 2-group
turns out to be a Lie(G)-valued 1-form A together with a Lie(H)-valued
2-form B. We show that a connection of this sort gives well-behaved
parallel transport along both curves and surfaces if a certain quantity
called the "fake curvature" vanishes. This quantity was introduced
by Breen and Messing. It equals
dA + A ^ A - dt(B)
When this is zero, we can define nonabelian parallel transport over
surfaces in a way that doesn't depend on the parametrization of the
surface. This was always the sticking point in attempts at higher
gauge theory.
You can see transparencies of my talk and also references at:
http://math.ucr.edu/home/baez/breen/