When we meet:    The Special Holonomy RAP will meet regularly on Mondays 4:00-5:00pm in 145 Altgeld,
and occasionally also on Wednesdays 4:00-5:00pm in 243 Altgeld.  See the Department Seminar Schedule, or
check the RAP Seminar Schedule  for more information.

News and announcements

Resources

Regular participants: Maarten Bergvelt, Rahul Biswas, Steve Bradlow(organizer), Rob Ghrist,
Sheldon Katz,  Eugene Lerman, Sean Nowling;  Thomas Rohwer; Eric Sharpe, John Sullivan;
Mehmet Sahin;  Walid Abou Salem ; Sue Tolman; Philippe Tondeur; Jeremy Wong

9/17/02:  A new preprint by Nigel Hitchen, on Generalized Calabi-Yau manifolds,
has been added to the Preprint list.

Introduction

Recent developments in string theory have stimulated new interest in two kinds of special geometric structure;
one related to Riemannian metrics with special holonomy, and one determined by differential forms of degree
2, 3 and 4.  The special geometries of the first sort include

· Kahler manifolds (U(n) holonomy)
· Calabi-Yau manifolds (SU(n) holonomy)
· Hyperkahler manifolds (Sp(n) holonomy)
· Spin(7)-manifolds, and
· G2 manifolds

Those of the second sort, which can also be described as geometries of metrics with special weak holonomy, include

· Special-Kahler manifolds in dimension 6 (weak SU(3) holonomy)
· 3-Sasakian manifolds in dimension 7 (weak G-2 holonomy)
· A new (not yet named!) special geometry in dimension 8

This RAP group will explore these geometric structures. As far as possible, it will look from both the mathematical
and the physical points of view.

After an introductory session, we will begin by reading

Stable forms and special metrics by Nigel Hitchin
[Global differential geometry: the mathematical legacy of Alfred Gray (Bilbao, 2000), 70-89,
Contemp. Math 288, 2001] (available as arXiv:math.DG/0107101)

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Schedule

September 3:  Mathematical Introduction to special holonomy (Sheldon Katz)  [4:00pm Henry 137]

September 9: Introduction to Special Holonomy in Physics (Eric Sharpe) [4:00pm Altgeld 145]

September 16:  Stable Forms and Special Metrics (I)  (Steven Bradlow ) [4:00pm Altgeld 145]

            Introduction to/overview of the paper ``Stable forms and special metrics'' by Nigel Hitchin
            [Global differential geometry: the mathematical legacy of Alfred Gray (Bilbao, 2000), 70-89,
            Contemp. Math 288, 2001] (available as arXiv:math.DG/0107101),  plus discussion of the
            first few sections of the paper.

From the Abstract to Hitchin's paper:``We show how certain diffeomorphism-invariant functionals in
dimensions 6,7, and 8 generate in a natural way special geometric structures in these dimensions:
metrics of holonomy G2 and Spin(7), metrics with weak holonomy SU(3) and G2, and a new
unexplored examples in dimension 8.''

Based on the approach in Robert Bryant's paper on `Metrics with exceptional holonomy'
[Annals of Math. 126 (1987), 525-576], we will give a description of the key basic facts about G2.

September 30:  Stable Forms and Special Metrics (II)  (Steven Bradlow ) [4:00pm Altgeld 145]

            Continuation  of discussion of ``Stable forms and special metrics'' by Nigel Hitchin
            [Global differential geometry: the mathematical legacy of Alfred Gray (Bilbao, 2000), 70-89,
            Contemp. Math 288, 2001] (available as arXiv:math.DG/0107101),  plus discussion of the
            first few sections of the paper.
 

October 7:  Stable Forms and Special Metrics (II)  (Steven Bradlow ) [4:00pm Altgeld 145]

            Continuation  of discussion of ``Stable forms and special metrics'' by Nigel Hitchin
            [Global differential geometry: the mathematical legacy of Alfred Gray (Bilbao, 2000), 70-89,
            Contemp. Math 288, 2001] (available as arXiv:math.DG/0107101),  plus discussion of the
            first few sections of the paper.

WEDNESDAY October 16 :  G-structures and holonomy(Eugene Lerman) [ 4:00 pm 243 Altgeld Hall]

           A freeze-dried version of section 2.6 of Joyce's "Contact manifolds with special holonomy"

October 21:    3.6 from Joyce on spinors and special holonomy  (Sue Tolman) [4:00pm Altgeld 145]

October 28:   Special holonomy and Ricci curvature (Jeremy Wong) [4:00pm Altgeld 145]
 

November 4:  Octonions: the eight-fold way  (John Sullivan)  [4:00pm 145 Altgeld Hall]
           I will give an introduction to (and several definitions of) the octonions,
            mainly following section 2 of Baez (math.RA/0105155).
 

November 11:  Octonions: the eight-fold way (cont.) (John Sullivan)  [4:00pm 145 Altgeld Hall]
           I will continue the introduction to (and several definitions of) the octonions,
            mainly following section 2 of Baez (math.RA/0105155).
 

November 18:  Spinors and holonomy  (Hui Li) [4:00pm 145 Altgeld Hall]

We will discuss the link between spin geometry and holonomy groups, and deduce some topological information about compact 4m-manifolds with Ricci-flat holonomy groups.

November 21: (BCDE Seminar): M-theory and special holonomy spaces. (Mirjam Cvetic, U. Penn)  [12:00 pm  358 Loomis Hall]

Explicit constructions of exceptional Spin (7) and G_2 holonomy metrics with asymptotically locally conical metric are reviewed. In particular, a connection of M-theory on a class of G_2 holonomy spaces to Type II string theory on a class of Calabi-Yau spaces (resolved and deformed conifolds) is highlighted. Some implications of M-theory on such spaces for the dynamics of four-dimensional N=1 supersymmetric vacua of M-theory are given.

December 2: (joint BCDE Seminar):
Fluxes, torsion, and the ground state of string theory.(Katrin Becker, Utah, Physics )  [4:00 pm  358 Loomis Hall]

In this lecture I will be discussing different compactifications of String Theory and of M-theory. These compactifications are characterized by the fact that they include non-vanishing expectation values for tensor fields. These expectation values will imply that many moduli fields of the internal manifold are stabilized. Stabilizing the moduli fields is one of the key issues in order to determine the ground state of string theory and is of vital importantce on order to make contact between string theory and the standard model of elementary particles.

We focus on Chapter 4 of Hitchin's paper, concerning 8 manifolds M with a stable 3 form fixed by PSU(3).  If the 3 form is critical for the volume functional within its cohomology class, the main result is that M admits a Rarita-Schwinger field (the massless spin 3/2 gravitino of supergravity theory in physics).  There is no covariant constant spinor, so M need not be Ricci flat as in many of the other cases we have investigated in this RAP; but 8 of the 36 components of the Ricci tensor turn out to vanish.

This talk will include some necessary background on Lie algebra representation theory as well as some unnecessary background on spin 3/2 fields as they occur in physics.

Results from Math. Reviews search on Special Holonomy   Exotic holonomy    ExceptionalHolonomy

Results from Math. Reviews search on Octonions [orOctonians]

Lectures by Robert Bryant on Manifolds with G2 Holonomy

Preprints