Exotic Smooth 4-Manifolds and Gerbes as Geometry for QG

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Exotic Smooth 4-Manifolds and Gerbes
as Geometry for QG
Jerzy Król
University of Silesia
Katowice, Poland
14 wrze±nia 2009
Matter to the Deepest USTRO 2009
Jerzy Król (US Katowice)
Ustro«: Exotics, Gerbes and QG
14 wrze±nia 2009
1 / 19
Motivations
Classical gravity is described by GR: spacetime is a smooth manifold
a pseudo-Riemannian metric
g
action for GR, in the absence of any other matter, is
where
(X , g )
R
X
with
satisfying Einstein's eld equations. The
S (X , g ) = X Rd volg
R
is the scalar curvature. The critical point for this action is just
which describes both, gravity and geometry.
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2 / 19
Motivations
g
where
(X , g )
R
X
with
R
What is geometry for quantum gravity?
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2 / 19
Motivations
g
where
(X , g )
R
X
with
R
∗
Assume that QG, at least in some limit, is given by superstring theory.
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2 / 19
Motivations
g
where
(X , g )
R
X
with
R
∗
Assume that QG, at least in some limit, is given by superstring theory.
Then, spacetime as a smooth manifold at the fundamental level is
meaningless.
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Instead, we have string backgrounds.
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These backgrounds are described by 2-dim. CFT and
σ -models.
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σ -models.
They have the classical limit where geometry emerges:
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σ -models.
They have the classical limit where geometry emerges:
Classical geometry from string theory are the triples (M , g , B ) where B
(B-eld) is a local 2-form.
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Conversely, given a classical string background
(M , g , B )
one has the
corresponding CFT and the string background. It is believed that every
string background emerges in this way:
UΣ ({γ1 , ..., γp }, {γp+1,...,γp+q }) =
where
γi
are the boundary of all
of the transportation along
Σ
R
{φ:Σ→M }
Σ's
e iS (φ,B )
in the integral and
from one set of
γ 's
UΣ
is the operator
to the other.
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4 / 19
B
eld on
M
is the same as
gerbe - the object from dierential geometry
M.
which is a generalization of bundles on
(Abelian) Gerbes are classied by
H 3 (M , Z).
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This talk (based on the joint work with Torsten
Asselmeyer-Maluga):
Exotic smooth
R4 's
are related to gerbes and backgrounds of string theory.
The geometry of QG refers to exotic 4-structures.
Exotic
R4 's
are as fundamental for QG as the standard
R4
for classical
gravity:
i.
ii.
Some eects of QG in spacetime can be locally cancelled by the suitable
choice of exotic smooth structure on R4 .
Exotic R4 's contain also quantum information of spacetime and gravity.
The quantization of electric charge can be explained with exotic
R4
instead
of magnetic monopoles.
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6 / 19
R4 :
the early days
1
Exotic Smooth
2
Exotic 4-smoothness in (Q)Gravity and QFT
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Exotic Smooth R4 : the early days
Early Days of Exotic Smooth
R4
Freedman (1982): there exists the smoothness of a topological
R4
which is
non-dieomorphic to the standard one (agreeing with the topological
product of axes).
Soon after Gompf showed that there are at least 3 such dierent smoothings.
Next, Taubes proved, via physical Yang-Mills theories techniques of
Donaldson, that there are innite
smoothings of
R4 .
In any other dimension than
Exotic
R4 's
continuum many such dierent
four there exists only one such smooth Rn .
are ordinary manifolds some of them have nite number of local
coordinate patches (say 3,5...).
The usual tools and techniques of dierential geometry and topology from
higher and lower dimensions, fail in dimension 4. We do not have clear idea
what the tools should be.
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8 / 19
4 is the physical dimension and general relativity of Einstein can be
formulated on any (exotic) smooth manifold. Brans (1994) conjectured and
Asselmeyer (1996) and Sªadkowski (1999) proved, that exotic smooth
R4
can act as sources for gravitational eld in 4-spacetime.
continuum many dierent smooth R4 's, nobody can give
even one example of exotic smooth function in any of the smooth structure.
There exist innite
This is a true limitation for physical applications and our mathematical
understanding.
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What to do?
It was proposed (JK, 2004) to use the tools of
model theory and topoi.
As the result the conjectures were formulated and analyzed:
i.
ii.
The suitable choice of exotic smooth R4 can compensate some local eects
(in spacetime) assigned to QG. Hence, exotic R4 's for QG are similarly
fundamental as the standard R4 for classical gravity and
(pseudo-)Riemannian 4-geometry.
In the AdS/CFT correspondence the exotic 4-structures can cause the
additional susy breaking (important for approaching the realistic QCD).
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Exotic 4-smoothness in (Q)Gravity and QFT: exact results
Still, the explicit calculations in local coordinate patches (and which are
exotic smooth) were absent.
Torsten Asselmeyer-Maluga, JK, hep-th/0904.1276:
Take special
S3
embedded in
R4
and measure the relative changes of
4-smoothness on various structures on this
S 3.
The structures are:
3
1. Hoeiger co-dimension-1 foliations of S .
2.
Z2
orbifolds of WZW model on
S 3.
3. Generalized Hitchin's structures and
4. Groupoids
U (1)-gerbes on S 3 .
SU (2) × SU (2) ⇒ SU (2).
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11 / 19
1. Hoeiger co-dimension-1 foliations of
S 3 , then:
Dierent smooth R4 's correspond to dierent classes of Hoeiger foliations
of S 3 , i.e. correspond to classes H 3 (S 3 , R).
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2. Given
S 3 ' SU (2) we have WZW SU (2) models on this S 3 , then in the case
3 3
of H (S , Z)):
Dierent smoothness of R4 correspond to the levels k of the Z2 orbifold of
WZW model on SU(2).
These appear as background solutions of some 5-branes congurations of
sustring theory, hence QG.
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3. Abelian gerbes on
H 3 (S 3 , Z)
' Z.
S3
correspond to the integral cohomology classes
Generalized Hitchin's structures on
S3
can be dened, then:
Dierent smoothness of R4 correspond to the deformed by U (1) gerbes
generalized Hitchin's structures on S 3 .
Again, the generalized complex structures and geometries of Hitchin play
important role in ux compactication in string theory.
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4. A groupoid structure thus emerges:
Dierent exotic smooth R4 's correspond to the deformed by S 1 -gerbes
equivariant cohomologies of the groupoid SU (2) × SU (2) ⇒ SU (2), where
SU (2) acts on itself by conjugations.
These cohomologies are just the Verlinde algebras of
SU (2) at the level k
(Freed, Teleman, Hopkins, 2002).
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Exotic smooth
R4 's
are surprisingly and deeply involved in the description of
quantum gravity. Geometry of string theory is based rather on locally exotic
than standard smooth 4-manifolds.
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QFT
The quantization of electric charge without magnetic monopoles:
Some exotic R4 's in a region of 4-spacetime can act as the sources of the
magnetic eld in this spacetime, hence give the condition for the
quantization of electric charge.
The theories with monopoles or monopolium give rise to the corrections of
certain cross sections of processes like
4
the contributions from exotic R 's.
e +e −
or
γγ .
These can be seen as
Some non-perturbative limits of a QCD-like YM theory can be reached when
the theory is formulated on 4-manifolds which are locally exotic
R4 .
(monopoles are non-perturbative objects in eld theory and the dual,
supersymmetric theory has monopoles as ground states (Witten, 2002)).
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Meta-remarks (philosophical)
Exotic
R4 's
should not be seen through their topological coordinate axes.
Rather, these are non-reducible states of the axes.
Due to this they contain also quantum information about spacetime, hence
quantum gravity.
Exotic 4-smoothness agrees rather with generalized topology than standard
product topology.
String theory favors geometry of manifolds with the presence of
which is connected with exotic
R4
B -eld,
rather than standard one.
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G. Segal, Topological structures in string theory,
London, A359, 1389 (2001).
Phil. Trans. Royal Soc.
SU (2) models and exotic
Comm. Math. Phys., hep-th/0904.1276.
T.Asselmeyer-Maluga, J.Król, 2009, Gerbes, WZW
smooth
R4 ,
submitted to
T. Asselmeyer, 1996, Generation of source terms in general relativity by
dierential structures.
Class. Quant. Grav., 14:749 758, 1996.
J. Sªadkowski, 2001, Gravity on exotic
Phys. D, 10, 311313 (2001).
R4
with few symmetries,
Int.J. Mod.
J. Król, Exotic smoothness and non-commutative spaces. The
model-theoretic approach.
Found. of Physics, 34:843869, 2004.
J. Król, 2005, Model theory and the AdS/CFT correspondence,
hep-th/0506003.
14 wrze±nia 2009
19 / 19

Exotic Smooth 4-Manifolds and Gerbes as Geometry for QG

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