Tadpole conditions in F theory and the Euler characteristic of … · 2007. 10. 13. · Tadpole...
Transcript of Tadpole conditions in F theory and the Euler characteristic of … · 2007. 10. 13. · Tadpole...
![Page 1: Tadpole conditions in F theory and the Euler characteristic of … · 2007. 10. 13. · Tadpole conditions in F theory and the Euler characteristic of singular varieties Mboyo Esole](https://reader035.fdocuments.us/reader035/viewer/2022071401/60ed5cf1e36213770b3eadb4/html5/thumbnails/1.jpg)
Tadpole conditions in F theory
and
the Euler characteristic
of singular varieties
Mboyo EsoleInstitute of Theoretical Physics
KU Leuven
P. Aluffi and M.E., Chern Class identities from tadpole matching in type IIB and F-
theory, to appear
P. Aluffi, A. Collinucci, F. Denef, and M.E,D-brane Deconstruction in type IIB orientifolds, to appear
![Page 2: Tadpole conditions in F theory and the Euler characteristic of … · 2007. 10. 13. · Tadpole conditions in F theory and the Euler characteristic of singular varieties Mboyo Esole](https://reader035.fdocuments.us/reader035/viewer/2022071401/60ed5cf1e36213770b3eadb4/html5/thumbnails/2.jpg)
Type IIB vs F-theory
Type IIB F-theory
![Page 3: Tadpole conditions in F theory and the Euler characteristic of … · 2007. 10. 13. · Tadpole conditions in F theory and the Euler characteristic of singular varieties Mboyo Esole](https://reader035.fdocuments.us/reader035/viewer/2022071401/60ed5cf1e36213770b3eadb4/html5/thumbnails/3.jpg)
D3 branes as a probe…
Type IIB F-theory?
…tadpole conditions as the test
![Page 4: Tadpole conditions in F theory and the Euler characteristic of … · 2007. 10. 13. · Tadpole conditions in F theory and the Euler characteristic of singular varieties Mboyo Esole](https://reader035.fdocuments.us/reader035/viewer/2022071401/60ed5cf1e36213770b3eadb4/html5/thumbnails/4.jpg)
In a non-compact space, fluxes can escape to
infinity.
In a compact space fluxes have no where to
go so the total flux has to vanish
Homework
Study the electrostatic potential for two point-like charges on a circle
of radius L.
Tadpole conditions
![Page 5: Tadpole conditions in F theory and the Euler characteristic of … · 2007. 10. 13. · Tadpole conditions in F theory and the Euler characteristic of singular varieties Mboyo Esole](https://reader035.fdocuments.us/reader035/viewer/2022071401/60ed5cf1e36213770b3eadb4/html5/thumbnails/5.jpg)
With Tadpole
cancellation:
Linear potential
Example: Electrostatic potential of
two charges on a circle
Without Tadpole
cancellation:
Parabolic potential!
When we ignore tadpole conditions,
we describe a very different physics…
![Page 6: Tadpole conditions in F theory and the Euler characteristic of … · 2007. 10. 13. · Tadpole conditions in F theory and the Euler characteristic of singular varieties Mboyo Esole](https://reader035.fdocuments.us/reader035/viewer/2022071401/60ed5cf1e36213770b3eadb4/html5/thumbnails/6.jpg)
Tadpole conditions with D-branes
We start from a Lagrangian of the type
The equation of motions are
Tadpole condition
for a D-brane
![Page 7: Tadpole conditions in F theory and the Euler characteristic of … · 2007. 10. 13. · Tadpole conditions in F theory and the Euler characteristic of singular varieties Mboyo Esole](https://reader035.fdocuments.us/reader035/viewer/2022071401/60ed5cf1e36213770b3eadb4/html5/thumbnails/7.jpg)
Chern-Simons terms and induced
chargesAnomaly cancellation required the presence of CS terms
that contributes to the tadpole: a brane has lower brane
charge !
![Page 8: Tadpole conditions in F theory and the Euler characteristic of … · 2007. 10. 13. · Tadpole conditions in F theory and the Euler characteristic of singular varieties Mboyo Esole](https://reader035.fdocuments.us/reader035/viewer/2022071401/60ed5cf1e36213770b3eadb4/html5/thumbnails/8.jpg)
F-theory
•A theory in 12 dimensions: 10 dimensions + an elliptic
curve (a two torus)
• Complex structure of the elliptic curve= type IIB
axion-dilaton
•F-theory on an elliptically fibered CY_4 = Type IIB on the
base
QuickTime™ and aTIFF (Uncompressed) decompressor
are needed to see this picture.
Vafa has proposed an elegant geometric
formulation of Type IIB with (p,q)-branes:
QuickTime™ and aTIFF (Uncompressed) decompressor
are needed to see this picture.
![Page 9: Tadpole conditions in F theory and the Euler characteristic of … · 2007. 10. 13. · Tadpole conditions in F theory and the Euler characteristic of singular varieties Mboyo Esole](https://reader035.fdocuments.us/reader035/viewer/2022071401/60ed5cf1e36213770b3eadb4/html5/thumbnails/9.jpg)
ACT I: The set up
![Page 10: Tadpole conditions in F theory and the Euler characteristic of … · 2007. 10. 13. · Tadpole conditions in F theory and the Euler characteristic of singular varieties Mboyo Esole](https://reader035.fdocuments.us/reader035/viewer/2022071401/60ed5cf1e36213770b3eadb4/html5/thumbnails/10.jpg)
Act II:The tadpole conditions
D7 tadpole:
In IIB
Type IIB :
D3 tadpole
F-theory:
D3 tadpole
![Page 11: Tadpole conditions in F theory and the Euler characteristic of … · 2007. 10. 13. · Tadpole conditions in F theory and the Euler characteristic of singular varieties Mboyo Esole](https://reader035.fdocuments.us/reader035/viewer/2022071401/60ed5cf1e36213770b3eadb4/html5/thumbnails/11.jpg)
Act III:Type IIB and F-theory
tadpoles matching condition
![Page 12: Tadpole conditions in F theory and the Euler characteristic of … · 2007. 10. 13. · Tadpole conditions in F theory and the Euler characteristic of singular varieties Mboyo Esole](https://reader035.fdocuments.us/reader035/viewer/2022071401/60ed5cf1e36213770b3eadb4/html5/thumbnails/12.jpg)
An embarrassing mismatch
One can prove that this is not
an exception… It is the norm!
The type IIB/F-theory tadpole
matching condition is not satisfied
![Page 13: Tadpole conditions in F theory and the Euler characteristic of … · 2007. 10. 13. · Tadpole conditions in F theory and the Euler characteristic of singular varieties Mboyo Esole](https://reader035.fdocuments.us/reader035/viewer/2022071401/60ed5cf1e36213770b3eadb4/html5/thumbnails/13.jpg)
Is it the end of the honeymoon
between string theory and
self-consistency?
Not so fast…
![Page 14: Tadpole conditions in F theory and the Euler characteristic of … · 2007. 10. 13. · Tadpole conditions in F theory and the Euler characteristic of singular varieties Mboyo Esole](https://reader035.fdocuments.us/reader035/viewer/2022071401/60ed5cf1e36213770b3eadb4/html5/thumbnails/14.jpg)
Seven branes in F-theory
Weierestrass form:
Discriminant:
The seven branes are located at the singularities of
the elliptic fibration is singular:
Elliptic fibration:
![Page 15: Tadpole conditions in F theory and the Euler characteristic of … · 2007. 10. 13. · Tadpole conditions in F theory and the Euler characteristic of singular varieties Mboyo Esole](https://reader035.fdocuments.us/reader035/viewer/2022071401/60ed5cf1e36213770b3eadb4/html5/thumbnails/15.jpg)
Sen’s weak coupling limit
of F-theory
Sen’s ansatz:
Calabi-Yau three-fold:
Orientifold O7-plane:
D7 branes:
![Page 16: Tadpole conditions in F theory and the Euler characteristic of … · 2007. 10. 13. · Tadpole conditions in F theory and the Euler characteristic of singular varieties Mboyo Esole](https://reader035.fdocuments.us/reader035/viewer/2022071401/60ed5cf1e36213770b3eadb4/html5/thumbnails/16.jpg)
Anything special
in
Sen’s weak
coupling limit?
I see …
Singularities!!!
![Page 17: Tadpole conditions in F theory and the Euler characteristic of … · 2007. 10. 13. · Tadpole conditions in F theory and the Euler characteristic of singular varieties Mboyo Esole](https://reader035.fdocuments.us/reader035/viewer/2022071401/60ed5cf1e36213770b3eadb4/html5/thumbnails/17.jpg)
Singularities of the D7 brane
•A double curve along the z axis (x=y=0)
•A pinch point at the origin (x=y=z=0)
QuickTime™ and aTIFF (Uncompressed) decompressor
are needed to see this picture.
the
![Page 18: Tadpole conditions in F theory and the Euler characteristic of … · 2007. 10. 13. · Tadpole conditions in F theory and the Euler characteristic of singular varieties Mboyo Esole](https://reader035.fdocuments.us/reader035/viewer/2022071401/60ed5cf1e36213770b3eadb4/html5/thumbnails/18.jpg)
Topological invariant and
singularities
•Singularities are an invitation to diversity:
There are many ways to extend the usual topological
invariants to singular varieties.
•There will be casualties :
Not all the properties are preserved in the process
•Hard choices:
what are the properties that really matter?
![Page 19: Tadpole conditions in F theory and the Euler characteristic of … · 2007. 10. 13. · Tadpole conditions in F theory and the Euler characteristic of singular varieties Mboyo Esole](https://reader035.fdocuments.us/reader035/viewer/2022071401/60ed5cf1e36213770b3eadb4/html5/thumbnails/19.jpg)
Example: Euler characteristic
of a singular elliptic curve
Fulton Euler
characteristic
Deformation
Stringy Euler
characteristic
(Crepant)
Resolution
![Page 20: Tadpole conditions in F theory and the Euler characteristic of … · 2007. 10. 13. · Tadpole conditions in F theory and the Euler characteristic of singular varieties Mboyo Esole](https://reader035.fdocuments.us/reader035/viewer/2022071401/60ed5cf1e36213770b3eadb4/html5/thumbnails/20.jpg)
Defining topological invariant
for singular spaces
• We work in homology
• We defined generalized Chern classes
• The topological invariants follow
For example, the Euler characteristic is defined from the “top Chern
class”, which in homology is the term of order zero.
![Page 21: Tadpole conditions in F theory and the Euler characteristic of … · 2007. 10. 13. · Tadpole conditions in F theory and the Euler characteristic of singular varieties Mboyo Esole](https://reader035.fdocuments.us/reader035/viewer/2022071401/60ed5cf1e36213770b3eadb4/html5/thumbnails/21.jpg)
The pinch points story
Smooth
prejudice
Pinch points
discrepancy
![Page 22: Tadpole conditions in F theory and the Euler characteristic of … · 2007. 10. 13. · Tadpole conditions in F theory and the Euler characteristic of singular varieties Mboyo Esole](https://reader035.fdocuments.us/reader035/viewer/2022071401/60ed5cf1e36213770b3eadb4/html5/thumbnails/22.jpg)
GeneralizationWe can upgrade the previous relation to a more general
Theorem :
•in any dimension
•Without the CY condition
•At the level of the total Chern classes
![Page 23: Tadpole conditions in F theory and the Euler characteristic of … · 2007. 10. 13. · Tadpole conditions in F theory and the Euler characteristic of singular varieties Mboyo Esole](https://reader035.fdocuments.us/reader035/viewer/2022071401/60ed5cf1e36213770b3eadb4/html5/thumbnails/23.jpg)
Recovering the tadpole
matching conditition
It is possible to define an Euler characteristic such that
we recover the relation expected from the smooth case:
is defined as the limit of as m goes to infinity.
is defined through the Hopf-Poincaré theorem from a
generalized Chern class .
m has to do with the definition of the “relative canonical divisor”.
![Page 24: Tadpole conditions in F theory and the Euler characteristic of … · 2007. 10. 13. · Tadpole conditions in F theory and the Euler characteristic of singular varieties Mboyo Esole](https://reader035.fdocuments.us/reader035/viewer/2022071401/60ed5cf1e36213770b3eadb4/html5/thumbnails/24.jpg)
ConclusionsSingularities are there and they can be very useful.
• In F-theory, we have to take them seriously into account since might
show up in the weak coupling limit
•This represents a challenge to define the topological invariant
•Can we get the same result from a direct string calculation or
different physical arguments? The answer seems to be yes,
but this is a different story…
![Page 25: Tadpole conditions in F theory and the Euler characteristic of … · 2007. 10. 13. · Tadpole conditions in F theory and the Euler characteristic of singular varieties Mboyo Esole](https://reader035.fdocuments.us/reader035/viewer/2022071401/60ed5cf1e36213770b3eadb4/html5/thumbnails/25.jpg)
D-brane Deconstruction
in Type IIB Orientifolds
The Euler characteristic formula for singular varieties that
we have presented here is confronted to
• K-theory
• Sen-Witten deconstuction of D-brane configuration,
•Dirac quantization
• moduli counting
• and more
It passes successfully several physical tests!
P. Aluffi, A. Colinnuci, F. Denef and M. E.