Algebroids, heterotic moduli spaces
and the Strominger system
James Gray, Virginia Tech
Based on work with:
Alexander Haupt and Andre Lukas 1303.1832
1405.????
and Lara Anderson, Andre Lukas and Burt Ovrut 1304.2704
1107.5076
1010.0255
and Lara Anderson and Eric Sharpe 1402.1532
See also Xenia de la Ossa and Eirik Svanes 1402.1725
The plan:
• Quick advert for a class of Calabi-Yau
fourfolds.
• Discussion of moduli in heterotic:
– Warm up with a well known, relatively simple
case.
– Move on to discuss moduli of some Strominger
system compactifications in terms of ordinary
bundle valued cohomology.
Calabi-Yau Fourfolds
A Class of Calabi-Yau Four-folds
• We would like to find a description of all Calabi-Yau fourfolds which are complete intersections in products of
projective spaces.
– Three-fold list has proved to be very useful:
•
•
•
•
Hübsch, Commun.Math.Phys. 108 (1987) 291
Green et al, Commun.Math.Phys. 109 (1987) 99
Candelas et al, Nucl.Phys. B 298 (1988) 493
Candelas et al, Nucl.Phys. B 306 (1988) 113
– There are 7890 elements in the data set.
• Generating the equivalent four-fold list involves
qualitative and quantitative changes…
Complete Intersections in Products of
Projective Spaces.
• An example of a
configuration matrix
(CICY four-fold 244)
• This represents a family of Calabi-Yau four-folds
defined by the solutions to the polynomials
• In the ambient space
• We can represent the projective factors by a
simple integer too
• Then generally we have:
With m rows and K columns.
• The dimension of the complete intersection is
• One can show that the configuration is Calabi-Yau
if:
Results:
• 921,497 different configurations
– Same code correctly reproduces three-fold list
– Product cases (15,813) are correct
• Some redundancies are still present
– Can prove cases are different with top. invariants
105
104
Number of matrices
There are at
least 36,779
different
topologies
Data and code in
arXiv submission
1000
100
10
500
1000
1500
2000
2500
Χ
Hodge Data
N
N
10 5
10 5
10 4
10 4
1000
1000
100
100
10
10
1
5
10
15
20
25
h 1,1
1
N
10
15
20
25
30
35
N
10
5
10 4
10 4
1000
1000
100
100
10
10
1
5
h 2 ,1
100
200
300
400
h 3 ,1
1
500
1000
1500
h 2 ,2
Elliptic Fibrations
• Consider configuration matrices which can be put in
the form:
Base:
• This is an elliptically fibred four-fold
• In our list of 921,497 matrices, 921,020 have such a
fibration structure.
104
Abundance
1000
100
10
1
0
50
100
150
200
250
300
350
Number of fibrations per configuration
• Total number of “different” elliptic fibrations:
50,114,908.
• All manifolds with
have at least one
such fibration.
• Some information about sections also provided…
Moduli in heterotic
Warm up: Bundles on Calabi-Yau
• The moduli space of a Calabi-Yau
compactification in the presence of a gauge
bundle is not described in terms of
–It is described in terms of a subspace of
these cohomology groups determined by
the kernel of certain maps
–Those maps are determined by the
supergravity data of the solution.
• To see this we can analyze the supersymmetry
conditions.
• The conditions for the gauge field to be
supersymmetric are the Hermitian Yang-Mills
equations at zero slope:
• Study perturbations obeying these equations:
Perturb the complex structure:
and the gauge field:
• Define
and rewrite our equation in a more usable form
• And work out the perturbed equation to first
order:
This equation is not of much practical use…
The Atiyah class:
• There is a description of this in terms of
cohomology of a certain bundle:
Define:
Atiyah states that the moduli are not
But rather:
How do we tie this in with our field theory analysis?
• Take
to vanish for simplicity
• Look at the long exact sequence in cohomology
where
• Thus we see Atiyah claims the moduli are given by
Concrete examples are known where
we can calculate all this:
• Take the case of a simple SU(2) bundle:
Extension is controlled by
• Over the Calabi-Yau
quotiented by
V. Braun:
arXiv:1003.3235
Take
(this is equivariant and manifold is favourable so
notation makes sense)
• Generically
and so no
such SU(2) bundle exists.
• At special loci in complex structure space this
cohomology can jump to non-zero values and
holomorphic SU(2) bundles can exist.
• The computation of which perturbations are allowed
around points on these loci precisely reproduces the
Atiyah calculation.
Our goal becomes to map the loci
where the Ext groups “jump”.
Most general defining relation for this example:
There are 10 complex structure parameters here (11
coefficients – the overall scale of the polynomial
does not matter)
cus in combined field space.
where do we get stabilized to in complex structure
erform
elimination
(projection)
to
the
complex
moduli space?
ructure moduli space for each piece.
• We find 25 distinct loci in complex structure moduli
es 25space
distinct
interesting
loci: to.
that you
can be restricted
me• of
these
are isolated
Some
of these
are
nts in
complex
isolated
pointsstructure
in
ce. complex structure
space.
ortunately we can not
• Unfortunately we have
declare
to be success
more careful:
wealso
mustcheck
also check
must
the the
CY is smooth on each of
oothness
of
the
CY
on
these loci
h locus.
Dim.
7
5
4
3
2
1
0
Num.
2
2
3
4
6
5
3
The degree of singularity of the loci:
• In this case only one of
the loci is smooth
(stabilizes 6 moduli):
• May ask if the singular
CY’s can be resolved
• The answer is definitely
yes, at least for some of
the cases (see
arXiv:1304.2704).
Non-Kähler Compactifications
Hull, Strominger
• The most general
heterotic
compactification with maximally symmetric 4d
space:
– Complex manifold
Gillard, Papadopoulos and Tsimpis
Non-Kähler Compactifications
Hull, Strominger
• The most general
heterotic
compactification with maximally symmetric 4d
space:
– Complex manifold
Gillard, Papadopoulos and Tsimpis
Non-Kähler Compactifications
Hull, Strominger
• The most general
heterotic
compactification with maximally symmetric 4d
space:
– Complex manifold
Gillard, Papadopoulos and Tsimpis
• Perturb all of the fields just as we did in the CalabiYau case:
• And look at what the first order perturbation to the
supersymmetry relations looks like…
In what follows I consider manifolds obeying the
-lemma
Lemma: Let
-closed
equivalent.
For some
be a compact Kähler manifold. For
form, the following statements are
,
,
,
and
a
.
• For the perturbation analysis the Atiyah
computation goes through unchanged.
• The other equations are somewhat more messy:
• Atiyah analysis:
• Totally anti-holomorphic part of
• Remaining components:
eqn:
• How do we interpret this result?
– Proceed by analogy with the Atiyah case:
Define a bundle
and a bundle
:
:
Baraglia and Hekmati 1308.5159
We claim the cohomology
precisely
encapsulates the allowed deformations.
• To make contact with the field theory we again
look at the associated long exact sequences in
cohomology.
and
Do the sequence chasing and you find…
• This is a subspace of
defined by maps determined by the supergravity
data.
• All maps are well defined, as are the extensions.
• This precisely matches the supergravity computation.
• Reduces to Sharpe,Melnikov arXiv:1110.1886 as
.
A few comments on the structure:
• One can easily generalize to the case where
.
• The overall volume is only a modulus in the CY case.
• Unlike in the Atiyah story, the bundle moduli are
constrained by the map structure here.
• Matter can be included in the analysis, simply by
thinking of it as the moduli of an E8 bundle.
• There seems to be a nice mathematical interpretation
of all of this…
Baraglia and Hekmati 1308.5159
Garcia-Fernandez 1304.4294
Conclusions
• For the case where a non-Kähler heterotic
compactification obeys the
-lemma:
– The moduli are given by subgroups of the usual sheaf
cohomology groups.
– The subgroups of interest are determined by kernels
and cokernels of maps determined by the
supergravity data
– This all has a nice mathematical interpretation in
terms of Courant algebroids (transitive and exact)
and, it seems, in terms of generalized complex
structures on the total space of certain bundles