Algebroids, heterotic moduli spaces and the Strominger system

James Gray, Virginia Tech

Based on work with:

Alexander Haupt and Andre Lukas 1303.18321405.????

and Lara Anderson, Andre Lukas and Burt Ovrut 1304.27041107.50761010.0255

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 four-

folds 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

There are at least 36,779 different topologiesData and code in arXiv submission

Hodge Data

Elliptic Fibrations• Consider configuration matrices which can be put in

the form:

• This is an elliptically fibred four-fold• In our list of 921,497 matrices, 921,020 have such a

fibration structure.

Base:

• 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:

• Over the Calabi-Yau

Extension is controlled by

quotiented by

V. Braun: arXiv:1003.3235

• 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.

Take

(this is equivariant and manifold is favourable so notation makes sense)

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)

• We find 25 distinct loci in complex structure moduli space that you can be restricted to.

where do we get stabilized to in complex structure moduli space?

• Some of these are isolated points in complex structure space.

• Unfortunately we have to be more careful: we must also check the CY is smooth on each of these 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).

The degree of singularity of the loci:

Non-Kähler Compactifications

• The most general heterotic compactification with maximally symmetric 4d space:– Complex manifold

Hull, Strominger

Gillard, Papadopoulos and Tsimpis

Non-Kähler Compactifications

• The most general heterotic compactification with maximally symmetric 4d space:– Complex manifold

Hull, Strominger

Gillard, Papadopoulos and Tsimpis

Non-Kähler Compactifications

• The most general heterotic compactification with maximally symmetric 4d space:– Complex manifold

Hull, Strominger

Gillard, Papadopoulos and Tsimpis

• Perturb all of the fields just as we did in the Calabi-Yau case:

• And look at what the first order perturbation to the supersymmetry relations looks like…

In what follows I consider manifolds obeying the -lemma

• For the perturbation analysis the Atiyah computation goes through unchanged.

• The other equations are somewhat more messy:

Lemma: Let be a compact Kähler manifold. For a -closed form, the following statements are equivalent.

For some , , , and .

• Totally anti-holomorphic part of eqn:

• Remaining components:

• Atiyah analysis:

• How do we interpret this result?– Proceed by analogy with the Atiyah case:

Define a bundle :

and a bundle :

We claim the cohomology precisely encapsulates the allowed deformations.

Baraglia and Hekmati 1308.5159

• 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