Calabi-Yau condition, moduli and Lichnerowicz equation

+ 2 like - 0 dislike
2512 views

I have a conceptual confusion about the metric moduli of Calabi-Yau manifolds, when I was reading Calabi-Yau compactification.

As I understand, the metric moduli is parametrized by infinitesimal deformation of the metric that preserve Calabi-Yau condition (Ricci-flatness or equivalently, admit one covariantly constant spinor). So we would have the Lichnerowicz equation of the deformation metric: $$\nabla^l \nabla_l\ \delta g_{mn} - [\nabla^l,\nabla_m]\ \delta g_{ln} - [\nabla^l, \nabla_n]\ \delta g_{lm} = 0$$

And moreover, the deformation is a deformation of complex structure: the compensating coordinate transformation is not holomorphic. Or more specifically, the metric can be no longer written in a Hermitian form. But since the deformation preserves Calabi-Yau condition, it should still be Calabi-Yau, so it is still Kahler.

Thus, is the deformed metric still Calabi-Yau? If it is, how to understand the emergence of non-Hermitian component in the metric?

This post imported from StackExchange Physics at 2014-08-12 18:14 (UCT), posted by SE-user Kevin Ye

+ 3 like - 0 dislike

The most general $\delta g_{mn}$ that preserve the Ricci-flatness on the original Calabi-Yau backgrounds is the sum of several components: the pure infinitesimal diffeomorphisms (I won't discuss those because they're physically vacuous and trivial); changes of the Kähler moduli; and changes of the complex structure.

A Calabi-Yau three-fold has $h_{1,1}$ real parameters describing the Kähler moduli (they become complexified when the $B$-field two-form is added in type II string theory) and $h_{1,2}$ complex parameters describing the complex structure moduli. These two integers are interchanged for the manifold related by the mirror symmetry.

The Kähler moduli describe different ways to choose the Ricci-flat metric on the manifold that are compatible with a fixed, given complex structure. The different solutions may locally be derived from the Kähler potential $K$ which can have many forms. These moduli effectively describe the proper areas of 2-cycles.

The complex structure deformations change the complex structure – and the corresponding spinor – but the new, deformed manifold still has a complex structure, just a different one! The metric after an infinitesimal variation would have a non-Hermitian component with respect to the old complex structure but with respect to the new, deformed complex structure, it is still perfectly Hermitian! Calabi-Yaus are always Kähler, $SU(3)$ (for 6 real dimensions) complex manifolds with a purely Hermitian metric in some appropriate complex coordinates, and a deformation still keeps the manifold in the set of Calabi-Yaus.

The simplest example to test your $\delta g_{mn}$ intuition is 1-complex-dimensional or 2-real-dimensional Calabi-Yaus, two-tori. The complex structure is changing the ratios of the two periods and the angle in between, namely the $\tau$ complex structure parameter. The Kähler modulus is the overall area of the two torus – the overall scaling of the whole 2-dimensional metric. You may easily see that all these transformation keep the manifold flat, and therefore Ricci-flat.

This post imported from StackExchange Physics at 2014-08-12 18:14 (UCT), posted by SE-user Luboš Motl
answered Aug 12, 2014 by (10,248 points)
Thank you Luboš! That is clear to me now. I realize that I mixed two concepts: complex structure and Kähler form. We should always talk about metric keeping in mind some specific complex structures implicitly.

This post imported from StackExchange Physics at 2014-08-12 18:14 (UCT), posted by SE-user Kevin Ye

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$y$\varnothing$icsOverflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.