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Flux Compactifications of String Theory and Kähler/Calabi–Yau manifolds

By Posted on In Cosmology, Grand Unification Physics, M Theory, Mathematical Physics, Mathematics, Physics, Super String Theory, Supersymmetry ,
Flux Compactifications of String Theory

In this post, I will analyse several deep issues pertaining to flux compactifications of string theory, and draw foundational conclusions in relation to Kähler and Calabi–Yau manifolds. I already discussed the connection with Hodge theory and Gromov-Witten invariants of Calabi-Yau 3-Folds. The key thing to realize is that fluxes present in 10/11-D string theory naturally stabilize the moduli of a compactification with a downward reduction to 4-dimensional Minkowskian manifolds with broken supersymmetry minimally affecting the quantum vacuum energy as well as a ‘natural’ non-fine-tuned solution to the hierarchy problem: that is their importance for phenomenology, not just theory. Start with the ten-dimensional type IIB supersymmetry transformations:

 

Untitledy1

and:

    \[\delta \lambda = \frac{1}{\kappa }{\Gamma ^M}{P_M}{B^{\left( {10} \right) * }}{\varepsilon ^ * } + \frac{1}{{24}}{\Gamma ^{MNP}}{G_{MNP}}\varepsilon \]

with the Gromov-Witten genus-one amplitude given by:

    \[{A^{g = 1}} = {\hat R^4}\int_{{{\hat F}_{\left( 1 \right)}}} {\frac{{{d^2}\tau }}{{\tau _2^2}}} \int_{\hat {\rm T}} {\prod\limits_{i = 1}^3 {\frac{{{d^2}{\nu ^{\left( i \right)}}}}{{{\tau _2}}}} } \,{e^{{D_{\left( 1 \right)}}}}\]

with the flux U-duality integral:

    \[\int {{d^{\left( {10} \right)}}} x\sqrt { - {g^{\left( {10} \right)}}} \varepsilon _{\left( {1,0} \right)}^{\left( {10d} \right)}{\hat \not D^4}{\hat R^4}\]

and:

    \[\varepsilon _{\left( {1,0} \right)}^{\left( {10d} \right)}{\hat \not D^4}{\hat R^4} = \upsilon _{10}^{ - 4/8}{E_{4/2}}\left( \Omega \right) + \frac{{2{\pi ^2}}}{3}\upsilon _{10}^{ - 6/8}\]

Fluxes in four-dimensional space–times are clearly central for phenomenological reasons, however, in six dimensions, flux backgrounds are solutions of string theory and induce internal four-dimensional manifolds with flux solutions that are uniquely constrained. Backgrounds with torsion in the connection appear analytically with the torsion being turned on by the three-form tensor field H with property:

    \[dh = {\rm{tr}}\,R \wedge R - \frac{1}{{30}}{\rm{tr}}\,F \wedge F\]

and since the fundamental form of the internal complex n-manifold

    \[{J_{a\bar b}} = i{g_{a\bar b}}\]

satisfies both:

    \[\partial \bar \partial J = \frac{i}{{30}}{\rm{tr}}\,F \wedge F - i\,{\rm{tr}}\,F \wedge F\]

and

    \[{d^\dagger }J = i\left( {\partial - \bar \partial } \right)\log \left\| \omega \right\|\]

we have supersymmetry-preservation, where \omega is a (n, 0)-holomorphic form. One further condition is that the Yang–Mills field must satisfy the Donaldson–Uhlenbeck–Yau equation in a torsional background, and the central metaplectic flux compactifications to four dimensions satisfy all conditions above.

In six-dimensional Minkowski space–time, the Picard-spinor satisfies:

    \[{\bar \nabla '_m}\,\varepsilon = 0\]

and thus the space–time metric will be conformal to a Calabi–Yau two-fold

Start with the compactification of type IIB supergravity to six dimensions in the presence of brane sources. Both tensor fields can be complex as opposed to the heterotic case.

I will show that the most general four-manifold describing the internal dimensions is conformal to a Kähler manifold, in contrast with the heterotic case where the four-manifold must be conformally Calabi–Yau

Type IIB string theory compactified to six dimension

 

As above, the ten-dimensional type IIB supersymmetry transformations are:

 

Untitledy1

 

and

    \[\delta \lambda = \frac{1}{\kappa }{\Gamma ^M}{P_M}{B^{\left( {10} \right) * }}{\varepsilon ^ * } + \frac{1}{{24}}{\Gamma ^{MNP}}{G_{MNP}}\varepsilon \]

with the ten-dimensional supersymmetry parameter being complex and satisfies the Weyl condition:

    \[{\Gamma ^{\left( {10} \right)}}\varepsilon = \varepsilon \]

with the configurations space satisfies six-dimensional Poincaré-invariance, with line element:

    \[d{s^2} = {e^{2D}}{\eta _{\mu \nu }}d{x^\mu }d{x^\nu } + {e^{ - 6D}}{g_{mn}}d{y^m}d{y^n}\]

Hence, supersymmetric configurations satisfy:

    \[\left( {{\nabla _m} - \frac{i}{2}{Q_m}} \right)\varepsilon - \frac{\kappa }{{24}}{e^{6D}}{\Gamma _m}^{pqr}{G^{\left( {10} \right) * }}{\varepsilon ^ * } = 0\]

    \[\not \partial D\varepsilon = \frac{\kappa }{8}{e^{6D}}\not G{B^{\left( {10} \right) * }}{\varepsilon ^ * }\]

and

    \[\not P{B^{\left( {10} \right) * }}{\varepsilon ^ * } = - \frac{\kappa }{4}{e^{6D}}\bar \not G'\varepsilon \]

with a spinor rescalling

    \[\varepsilon \to {e^{ - 3D/2}}\varepsilon \]

And so, we have a non-constant warp factor D(y) requiring at least one component of {G_{mnp}} being non-zero.

Hence, a decomposition of the ten-dimensional spinor allows us to represent an anti-commuting six-dimensional spinor as a pair of Weyl spinors {\xi _i},\,\quad i = 1,2 obeying the symplectic Majorana–Weyl condition:

    \[{\Gamma ^{\left( 6 \right)}}\xi _i^ \pm = \pm \,\xi _i^ \pm \]

    \[{\varepsilon _{ij}}{B^{\left( 6 \right) * }}{\left( {\xi _j^ \pm } \right)^ * } = \xi _i^ \pm \]

Given the decomposition of the Lorentz algebra

    \[SO\left( {9,1} \right) \to SO\left( {5,1} \right) \times SO\left( 4 \right)\]

a positive chirality spinor decomposes as:

    \[{16_ + } \to \left( {{4_ + },{2_ + }} \right) + \left( {{4_ - },{2_ - }} \right)\]

and for the supersymmetry parameter \varepsilon this decomposition reduces to:

    \[\varepsilon = {\varepsilon ^{ij}}\left( {\xi _i^ + \otimes \eta _j^ - + \xi _i^ - \otimes \eta _j^ - } \right)\]

The decomposition is hence invariant under an SU(2) transformation acting on the spinor labels i and j and \varepsilon becomes a ten-dimensional Majorana–Weyl spinor.

Now let me set:

    \[\varepsilon = {\varepsilon ^{ij}}{\xi _i} \otimes {\eta _j}\]

and

    \[{B^{\left( {10} \right) * }}{\varepsilon ^ * } = {\varepsilon ^{ij}}{\xi _i} \otimes {\tilde \eta _j}\]

and by the above decomposition into the supersymmetry variations,

one gets the result that supersymmetry is preserved if and only if the following conditions hold

    \[\left( {{\nabla _m} - \frac{i}{2}{Q_m}} \right){\eta _j} + {g_m}\tilde \eta = 0\]

    \[\not \partial D{\eta _j} = - \frac{1}{2}\not \bar g{\tilde \eta _j}\]

and

    \[\not P{\tilde \eta _j} = \not \bar g{\tilde \eta _j}\]

with the dualized one-form field being:

    \[g = - \frac{\kappa }{4}{e^{6D}}\left( { * G} \right)\]

hence, from:

    \[{\Gamma ^{\left( 6 \right)}}\xi _i^ \pm = \pm \,\xi _i^ \pm \]

    \[{\varepsilon _{ij}}{B^{\left( 6 \right) * }}{\left( {\xi _j^ \pm } \right)^ * } = \xi _i^ \pm \]

we can derive:

    \[{\eta ^{\dagger i}}{\tilde \eta _j} = \frac{1}{2}{\delta ^i}{j^\nu }\]

where

    \[{\eta ^\dagger } = {\left( {{\eta ^ * }} \right)^T}\]

holds and

    \[\nu = {\eta ^{\dagger k}}{\tilde \eta _k}\]

is a complex function. One then obtains that the covariant derivative on the bilinear spinor takes the following form:

    \[\begin{array}{l}{\nabla _m}\left( {{\eta ^{\dagger i}}{\eta _j}} \right) = \frac{1}{2}{\delta ^i}_j\left( {{g_m}\nu + g_m^ * {\nu ^ * }} \right)\\ = {\delta ^i}_j\left( {{\eta ^{\dagger k}}{\gamma _m}\not \partial {D_{{\eta _k}}} + {\eta ^{\dagger k}}\not \partial D{\gamma _m}{\eta _k}} \right)\\ = 2{\delta ^i}_j{\partial _m}D{\eta ^{\dagger k}}{\eta _k}\end{array}\]

by solving, one can infer:

    \[{\eta ^{\dagger i}}{\eta _j} = \frac{1}{2}{e^{4\left( {D + {D_0}} \right)}}{\delta ^i}_j + {A^i}_j\]

{D_0} being a normalization constant and {A^i}_j is a constant traceless Hermitian matrix.

Thus, we can diagonalize on the SU(2) indices to yield:

    \[{A^i}_j = \frac{1}{2}{e^{4A}}{\left( {{\sigma _3}} \right)^i}_j\]

Now, I can renormalize the spinors by defining:

    \[\left\{ {\begin{array}{*{20}{c}}{{\eta _1} = {\alpha _ + }{\lambda _1}}\\{{\eta _2} = {\alpha _ - }{\lambda _2}}\\{{{\tilde \eta }_1} = {\alpha _ - }{{\tilde \lambda }_1}}\\{{{\tilde \eta }_2} = {\alpha _ + }{{\tilde \lambda }_2}}\end{array}} \right.\]

with

    \[\alpha _ \pm ^2 = \frac{1}{2}\left( {{e^{4\left( {D + {D_0}} \right)}} \pm {e^{4D}}} \right)\]

and the normalized spinors obey:

    \[{\lambda ^{\dagger i}}{\lambda _j} = {\delta ^i}_j\]

as well as

    \[{\lambda ^{\dagger i}}{\tilde \lambda _j} = {e^{i\varphi }}{\delta ^i}_j\]

with \varphi a y-dependent function. From the last two equalities above, it follows that:

    \[{\tilde \lambda _i} = {\varepsilon _{ij}}{B^{\left( 4 \right) * }}{\left( {{\lambda _j}} \right)^ * } = {e^{i\varphi }}{\lambda _i}\]

hence, the above supersymmetry conditions become:

    \[\begin{array}{l}\left( {{\nabla _m} - \frac{i}{2}{Q_m} + {\partial _m}{\rm{In}}{\alpha _ + }} \right){\lambda _1} = \\ - \frac{{{\alpha _ - }}}{{{\alpha _ + }}}{g_m}\,{e^{i\varphi }}{\lambda _1}\end{array}\]

and

    \[\begin{array}{l}\left( {{\nabla _m} - \frac{i}{2}{Q_m} + {\partial _m}{\rm{In}}{\alpha _ - }} \right){\lambda _2} = \\ - \frac{{{\alpha _ + }}}{{{\alpha _ - }}}{g_m}\,{e^{i\varphi }}{\lambda _2}\end{array}\]

with the following holding:

    \[\not \partial D{\lambda _1} = - \frac{1}{2}\frac{{{\alpha _ - }}}{{{\alpha _ + }}}\not \bar g\,{e^{i\varphi }}{\lambda _1}\]

    \[\not \partial D{\lambda _2} = - \frac{1}{2}\frac{{{\alpha _ + }}}{{{\alpha _ - }}}\not \bar g\,{e^{i\varphi }}{\lambda _2}\]

as well as:

    \[\not P{e^{i\varphi }}{\lambda _1} = \frac{{{\alpha _ + }}}{{{\alpha _ - }}}\not \bar g{\lambda _1}\]

    \[\not P{e^{i\varphi }}{\lambda _2} = \frac{{{\alpha _ - }}}{{{\alpha _ + }}}\not \bar g{\lambda _2}\]

which combined, yield the constraint:

    \[{\nabla _m}\left( {{\lambda ^{\dagger i}}{\lambda _j}} \right) = 0\]

giving us the following:

    \[{\nabla _m}{\lambda _1} = \left( { - \frac{i}{2}{{\not \partial }_m}\varphi + \frac{i}{2}{W_m}} \right){\lambda _1}\]

    \[{\nabla _m}{\lambda _2} = \left( { - \frac{i}{2}{{\not \partial }_m}\varphi + \frac{i}{2}{W_m}} \right){\lambda _2}\]

where {W_m} is:

    \[{W_m} = \frac{1}{{2i}}\left( {\frac{{{\alpha _ + }}}{{{\alpha _ - }}} - \frac{{{\alpha _ - }}}{{{\alpha _ + }}}} \right)\left( {{g_m}{e^{i\varphi }} - {g^ * }_m{e^{ - i\varphi }}} \right)\]

The complex structure of the four-manifold

Now consider

    \[{\left( {{J_A}} \right)_m}^n = \frac{i}{2}{\left( {{\sigma _A}} \right)^j}_i{\lambda ^{\dagger i}}{\gamma _m}^n{\lambda _j}\]

then, from

    \[{\tilde \lambda _i} = {\varepsilon _{ij}}{B^{\left( 4 \right)*}}{\left( {{\lambda _j}} \right)^*} = {e^{i\varphi }}{\lambda _i}\]

we can derive:

    \[{\left( {{J_A}} \right)_m}^k{\left( {{J_A}} \right)_n}^l{g_{kl}} = {g_{mn}},\;A = 1,2,3\]

Hence, I just defined an almost hyper-Kähler structure on the four-manifold

So, with respect to J, we have the (p, q)- forms:

    \[\left\{ {\begin{array}{*{20}{c}}{{\Omega _{mn}} = {{\left( {{J_2} + i{J_1}} \right)}_{mn}},\quad \left( {2,0} \right)}\\{{J_{mn}} = {{\left( {{J_3}} \right)}_{mn}},\quad \left( {1,1} \right)}\\{{{\bar \Omega }_{mn}} = {{\left( {{J_2} + i{J_1}} \right)}_{mn}},\quad \left( {0,2} \right)}\end{array}} \right.\]

The covariant derivatives of the Hermitian form, J, and the complex 2-form, \Omega are now:

    \[\left\{ {\begin{array}{*{20}{c}}{{\nabla _p}{J_{mn}} = 0}\\{{\nabla _p}{\Omega _{mn}} = - i{W_p}{\Omega _{mn}}}\end{array}} \right.\]

with

    \[{\Omega _{mn}} = - {\lambda ^{\dagger i}}{\gamma _{mn}}{\lambda _2}\]

 

Hence, it follows that the four-manifold is not only complex, but also Kähler, yet, the manifold is Calabi–Yau if and only if 

    \[{\nabla _p}{J_{mn}} = {\nabla _p}{\Omega _{mn}} = 0\]

and so we have a hyper-Kähler structure

The complex structure allows one to introduce holomorphic and anti-holomorphic coordinates a,b...;\;\bar a,\bar b,... and set:

    \[\left\{ {\begin{array}{*{20}{c}}{J_a^b = i\delta _a^b}\\{J_{\bar a}^{\bar b} = - i\delta _{\bar a}^{\bar b}}\end{array}} \right.\]

The Kähler form and the metric are related as:

    \[{J_{a\bar b}} = i{g_{a\bar b}}\]

entailing:

    \[\left\{ {\begin{array}{*{20}{c}}{{\gamma _a}{\lambda _1} = {\gamma ^{\bar a}}{\lambda _1} = 0}\\{{\gamma _{\bar a}}{\lambda _2} = {\gamma ^a}{\lambda _2} = 0}\end{array}} \right.\]

applied to:

    \[\not \partial D{\lambda _1} = - \frac{1}{2}\frac{{{\alpha _ - }}}{{{\alpha _ + }}}\not \bar g\,{e^{i\varphi }}{\lambda _1}\]

    \[\not \partial D{\lambda _2} = - \frac{1}{2}\frac{{{\alpha _ + }}}{{{\alpha _ - }}}\not \bar g\,{e^{i\varphi }}{\lambda _2}\]

as well as:

    \[\not P{e^{i\varphi }}{\lambda _1} = \frac{{{\alpha _ + }}}{{{\alpha _ - }}}\not \bar g{\lambda _1}\]

    \[\not P{e^{i\varphi }}{\lambda _2} = \frac{{{\alpha _ - }}}{{{\alpha _ + }}}\not \bar g{\lambda _2}\]

gives us:

    \[\left\{ {\begin{array}{*{20}{c}}{{g_a} = - 2\frac{{{\alpha _ + }}}{{{\alpha _ - }}}{e^{ - i\varphi }}{\partial _a}D}\\{{g_{\bar a}} = - 2\frac{{{\alpha _ - }}}{{{\alpha _ + }}}{e^{ - i\varphi }}{\partial _{\bar a}}D}\end{array}} \right.\]

as well as:

    \[\left\{ {\begin{array}{*{20}{c}}{{{\not P}_a} = - 2{{\left( {\frac{{{\alpha _ + }}}{{{\alpha _ - }}}} \right)}^2}{e^{ - 2i\varphi }}{\partial _a}D}\\{{{\not P}_{\bar a}} = - 2{{\left( {\frac{{{\alpha _ - }}}{{{\alpha _ + }}}} \right)}^2}{e^{ - 2i\varphi }}{\partial _{\bar a}}D}\end{array}} \right.\]

thus, we can infer:

    \[{g_a}{e^{i\varphi }} - {g^ * }_a{e^{ - i\varphi }} = - 2\left( {\frac{{{\alpha _ + }}}{{{\alpha _ - }}} - \frac{{{\alpha _ - }}}{{{\alpha _ + }}}} \right){\partial _a}D\]

Without loss of generality, consider now the class of solutions:

    \[\left\{ {\begin{array}{*{20}{c}}{{C_0} = 0}\\{{\mathop{\rm Im}\nolimits} \left[ {{g_m}{e^{i\varphi }}} \right] = 0}\end{array}} \right.\]

with {C_0} the R–R 1-form and the second condition entailing {\alpha _ + } = {\alpha _ - }. In such backgrounds, the following holds:

    \[\left\{ {\begin{array}{*{20}{c}}{{Q_m} = 0}\\{{{\not P}_m} = \frac{1}{2}{\partial _m}\,\phi }\end{array}} \right.\]

and since the sourceless Bianchi identity for {G_{mnp}} is:

    \[{\partial _{\left[ {_m{G_{npq}}} \right]}} = - {\not P_{\left[ {_mG_{npq}^ * } \right]}}\]

we can derive the following:

    \[\begin{array}{l}0 = {\left( { - \partial } \right._{\left[ {m\,\left( {{e^{ - 8D}}{{\left. {{\varepsilon _{npq}}} \right]}^r}} \right)} \right.}}\left. {{\partial _r}D} \right) = \\ - {\partial _{\left[ {m\,\left( {{e^{ - 8D}}{{\left. {{{\bar \varepsilon }_{npq}}} \right]}^r}} \right)} \right.}}\left. {{\partial _r}\bar D} \right) = \tilde \bigcirc 'D\end{array}\]

where the rescalling of the metric:

    \[{\bar g_{mn}} = {e^{ - 8D}}{g_{mn}}\]

gives us:

    \[{\tilde \varepsilon _{mnpq}}{\tilde g^{qr}} = {e^{ - 8D}}{\tilde \varepsilon _{mnpq}}{g^{qr}}\]

Hence, for:

    \[\left\{ {\begin{array}{*{20}{c}}{\varphi = 0}\\\pi \\{{{\tilde g}_{mn}}}\end{array}} \right.\]

we have a correspondence to the metric of the four-manifold in the string frame and this is the result for NS 5-branes

thus,

    \[\tilde \bigcirc 'D = * {\rho _5}\]

with {\rho _5} the density distribution of the 5-branes.

hence, type IIB solution are two copies of analogous six-dimensional heterotic backgrounds and the four-manifold must be conformally Calabi–Yau in the heterotic case.

Informal proof: 

take the heterotic dilatino and gravitino supersymmetry constraints in the string frame:

    \[\delta \lambda = {\Gamma ^M}{\partial _M}\phi \varepsilon - \frac{1}{6}{H_{MPQ}}{\Gamma ^{PQR}}\varepsilon = 0\]

    \[\delta {\psi _M} = {\nabla _M}\varepsilon - \frac{1}{4}{H_{MPQ}}{\Gamma ^{PQ}}\varepsilon = 0\]

 

Given that the string frame metric has the form:

    \[d{s^2} = {\eta _{\mu \nu }}d{x^\mu }d{x^\nu } + {g_{mn}}\left( y \right)d{y^m}d{y^n}\]

the supersymmetry transformations become:

    \[\delta \lambda = {\Gamma ^m}\left( {{\partial _m}\phi - {h_m}{\Gamma ^{\left( 4 \right)}}} \right)\varepsilon = 0\]

    \[\delta {\psi _m} = \left( {{\nabla _m} - \frac{1}{2}\Gamma _m^r{h_r}{\Gamma ^{\left( 4 \right)}}} \right)\varepsilon = 0\]

rescalling the internal metric as:

    \[{g_{mn}} = {e^{2\phi }}{g'_{mn}}\]

we get:

    \[{\nabla '_m}\varepsilon = 0\]

which is deep, as {g'_{mn}} is necessarily a Calabi–Yau metric and the internal four-manifold is conformally Calabi–Yau

previously

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Hodge Theory and Gromov-Witten Invariants of Calabi-Yau 3-Folds
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M-Theory and how the Quantum Worldsheet Knows 4-D Spacetime Physics
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