The String-String Duality, K3 Geometry, and Dimensional Reduction

This post is on the String-String Duality. In particular, the D=6 string-string duality, which is crucial since it allows interchanging the roles of 4-D spacetime and string-world-sheet loop expansion, and this is mathematically essential for phenomenologically adequate string-compactifications. Here I will prove an equivalence between K3 membrane action and {T^3} \times {S^1}/{Z^2} orbifold action and show how it entails D=6 string-string duality. Working in the bosonic sector, the membrane action is:

    \[\begin{array}{l}S = {S_M} + \int_{\partial {M^3}} {\left\{ {\frac{1}{2}} \right.} \left( {{g_{mn}}{\eta ^{ij}} + {b_{mn}}{\varepsilon ^{ij}}} \right)\\{{\not \partial }_i}{x^m}{{\not \partial }_j}{x^n} + \frac{1}{2}\left( {{g_{IJ}}{\eta ^{ij}} + {b_{IJ}}{\varepsilon ^{ij}}} \right)\\{{\not \partial }_i}{x^I}{{\not \partial }_j}{x^J} + {\varepsilon ^{ij}}{{\not \partial }_i}{x^J}{{\not \partial }_j}{x^m}\left. {A_m^J(x)} \right\}\end{array}\]


    \[\begin{array}{l}{S_M} = \int_{{M^3}} {\left( {\sqrt { - {g_{mn}}{{\not \partial }_i}{x^m}{{\not \partial }_j}{x^n}} } \right.} + \\\frac{1}{6}{\varepsilon ^{ijk}}{{\not \partial }_i}{x^m}{{\not \partial }_j}{x^n}{{\not \partial }_k}{x^p}\left. {{B_{mnp}}} \right)\end{array}\]

Recall I derived, via Clifford algebraic symmetry, the total action:

    \[\begin{array}{l}{S^{Total}} = \frac{1}{{2\pi {\alpha ^\dagger }12}}\int\limits_{{\rm{world - volumes}}} {{d^{26}}} x\,d\,\Omega {\left( {{\phi _{INST}}} \right)^2}\sqrt {\frac{{ - {g_{\mu \nu }}}}{{ - \gamma }}} \,{e^{ - {c_{2n}}/{\Upsilon _\kappa }(\cos \varphi )}} \cdot \\\left( {{R_{icci}} - 4{{\left( {{{\not D}^{SuSy}}\left( {{\phi _{INST}}} \right)} \right)}^2}} \right) + \frac{1}{{12}}H_{3,\mu \nu \lambda }^bH_3^{b,\mu \nu \lambda }/A_\mu ^H + \sum\limits_{D - p - branes} {S_{Dp}^{WV}} \end{array}\]

which is deep since Clifford algebras are a quantization of exterior algebras, and applying to the ‘Einstein-Minkowski’ tangent bundle, we get via Gaussian matrix elimination, an expansion of {\not D^{SuSy}} via Green-functions, that yields M-Theory’s action:

    \[{S_M} = \frac{1}{{{k^9}}}\int\limits_{{\rm{world - volumes}}} {{d^{11}}} \sqrt {\frac{{ - {g_{\mu \nu }}}}{{ - \gamma }}} {T_p}^{10}{\mkern 1mu} d{\mkern 1mu} \Omega {\left( {{\phi _{INST}}} \right)^{26}}\left( {{R_{icci}} - A_\mu ^H\frac{1}{{48}}G_4^2} \right) + \sum\limits_{Dp} {\not D_\mu ^{SuSy}} {e^{ - H_3^b}}/S_{Dp}^{WV} + \sum\limits_{Dp} {\not D_\nu ^{SuSy}} {e^{H_3^b}}/S_{Dp}^{SV}\]

with k the kappa symmetry term. With {g_{mn}} the metric on {M^{11}}, and {x^m} the corresponding coordinates with {B_{mnp}} an antisymmetric 3-tensor. The worldvolume {M^3} is:

    \[R \times {S^1} \times {S^1}/{Z_2}\]

The bosonic sector lives on the boundary of the open membrane: two copies of R \times {S^1}, which naturally couple to the U(1) connections {A^J}.

Now, double dimensional reduction of the twisted supermembrane on:

    \[{M^{10}} \times {S^1}/{Z_2}\]


    \[\begin{array}{l}S = {S_M} + \int_{\partial {M^3}} {\left\{ {\frac{1}{2}} \right.} \left( {{g_{mn}}{\eta ^{ij}} + {b_{mn}}{\varepsilon ^{ij}}} \right)\\{{\not \partial }_i}{x^m}{{\not \partial }_j}{x^n} + \frac{1}{2}\left( {{g_{IJ}}{\eta ^{ij}} + {b_{IJ}}{\varepsilon ^{ij}}} \right)\\{{\not \partial }_i}{x^I}{{\not \partial }_j}{x^J} + {\varepsilon ^{ij}}{{\not \partial }_i}{x^J}{{\not \partial }_j}{x^m}\left. {A_m^J(x)} \right\}\end{array}\]

entails that the bosonic sector is that of the heterotic string:

    \[\begin{array}{l}{S_h}\int {{d^2}} \sigma \left\{ {\frac{1}{2}} \right.\left( {{g_{mn}}{\eta ^{ij}} + {b_{mn}}{\varepsilon ^{ij}}} \right){{\not \partial }_i}{x^m}{{\not \partial }_j}{x^n}\\ + \frac{1}{2}\left( {{g_{IJ}}{\eta ^{ij}} + {b_{IJ}}{\varepsilon ^{ij}}} \right){{\not \partial }_i}{x^I}{{\not \partial }_j}{x^I} + \\{\varepsilon ^{ij}}{{\not \partial }_i}{x^I}{{\not \partial }_n}{x^m}\left. {A_m^{(I)}(x)} \right\}\end{array}\]

with gauge group indices I = 1, …, 16.

It gets interesting when we consider:

    \[{M^{10}} = {T^3}{\rm{ }} \times {\rm{ }}{M^7}\]

with dimension:

    \[dim{\rm{ }}{H^1}\left( {{M^7}} \right) = 0\]

since the worldsheet action:

    \[{S_{het}} = {S_{st}} + {S_{KK}} + {S_{\bmod }}\]

is now just a sum of three terms:

    \[{S_{st}} = \int {{d^2}} \sigma \frac{1}{2}\left( {{g_{mn}}{\eta ^{ij}} + {b_{mn}}{\varepsilon ^{ij}}} \right){\not \partial _i}{x^m}{\not \partial _j}{x^n}\]

    \[{S_{KK}}\int {{d^2}} \sigma {\varepsilon ^{ij}}{\not \partial _i}{x^I}{\not \partial _j}{x^m}A_m^I\]

    \[{S_{\bmod }} = \int {{d^2}} \sigma \frac{1}{2}\left( {{g_{IJ}}{\eta ^{ij}} + {b_{IJ}}{\varepsilon ^{ij}}} \right){\not \partial _i}{x^J}{\not \partial _j}{x^I}\]

and the index I = 1, …, 22 labels 22 gauge fields: 16 coming from the internal dimensions of the heterotic string, and the other 6 gauge fields are the KK modes of the metric and antisymmetric tensor. The action {S_{\bmod }} has a massless spectrum given by moduli fields corresponding to deformations of the Narain lattice and thus take values in the group manifold:

    \[\frac{{SO\left( {19,3} \right)}}{{SO\left( {19} \right) \times SO\left( 3 \right)}}\]

Now, something fundamentally deep has occurred: all the gauge fields of the action {S_{het}} have appeared within a two-dimensional theory, and not a three-dimensional theory

This is precisely the long wavelength limit behavior of the open membrane:

the gauge fields are defined in terms of fields which live on 10-dimensional boundaries of M-theory

In the closed membrane case:

the gauge fields are defined in terms of 11-dimensional fields

Hence, the gauge fields of the closed membrane to be defined over M3 and not over its boundary, unlike the closed membrane, whose action on K3 \times {M^7} is:

    \[\begin{array}{l}{{S'}_M} = \int_{{M^3}} {{d^3}} \zeta \left( {\sqrt { - {g_{mn}}{{\not \partial }_i}{x^m}{{\not \partial }_j}{x^n}} } \right.\\ + \frac{1}{6}{\varepsilon ^{ijk}}{{\not \partial }_i}{x^m}{{\not \partial }_j}{x^n}\left. {{{\not \partial }_k}{x^p}{B_{mnp}}} \right)\end{array}\]

where {M^3} is {T^2} \times R with the spacetime being {M^7} \times K3.

Hence, the closed membrane action {S'_M} on {M^7} \times K3 reduces to:

    \[{S'_M} = {S'_{st}} + {S'_{KK}} + {S'_{\bmod }}\]


    \[{S'_{st}} = \int {{d^3}} \sigma \sqrt { - {g_{mn}}{{\not \partial }_i}{x^m}{{\not \partial }_j}{x^n}} + \frac{1}{6}{B_{mnp}}{\varepsilon ^{ijk}}{\not \partial _i}{x^m}{\not \partial _j}{x^n}{\not \partial _k}{x^p}\]

    \[{S'_{KK}} = \frac{1}{6}\int {{d^3}} \sigma {\varepsilon ^{ijk}}{\not \partial _i}{x^a}{\not \partial _j}{x^b}{\not \partial _k}{x^m}{B_{abm}}\]


    \[{S'_{\bmod }} = \int {{d^3}} \sigma \sqrt { - {g_{ab}}{{\not \partial }_i}{x^a}{{\not \partial }_j}{x^b}} + \frac{1}{6}{\varepsilon ^{ijk}}{\not \partial _i}{x^a}{\not \partial _j}{x^b}{\not \partial _k}{x^c}{B_{abc}}\]

and since K3 surfaces have no one-cycles, it follows that the three-form potential that appears in {S'_{KK}} of the action:

    \[{S'_{\bmod }} = \int {{d^3}} \sigma \sqrt { - {g_{ab}}{{\not \partial }_i}{x^a}{{\not \partial }_j}{x^b}} + \frac{1}{6}{\varepsilon ^{ijk}}{\not \partial _i}{x^a}{\not \partial _j}{x^b}{\not \partial _k}{x^c}{B_{abc}}\]

can be expanded in terms of the cocycles of K3.

For the 22 2-cocycles of K3, one can decompose B in a similar way for the two-form potential:

    \[{B_{abm}} = b_{ab}^I\left( {{x^a}} \right)C_m^I\left( {{x^r}} \right)\]

with I = 1, …, 22 labeling the two-cycles of K3. So after insertion into {S'_{KK}}, we can derive:

    \[\int_{{M^3}} {{\varepsilon ^{ijk}}} {\not \partial _i}{x^m}{\not \partial _j}{x^b}{\not \partial _k}{x^a}b_{ab}^I\left( {{x^c}} \right)C_m^I\left( {{x^r}} \right)\]

Applying reparametrization invariance, one can set:

    \[\rho = {x^{11}}\]

where \rho is a worldvolume coordinate, and now one performs a dimensional reduction of:

    \[\int_{{M^3}} {{\varepsilon ^{ijk}}} {\not \partial _i}{x^m}{\not \partial _j}{x^b}{\not \partial _k}{x^a}b_{ab}^I\left( {{x^c}} \right)C_m^I\left( {{x^r}} \right)\]

Here are the key propositions relevant to the membrane/string duality of the low energy theory in D=7.

  • the kinetic terms for the gauge fields in D=7 supergravity are:

    \[\int_{{M^7}} {\sqrt { - {g^{\left( 7 \right)}}} } {a_{IJ}}F_{mn}^I{F^{Jmn}}\]

derived by a split of the 4-4 field strength H = dB, of the 11-dimensional supergravity action:

    \[{H_{abmn}} = b_{ab}^IF_{mn}^I\]

from the the following term:

    \[\begin{array}{l}\int_{{M^{11}}} {\sqrt { - {g^{\left( {11} \right)}}} } {H^2} = \int_{{M^7}} {\sqrt { - {g^{\left( 7 \right)}}} } F_{mn}^I{F^{Jmn}}\\\int_{K3} {\sqrt { - {g^{\left( {K3} \right)}}} } b_{ab}^I{b^{Jab}}\end{array}\]

  • Membrane/string duality in D=7 requires the existence of a point in the moduli space of K3 where all the 22 gauge fields are enhanced via