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String Field Theory, 4-D Space-Time and the Superstring

By Posted on In Grand Unification Physics, M Theory, Mathematical Physics, Mathematics, Philosophy Of Physics, Philosophy Of Science, Physics, Super String Theory, Witten Theory ,
String Field Theory

Let us see how deeply interconnected string field theory, 4-D space-time and the topological superstring are. In the context of topological sigma models with Calabi-Yau target space, the BRST action is:

    \[S = it\int_\Sigma {{d^2}} z\left\{ {\tilde Q',V} \right\} + t\int_\Sigma {{\Phi ^ * }} \left( {\hat K} \right)\]

with:

    \[V = {g_{i\bar j}}\left( {\psi _z^{\bar i}{{\not \partial }_{\bar z}}{\phi ^j} + {{\not \partial }_z}{\phi ^{\bar i}}\psi _{\bar z}^j} \right)\]

and

    \[\int_\Sigma {{\Phi ^*}} \left( {\hat K} \right)\]

is the integral of the pullback of the Kähler form \hat K, and is:

    \[\int_\Sigma {{\Phi ^*}} \left( {\hat K} \right) = \int_\Sigma {{d^2}} z\left( {{{\not \partial }_z}{\phi ^i}{{\not \partial }_{\bar z}}{\phi ^{\bar j}}{g_{i\bar j}} - {{\not \partial }_{\bar z}}{\phi ^i}{{\not \partial }_z}{\phi ^{\bar j}}{g_{i\bar j}}} \right)\]

In the A model, given that:

    \[\begin{array}{l}W = \int_\Sigma {\left( { - {\theta _i}\not D{\rho ^i} - \frac{i}{2}{R_{i\overline i j\overline j }}{\rho ^i}} \right.} \\\left. { \wedge {\rho ^j}{\eta ^{\overline i }}{\theta _k}{g^{k\overline j }}} \right)\end{array}\]

and:

    \[V = {g_{i\overline j }}\left( {\rho _z^i{{\not \partial }_z}{\phi ^{\bar j}} + \rho _{\bar z}^i{{\not \partial }_z}{\phi ^{\bar j}}} \right)\]

the action above can be expressed as:

    \[S = it\int {\left\{ {Q,V} \right\}} + tW\]

where Q is the BRST holomorphic operator. Since Edward Witten showed in ‘Chern-Simons Gauge Theory as a String Theory‘ that on-shell coupling between the open string and the closed string is zero, topologically, one need only work with open string field theory. Let us build it from the ground-up. In this post, I will only analyse the topological model A.

Let {\rm A} be the string fields with ghost number 1, with multiplication * ' and the BRST operator Q with ghost number 1 and {Q^2} = 0 with a functional \int {...} of ghost number -3 where the following 2 conditions hold, for fields {b_{\alpha \beta }}, {a_{\alpha \beta }}:

    \[\int {a * '} b = {\left( { - 1} \right)^{\deg \left( a \right)\deg \left( b \right)}}\int {b * 'a} \]

    \[\int {Qb = 0} \]

with:

    \[\deg \left( a \right) = gh{t_\# }\left( a \right)\]

The open string field theory action is hence:

    \[S = \frac{1}{2}\int {\left( {{\rm A} * 'Q{\rm A} + {\rm A} * '{\rm A} * '{\rm A}} \right)} \]

which is invariant under the following gauge tranformation:

    \[\delta {\rm A} = Q\varepsilon - \varepsilon * '{\rm A} + {\rm A} * '\varepsilon \]

Let me work with a topological sigma model with a Calabi-Yau target spaceX of dimension 6 as the world-sheet theory

Note, in string field theory, the Feynman diagram expansions generate all the possible \Sigma‘s, and the topological string theory consists of two parts: the instantons with target space X and boundary values in M such that

any neighborhood of M in X is equivalent topologically to a neighborhood of M in its cotangent bundle {T^ * }M

and the Chern-Simons theory with target space M.

Proposition:

the instantons mapping \Sigma to {T^ * }M and \partial \Sigma to M are constant

Since {T^ * }M is symplectic, we have:

    \[\omega = \sum {d{p_a}} \wedge d{q^a}\]

with {p_a} the coordinates in the fibers that vanish on M, thus we have:

    \[\left\{ {\begin{array}{*{20}{c}}{\omega = d\rho }\\{{g_{i\overline j }} = {g_{j\overline i }} = - {\omega _{i\overline j }}}\end{array}} \right.\]

and so

    \[\rho = \sum {{p_a}} d{q^a}\]

vanishes on M!

Now, an instanton is by definition a map

    \[\Phi :\Sigma \to X\]

with:

    \[\bar \partial {\phi ^i} = 0\]

and the bosonic part of the action is given as:

    \[\begin{array}{l}I = i\int_\Sigma {dz} \wedge d\bar z{g_{IJ}}{{\not \partial }_z}{\phi ^I}{{\not \partial }_{\bar z}}{\phi ^J} \cdot \\ = 2i\int_\Sigma {dz} \wedge d\bar z{g_{i\overline j }}{{\not \partial }_z}{\phi ^{\bar i}}{{\not \partial }_{\bar z}}{\phi ^j} - \\i\int_\Sigma {dz} \wedge d\bar z{g_{i\overline j }}\left( {{{\not \partial }_z}{\phi ^{\bar i}}{{\not \partial }_{\bar z}}{\phi ^j} - {{\not \partial }_z}{\phi ^j}{{\not \partial }_{\bar z}}{\phi ^{\bar i}}} \right)\\ = 2i\int_\Sigma {dz} \wedge d\bar z{g_{i\overline j }}{{\not \partial }_z}{\phi ^{\bar i}}{{\not \partial }_{\bar z}}{\phi ^j} + \\\int_\Sigma {{\Phi ^ * }} \left( \omega \right)\end{array}\]

where:

    \[I = i\int_\Sigma {dz} \wedge d\bar z{g_{IJ}}{{\not \partial }_z}{\phi ^I}{{\not \partial }_{\bar z}}{\phi ^J}\]

vanishes for the instantons and so:

    \[S = it\int_\Sigma {{d^2}} z\left\{ {\tilde Q',V} \right\} + t\int_\Sigma {{\Phi ^ * }} \left( {\hat K} \right)\]

reduces to:

    \[S = \frac{1}{2}\int {\left( {{\rm A} * 'Q{\rm A} + {\rm A} * '{\rm A} * '{\rm A}} \right)} \]

due to the following identity:

    \[\int_\Sigma {{\Phi ^ * }} \left( \omega \right) = \int_{\partial \Sigma } {{\Phi ^ * }} \left( \rho \right) = 0\]

Note that the low lying topological string modes are functions with the form {\rm A}\left( {{q^a},{\chi ^b}} \right) and the fields are linear in \chi, so the following expansion is valid:

    \[{\rm A} = {\chi ^a}{A_a}\left( q \right)\]

Hence, from

    \[\left\{ {\begin{array}{*{20}{c}}{\omega = d\rho }\\{{g_{i\overline j }} = {g_{j\overline i }} = - {\omega _{i\overline j }}}\end{array}} \right.\]

it follows that in the limit t → ∞, the first part of the string field action is

    \[\frac{1}{2}\int_M {{\rm{Tr}}} A \wedge dA\]

Let us move to cubic analysis of the action

With the gauge field mode {A^{\left( j \right)}} of A, the vertex operator is:

    \[{V^{\left( j \right)}} = {\chi ^a}A_a^{\left( j \right)}\left( q \right)\]

Given the superstring SL(2, R) symmetry, one need only factor in:

    \[\left\langle {{V^{\left( 1 \right)}}\left( 0 \right){V^{\left( 2 \right)}}\left( 1 \right){V^{\left( 3 \right)}}\left( \infty \right)} \right\rangle \]

then given that the pullback of the Kähler form \hat K is:

    \[\int_\Sigma {{\Phi ^*}} \left( {\hat K} \right)\]

the path integral reduces to an integral over zero modes in the large t limit, and we get the

the action for the Chern-Simons gauge theory

    \[\begin{array}{l}S = \frac{1}{2}\int_M {{\rm{Tr}}} \left( {A \wedge dA + \frac{2}{3}A \wedge A \wedge A} \right)\\ = \frac{1}{2}\int_M {{d^3}} q{\rm{Tr}}\left( {{\varepsilon ^{\mu \nu \lambda }}\left( {{A_\mu }{{\not \partial }_\nu }{A_\lambda }} \right) + {A_\mu }{A_\nu }{A_\lambda }} \right)\end{array}\]

In topological string field theory, the path integral reduction, unlike non-topological string theory, is exact, and thus via Feynman path summation we made contact with 4-D space-time physics

I will follow up with link-back on B-model analysis.

Untitled

Hardy’s comment now applies, and never truer, to M-theory!

previously

previously

M-Theory and how the Quantum Worldsheet Knows 4-D Spacetime Physics
up next

up next

The Superstring and the Duality of Topological Strings
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1 Response
  • The Superstring and the Duality of Topological Strings
    Wednesday, September 7, 2016

    […] will finish my analysis of the interconnectedness of string field theory, 4-D space-time and the topological superstring here and derive a crucial duality between topological strings. Aside: the book on the cover is a […]