2.5 Genus-one fiber with discrete symmetries
3.1.2 Coupling to gravity: Topological string theory
As a result we find the following commutation relations with the supersymmetry generators [MA,Q+]=−2Q+ [MB,Q+]=−2Q+
[MA,Q−]= 0 [MB,Q−]=2Q−
[MA,Q¯+]= 0 [MB,Q¯+]=0 [MA,Q¯−]= 2 ¯Q− [MB,Q¯−]=0.
(3.19)
We can see that for the Lorentz generatorMA the operatorQAhas vanishing commutator and is thus a scalar and forMBthe operatorQB. This allows the definition on arbitrarily curved manifolds and hence leads to two topological field theories.
leading to nontrivial correlation functions we have to contract them like in bosonic string theory with the ghosts which are in this case theQpartnersGof the energy-momentum tensorT [93]. We then find the following possible amplitudes with no insertions [98]
Fg= Z
Mg
*3g−3
Y
i=1
dmid ¯mi Z
Σg
Gzz(µi)z¯z Z
Σg
G¯z¯z( ¯µi)¯zz
+
, g≥2, (3.23)
wheremi are the 3(g−1) complex structure moduli we have to integrate over. Besides we find then point functions for a genusgworldsheet [98]
Cgi
1...in = Z
Mg
* n
Y
r=1
Z
Σg
Oir
3g−3
Y
i=1
dmid ¯mi Z
Σg
Gzz(µi)z¯z Z
Σg
G¯z¯z( ¯µi)¯zz
+
. (3.24)
Here a genus zero worldsheet requires at leastn=3 insertions and a genus-one worldsheet at leastn=1 insertion to fix the remaining continuous symmetries of the worldsheet.
For a CY threefold as target space geometry the free energy is even nonzero for all genera which makes this dimension also special in topological string theory (see [93] for more details). The free energiesFgcan then be combined to the generating function
F(t)=
∞
X
g=0
g2g−2s Fg(t), (3.25)
wheregsis the coupling constant of the topological string.
We have now constructed two possible topological string theories stemming from the two possible twists of the operator algebra. These theories are called the A-model and the B-model. In the following we want to have a closer look at these theories.
A-model
Let us first discuss the A-model topological string. After the A twist given in (3.18) the action of the field theory can be written as [93]
S =−it Z
Σg
d2z{QA,V}+2t Z
Σg
d2zgi¯j
∂zφi∂z¯φ¯¯j−∂z¯φi∂zφ¯¯j
. (3.26)
Heretis the coupling constant of the theory and theQAexact term is given by V=gi¯j
ψiz∂z¯φ¯¯j+∂zφiψz¯¯j
. (3.27)
For the fields we used the notation thatψiz = ψi+and ¯ψ¯iz¯ = ψ¯¯i−in terms of the notation of (3.7). The second part in (3.26) which is notQAexact can be reformulated as a pullback of the target space Kähler
formω=2igi¯jdzi∧d¯z¯j[93]
2t Z
Σg
d2zgi¯j
∂zφi∂¯zφ¯¯j−∂z¯φi∂zφ¯¯j
=t Z
Σg
φ∗(ω)
=t Z
φ(Σg)
ω
=tωβ ,
(3.28)
whereβ ∈ H2(X,Z) is the homology class ofφ(Σg). Since this term does not depend anymore on the fields or the metric it can be factored out of the path integral and just contributes a prefactor. The rest of the action isQA exact which leads up to the prefactor to aQAexact energy-momentum tensor and thus a topological theory.
Apart from that we notice that the left over path integral after factoring out the prefactorse−tωβ is also independent of t. This can be seen by taking the derivative with respect totwhich leads to the expectation value of dSdt0 which is also QA exact. Thus this expectation value vanishes and the path integral is independent oft. As usually in localization the integral can then be evaluated at any value for tespeciallyt → ∞. This then allows a classical treatment with theQA exact part of the action (3.26).
With (3.27) we then see from the classical equations of motion that [93]
∂z¯φi=∂zφ¯i =0. (3.29)
This means that the partition function localizes up to prefactors onto holomorphic mapsφ:Σg →Xto the target space. For the genusgfree energy (3.23) we find up to classical terms
Fg(t)= X
β∈H2(X,Z)
Ng,βqβ, qi=e−ti. (3.30)
Heretiare the complexified Kähler parameters andNg,βare the Gromov-Witten (GW) invariants which count the number of holomophic curves of genusgand homology classβ.
Furthermore we observe that the non QA exact part of the action (3.28) does not depend on the complex structure of the target space manifoldX. As discussed above for the example of atdependency the observables do not depend on quantities which only show up in V (3.27). Hence we find that the free energy does not depend on the complex structure moduli but only on the Kähler moduli.
B-model
In the case of the B-model we find for the action after the B twist in (3.18) [93]
S =−it Z
Σg
d2z{QB,V} −t Z
Σg
d2z iθi
Dz¯ρiz−Dzρiz¯ + 1
2Ri¯jk
lρizρk¯zη¯jθl
!
. (3.31)
HereDzdenotes the convariant derivative of the target spaceX,Ri jklis the target space Riemann tensor and we used the following notation for the fields in terms of the notation of (3.7)
η¯i=ψ¯i++ψ¯i−, θi =gi¯j
ψ¯+j −ψ¯−j ,
ρiz=ψi+, ρi¯z=ψi−. (3.32)
TheQBexact part of the action (3.31) is given by V =gi¯j
ρiz∂¯zφ¯j+ρi¯z∂zφ¯j
. (3.33)
Note that the nonQBexact part of the action (3.31) is antisymmetric inzand ¯z. This allows us to rewrite it as a (1,1) form which is independent of the metric after integrating over a Riemann surface. Thus we again see that the energy-momentum tensor isQBexact and the theory is topological [93].
To use again localization techniques on the path integral it must be independent of the coupling constantt. In the case of the A-model this was achived by factoring out the nonQA exact part of the action and using localization on the left over path integral. In the case of the B-model we cannot factor the nonQBexact part out since it still depends on the fields. Nevertheless we observe that only the non QBexact part depends on the fieldθi and that it does so only linear. Hence we can absorb the coupling constanttin the definition of the fieldθand make the nonQBexact parttindependent [93].
After this we can again apply localization and evaluate the path integral fort → ∞. The classical equations of motion of thetdependent part of the action are obtained using (3.33) [93]
∂zφi=∂¯zφi =∂zφ¯i =∂z¯φ¯i=0. (3.34) Hence we see that the path integral localizes to constant mapsφ:Σg →X. The corresponding moduli space we still have to integrate over is just the target space manifold itself M = X. This makes the evaluation of B-model correlation functions much easier than the evaluation of A-model correlation functions where we still have to integrate over a more complicated moduli space.
Since the operatorQBacts differently on the complex conjugated fieldsφiand ¯φ¯ithe theory explicitly depends on the choice of complex structure of the target spaceX. Additionally it can be shown that the theory does not depend on the Kähler parameters [93]. So in total we see that the dependence of the B-model on the moduli is opposite to the one of the A-model which only depends on the Kähler moduli.
Mirror symmetry reflects this by identifying the Kähler moduli space of a CY X with the complex structure moduli space of a mirror CYY. The observables calculated on the A-model topological string with target space CYXare the same as the observables calculated on the B-model side on the CYY.