Closed string mirror symmetry
2.2. Topological field theories
2.1.2. Chiral rings
There is an interesting ring structure coming with anN = 2 superconformal symmetry algebra [21]. For this let|φi be a primary state in the NS-sector which furthermore fulfills
chiral primary: G+−1/2|φi= 0 ,
anti-chiral primary: G−−1/2|φi= 0 .
(2.5)
By the superconformal algebra one can show that the space of chiral fields is finite dimensional as for such a state [21]
hφ≤ c
6, and hφ=
qφ
2 . (2.6)
Under the OPE the chiral fields form a closed finite ring, called the chiral ring [21]
φi·φj =Cijk φk , (2.7)
wherei, j, k label the finite number of different fields. The structure constantsCk
ij are related
to the three-point functionsCijk of three chiral primaries on a genus zero worldsheet
Cijk ≡ hφiφjφki0 =hφiφmi0Cjkm ≡ηimCjkm . (2.8)
Here the two-point function hφiφmi0 = ηim is called the topological metric and becomes
important once one considers deformations of a given theory.
As theU(1)-charge is additive the chiral ring comes with a charge filtration [21, 35]. This becomes also apparent when looking at the operator-state correspondence. By spectral flow the R- and the NS-sector can be mapped to each other and the chiral primaries are mapped to the supersymmetric ground states in the R-sector [21]. There is a canonical vacuum |0i from which one can construct the other vacua by the action of the chiral primaries φi [35]3
|ii=φi|0i . (2.9)
Therefore the states have a definite U(1)-charge. The same steps can also be carried out for the anti-chiral fields. If one considers a theory of closed strings one has a left- and a right moving N = 2 superconformal algebra. This will lead to combined chiral rings denoted by (c, c) and (a, c), where c denotes the chiral- andathe anti-chiral ring.4
2.2. Topological field theories
Up to now implicitly only flat worldsheets were considered. To define the theory on a closed worldsheet one has to make sure that there exist covariantly constant sections of the spin
3
Strictly speaking this is not a one-to-one map in general. However for the concrete models we will be interested in (theA-model on a Calabi-Yau manifold andB-twisted LG-models) this statement is true [35].
4
bundle. For higher genus surfaces this is however not guaranteed. One way to solve this problem is to construct a new theory out of the original one by the so-called topological twist procedure [18, 19, 20].5
The idea is to combine the local Lorentz/euclidean symmetry with the R-symmetries of the worldsheet supersymmetry and to declare a diagonal subgroup to be the new local Lorentz/euclidean symmetry group. This changes the bundles the original fields were sec- tions of and thus their spins. For the supercharges this procedure generically leads to some fermionic scalar charges and one-form fields. Thus one obtains scalar chargesQwhich square to zero and lead in particular to the simplifications one knows from supersymmetric theories also for higher genus worldsheets.
2.2.1. Properties of topological field theories
A topological field theory (TFT) obtained from twisting has some very remarkable properties:
• The operators and states of the topological and the superconformal theory do not differ. However the notion of a physical state is changed. In the topological theory the Q- cohomology classes are the physical states and they correspond to the ground states of the supersymmetric theory.
• The energy-momentum tensor T = {Q, G} is Q-exact. As a consequence the correla- tion functions of physical operators do not depend on the metric on the worldsheet. Furthermore the correlation functions are independent of the position of the inserted operators.
• The semi-classical approximation is exact. If the action of the theory is furthermore
Q-exact the correlation functions get contributions only from the fixed-points of the
Q-symmetry. This important property is called localization.
• The correlation functions are independent of a large class of deformations of the under- lying QFT. For example they generically do not depend on D-term deformations. Naive arguments leading to these kind of statements can however have important subtleties, as in the case of the holomorphic anomaly equation of topological string theory at higher genus [64, 22].
In addition a topological theory obtained by twisting anN = 2 superconformal theory has vanishing central charge and is thus automatically conformal at the quantum level. In contrast to physical string models the quantum consistency of the model therefore does not single out a particular space-time dimension anymore. However for the special case of topological non-linear σ-models on Calabi-Yau spaces the correlation functions are non-vanishing only if the ghost number anomaly is cancelled [18, 19, 20] and this condition singles out the
5
2.2 Topological field theories 21
complex dimensions three and four as particular interesting dimensions for target spaces of the topological string.6
2.2.2. Deformations of topological theories
Deformations of topological theories can be constructed via the so-called descent procedure. This procedure is analogous to the construction of marginal operators from chiral fields of charge (q,q¯) = (1,1) in the superconformal theory.7
By the topological twist some of the supercharges become scalars, while the others become one-form operators. Now given a Q-closed operatorφ(0) one can find one-form and two-form operatorsφ(1) and φ(2) such that [20]8
0 = [Q, φ(0)], dφ(0)={Q, φ(1)},
dφ(1) = [Q, φ(2)], dφ(2)= 0 . (2.10)
These equations are called descent equations and can be formulated in any dimension. In- tegrating φ(1) over a curve γ and φ(2) over the whole worldsheet Σ one can construct new non-local topological observables by virtue of the descent equations (2.10). The second type of operator with the appropriate U(1)-charges can be used to deform the action
S(t,¯t) =S0+ti Z Σ φ(2)i + ¯t¯j Z Σ ¯ φ(2)¯j , (2.11)
where we included for convenience the deformations of the anti-topological theory obtained by twisting for example the (a, a)-ring. The indices i,¯j = 1, . . . , N run over all physical operators with U(1)-charges (q,q¯) = (1,1). The parameters (ti,¯t¯j) are local coordinates on the deformation space Mof the theory, which is called the moduli space.9
The deformations can be used to construct perturbed correlation functions. For example the three-point correlator in the deformed theory can be written as
Cijk(t) =hφiφjφk exp tn Z Σ φ(2)n i0 . (2.12)
Although this expression looks asymmetric with respect to its insertions it can be shown to fulfill an integrability condition [77]
∂iCjkl=∂jCikl (2.13)
6At least if the theory is coupled to worldsheet gravity and one considers also higher genus worldsheets [78].
For the ghost number anomaly without worldsheet gravity see also section 2.3.
7
See e.g. [74].
8
See also e.g. [76] for a pedagogical account.
9For non-linearσ-models on Calabi-Yau target spaces, in which we will be mainly interested in the following,
this deformation space is classically unobstructed and one therefore obtains a true classical moduli space, see also section 2.4.1.
and thus there exists at least locally a functionF(t) such that
Cijk(t) =∂i∂j∂kF(t). (2.14)
The functionF(t) is called the prepotential and can be considered as the generating function of tree-level correlators of the topological field theory. The integrability condition furthermore leads to the identity10
∂iηmn=∂iC0mn =∂0Cimn = 0 , (2.15)
which shows that in the flat TFT coordinates (ti,t¯¯j) in (2.11) the topological metric is locally constant. In addition it follows from crossing-symmetry that the deformed chiral ring is associative
CijnCnkl =CiknCnjl . (2.16)
These equations are called the WDVV-equations [82, 83] and they yield infinitely many condi- tions on the expansion coefficients of the prepotentialF(t) and therefore between the various
n-point correlation functions. For open topological strings there exist analogous conditions, which lead to the very rich mathematical structure of A∞-categories [84, 85].
The above conditions (2.14)-(2.16) abstractly define what is called a Frobenius algebra [86, 77]. This is a space Mtogether with a vector bundleV, which is equipped with a locally constant bilinear formη. Furthermore on each fiber there is an associative multiplication with totally symmetric structure constantsCijk, which can be integrated to a generating function F(t). From this viewpoint a two-dimensional topological field theory can be considered as a particular representation of a Frobenius algebra [86, 77].
2.2.3. The vacuum bundle and tt∗-geometry
The physical operators of the twisted topological theory are in one-to-one correspondence to the RR ground states of the original N = (2,2) theory11 and furthermore by spectral flow to the chiral primaries of the NSNS-sector. When considering deformations of the theory the space of RR ground states, denoted by H(t), gets fibered over the moduli space Mand one obtains a bundle V of ground states
H ⊃H(t) −→ V ↓ M
(2.17)
Here we denoted by Hthe full Hilbert-space of the superconformal theory in which, in view of (2.9), the subspace H(t) sits. The bundleV is called the vacuum bundle of the theory and
10Note thatφ
0 denotes the identity operator. 11
2.2 Topological field theories 23
it can be written as a direct sum of sub-bundlesVq with definite U(1)-charges
V =⊕qVq . (2.18)
There is an interesting catch in the above [87]. As one varies the parameters of the theory the number of ground states is known to stay constant, however the chiral ring (2.7) changes rather drastic, e.g. the structure constants Cijk(t) will depend on t. This means that the variation of the ground states under a change of parameters should encode the structure of the chiral ring. This turns out to be true and the dependence of the ground states on the deformation parameters is captured by the so-calledtt∗-equations [87, 35], which we describe in the following.
For this let|¯iidenote the ground states of the anti-topological theory obtained from acting with the anti-chiral fields ¯φ¯i on the canonical ground state |0i. Using the topological and
anti-topological theory one can define thett∗-metric [87]
g¯ij =h¯i|ji . (2.19)
The tt∗-metric induces a connection on the bundle V which is compatible with the natural complex structure of M. There exists a basis, called the holomorphic or topological basis, which makes this explicit. In this basis the connectionDi= 11∂i−Ai is given by
(Ai)ab =ga¯c∂igb¯c, (A¯i)ab = 0 . (2.20)
The structure constants Ci of the chiral ring and the tt∗-connection are related by the tt∗- equations [87] [Di, Dj] = 0, [ ¯D¯i,D¯¯j] = 0, [Di,C¯¯j] = 0, [ ¯D¯i, Cj] = 0, [Di, Cj] = [Dj, Ci], [ ¯D¯i,C¯¯j] = [ ¯D¯j,C¯¯i], [Di,D¯¯j] =−[Ci,C¯¯j], (2.21)
where [Di,D¯¯j] =∂iA¯¯j−∂¯jAi−[Ai,A¯¯j]. Thett∗-equations imply that the improved connection
∇i =Di−Ci is flat [87]
[∇i,∇j] = [∇i,∇¯j] = [∇¯i,∇¯j] = 0. (2.22)
In a certain non-linear σ-model realization of the above structures, the so-called B-model which we will consider in a moment, the improved connection can be given a geometrical interpretation and is in this context known as the Gauss-Manin connection.
2.2.4. Geometry of the moduli space
There exists a preferred basis of sections of the bundleV which can be used to define a metric on TM. For this one has to remember that the elements φa, for a = 1, . . . , N with U(1)
charge qφa = 1, can be used to deform the theory. Thus these elements can be considered
as sections of TM. Denoting the operator which generates the canonical ground state of charge zero by φ0 one can define a basis of the chiral (sub-)ring by taking the fields φ0 and
φa and furthermore their duals with respect to either the topological or thett∗-metric.12 We
therefore obtain two distinct splits of the vacuum bundle. For the case ofc= 9, corresponding to Calabi-Yau three-fold, the vacuum bundle can be split for example in a non-holomorphic way as
VC =L ⊕(L ⊗TM)⊕(L ⊗TM)∗⊕ L∗ , (2.23)
where the ground state is a section of the line bundleL and∗ denotes the dual space. Using instead the topological metric leads to a holomorphic split of the bundle and the basis elements can be constructed by successive OPE-multiplication of the operatorsφ0, φa. Restricting the
tt∗-metric to the directions tangent toMone obtains the Zamolodchikov metric, denoted by
Ga¯b and given by
Ga¯b=
ga¯b
g0¯0 . (2.24)
Using the tt∗-equations one can furthermore show that [87]
Ga¯b =−∂a∂¯blng0¯0
and thus the Zamolodchikov metric is K¨ahler with K¨ahler potential given by K =−lng0¯0. The tt∗-equations also imply a certain relation for the curvature of the metric Ga¯b, see e.g. [22, 35]
Ra¯bcd≡ −∂¯¯bΓacd =Gc¯bδda+Gab¯δdc−e2KCacnGn¯nC¯¯bm¯¯nGmd¯ . (2.25)
This condition together with the K¨ahler property turn the manifoldMinto a special K¨ahler manifold and the geometry of such spaces is called special geometry [50, 22].