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Topological Field Theory
In this and the next chapter, our goal is to partially answer the question: what is topological string theory? In order to achieve that, we need to first introduce the necessary theoretical backgrounds. Section 2.1 is a quick summary of two dimensional N = (2, 2) supersymmetry which is the correct language to describe actions of the string worldsheet. Section 2.2 quickly introduces the notion of a topological field theory (TQFT) and in particular, of Witten type. In section 2.3, we first present a powerful way to produce TQFTs, known as topological twisting, then apply it to the low energy effective theory of the worldsheet. It turns out that there are two possibilities to do the twisting which are both of importance, so we devote subsections 2.3.1 and 2.3.2 to each of them in turn. Section 2.4 serves as an interlude, introducing some basics of moduli space of complex structures in order to answer a question raised at the end of subsection 2.3.2. Useful references for this chapter include [103, 174, 145, 136, 183, 76].
Two dimensional N = (2, 2) supersymmetry
Considering superstrings propagating inside a target space leads to the notion of super-symmetric nonlinear sigma model mapping a two dimensional worldsheet to a manifold X. In this thesis we consider models having N = (2, 2) supersymmetry. Let’s first introduce some rudiments of two dimensional N = (2, 2) supersymmetry, following closely chapter 12 of [103].
We start by constructing the supersymmetry algebra. It necessarily contains the Poincar´e algebra, H, P, M, (2.1.1) corresponding to the Hamiltonian, momentum and angular momentum respectively. More-over, supersymmetry gives us fermionic Noether charges. As the name suggests, we have four supercharges: Q+, Q−, Q+, Q− , (2.1.2) where the subscript + or − means left or right chiralities and the bar means complex conjugation. Since the supersymmetry is extended, we naturally have R-symmetries acting on supercharges. Here, we have two U(1) R-symmetries, whose Noether charges are denoted by FL and FR. For convenience, we recombine them into vector and axial R-symmetries, where we only integrate over half of the four θ: d2θ = dθ−dθ+. Thanks to the constraints (2.1.18), it’s easy to verify that the Lagrangian is still supersymmetric. The W (Φi) is also called a superpotential. In general, the total Lagrangian is a sum of two terms Eqs. (2.1.21) and (2.1.22).
Next, we would like to know if the two U(1) R-symmetries are preserved or not. Let’s start our discussion from the classical level. Since their actions (2.1.16) only change the overall phase hence do not mix the D-term and F-term, we can treat Eqs. (2.1.21) and (2.1.22) separately.
First let’s discuss the D-term. Since θ4 is invariant under both R rotations, Skin is ¯¯ invariant under both both symmetries if both charges of the K¨ahler potential K(Φi, Φi) vanish. Often this is possible just by demanding the qV and qA of Φi to be zero.
However, if we look at the F-term, the situation is different. Because θ2 has U(1)A R-charge 0, setting qA of Φi to zero is still consistent. However, θ2 has U(1)V R-charge 2, the superpotential must have qV = 2 if we want to preserve the U(1)V symmetry. For general W this is impossible to achieve2.
To summarize, the lesson we draw is as follows. Classically, U(1)A is always a good symmetry by assigning qA to zero for all the chiral fields Φi. On the other hand, U(1)V is a symmetry only for very special types of W .
This finishes our discussion at the classical level. What could go wrong in the quantum world? At the quantum level, it’s a well-known fact that a chiral symmetry could be anomalous. A detailed analysis requires a full chapter of its own3, so here we only quote the final result in chapter 13 of [103]: to make sure that the U(1)A is not anomalous, the target space must obey c1(X) = 0 . (2.1.23)
In other words, X must be a Calabi-Yau manifold.
Another useful subclass of superfields is the twisted chiral superfield U, satisfying D+U = D−U = 0. (2.1.24)
We can repeat the whole story, introducing the K¨ahler and superpotential and writing down the D-term and F-term. The only difference is that in the latter case, we integrate 2˜ ¯− + 2 over d θ = dθ dθ instead of d θ.
Topological field theories of Witten type
There exist many definitions of a topological field theory (TQFT). In the mathematics literature, it was M. Atiyah who first axiomatized the topological field theories [14], inspired by works in two dimensional conformal field theories. Being physicists, we prefer to use less rigorous but more physical definitions, which are given below.
Let’s first set the stage. We put our physical system on a manifold X with a given metric g, which is usually not flat. In physics, we are interested in partition function Z, physical operators Oi as well as various correlation functions hO1(x1) • • • On(xn)ig , (2.2.1) where we use the subscript g to emphasis that they are computed in that given metric. A TQFT simply means that all these quantities are independent of g therefore should be topological.
In the physics literature, there are roughly speaking two types of TQFTs: one is known as the Schwarz type, where there is no explicit dependence on g in the Lagrangian and physical operators. Therefore, it’s natural to expect that the theory should be topological. Examples of this sort include, e.g., Chern-Simons theory in three dimensions [182].
In this chapter, we are interested in the second kind of TQFT, known as the Witten type or cohomological type. In this set-up, the Lagrangian and operators do depend explicitly on the metric, but as soon as we pass into the cohomology, the g dependence drops out.
Formally speaking, a TQFT of Witten type has a special fermionic symmetry, whose Noether charge is denoted as Q. The anti-commutator of Q with operators generates the symmetry transformation δOi = i {Q, Oi} . (2.2.2)
The first condition we impose is the following: Q2 = 0 . (2.2.3)
This may look strange at the first sight, but for readers familiar with algebraic topology, this hints at the possibility of defining cohomology. We also assume that the vacuum |vaci of our system is invariant under the symmetry hence annihilated by Q.
The reason why it is also called a cohomological TQFT is clear: since the operator Q squares to zero, an observable which is Q of another observable (Q-exact) is also annihilated by Q (Q-closed). Furthermore, repeating the argument for the energy-momentum tensor, it is easy to verify that the correlation function vanishes if any of the operators is Q-exact. Thus, as far as correlation functions are concerned, the physical operators are in one-to-one correspondence with the elements in the Q-cohomology, HQ = {Q − closed operators} . (2.2.8) {Q − exact operators} Up to now, we only discussed the definitions and some consequences of a TQFT. Beau-tiful as it is, we still haven’t shown how to construct a TQFT in practice. Below we will introduce the powerful topological twisting method first introduced in [180] and show two possible ways to construct a TQFT out of an N = (2, 2) supersymmetric non-linear sigma model [181, 171].
Topological twisting
Before discussing topological properties, there are in fact issues with the non-linear sigma model itself when placed on a curved Riemann surface Σ, i.e., the Lagrangian is not nec-essarily supersymmetric. Under supersymmetry transformation, the D–term action gives a total differential, which integrate to zero on a flat space, but may not integrate to zero on a curved Σ. Another way to look at this problem is to write down the variation of the action under a supersymmetric transformation (2.1) δS = (rµ +Gµ− − rµ −Gµ+ − rµ +Gµ− + rµ −Gµ+) h d2x . (2.3.1)
Here ± and ± are the variational parameters that are spinors on Σ. If Σ is flat, they can be chosen to be constant spinors the above equation tells us that the Lagrangian is su-persymmetric. However, for a curved Σ, covariantly constant spinors (satisfying equations rµ ± = rµ ± = 0) may simply not exist4! In other words, we can still formulate a the-ory with equal amount of bosonic and fermionic degrees of freedom on a curved Riemann surface, but the supersymmetric invariance of the action of may no longer be preserved.
Still, we would like to preserve a fermionic symmetry on Σ, out of the original su-persymmetry. One possible solution is topological twisting. Naively, if we can somehow change the spinor to the scalar, then it’s possible to find at least one non-trivial covariantly constant solution, namely the constant hence preserve the modified symmetry. Motivated by this observation, our next goal is to learn how to change a spinor into a scalar by the twisting procedure5.
From now on, we consider the Euclidean version of the theory obtained by performing the Wick rotation x0 = −ix2. We also define complex coordinate z = x1 + ix2. Then the 2d Lorentz group becomes the Euclidean rotation group SO(2)E = U(1)E with the generator ME = iM . (2.3.2)
Another useful point of view is to regard the twisting as modifying the Lagrangian by adding either the U(1)V or the U(1)A R-symmetry current into the spin connection, as elucidated in chapter 3 of [145].
Whichever way we choose, the consequence of topological twisting is to change the flavor index to the spinor index, thus change the spin of various fields. For example, if we consider a chiral superfield Φ whose R-charges are both trivial Φ=φ+θ+ψ++θ−ψ−+ + ++••• . (2.3.8) θ ψ
Then we know that the Weyl fermion ψ+ has ME charge −1, U(1)V charge qV = −1 and U(1)R charge qR = −1. The fact that ME charge is −1 means exactly that it is a anti-spinor, or a section of the anti-spinor bundle S over Σ. Now let’s see how does it change under two twists. If we perform the A-twist, it has ME0 charge −1−1 = −2 and it becomes a vector field or an anti-holomorphic one-form; If we perform the B-twist, it has ME0 charge also equal to −1 − 1 = −2 and still becomes an anti-holomorphic one-form.
More dramatically, we consider another Weyl fermion ψ−. After the A-twist it has ME0 charge 1 − 1 = 0 and becomes a scalar field. Namely we successfully change a spinor to a scalar which is our goal! Similarly, we can show that after the B-twist, it is the fermion ψ+ that becomes a scalar. For sake of completeness, the result for all possible cases after the twisting is shown in Table 2.1.
Now comes the magic: after either twisting, the modified theory has a fermionic sym-metry and turns out to satisfy all the conditions of a TQFT! Let’s spell out all the details in turn.
A-Model
The field theory obtained from an A-twist is dubbed the A-model6. From Table 2.1, Q+ and Q− become scalars after the A-twist. We choose the fermionic symmetry charge to be QA =Q++Q−.
Again from Table 2.1 we know that the Weyl fermions ψ− and ψ+ become scalars, while ψ− and ψ+ which were spinors and anti-spinors are now holomorphic and anti-holomorphic one-forms. We rename the fields to make this point manifest.
Interlude: moduli space of complex structures
Recalled that our target space X is a Calabi-Yau threefold. By definition, the moduli space of complex structures MX encodes all possible complex deformations on X. A point on MX corresponds to a fixed complex structure on X, where there is a nowhere vanishing holomorphic top form Ω which generates H3,0(X).
What will happen if we start to move on MX , i.e., vary the complex structure? First of all Ω will still be closed since the exterior derivative does not depend on the complex structure. However, it is no-longer holomorphic and we need to define a new holomorphic form Ω0. In other words, we have a line bundle L on MX , where the one-dimensional space above each point is generated by the Ω determined by that complex structure. Moreover, we can define a metric on it, h = Ω = i Ω Ω . (2.4.1) Ω Ω is a (3, 3) form so this integral is not trivially zero. Also since Ω is defined up to non-zero scaling at each point, we can always multiply Ω by a nowhere vanishing holomorphic function ef on MX , hence h → |ef |2 h.
After all these preliminaries, we are finally able to define topological string theory in section 3.1. Furthermore, in subsection 3.1.2 we discuss briefly a recent generalization of ordinary topological string theory. In section 3.2, we introduce the important notion of mirror symmetry. It states that the A model topological string theory on a Calabi-Yau threefold X is equivalent to type B topological string theory on a mirror Calabi-Yau threefold X. We first discuss Batyrev’s construction of mirror pairs in subsection 3.2.1, then we extend to non compact situation in subsection 3.2.2. Useful references for this chapter are, for example, [103, 152, 174, 145].
Topological string theory
In the discussion above, we always assume that in the topological nonlinear sigma models the worldsheet metric g is fixed, even though after topological twisting we argue that the models are independent of continuous deformation of the metric. An immediate drawback of A and B models is that according to the selection rules Eqs. (2.3.22) and (2.3.41), Σ must be a sphere to allow for non trivial correlation functions. From a string theorist point of view, this is not at all satisfactory. In string theory, we are all familiar with summation over all possible genera in a scattering diagram. Therefore, it would better if the worldsheet Σ of higher genus also plays a role. In other words, we are interested in making the metric on Σ dynamical and coupling the topological sigma model to gravity. Then we should integrate over the space of all possible metrics on Σ in the path integral, just like what we did in ordinary string theory.
The theory constructed from A–twisted and B–twisted topological nonlinear sigma models are called type A and type B topological string theories respectively. As their names suggest, they are related to type IIA and type IIB superstring theories. In fact, the moduli space of type A (type B) topological string are identified with the vector multiplet moduli space of the type IIA (type IIB) superstring compactified on the Calabi–Yau threefold.
At first glance, this may seem trivial: since the theory is topological, the integrand should be independent of the metric and naively we would simply get Dg Z[g] = Vol(G) Z[g0] , (3.1.1) where the Vol(G) is formally the volume of the “gauge group” which generates the space of metrics. However, even at the level of physical rigor, there are several issues with this line of reasoning:
• The topological symmetry could be anomalous at the quantum level invalidating the conclusion that all the configurations in a gauge group orbit are equivalent.
• Although our theory is invariant under continuous deformation of the metric, there could be metric configurations that cannot be reached from a given metric by con-tinuous changes.
Let’s be more careful when talking about the integration over the space of all possible metrics. First of all, just like in ordinary string theory, the two-dimensional sigma models become conformal when we integrate the metric in the path integral, making the energy-momentum tensor traceless. Notice that this has nothing to do with topological twisting. This also means that we can divide the integration over the space of metrics into two steps: we first integrate over all conformally equivalent metrics, then integrate over the quotient space. As we shall see below, since the two-dimensional conformal group is rather large, the quotient is actually finite dimensional.
The first step could in principle leads to problems. From the study of bosonic strings, on a curved worldsheet, there is the notorious conformal anomaly, with coefficient proportional to the central charge. In ordinary string theory, the central charge is canceled by choosing the correct dimension. In our situation, this is done automatically by topological twisting. As detailed in [138], topological twisting amounts to adding a conserved current into the Lagrangian, and the energy-momentum tensor is modified such that the new central charge is zero. The upshot is that a topological nonlinear sigma model coupled to gravity is free of conformal anomaly.
The more interesting part is the second step. By construction, the quotient space is the space of conformally equivalence classes. Since we can associate to each equivalence class a complex structure, it is the same as the moduli space of complex structures on Σ, denoted as Mg, where g is the genus of Σ1. It’s a known fact that M0 consists of one point and M1 is the fundamental domain of a torus, while for g > 1, Mg has complex dimension 3g − 3 and is non-compact. They can be compactified, by adding new points that have mild singularities, corresponding to the so-called “stable curve”.
• Its only singularities are simple nodes.
• It has a finite number of automorphisms. This means that its genus zero part should have at least three nodal points and genus one part should have at least one nodal point2.
The passage to its boundary can be represented figuratively, e.g., for genus two, as Figure 3.1. The space Mg is the famous Deligne–Mumford compactification of the moduli space Mg of Riemann surfaces [52]. Moreover, if we consider correlation functions, we need to consider the complex structure of Riemann surfaces having n marked points, whose compactification leads to Mg,n.
Note that by borrowing ideas from bosonic string theory, we bypass the first issue and find that the second issue does not occur. We also understand better the space that we integrate over. Now let’s look at what quantities to integrate. Recall the first condition (2.2.5) of a TQFT. For the sake of reader’s convenience, let’s record it here, Tµν = δS = {Q, Gµν}. (3.1.2)
In addition, the energy-momentum tensor Tµν is traceless because the theory is confor-mal. Therefore, the only nonzero components of Tµν are Tzz and Tz¯z¯ and they satisfy Tzz = {Q, Gzz},Tz¯z¯ = {Q, Gz¯z¯} . (3.1.3)
Since Tµν has axial charge 0 and Q axial charge 1, the G’s have axial charge −1. They can be used to define a measure on the moduli space of Mg. The tangent space to Mg at 2 Recall that the automorphic group of a sphere is P SL(2, C) and that of a torus can be identified with itself, acting by translations. While for g > 1, it is finite a given point Σ corresponds to a choice of Beltrami differential (2.3.43) on the Riemann surface Σ. Let µi denote a basis. For genus one there is only one while for higher genera there are 3g − 3 of them, which correspond the dimension of Mg. Let’s start from the genus one case.
Table of contents :
1 Introduction
1.1 String theory as a candidate for a “theory of everything”
1.2 Topological string theory as a bridge between mathematics and physics
2 Topological Field Theory
2.1 Two dimensional N = (2, 2) supersymmetry
2.2 Topological field theories of Witten type
2.3 Topological twisting
2.4 Interlude: moduli space of complex structures
3 Topological String Theory: Coupling to Gravity
3.1 Topological string theory
3.2 Mirror symmetry
4 Resurgence and Quantum Mirror Curve: A Case Study
4.1 The Harper-Hofstadter problem
4.2 Trans-series expansion and one-instanton sector
4.3 Two-instanton sector
4.4 Instanton fluctuation from topological string
4.5 Interlude: holomorphic anomaly at work
5 Elliptic Genera and Topological Strings: Overview
5.1 Elliptic fibrations and four-cycles
5.2 The base degree k partition function as Jacobi form
6 Geometries without codimension-one singular fibers: Reconstruction
6.1 The structure of the topological string free energy
6.2 Zk of negative index
6.3 Partition function from genus zero GW Invariants
7 Geometries with codimension-one singular fibers: Higgsing Trees
7.1 Higgsing trees
7.2 a2 model
7.3 g2 model
8 Conclusion and Outlook
A Modular Forms
A.1 Elliptic modular forms
A.2 Jacobi modular forms
A.3 Weyl-invariant Jacobi forms
B Toric Geometries
B.1 Fans
B.2 Polytopes
B.3 Blow-ups
B.4 Toric Diagrams
C Computations in One-Instanton Sector
C.1 Instanton solution
C.2 The moduli-space metric
C.3 The one-instanton determinant
D Principal Parts of Z3 and Z4 for the Massless E-string
D.1 Base degree 3
D.2 Base degree 4
E Some Enumerative Invariants
E.1 Genus zero GW invariants for massless E strings
E.2 GV invariants for a2 and g2 models
Bibliography
