Posted: Jun 28, 2011 7:38 am
Post #10 Topological String Theory

Twistor string theory is an example of a topological string theory, so I’ll try to explain this concept first. To do this, I’ll start with a description of 𝒩=(2,2) supersymmetric theories, and the A and B model twists that convert them to topological models. To start this process I’ll mention a few things about spinors in even numbers of dimensions and worldsheet supersymmetry.

Spinors in D dimensions

We’ll assume that D is even and that the space we’re working on has a metric ηmn with Minkowski signature (+,-,-....-). We start with the relation defining the Clifford algebra, i.e.
𝛾m𝛾n+ 𝛾n𝛾m = 2ηmn m,n = 0..D-1

The 𝛾s are just generalizations of the Dirac gamma matrices to D dimensions. Multiplying them together in various combinations gives you a finite group with 2D+1 distinct elements. This group has a representation of dimension 2D/2, namely the spinor representation.

The complex conjugate gamma matrices 𝛾m* also form a representation, and there is a matrix B such that

𝛾m* = B𝛾mB-1

The transposed gamma matrices 𝛾mT also form a representation, and there is a matrix C (actually representing the well known charge conjugation in the case of 4D spacetime) such that

𝛾mT = -C𝛾mC-1
The B and C operators satisfy

BT = εB
CT = -εC

Where
ε = +1 if D = 2 or 4 mod 8
.....-1 if D = 6 or 8 mod 8

For D = 2 or 4 mod 8, B is symmetric and unitary and this can be used to show that in those cases 𝛾m can be chosen to be real, and C = 𝛾0.

Under a (infinitesimal) Lorentz transformation, a spinor λ transforms as
𝛿λ = (1/4) 𝛿ωmnγmnλ

here 𝛿ωmn are the antisymmetric Lorentz generators I introduced in the appendix to post #5 and γmn means γmγn.

We can define two conjugation operations on spinors:

Dirac conjugation: λ-> λbarD = λ γ0

Majorana conjugation: λ-> λbarM = λTC

A Majorana spinor is defined to be one whose Majorana conjugate is equal to its Dirac conjugate: λbarM = λbarD

In D dimensions define γD+1 = γ0 γ1.... γD-1 (this is the generalisation of the famous γ5 matrix which appears in many elementary QFT books in 4 dimensions). Then a Weyl spinor 𝜒 is one for which

γD+1𝜒 = ±𝜒 if D = 2mod4
I γD+1𝜒 = ±𝜒 if D = 4mod4

A Majorana-Weyl spinor is (surprise) one which is both Majorana and Weyl, and can only exist if D=2mod8.

The spinors I was using in posts 1 and 2, in 4 dimensions, were Weyl spinors.

If anybody is burning to know more about spinors, Clifford algebra and the relevant group theoretic mumbo jumbo, I can recommend this.

Worldsheet Supersymmetry

As a one-dimensional string moves through space, it sweeps out a two-dimensional surface in spacetime, called the worldsheet. (Alternatively you can forget about spacetime and think of what were spacetime coordinates as a bunch of fields on the worldsheet). The worldsheet is traditionally given a time coordinate 𝜏 and a space coordinate σ. An alternative which proves to be very convenient is to use, instead of 𝜏 and σ, the lightcone coordinates
ξ++ = 𝜏 + σ
ξ= = 𝜏 - σ

Rather surprising to see ++ and = when you’re used to seeing components as labelled ‘0’, ‘1’, but that’s the convention !

In post #6 I described superfields on spacetime, whereby you add some Grassmann variables to your usual spacetime coordinates. These Grassmann variables have spinor indices. You can do the same on the worldsheet, the difference being that the base manifold is now two dimensional instead of four. Fortunately, however, we’ve just learned how to define spinors in two dimensions. In spacetime, where we used indices like ‘A’, ‘B’ for spinors, each index took the values 0 or 1. On the worldsheet we’ll use α, β as spinor indices, each of which takes the value +, -. So a worldsheet spinor would be Ψα = (Ψ+, Ψ-). In spacetime, the conjugate spinors are written with primed indices A’, B’ etc. On the world sheet, we’ll do the same i.e. indices are α’, β’ etc.

Just as we think of the spacetime coordinates as a bunch of functions on the worldsheet xμ( 𝜏, σ), so for the purposes of defining worldsheet supersymmetry, we add a bunch of worldsheet spinor functions Ψμβ(𝜏, σ). Just as in the spacetime case, they will be anticommuting i.e. Ψμ+ Ψμ- = -Ψμ- Ψμ+.

To define the worldsheet supersymmetry, we take a constant Grassman valued spinor
𝞮 and define the SUSY transformations as

𝛿(𝞮)Xμ = 𝞮bar σμ
𝛿(𝞮)σμ = -(1/2)βγa𝜕aXμ 𝞮.............................(1)

In the latter equation the index a runs over the coordinates ‘0’ and ‘1’ on the worldsheet, meaning (𝜏 and σ). A representation of the 2d gamma matrices is

𝛾0 = (0......-1)
......(1......0)

𝛾1 = (0......1)
......(1......0)

Note 𝞮bar σμ means = 𝞮β (𝛾0)βγσμγ

The SUSY transformations defined above leave invariant the action

S = -(1/2)d2σ(∂aXμaXμ + Ψbarμ 𝛾aaΨμ)

However this invariance is rather messy to show. To make the supersymmetric action more manifest (i.e obvious), we recast things using the superfield formalism that I mentioned in previous post in the context of spacetime supersymmetry. This works by taking the original manifold (in this case the two dimensional worldsheet), and “thickening” it by adding some extra dimensions – in this case two fermionic Grassman Majorana spinor coordinates θα = +, -. SuperWorldsheet.jpg (11.9 KiB) Viewed 2467 times

We then consider “superfields” on the thickened worldsheet. These are maps into the spacetime manifold Φμ(𝜏a, θα). The most general worldsheet superfield can be expanded as:

Φμ(𝜏a, θα) = Xμ(𝜏a) + θbarΨμ (𝜏a) + (1/2) θbarθBμ(𝜏a)

Whenever we have a product of a barred spinor with a non-barred one, we omit the indices and gamma matrices, so θbarΨμ is short for θα( 𝛾0)αβΨμβ and θbarθ for θα(𝛾0)αβθβ.

So we have our two sets of fields – the bosonic one Xμ(𝜏a) and the fermionic one Ψμ (𝜏a), but there’s also an extra set Bμ(𝜏a). This is an auxiliary field, which is needed to make the supersymmetry manifest, but has no physical content and ends up with an equation of motion Bμ(𝜏a) = 0.

It can be shown that the supersymmetry transformations (1) above are represented in the superspace/superfield formalism by generators (supercharges):

Qα = ∂/∂θα – (𝛾aθ)αa

It’s possible to extend the existing model to add more supercharges. In general, if I have p ‘+’ supercharges Qm+ m=1..p and q ‘-‘ supercharges Qm- m=1..q, then the worldsheet is said to have 𝒩=(p,q) supersymmetry. Interestingly, the values of p and q place restrictions on the geometry of the target spacetime:

𝒩=(0,0), (1,0), (1,1)..................Riemannian
𝒩=(1,0), (2,1),........................Kȁhler
𝒩=(2,2).................................Kȁhler or bi-Kȁhler

For our purposes of describing a topological string theory that will be relevant for twistor strings, we are interested in the 𝒩=(2,2) case.

𝒩=(2,2) Supersymmetry

It’s convenient to work with complex coordinates for the bosonic directions on the worldsheet. We define z = ½(𝜏+iσ), zbar = ½(𝜏-iσ). For 𝒩=(2,2) supersymmetry, we’ll need an extra fermionic direction on superspace, which we denote by θbar. θbar is an independent coordinate, not any sort of conjugate of θ. The worldsheet coordinates are now (z, zbar, θα, θbarα). Acting on this superspace, the SUSY generators are:

Q+ = ∂/∂θ+ –θbar+∂/∂z
Q- = ∂/∂θ- –θbar-∂/∂zbar
Qbar- = -∂/∂θbar--∂/∂zbar
Qbar]+ = -∂/∂θbar+-∂/∂zbar

If you try to write down all the possible SUSY generators you could think of, in addition to the above you might also write down

D+ = ∂/∂θ+bar+∂/∂z
D- = ∂/∂θ-bar-∂/∂zbar
Dbar- = -∂/∂θbar--∂/∂zbar
Dbar]+ = -∂/∂θbar+-∂/∂zbar

Looks like we end up with 𝒩=(4,4) supersymmetry. However all the D’s anticommute with all the Q’s. If, therefore, we try to generate superfields by writing down any old functions Φμ(z, zbar, θα, θbarα ) on the superspace worldsheet, then will end up with a reducible representation of the SUSY algebra.

The situation is analogous to what you’d get if you tried to represent SO(2) by rotations of ℝ3 about the z axis. To get an irreducible representation, I’d need to constrain myself to looking at, say, the just the xy plane. The cause of the reducibility is the existence of the z translations which commute with the rotation generators. The solution is to constrain the z freedom. So, here, we take superfields which are annihilated by the D’s. The options are:
Dbar+Φ = 0 & Dbar-Φ “chiral superfields”, and

D+Φ = 0 & D-Φ “antichiral superfields”.

These give us irreducible representations of the SUSY algebra.

Now, suppose we perform a “Lorentz” transformation of the 1 space and 1 time coordinate on the worldsheet. The quotes are because we’re working in Wick rotated coordinates, so the transformation is a straightforward rotation in the 𝜏 σ plane. This is generated by the operator z∂/∂z – zbar∂/∂zbar. If we ask what the effect of this transformation is on the worldsheet “thickened” with the fermionic coordinates, just note that the fermionic coordinates transform in the same way as the bosonic ones, but at half the rotation rate (since they’re spinor components). So, the overall rotation operator on the super worldsheet is

M = 2z∂/∂z – 2zbar∂/∂zbar + θ+ ∂/∂θ+ - θ- ∂/∂θ- + θbar+ ∂/∂θbar+ - θbar- ∂/∂θbar-

(remember also the barred ones are independent fermionic coordinates, not complex conjugates). So we have, in M, a nice consistent transformation on the super worldsheet. Interestingly, though, it’s not the only one. We can also mix up the supersymmetry generators amongst themselves by applying transformations generated by

Fv = -θ+∂/∂θ+ - θ-∂/∂θ- + θbar+∂/∂θbar+ + θbar-∂/∂θbar-
and
FA = -θ+∂/∂θ+ + θ-∂/∂θ- + θbar+∂/∂θbar+ - θbar-∂/∂θbar-

The U(1) symmetries generated by Fv and FA are called “R-Symmetries”.

Twisting the Sigma Models

When we describe infinitesimal symmetry transformations, we parametrise them by an infinitesimal parameter (traditionally ϵ). A simple analog would be a good old-fashioned infinitesimal space-time translation on Minkowski space, giving rise to a change in a field

𝛿φ(x) = ϵμPμ φ(x)

Here, ϵμ is a constant vector on Minkowski space. Now, on the worldsheet we’d like to do something similar, including transformations in the fermionic dimensions. So, for example, we might like to write a transformation generated by a supersymmetry generator:
𝛿φ = ϵ+Q+φ

Now ϵ+ is a constant spinor on the worldsheet. Of course “constant” has to mean “covariantly constant”. However, for an arbitrary worldsheet metric, it is not possible to find covariantly constant spinor fields. It would be nice if, instead, we were not dealing with spinors, but scalars, because then defining constant ones would be no trouble. We were forced into using spinors because the Q’s are spinors. If we could find a way of making the Q’s transform as scalars we would be OK. There is a way (in fact there are two ways) to do this.

The way we do it is to use an idea due to Ed Witten where we replace the existing Lorentz symmetry, which is defined by the operator M above, with either MA = M – FV or with MB = M – FA where FV and FA are the R-symmetry generators defined above. The two models thus obtained are called the “A model” and the “B model”. The one we want for twistor string theory is the B model.

The supersymmetry generator for the B model is QB = Qbar+ + Qbar-. In fact, the commutator [MB, QB] = 0, so QB is a scalar as far as the modified Lorentz transformation MB.

Topological Theories in General

What does it mean to say that a theory is “topological” ?

The quantum field-theoretical content of a theory is the set of correlation values, computed via the path integral

<𝒪1𝒪2...𝒪n>=[𝒟φ]𝒪1(φ) 𝒪2(φ)... 𝒪n(φ) eiS[φ]/ℏ
Here the action functional S[φ] is some sort of map from the classical fields φ to a number. Typically it will be an integral of some function of the fields, their space time derivatives and also the spacetime metric. So, just changing the metric a little bit will result in a different value for the correlation functions. For the path integral, you have to integrate over ALL of the classical field configurations, including all the metrics. Now suppose you lived in a universe (defined by the action functional) where just changing the metric a little bit didn’t make any difference to the action. Then I’m sure you would agree that the path integral would be a hell of a lot easier to do, since there would be far fewer configurations to integrate over.

We can express this formally as saying 𝛿/𝛿gμν <𝒪1𝒪2...𝒪n> = 0, i.e. the functional derivative with respect to the metric vanishes.

“Schwarz-type” topological field theories are ones in which the action (and the operators 𝒪i) are simply independent of the metric.

For the other type of topological field theory “Witten-type”, we have a global symmetry whose infinitesimal form 𝛿 has the properties

𝛿𝒪i(φ) = 0
Tμν(φ) = 𝛿Gμν(φ) for some tensor Gμν

With these properties it can be shown that the functional derivative of the correlation values with respect to the metric vanishes.

The B model turns out to be indeed a (Witten-type) topological theory and forms the basis of the topological string theory which motivated the development of twistor string theory, which I might finally get to describe in the next post.