PHYS 652: String Theory II Homework 2

Due 2/27/2017 (Assigned 2/15/2017)

1 Heterotic Strings

Last semester, you constructed the closed bosonic string by combining left- and right-moving copies of the open bosonic string subject to the level-matching condition. In this class we repeated that construction to obtain the type-II closed superstrings from two copies of the open type-I superstring. In the RNS formalism we employed a GSO projection to remove the tachyon and computed the massless spectrum in lightcone gauge. In the GS formalism spacetime supersymmetry extends the lightcone gauge to superspace and there is no tachyon. Despite the differences between the two formulations, the massless spectra agree. The heterotic strings are closed superstrings obtained by combining the bosonic string for the left-movers with the type-I superstring for the right-movers, again subject to level-matching. Split the 26 coordinates of the critical bosonic string into 10 X’s and 16 Y ’s and treat the X’s as the left-moving side of the spacetime coordinate with the right-moving side identified with the 10 X’s of the critical type-I superstring. Using level-matching, construct the massless spectrum in both the RNS and GS formalisms and interpret it in terms of gauge and gravity multiplets. At this point, it is not possible to determine the gauge group of this theory but you can deduce the rank. If, in addition, you were told that the gauge algebra is semi-simple and its dimension is 496, what are the possibilities for the heterotic strings? Why is this string called “heterotic”?

2 Fermionization and Picture

Fixing reparameterization-invariance of the bosonic string to conformal gauge requires the in- clusion of the bc ghost system. This introduces a degeneracy of the ground state since the 0-mode of the c ghost forms an abelian super-algebra

{c0, c0} = 0 (2.1) of which, the ground state should be a representation. In this simple case, the representation is two-dimensional |↑↓i with

|↑i = c0 |↓i . (2.2)

1 Vertex operators U constructed from the former are called “unintegrated”. A vertex operator of the form Z d2σI (2.3) is called “integrated”. The two are related by I = b(U) where b(U) means the simple pole in the OPE of b with U. Use this to show that the integrated vertex operator is BRST-invariant if the unintegrated vertex operator is. Recall that both integrated and unintegrated operators were used to define the bosonic string scattering matrix. Worldsheet supersymmetry of the RNS superstring requires us to include the βγ ghost system. The NS ground state satisfies

βr≥1/2 |0iNS = 0 and γr≥1/2 |0iNS = 0. (2.4)

The operator-state correspondence related this state to an operator traditionally denoted δ(γ). Show that this implies that the operator products

γ(z)δ(γ(0)) = O(z) and β(z)δ(γ(0)) = O(z−1). (2.5)

Friedan-Martinec-Shenker ghosts allow us to write an explicit expression for δ(γ). Consider the odd ghost η(z), its conjugate ξ(z) and a holomorphic scalar φ(z): 1 η(z)ξ(0) ∼ , φ(z)φ(0) ∼ − log(z). (2.6) z Show that the combinations

β ↔ e−φ∂ξ and γ ↔ eφη (2.7) satisfy the same operator products as the βγ system. Construct the ghost number current JFMS and stress-energy tensor TFMS to have the correct OPEs as those in the βγ representation. Show that

δ(γ) ↔ e−φ (2.8) satisfies (2.5). What is the conformal weight of this operator? Define the “picture-changing operators”

Z = {Q, ξ} and Y = ic∂ξe−2φ, (2.9) and show that if U is annihilated by Q then so is ZU. Also show that if ZU = {Q, Λ} for some Λ then U = {Q, Y Λ}. Conclude that if U is in the cohomology of Q then so are ZnU and Y nU for any n = 1, 2,... . This is the picture-changing ambiguity of the RNS superstring vertex operators that needs to be addressed in the construction of superstring scattering amplitudes.

2 3 BRST vs. Cohomology

Let g be a Lie algebra which we assume to be real for definiteness. Picking a basis ta for the Lie c algebra with a = 1,..., dim(g) defines the structure constants fab of the Lie algebra in this basis. a Introduce the ghosts c as anti-commuting vectors in this basis for the Lie algebra and let ba denote the conjugate ghost momenta. Anti-symmetric tensors of the Lie algebra define the analog of differential forms. Replacing the basis of 1-forms of the Lie algebra with ghosts, we can take a form of degree p

ap a1 ω ↔ c . . . c ωa1...ap (3.1) to be identified with a monomial of ghost-number p. Consider the collection of all such forms. This is a vector space graded by ghost number

a N = c ba (3.2) with a p-form corresponding to ghost-number p.(I.e. an element of this vector space is a formal sum of p-forms of degrees p = 0, 1,..., dim(g).) It is also a (graded abelian) algebra since products of sums of elements of the form (3.1) are still sums of elements of the form (3.1). Show that the Chevalley-Eilenberg differential

a 1 a b c Q = c ta − 2 c c fab bc (3.3) acts on this algebra, squaring to 0. Note that Q has ghost-number 1. Define the cohomology H∗(g; R) = ker(Q)/im(Q), (3.4)

∗ Ldim(g) p and note that is also a graded algebra. This again splits by ghost number H = p=0 H . What does H0(g; R) the cohomology at ghost number 0 compute? Now take g to be the supersymmetry algebra. This is almost a graded (!) abelian algebra except that there is the relation

c {Qa,Qb} = fab Pc, (3.5) where Qa are the fermionic generators and P are the usual translation generators. Introduce the appropriate ghost system and construct the Chevalley-Eilenberg differential. It should depend on Clifford/Dirac/Pauli matrices. Under what assumption on the latter does this differential square to 0? This construction can also be carried out for affine Lie (super-)algebras. The Chevalley-Eilenberg differential for the Virarsoro algebra is worked out in GSW volume 1 where it is shown that can- cellation of the anomaly is equivalent to the condition Q2 = 0.

4 Pure Spinor Superparticle

The Brink-Schwarz action for the ten-dimensional superparticle is Z µ µν S = dτ (PµΠ + ePµη Pν) . (4.1)

3 .µ . Here Πµ = x − ΘγµΘ is the supersymmetric velocity and e(τ) is an ein-bein for the world-line. This is a first-order formalism in which the momentum Pµ is an independent variable. (The square is taken with the target space Minkowski metric.) It is manifestly super-Poincar´einvariant in the target space (assuming e and P are invariant under global supersymmetry) and reparameterization invariant on the worldline. Show that it is invariant under the κ symmetry . µ µ a µ a a δX = Θγ δΘ , δΘ = (γ κ) Pµ , δPµ = 0 , δe = Θ κa. (4.2) a Compute the momentum pa canonically conjugate to Θ to conclude that there is a constraint on the phase space variables of the form

µ da = pa − (γ Θ)Pµ. (4.3) Show that they satisfy the supersymmetry algebra

µ {da, db} = −2i(γ )abPµ, (4.4) where the bracket is the graded Poisson bracket. Since the Hamiltonian P 2 is also a constraint (it generates reparameterizations), this implies the spinorial constraints (4.3) are not all first class. Introduce a bosonic spinor “ghost” λa, define the Berkovits differential

a Q = {λ da, ·} (4.5) acting on the phase space through the Poisson bracket. Find the condition on λ that makes Q2 = 0 (cf. problem3). “Wave functions” of ghost number 1 are of the form

a U = λ Aa(X, Θ). (4.6) By definition, they are in the cohomology of the Berkovits differential when they are closed QU = 0 but not exact, that is, when U cannot be written as Qf for any scalar superfield f(X, Θ). Recall that exact parts get shifted by the gauge transformation δU = Qα for a gauge parameter superfield α(X, Θ). Expand

(0) a (1) a b (2) α(X, Θ) = α (X) + θ αa (X) + θ θ αab (X) + ..., (4.7) and use this to show that there is a “Wess-Zumino gauge” in which1

µ µνλ b Aa = (γ θ)aAµ(X) + (θγ θ)(γµνλ)abχ (X) + .... (4.8) Not all of the components of the gauge parameter superfield α are used up to get to this gauge. (0) In particular, the lowest component α(X) = α (X) = α(X, Θ)|Θ=0 survives. Show that in this gauge, the “photon” component has the correct gauge transformation

δAµ(X) = ∂µα(X). (4.9)

1Recall that this means that certain components in the Θ expansion of U and Qα are of the same form. This means that those components of α can be used to remove the analogous components of A.

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