4.9 Applications to String Theory
4.9.3 String quantisation ` a la CFT
Having reviewed the two ingredients in the bosonic path integral in CFT language, let us take a fresh look at how the two pieces come about.
• The starting point of the bosonic string quantization is the action SBDH[Xµ, hab] with Xµ
taking values in R1,d−1. The action enjoys local Weyl and diffeomorphism invariance on
the worldsheet.
• Gauge fixing `a la Faddeev-Popov leads to a theory defined on11C (for closed strings at tree
level). The remnant of the original Weyl and diffeomorphism symmetry is the conformal symmetry on C. The full gauge fixed theory is described by Stot= S(Xi)+S(bc), where S(Xi)
describes d copies of the free-boson-CFT, each with central charge c(Xi)= 1, i = 1, . . . , d,
and S(bc) describes the λ = 2-(bc)-CFT with c(bc)= −26.
• Self-consistency of the Faddeev-Popov procedure requires absence of the total conformal anomaly,
ctot=X
i
c(Xi)+ c(bc) != 0. (4.122) Thus the bosonic string in d = 26 dimensions is fully consistent as a quantum theory.
• We are now in a position to take a more abstract perspective: The requirement ctot = 0
can also be met by considering more general CFTs and replacing (some of) the 26 copies of the X-CFT therewith. The non-free CFTs are called ”internal” sectors. E.g. one can take the tensor product of only 4 copies of the X-CFT (corresponding to propagation of the string in R1,3) and a tensor product of more complicated CFTs with c = 22. Later
on we will discuss the idea of compactification by considering as target space not flat 26 dimensions, but a space-time of the form R1,3× M with M some internal compact manifold (here of dimension 22). The ”internal” CFT can then be thought of as describing the string propagation on this internal manifold M. Celebrated examples of such CFTs include so-called Gepner models, which are known to describe the string propagation on certain Calabi-Yau manifolds appearing compactifications of the superstring.
Even though this is not yet apparent from what we have learned, the secret of the worldsheet approach to string theory is the insight that space-time is only a manifestation of an abstract CFT on the worldsheet. What is important is not that space-time is well-defined in the sense of a smooth manifold, but that the CFT that describes the propagation of the string on that space is well-defined. This implies that strings can consistenly propagate even on certain singular spaces, e.g. orbifolds of the form M/G with G the action of a finite group, as long as the underlying CFT is non-singular. If you want to learn more about this fascinating way to think about the nature of space-time within string theory, you can try already at this stage to read to the introduction to Brian Greene, String Theory on Calabi-Yau manifolds, http://arXiv.org/pdf/hep-th/9702155, even though you might wish to wait until we have introduced the superstring later in this course.
• For the X-CFT the requirement of BRST-invariance gives the physical state condition Lm|φ i = 0, L˜m|φ i = 0, ∀m > 0 (4.123)
(L0− 1) |φ i = 0, ( ˜L0− 1) |φ i = 0. (4.124)
This establishes the central insight:
Physical string states are 1-1 to primary fields of weight h = 1 = ¯h.
It is crucial to appreciate that this restriction to primaries and to h = 1 = ¯h does not follow from the CFT itself, but is information ”prior to CFT”, i.e. it arises as an extra consistency condition for the X-CFT to make sense as part of the the gauge fixed version of the original BDH-action. By contrast, in a general CFT the Hilbert space consists of all Verma modules over all primary fields, not just the primaries of h = ¯h = 1.
• This leads to the concept of a vertex operator.
Definition 4.3. A vertex operator is a primary field of dimension (h, ¯h) = (1, 1).
Its insertion at z = 0 creates a physical state from the PSL(2, C) × PSL(2, C)-invariant vacuum.
We could also have approached the construction of the string spectrum entirely by constructing the various vertex operators. Let us demonstrate this in two examples:
i) The primary field : eikµXµ(z,¯z): acting on the vacuum creates the state
|k i = lim z,¯z→0: e ikµXµ(z,¯z): |0 i, h = ¯h = α 0 4 k 2. (4.125)
• The field : eikµXµ(z,¯z): is always primary as shown above.
• In order to satisfy h = 1 we need k2= −m2= 4
α0. This gives the mass shell condition
for the lowest-lying state.
Indeed one can rigorously show that the so-defined state |k i is the momentum eigenstate with momentum k, αµ0|k i = r 2 α0i I dz 2πi∂X µ(z) : eikνXν(0,0): |0 i OPE = r 2 α0i I dz 2πi −i 2α 0kµ1 z : eikνXν(0,0): |0 i = r α0 2k µ |k i. (4.126)
Here we made use of the OPE (4.115).
ii) The fields at the first excited level are created by the following vertex operator: |k, ξ i = lim z,¯z→0ξµν : ∂X µ(z) ¯∂ ¯Xν(¯z)eik·X(z,¯z): | {z } =:V1(k,ξ;z,¯z) |0 i. (4.127)
The physical state conditions are as follows: • The dimension of V1 is hV1 = 1 +
α0
4k 2= ¯h
V1 To achieve hV1 = 1 we must set k
2 != 0.
• Higher constraints arise from demanding that V1(k, ξ; z, ¯z) be a primary field.
Again with the help of Wick’s theorem we compute the OPE T (z)V1(k, ξ; w, ¯w) = const × kµξ µν (z − w)3 : ¯∂ ¯X ν( ¯w)eik·X(w, ¯w): + α 0 4 k 2+ 1 (z − w)−2+ ∂w (z − w) V1(k, ξ; w, ¯w) +...
This has the form of the OPE of a primary field if kµξ µν
!
= 0, thereby reproducing the known transversality constraint of momentum and polarization.
Summary:
Chapter 5
String Interactions
5.1
Perturbative Expansion
We have finally gathered the technology required to approach the important question of string interactions.
• The basic object to compute in string perturbation theory is the S-matrix, defined as the amplitude for the scattering of asymptotic in- and out-states. E.g. scattering of two in-states into two out-states is described by worldsheets of the following form for the closed and open string, respectively:
• Just from drawing the worldsheets we note a key property of string scattering - the ab- sence of definite local interaction vertices. This fundamentally distinguishes string scattering from the scattering of point particles in QFT as is evident by comparing the above worldsheets with a 2-to-2 Feynman graph, e.g. in QFT. The string worldsheet always looks locally like the worldsheet of a freely propagating string, and only global properties of the worldsheet capture interactions.
Put differently, the string interactions are encoded already in the free two-dimensional CFT without adding arbitrary further terms in the worldsheet action.
This crucial difference compared to point particle QFTs in target space cannot be overes- timated - after all the specification of interactions by a QFT Lagrangian adds a degree of arbitrariness into the theory that makes it hard to accept this as a fundamental theory. • In the following we consider the path integral with a Euclidean metric on the worldsheet.
In fact, the only worldsheets for which a globally non-singular Lorentzian signature metric exists are the torus T2, which has no boundaries, or the cylinder. This is because a
globally-defined Lorentzian metric requires a globally non-singular Killing vector field that 93
distinguishes the time coordinate from the spatial ones. Such Killing vector fields exist only for the mentioned surfaces. Consider e.g. the surfaces above arising in 2-to-2 scattering: Due the merging and splitting of the in- and out-going pants there is a bifurcation of the ”time”-coordinate and thus no globally defined time-like Killing vector field.
• By means of the operator-state correspondence - the central lesson from the previous CFT chapter - the in- and out-states are encoded in the path integral via insertion of the corre- sponding vertex operators on the worldsheet.
As the simplest example consider the worldsheet describing 1-to-1 closed string scattering, with Euclidean coordinates and metric
w = τ − iσ, ds2= dw d ¯w. (5.1) It has the topology of a cylinder, with the spatial slices at τ = ±∞ corresponding to the in- and out-states:
The conformal map z = e2π` w maps this to a sphere, with the in- and out-states inserted
at the two poles, or via the stereographic projection to C ∪ ∞ with insertions at the origin and at z = ∞:
∼ =
• By similar conformal maps the above closed 2-to-2 scattering process can be mapped to an S2 with 4 marked points corresponding to the insertion of the 4 vertex operators.
Analogously for open strings the scattering worldsheet is mapped to a disk with insertions on the boundary:
Equivalently we can consider the upper half plane with 4 vertex operators inserted on the real line.
Summary: By conformal symmetry string scattering is described by compact worldsheets with insertions of vertex operators for all in- and out-states.
• Our notation will be to describe by Vj(k) the vertex operator associated with the j-th state
with momentum kµin= (E, ~k) or, equivalently, kµout= −(E, ~k).
• The above examples include only the simplest compact worldsheets with 4 insertions, but in the path integral we must sum over all possible topologies.
In the oriented string all worldsheets are orientable.1 In characterising the types of
worldsheets we need to consider we make use of the following
Theorem 5.1. Every compact, connected, oriented2 two-dimensional manifold is topolog-
ically equivalent to a sphere with g handles and b holes representing the boundaries. As some examples consider the following closed worldsheets
S2: (g, b) = (0,0) torus T2: (g, b) = (1,0) double torus (g, b) = (2,0)
or open worldsheets
disk: (g, b) = (0,1) annulus: (g, b) = (0,2) pants: (g, b) = (0,3)
Note that the disk is topologically equivalent to a sphere S2 with one hole, i.e. with one
boundary,
∼ =
1By contrast in orientifold theories we include non-oriented worldsheets. Recall that such unoriented string
theories were introduced and discussed in the tutorial.
and likewise the cylinder can be viewed as a sphere with two holes etc. • A topological invariant of two-dimensional oriented surfaces3is the
Euler number χ = 2 − 2g − b. (5.2) By the famous Riemann-Roch-theorem this topological invariant is computed by the expression χ = 1 4π Z Σ d2ξ√hR(2)+ 1 2π Z ∂Σ ds k (5.3)
with R(2) the Ricci scalar of the surface and k the geodesic curvature of the boundary.
• Recall that we can add to the string action the term λ 1 4π R Σd 2ξ√hR + 1 2π R ∂Σds k with λ ∈ R without affecting the dynamics. This term then keeps track of the topology of the worldsheet in the path integral.
Putting everything together we arrive at the following heuristic expression for the S-matrix describing the scattering of n string states (viewed as an expression before gauge fixing):
Sj1...jn(k1, . . . , kn) = X compact topologies R DXDh Voldiff.×Weyl e−SX−λχ n Y i=1 Vji(ki). (5.4)
Note that we divide by the volume of the group of diffeomorphisms and Weyl transformations as this group will factor out in the process of the Faddeev-Popov procedure. The particulars of this gauge fixing, however, have to be reconsidered in the presence of vertex operators.
Comments:
1) Let us define the quantity
gs= eλ. (5.5)
Then the expansion in terms of worldsheets of different topology is governed by factors of g−χ
s = g
−(2−2g−b)
s (for the oriented case and correspondingly g−χs = g
−(2−2g−b−c)
s for the
non-oriented theory) in the path integral. For λ << 0 we observe that gs << 1. Then the
sum over topologies defines a perturbative series.
2) Consider a closed string worldsheet and add a handle to it, e.g.
Since this adds two boundaries the Euler number decreases as χ −→ χ − 2.
Physically adding a handle describes emission and reabsorption of a closed string. If we think in terms of Feynman diagrams this would corresond to two ”vertices”, each coming with ”coupling constant” gc. Therefore, we interpret gsas the
closed string coupling gc= eλ= gs. (5.6)
3) Similarly add a boundary to an open worldsheet, thereby decreasing the Euler number by 1,
The physical interpretation - emission and reabsorption on an open string - indentifies the open string coupling go= eλ/2= gc1/2. (5.7)
4) The object Vji(ki) is the integrated vertex operator
Vji(ki) = const × Z d2ξi p h(ξi)Vji(ki; ξi) ∼= const × Z d2ziVji(ki; zi) in flat gauge. (5.8)
Integration over the insertion of the vertex operator, R d2z
i, ensures invariance of the full
amplitude under diffeomorphisms on the worldsheets.
The integrated vertex operator is normalised such as to carry one factor of goor gc for open/
closed states.
E.g. the integrated vertex operator for a closed tachyon is VT(ki) = gs
Z
d2zi : eiki·X(zi,¯zi): . (5.9)
For a general closed state we oftentimes employ the notation Vji(ki) = gs
Z
d2zi Vji(zi, ¯zi) : e
iki·X(zi,¯zi): . (5.10)