So w is the ratio ("projection") between the 2 parts of z (square part in the denominator) that's gauge invariant. (ū is pure gauge.) For various purposes, it's also useful to define the orthogonal z_{M}^{A}, with gauge group U(N-n|2),
z̄_{A' }^{M} z_{M}^{B} = 0 ⇒ z_{M}^{A} = |
⎧ ⎩ |
δ_{M}^{N} - w_{M' }^{N} |
⎫ ⎭ |
u_{N}^{A} |
g = |
⎧ ⎩ |
a c |
b d |
⎫ ⎭ |
, g^{-1} = |
⎧ ⎩ |
d̃ -b̃ |
-c̃ ã |
⎫ ⎭ |
where z̄_{A' }^{M} transforms with g and thus z_{M}^{A} with g^{-1}, we have 2 forms of the symmetry transformation as fractional linear transformations:
These should be familiar as SL(2) (the scale part cancels) for the projective space RP(1) (or CP(1) for complex) when g is 2×2 and w is 1×1 (just a number). Much less familiar is conformal transformations (N=0) in D=4, where g is 4×4 and w is 2×2, describing spacetime as HP(1) (a quaternion). Then this is the simplest way to write conformal transformations, finite or infinitesimal (again proving spinor indices are good for more than just fermions). A useful application is the ADHM construction for instantons (and its supersymmetric generalization, for n=0 & N>0, on HP(1|½N)).
Superconformal invariants are easiest to derive in this approach: We 1st construct superconformal, but not gauge, invariant objects of the form (in matrix notation)
The u's transform linearly under the gauge group, so it's then easy to construct invariants by canceling them (e.g., str(z_{12}z_{32}^{-1}z_{34}z_{14}^{-1})).
For N>0, w has the usual 4 spacetime coordinates x, half the full set of anticommuting coordinates θ (i.e., 2N), and varying numbers of internal coordinates y: Writing M=(m,μ), M'=(m',μ̇), where μ,μ̇ are Weyl spinor indices and m,m' are R-indices labeling which supersymmetry,
w_{M' }^{M} = |
m' μ̇ |
⎧ ⎩ |
m y_{m' }^{m} θ̄_{μ̇}^{m} |
μ θ_{m' }^{μ} x_{μ̇ }^{μ} |
⎫ ⎭ |
For example, n=0 & N describe chiral and antichiral superspace, with no y's. Note that the rest of the θ 's of the usual full superspace are contained in u & ū; they are treated as nondynamical, but appear in such things as covariant derivatives. (Consider again the example of chiral superspace, for N=1.)
We’ll mostly be interested in n=N/2, so w is a square matrix, because these cases allow the deﬁnition of a reality condition. This follows from the modiﬁed unitarity of the superconformal group elements,
Υ^{ṀN} = |
μ̇ ṁ μ ṁ' |
⎧ | | ⎩ |
ν 0 0 iC 0 |
n 0 I 0 0 |
ν̇ -iC 0 0 0 |
n' 0 0 0 I |
⎫ | | ⎭ |
, C^{μν} = |
⎧ ⎩ |
0 -i |
i 0 |
⎫ ⎭ |
, (I)^{ṁn} ≡ δ_{m}^{n} |
(Cw)^{†} = |
m μ̇ |
⎧ ⎩ |
m' -y^{-1}_{m}^{m'} -iC^{μ̇ν̇}θ̄_{ν̇}^{n}y^{-1}_{n}^{m'} |
μ iy^{-1}_{m}^{n'}θ_{n' }^{ν}C_{νμ} C^{μ̇ν̇}(x_{ν̇ }^{ν}-iθ̄_{ν̇}^{n}y^{-1}_{n}^{n'}θ_{n' }^{ν})C_{νμ} |
⎫ ⎭ |
This allows deﬁnition of a superconformally invariant (4D extension of the Hilbert space) inner product as
General oﬀ-shell theories have been written in projective superspace for the case of N=2 (Lindström, Roček, ..., ’84-).
OSp(N|4) |
OSp(n|2)OSp(N−n|2) |
which leads to exactly the same coordinates w, but diﬀerent “u”. We can also use the contraction (to a diﬀerent superconformal subgroup)
I[OSp(n|2)OSp(N − n|2)] |
OSp(n|2)OSp(N−n|2) |
which has a ﬂat & torsion-free coordinate space (w), but a “curved” tangent space (u): Translations of w include ½ the supersymmetries, while rotations include the sum of the other ½ plus ½ the S-supersymmetries (not the direct sum). Also, the general set of 1st-class constraints (ﬁeld equations) for supersymmetric theories has too many free indices, so the ghosts have more and more indices at each ghost level, making a mess for quantization. The superconformal constraints Ĝ can be written in terms of its generators Ĝ as
The net result is that the complete minimal BRST operator can be written in the simple form (matrix multiplication with metric, and trace, implied)
∞ ∑ m,n=0 |
where we have replaced the constraints with the “dual” ones in terms of covariant derivatives d. Here “f ...” denotes structure-constant terms. (We won’t need those for the contracted projective case, where we can replace d with ∂/∂w.)
In terms of the graded transpose “^{T} ”, we have
(dηd)^{T} = +(dηd)
G_{3} ≡ dηG_{2} + G_{2}ηd = -G_{3}^{T} = 0
Using this construction for the BRST operator, and including terms for closure (Q^{2} = 0) leads to the above expression for the BRST operator, where
We now specialize to our coset representation by setting to 0 the derivatives that are isotropy constraints, so we can drop some terms, & some constraints altogether. The above result for Q can then be applied directly by dividing the ranges of the indices in half and dropping irrelevant blocks. The result is as above for odd n, except both indices are primed or both unprimed, while for even n we have mixed indices:
⎧ ⎨ ⎩ |
c_{2n+1,AB} , c_{2n+1,A′B′} where c_{2n+1} =
(−1)^{n}c_{2n+1}^{T} c_{2n,AB′} |
Thus the symmetry has a cycle of 4, going as asymmetric, (twice) graded symmetric, asymmetric, (twice) graded antisymmetric.
Note that for N=4 the number of bosons & fermions is equal at each ghost level; this suggests ghost 0-modes could cancel each other without any type of insertion. Furthermore, the use of super-AdS or its contraction suggests all momenta should appear in the propagator (because of OSp symmetry), preventing momentum 0-modes.
N=4 is the only projective case where the ∇_{u} algebra has ﬁeld strengths, but these can be absorbed by (the gauge ﬁelds of) the SO(2) derivatives. (A similar procedure works for the N=2 chiral case, but not for N=4 chiral.) Examining the relations
{∇_{a′α̇} , ∇_{b′β̇}} = −C_{α̇β̇} ( ∇_{a′b′} + φ̄_{a′b′} ) − η_{a′b′} ∇_{α̇β̇}
φ_{ab} = C_{ab} φ, φ̄_{a′b′} = C_{a′b′} φ, [∇_{u} , φ] = 0
as a closed algebra. (This is equivalent to a redeﬁnition of the SO(2) derivatives.) In particular, we can choose the gauge where the above isotropy constraints reduce to just d_{u} = 0. In this gauge, there is a residual gauge invariance with d_{u} λ = 0; i.e., the gauge parameter λ is projective. At that point we can work exclusively in terms of ∇_{w}. Here φ is the projective ﬁeld strength, which contains all component ﬁeld strengths in its expansion.
Some interesting features of this required modiﬁcation are: (1) It involves only the SO(n)SO(N−n) isotropy derivatives, and hence requires the super anti-de Sitter construction. (The analogous derivatives in ﬂat superspace would be central charges, which would break superconformal invariance. However, we can still use our contracted coset, since the isotropy group is unchanged.) (2) The modiﬁcations must involve only a single ﬁeld strength (φ) to avoid generation of ﬁeld-strength commutator terms (and hence nonclosure) in the algebra of isotropy constraints, and hence both n and N−n ≤ 2. This shows that chiral superspace does not exist for N=4 Yang-Mills.
As an aside, note that the apparently excessive Lorentz derivatives can have their uses: For example, even in the N=0 case, these derivatives are useful for selfdual Yang-Mills. In the lightcone gauge for this theory, we separate the + and − components of the undotted spinor index (but not the dotted one) to solve some of the selfduality conditions as
[∇_{+[α̇}, ∇_{−β̇]}] = 0 ⇒ A_{−α̇} = ∂_{+α̇} A_{−−}
Pure spinors are also related to (coset) Lorentz coordinates.
The usefulness of projective superspace for scattering amplitudes can be seen in the simple example of the 4-point (although there is not yet an oﬀ-shell N=4 covariant derivation). It has the same kinematic factor at all loops, which is multiplied by a purely x-space expression. This factor has a very simple form (Kallosh, ’07), especially because the projective ﬁeld strength φ introduced above is a single scalar. For example, for the tree graph, the θ dependence is the local product, and the y dependence evaluates at y = 0:
δ^{4}(x_{1} − x_{2} + x_{3} − x_{4}) |
x_{12}^{2} x_{23}^{2} |
On shell this amplitude also has a simple supertwistor expression: Besides conservation δ-functions for conjugate momenta for both x and (the 2N projective) θ’s (but not redundant ones for y), it’s simply 1/st. Unlike the chiral or antichiral (MHV or MHV) cases, this depends only on momenta, without twistor phases. This is related to the fact that there are no chiral scalars in N=4 Yang-Mills.