Landau-Lifshitz sigma-models, fermions and the AdS/CFT correspondence
arXiv:0704.1460v1  [hep-th]  11 Apr 2007
hep-th/yymmnnn
MIT-CTP-nnnn
Imperial/TP/2-07/nn
Landau-Lifshitz sigma-models, fermions and
the AdS/CFT correspondence
B. Stefański, jr.1,2
1 Center for Theoretical Physics
Laboratory for Nuclear Science,
Massachusetts Institute of Technology
Cambridge, MA 02139, USA
2 Theoretical Physics Group, Blackett Laboratory,
Imperial College,
London SW7 2BZ, U.K.
Abstract
We define Landau-Lifshitz sigma models on general coset space G/H, with H a maximal
stability sub-group of G. These are non-relativistic models that have G-valued Nöther
charges, local H invariance and are classically integrable. Using this definition, we con-
struct the PSU(2, 2|4)/PS(U(2|2)2) Landau-Lifshitz sigma-model. This sigma model
describes the thermodynamic limit of the spin-chain Hamiltonian obtained from the com-
plete one-loop dilatation operator of the N = 4 super Yang-Mills (SYM) theory. In the
second part of the paper, we identify a number of consistent truncations of the Type IIB
Green-Schwarz action on AdS5×S5 whose field content consists of two real bosons and 4,8
or 16 real fermions. We show that κ-symmetry acts trivially in these sub-sectors. In the
context of the large spin limit of the AdS/CFT correspondence, we map the Lagrangians of
these sub-sectors to corresponding truncations of the PSU(2, 2|4)/PS(U(2|2)2) Landau-
Lifshitz sigma-model.
http://arxiv.org/abs/0704.1460v1
1 Introduction
The gauge/string correspondence
adscft
[1] provides an amazing connection between quantum gauge
and gravity theories. The correspondence is best understood in the case of the maximally
supersymmetric dual pair of N = 4 SU(N) super-Yang-Mills (SYM) gauge theory and Type
IIB string theory on AdS5 × S5. Recent progress in understanding this duality has come from
investigations of states in the dual theories with large charges
bmn,gkp2,ft
[3, 4, 5]. In these large-charge
limits (LCLs) it is possible to test the duality in sectors where quantities are not protected by
supersymmetry. Typically, one compares the energy of some semi-classical string state with
large charges (labelled schematically J) to the anomalous dimensions of the corresponding op-
erator in the dual gauge theory, using 1/J as an expansion parameter which supresses quantum
corrections. A crucial ingredient, which made such comparisons possible, was the observation
that computing anomalous dimensions in the N = 4 SYM gauge theory is equivalent to find-
ing the energy eigenvalues of certain integrable spin-chains
andim
[6] (following the earlier work on
more generic gauge theories
oandim
[7]). At the same time the classical Green-Schwarz (GS) action
for the Type IIB string theory on AdS5 × S5 was shown to be integrable
[21]. The presence of
integrable structures has led to an extensive use of Bethe ansatz-type techniques to investigate
the gauge/string duality
[8]. In particular, impressive results for matching the world-sheet S-
matrix of the GS string sigma-model with the corresponding S-matrix of the spin-chain have
been obtained
smatrix
[10].
The matching of anomalous dimensions of gauge theory operators with the energies of semi-
classical string states was shown to work up to and including two loops in the ’t Hooft coupling
λ. At three loops it was shown that the string and gauge theory results differ. As has been
noted many times in the literature, this result should not be interpretted as a falsification of
the gauge/string correspondence conjecture. Indeed, while the (perturbative) gauge theory
computatons are done at small values in λ, they are compared to dual string theory energies
which are computed at large values of λ and as such are not necessarily comparable. It has
then been a fortunate coincidence that the one- and two-loop results do match.
This match was first established in a number of particular semi-classical string solutions
and corresponding single-trace operators
[5]. Later it was shown that, to leading order in the
LCL, for some bosonic sub-sectors the string action reduced to a generalised Landau-Lifshitz
(LL) sigma model, which also could be obtained as a thermodynamic limit of the corresponding
spin-chain
k1,k2,hl1,st1,k3
[13, 14, 16, 18, 15] (see also
[20]). In this way, by matching Lagrangians on both
sides one can establish that energies of a wide class of string solutions do indeed match with
the corresponding anomalous dimensions of gauge theory operators without having to compute
these on a case-by-case basis.
A natural extension of this programme is to match, to leading order, the LCL of the full
GS action of Type IIB string theory on AdS5 × S5 to the thermodynamic limit of the spin-
chain corresponding to the dilatation operator for the full N = 4 SYM gauge theory; including
fermions on both sides of the map is interesting given the different way in which they enter the
respective actions. On the spin-chain side fermions are on equal footing to bosons
st2,hl2
[19, 17] - the
LL equation, which describes the thermodynamic limit of the system, relates to a super-coset
manifold when fermions are included, as opposed to a coset manifold when there are no fermions.
In particular, both fermions and bosons satisfy equations which are first order in τ and second
order in σ. On the other hand, fermions in the GS action possess κ-symmetry
ws,gs,hm,mt
[22, 23, 24, 25]
and their equations of motion are first order both in τ and σ. Previous progress on this
question was able to match string and spin chain actions in a LCL up to quadratic level in
fermions
mikh,hl2,st2
[20, 17, 19]. Roughly speaking, on the string side, κ-gauge fixed equations of motion
for fermions typically come as 2n first order equations. From these one obtains n second-order
equations for n by ’integrating out’ half of the fermions. Taking a non-relativistic limit on
the worldsheet one ends up with equations which are first order in τ and second order in σ
which can be matched with the corresponding LL equations obtained from the spin chain side.
Matching the terms quartic and higher in the fermions had so far not been achieved, though
it is expected that this should be possible given the results of
fullalgcurve
[9]. However, finding a suitable
κ-gauge in which this matching could be done in a natural way remained an obstacle. Below
we propose a κ gauge which appears to be natural from the point of view of the dual spin-chain
and allows for a matching of higher order fermionic terms in the dual Lagrangians.
In this paper we first present a compact way of writing LL sigma models for quite general
(super-)cosets G/H ; in particular we write down the full PSU(2, 2|4)/PS(U(2|2)2) LL sigma
model which arrises as the thermodynamic limit of the one-loop dilatation operator for the
full N = 4 SYM theory. This generalises earlier work by
[11], and allows one to write down
LL-type actions without having to go through the coherent-state
[12] thermodynamic limit of
the spin chain. We then identify a number of sub-sectors of the classical GS action 1 all of
which have two real bosonic degrees of freedom and a larger number of fermionic degrees of
freedom (specifically 4,8 and 16 real fermionic d.o.f.s 2). Finally, we define a LCL in which
1By a sub-sector we mean that the classical equations of motion for the full GS superstring on AdS5 × S5
admit a truncation in which all other fields are set to zero in a manner which is consistent with their equations
of motion. This is quite familiar in two cases: (i) when one sets all fermions in the GS action to zero and, (ii)
when one further restricts the bosons to lie on some AdSp × Sq sub-space (1 ≥ p , , q ≥ 5).
2The 4 fermion model was previously postulated to be a sub-sector of the classical GS action in
[28] and
represents a starting point for our analysis.
the GS actions for these fermionic sub-sectors reduce to corresponding LL actions. In this way
we match the complete Lagrangians for these sub-sectors and not just the terms quadratic in
fermions. Since the largest of these sectors contains the maximal number of fermions (sixteen)
for a κ-fixed GS action the LCL matching to a LL model gives a clear indication of what the
natural κ-gauge is from the point of view of the dual spin-chain.
The fermionic sub-sectors of the GS action that we find are quite interesting in themselves
because on-shell κ-symmetry acts trivially on them - in particular the sub-sector containing 16
fermionic degrees of freedom contains the same number of fermions as the κ-fixed GS superstring
on AdS5×S5. Since κ-symmetry acts trivially in this case one cannot use it to eliminate half of
the fermions as one does in more conventional GS actions. Further, these fermionic sub-sectors
naturally inherit the classical integrability of the full GS superstring on AdS5×S5 found in
[21].
Integrating out the metric and the two bosonic degrees of freedom one then arrives at a new
class of integrable differential equations for fermions only.
This paper is organised as follows. In section
2 we give a prescription for constructing a LL
sigma model on a general coset G/H . We also present a number of explicit examples of LL
sigma models most relevant to the gauge/string correspondence there and in Appendix
A. In
section
3 we identify the fermionic sub-sectors of the GS superstring on AdS5×S5. In section
we define a LCL in which the GS action of the fermionic sub-sectors reduces, to leading order in
J , to the LL sigma models for the corresponding gauge-theory fermionic sub-sectors. Since the
GS action for the four fermion subsector is quadratic in the remaining appendices to this paper
we present a more detailed discussion of it including a light-cone quantisation in Appendix
a discussion of its conformal invariance in Appendix
C and a T-dual form of the action in
Appendix
2 Landau-Lifshitz sigma models
In this section we construct the Lagrangian for a Landau-Lifshitz (LL) sigma model on a coset
G/H . 3 The Lagrangian will typically be first (second) order in the worldsheet time (space)
coordinate, and so is non-relativistic on the worldsheet. We refer to such models as LL sigma
models because in the case of G/H = SU(2)/U(1) the equations of motion reduce to the usual
LL equation
∂τni = εijknj∂
σnk , where nini = 1 . (2.1)
3For earlier work on this see
[11].
The construction of LL Lagrangians is closely related to coherent states |ω,Λ〉. Recall 4 that
to construct a coherent state |ω ,Λ〉 we need to specify a unitary irreducible representation Λ
of G acting on a Hilbert space VΛ and a vacuum state |0〉 on which H is a maximal stability
sub-group, in other words for any h ∈ H
Λ(h) |0〉 = eiφ(h) |0〉 , (2.2) vacuumphase
with φ(h) ∈ R. Given such a representation Λ and state |0〉 we define the operator Ω as
Ω ≡ |0〉 〈0| . (2.3) Omega
The LL sigma model Lagrangian on G/H is defined as
LLL G/H = LWZLL G/H + LkinLL G/H (2.4) LLsigmamodel
where
LWZLL G/H = −iTr
Ωg†∂τg
, (2.5) LLLWZ
LkinLL G/H = Tr
g†Dσgg
. (2.6)
Above, g†Dσg ≡ g†∂σg− g†∂σg|H is just the standard H-covariant current. It is then clear that
LkinLL G/H is invariant under gauge transformations
g → gh , (2.7)
for any h = h(τ , σ) ∈ H . We may also show that the same is true of LWZLL G/H. To see this note
that the gauge variation of LWZ
LL G/H
, using equation (
vacuumphase
2.2), is given by
δHLWZLL G/H = e−iφ(h) 〈0| ∂τh |0〉 = e−iφ(h)∂τ (〈0|h |0〉) = i∂τφ(h) . (2.8)
This in turn is a total derivative; and so the full action is invariant under local right H action.
The Lagrangian also has a global G symmetry
g → g0g , (2.9)
for any g0 ∈ G with ∂τg0 = ∂σg0 = 0, and the corresponding Nöther current is given by
(jτ , jσ) = (gΩg
†, 2iDσgg
†) . (2.10)
4For a detailed exposition of coherent states see
[12]; a brief summary, using the same notation as in this
paper, is also presented in Appendix A of
[18].
st1,st2
[18, 19] LL actions were written down in terms of Lie algebra matrices denoted typically
by N . To make contact with the present notation we note that 5
N ≡ gΩg† − 1
In , (2.11)
where the second term on the right hand side is included since N is traceless. Finally, let us
note that these LL sigma models admit a Lax pair representation and as a result are integrable.
This is most easily seen in terms of the matrix N for which the equations of motion are the LL
matrix equation
∂τN =
N , ∂2σN
. (2.12)
This is equivalent to the zero-curvature condition on the following Lax pair
L −→ ∂σ −
, (2.13)
M −→ ∂τ −
4π2x2
− [N, ∂σN}
, (2.14)
where [· , ·} is the (super)-commutator. In the remainder of this section we construct a number
of explicit examples of LL sigma models. Further examples of interest in the gauge/string
correspondence are relagated to Appendix
A. The reader who is not interested in the details
of these examples should skip the remainder of this section.
2.1 The U(1|1)/U(1)2 model
This is one of the simplest LL sigma models, 6 in that the Lagrangian is quadratic
LLL U(1|1)/U(1)2 = iψ̄∂τψ + ∂σψ̄∂σψ , (2.15) LLu11
with ψ a complex Grassmann-odd field and ψ̄ its complex conjugate. Notice that this result
can be obtained using the explicit 2×2 supermatrix representation of U(1|1), with the vacuum
state |0〉 being the super-vector (0, 1).
2.2 The SU(3)/S(U(2)× U(1) model
Before proceeding to our main example - the PSU(2, 2|4) model - in this subsection we show how
the above formal prescription applies to the well known SU(3) Landau-Lifshitz model
st1,hl1
[18, 16].
5The following equation is due to Charles Young.
6There is also the equally simple bosonic U(1) LL sigma model.
Recall that the Lagrangian for this is
LSU(3)/S(U(2) × U(1)) = −iU i∂τUi −
|DσUi|2 + Λ(UiU i − 1) , (2.16) gtsu3
where
DµUi ≡ ∂µ − iCµ , Cµ = −iU i∂µUi , (2.17)
for µ = τ, σ and U i ≡ U∗i . To show that we can obtain this from our general expression (
LLsigmamodel
we write elements of the group SU(3) as 3 × 3 matrix g, split into a 3 × 2 matrix X and a
vector Y
g = (X, Y ) , (2.18)
and because g is in SU(3) (i.e. g†g = 1) we have
X†X = 12 , Y
†Y = 11 , X
†Y = 0 , Y †X = 0 , (2.19)
XX† + Y Y † = 13 . (2.20)
The kinetic part of the Lagrangian (
LLsigmamodel
2.4) is then given by
Lkin SU(3)/S(U(2) × U(1)) =
(g−1D1g)(g
−1D1g)
X†D1X X
Y †D1X Y
0 X†∂1Y
Y †∂1X 0
X†∂1Y Y
†Y Y †∂1X
†XX†∂1Y
†(1−XX†)∂1X
i − YiY j)∂1Yj
iD1Yi . (2.21)
The final expression is the same as the kinetic term of the usual SU(3) Landau-Lifshitz La-
grangian (
gtsu3
2.16) upon identifying Yi with Ui (above Y
i ≡ Y †). Above, we have defined
D1Yi = ∂1Yi − YiY j∂1Yj , D̄1Y i ≡ (D1Yi)† (2.22)
D1X = ∂1X −XX†∂1X , D̄1X ≡ (D1X)† . (2.23)
The WZ term of the Lagrangian is given by equation (
LLLWZ
2.5) and can be written as
LWZ SU(3)/S(U(2) × U(1)) = iTr(X†∂0X) = −iY i∂0Yi . (2.24)
This follows from the fact that g−1∂0g is traceless and so
Tr(X†∂0X) = −Y i∂0Yi . (2.25)
Upon identifying Yi with Ui, the WZ term above is the same as the usual SU(3) Landau-
Lifshitz one (
gtsu3
2.16). Notice that we have also given an alternate parametrisation of the SU(3)
Landau-Lifshitz model in terms of X
LSU(3)/S(U(2) × U(1)) = iTr(X†∂0X)−
+ Λ(X†X − 12) , (2.26)
which has an explicit SU(2) gauge invariance.
Finally, out of X and Y we may define a matrix which takes values in the SU(3) Lie algebra
N ij = 3Y
iYj − δij = −3XjaXai + 2δij , (2.27)
where a = 1, 2. This matrix is however, not a general SU(3) matrix but rather satisfies the
identity
N2 = N + 2 . (2.28)
In terms of N the equations of motion take the form of the matrix Landau-Lifshitz equation
∂0N = −
[N , ∂21N ] . (2.29)
These are equivalent to the consistency of the following linear problem
ψ = 0 , (2.30)
4π2x2
N − b
[N, ∂1N ]
ψ = 0 . (2.31)
2.3 The SU(2, 2|4)/S(U(2|2)× U(2|2)) model
In this sub-section we present an explicit Lagrangian for the complete PSU(2, 2|4) Landau-
Lifshitz sigma model Lagrangian following the general discussion at the start of the present
section. The action we are interested in is the Landau Lifshitz model as defined in equation (
LLsigmamodel
on the coset
PSU(2, 2|4)
PS(U(2|2)× U(2|2)) , (2.32)
or on the coset
SU(2, 2|4)
S(U(2|2)× U(2|2)) , (2.33)
both of which have 32 real components. The derivation is very similar to the SU(3) Lagrangian
derived in the previous sub-section, and so we will simply state our results. A general group
element g can be written as (X, Y ) where now X (Y ) is a 8× 4 supermatrix, with the diagonal
4 × 4 blocks bosonic (fermionic) and the off-diagonal 4 × 4 blocks fermionic (bosonic). The
Lagrangian is then given by
LLL PSU(2, 2|4)/PS(U(2|2) × U(2|2)) = iSTr(X†∂0X)−
STr(D̄1X
†D1X) + Λ(X
†X − 1) . (2.34)
Note that there are 32 complex degrees of freedom in X , which the constraints reduce to 48
real degrees of freedom. The action also has a local U(2|2) gauge invariance, so in total the
above Lagrangian has 32 degrees of freedom - the same as the coset.
In fact we may write X as
X = (Ũa, Ṽa, Ua, Va) , X
† ≡ (Ũa, Ṽ a, Ua, V a) , (2.35)
where a = 1, . . . , 8, and
ŨaŨa = −1 , Ṽ aṼa = −1 , Ṽ aŨa = 0 , ŨaṼa = 0 , (2.36)
UaUa = 1 , V
aVa = 1 , V
aUa = 0 , U
aVa = 0 , (2.37)
UaŨa = 0 , U
aṼa = 0 , V
aŨa = 0 , V
aṼa = 0 , (2.38)
ŨaUa = 0 , Ũ
aVa = 0 , Ṽ
aUa = 0 , Ṽ
aVa = 0 . (2.39)
Above we have defined
Ua = U∗bC
ba , V a = V ∗b C
ba , Ũa = −Ũ∗bCba , Ṽ a = −Ṽ ∗b Cba , (2.40)
where Cab = diag(−1,−1, 1, 1, 1, 1, 1, 1).
The Lagrangian (
psu224
2.41) written in terms of Ũa, Ṽa, Ua, Va is
LLL PSU(2, 2|4)/PS(U(2|2)2) = −iŨa∂0Ũa − iṼ a∂0Ṽa − iUa∂0Ua − iV a∂0Va
a∂1Ũa + ∂1Ṽ
a∂1Ṽa + ∂1U
a∂1Ua + ∂1V
a∂1Va
−Ũa∂1ŨaŨ b∂1Ũb − Ṽ a∂1ṼaṼ b∂1Ṽb
+V a∂1VaV
b∂1Vb + U
a∂1UaU
b∂1Ub
+2V a∂1UaU
b∂1Vb − 2Ṽ a∂1ŨaŨ b∂1Ṽb + 2Ũa∂1UaU b∂1Ũb
+2Ũa∂1VaV
b∂1Ũb + 2Ṽ
a∂1UaU
b∂1Ṽb + 2Ṽ
a∂1VaV
b∂1Ṽb
. (2.41) psu224
One can check explicitly that this action has local U(2|2) invariance
(Ũa, Ṽa, Ua, Va)→ (Ũa, Ṽa, Ua, Va)U(τ, σ) , (2.42)
for U a U(2|2) matrix.
2.3.1 Subsectors of the the SU(2, 2|4)/S(U(2|2)× U(2|2)) model
In the above Lagrangian we may set
Ũa = (1, 0
7) , Ṽa = (0, 1, 0
6) , Ua = (0
2, U3, . . . , U8) , Va = (0
2, V3, . . . , V8) , (2.43)
where
UaUa = 1 , V
aVa = 1 , V
aUa = 0 , U
aVa = 0 . (2.44)
The resulting Lagrangian is that of the SU(2|4) sector. If we further set
0 = U3 = U4 = V3 = V4 , (2.45)
we can recover the SO(6) Lagrangian (
[18]). Details of this are presented in Appendix
B. We
may further consistently set
0 = U8 = V3 = V4 = V5 = V6 = V7 , V8 = 1 , (2.46)
in which case we obtain the SU(2|3) Lagrangian (
[19]), with the identification (U3, U4) ≡
(ψ1, ψ2).
We may instead set
Ua = (0
7, 1) , Va = (0
6, 1, 0) , Ũa = (U1, . . . , U6, 0
2) , Ṽa = (V1, . . . , V6, 0
2) , (2.47)
where
ŨaŨa = −1 , Ṽ aṼa = −1 , Ṽ aŨa = 0 , ŨaṼa = 0 . (2.48)
The resulting Lagrangian is that of the SU(2,2|2) sector. If we further set
0 = U3 = U4 = V3 = V4 , (2.49)
we recover the SO(2,4) Lagrangian, which is the Wick rotated version of the SO(6) La-
grangian (
[18]). In Appendix
B we write out this Lagrangian explicitly.
A final interesting choice is to set
Ua = (0
7, 1) , Va = (0, V2, . . . , V7, 0) , Ũa = (0, U2, . . . , U7, 0) , Ṽa = (1, 0
7) , (2.50)
where
ŨaŨa = −1 , V aVa = 1 , V aŨa = 0 , ŨaVa = 0 . (2.51)
The resulting Lagrangian is that of the SU(1,2|3) sector. If we further set
0 = V2 = V7 = Ũ2 = Ũ7 , (2.52)
we get the SU(2|2) Lagrangian. In Appendix
B we write out this Lagrangian explicitly.
3 Green-Schwarz actions and fake κ-symmetry
in this section we construct GS sigma model actions whose field content are two real bosons
and 4,8 or 16 real fermions. These models all come from consistent truncations of the equations
of motion for the full Type IIB GS action on AdS5 × S5. Just as any GS sigma model these
fermionic actions have a κ-symmetry. However, we show that for these models κ-symmetry is
trivial on-shell. As a result one cannot use it to reduce the fermionic degrees of freedom of
these models by fixing a κ-gauge as one does in more conventional GS actions.
Let us briefly recall the construction of the GS action on a super-coset G/H . We require
that: (i) H be bosonic and, (ii) G admit a ZZ4 automorphism that leaves H invariant, acts
by −1 on the remaining bosonic part of G/H , and by ±i on the fermionic part of G/H . The
currents jµ = g
†∂µg can then be decomposed as
jµ = j
µ + j
µ + j
µ + j
µ , (3.1) currz4dec
where j(k) has eigenvalue ik under the ZZ4 automorphism. In terms of these the GS action can
be written as
LGS G/H =
−ggµνStr(j(2)µ j(2)ν ) + ǫµνStr(j(1)µ j(3)ν ) , (3.2) z4gs
from which the equations of motion are
0 = ∂α(
−ggαβj(2)β )−
−ggαβ
j(0)α , j
j(1)α , j
j(3)α , j
, (3.3) eom1
−ggαβ + ǫαβ
j(3)α , j
, (3.4) eom2
−ggαβ − ǫαβ
j(1)α , j
. (3.5) eom3
3.1 Fermionic GS actions
Having briefly reviewed the general construction of GS actions on G/H super-cosets, we now
turn to the main focus of this section which is identifying GS actions with a large number of
fermionic degrees of freedom, which are consistent truncations of the full AdS5×S5 GS action.
To do this consider the following sequence of super-cosets
U(1|1)× U(1|1)
U(1)× U(1) ⊂
U(2|2)
SU(2)× SU(2) ⊂
PS(U(1, 1|2)× U(2|2))
SU(1, 1)× SU(2)3 ⊂
PSU(2, 2|4)
SO(1, 4)× SO(5) . (3.6)
The ⊂ symbols are valid both for the numerators and denominators and hence for the cosets
as written above. Notice that the right-most of these cosets is just the usual Type IIB on
AdS5 × S5 super-coset. Further, it is easy to convince onself that each of the cosets above
admits a ZZ4 automorphism which is compatible with the ZZ4 automorphism of the Type IIB
on AdS5 × S5 super-coset. The ZZ4 automorphisms may be used to write down GS actions
for each of these cosets. The fact that the cosets embed into each other as shown above in a
manner compatible with the ZZ4 automorphism implies that their GS actions can be thought
of as coming from a consistent truncation of the GS action of any coset to the right of it in
the above sequence. In particular this reasoning shows that the GS actions for U(1|1)2/U(1)2,
U(2|2)/SU(2)2 and U(1, 1|2)×U(2|2)/(SU(1, 1)×SU(2)3 can all be thought of as coming from
consistent truncations of the Type IIB GS action on AdS5 × S5.
Counting the number of bosonic and fermionic components of the three cosets U(1|1)2/U(1)2,
U(2|2)/SU(2)2 and U(1, 1|2)× U(2|2)/(SU(1, 1)× SU(2)3 we see immediately that they each
have 2 real bosonic components and, respectively, 4,8 and 16 real fermionic components - which
is why we refer to these actions as fermionic GS actions. We might expect that some of the
femrionic degrees of freedom could be eliminated from the GS actions by fixing κ-symmetry.
In fact, it turns out that for these models κ-symmetry acts trivially on-shell and so cannot
be used to eliminate some of the fermionic degrees of freedom. Indeed, the GS actions on the
above-mentioned cosets do have 4,8 and 16 real fermionic degrees of freedom, respectively.
In the remainder of this sub-section we write down explicitly the GS actions for U(2|2)/SU(2)2
and U(1|1)2/U(1)2 and discuss their κ and gauge transformations; the GS action for U(1, 1|2)×
U(2|2)/SU(1, 1)× SU(2)3 may also be written down in an analogous fashion but since we will
not need its explicit form later we refrain from writing it out in full.
3.2 The GS action on U(2|2)/SU(2)2
The GS action on action on U(2|2)/SU(2)2 can be written down in terms of the parametrisation
of the U(2|2) supergroup-valued matrix written as
g = (X, Y ; X̃, Ỹ ) , (3.7) ads2s2param
whereX , Y (X̃ , Ỹ ) are four-component super-vectors with the first (last) two entries Grassmann
even and the last (first) two entries Grassmann odd. Since the matrix g is unitary we must
1 = X†X = Y †Y = X̃†X̃ = Ỹ †Ỹ ,
0 = X†Y = Y †X = X†X̃ = X̃†X = X†Ỹ = Ỹ †X
= Y †X̃ = X̃†Y = Y †Ỹ = Ỹ †Y = X̃†Ỹ = Ỹ †X̃ ,
1(2|2) = XX
† + Y Y † + X̃X̃† + Ỹ Ỹ † , (3.8) ads2s2const
where the matrix 1(2|2) is just the 4× 4 identity matrix. The ZZ4 automorphism is given by
Ω : M =
−AT CT
−BT −DT
, (3.9) Z4autu22
which acts on the current as
Ω(jµ) =
−Y †∂µY X†∂µY −Ỹ †∂µY X̃†∂µY
Y †∂µX X
†∂µX Ỹ
†∂µX −X̃†∂µX
Y †∂µỸ −X†∂µỸ −Ỹ †∂µỸ X̃†∂µỸ
−Y †∂µX̃ X†∂µX̃ Ỹ †∂µX̃ −X̃†∂µX̃
. (3.10)
The Green-Schwarz action then is
LGS U(2|2)/(SU(2)×SU(2)) =
(X†∂µX + Y
†∂µY )(X
†∂νX + Y
†∂νY )
−(X̃†∂µX̃ + Ỹ †∂µỸ )(X̃†∂νX̃ + Ỹ †∂ν Ỹ )
+2iǫµν
X†∂µX̃Y
†∂νỸ + Ỹ
†∂µY X̃
−X†∂µỸ Y †∂νX̃ − X̃†∂µY Ỹ †∂νX
. (3.11) u22gs
One can easily check that this action has a local SU(2)× SU(2) invariance which acts on the
doublets (X, Y ) and (X̃, Ỹ ). The action also has κ-symmetry which acts on the fields as 7
δκX = −X̃(ǭ1 + ¯̃ǫ1)− Ỹ (ǭ2 + ¯̃ǫ2)
δκY = iX̃(ǫ2 − ǫ̃2)− iỸ (ǫ1 − ǫ̃1)
δκX̃ = X(ǫ1 + ǫ̃1) + iY (ǭ2 − ¯̃ǫ2)
δκỸ = X(ǫ2 + ǫ̃2)− iY (ǭ1 − ¯̃ǫ1) , (3.12) ku22coord
where
ǫi = Π
†∂αX + Y
†∂αY + X̃
†∂αX̃ + Ỹ
†∂αỸ )κi , β
ǫ̃i = Π
†∂αX + Y
†∂αY + X̃
†∂αX̃ + Ỹ
†∂αỸ )κ̃i , β , (3.13) ku22eps
for i = 1 , 2 with κi , β and κ̃i , β local Grassmann-odd parameters. The world-sheet metric also
varies as
−ggαβ) = Παγ+
1 ,+(X̃
†∂γX − iY †∂γ Ỹ ) + κβ2 ,+(Ỹ †∂γX + iY †∂γX̃) + c.c.
+ α↔ β
1 ,−(X̃
†∂γX + iY
†∂γỸ ) + κ̃
2 ,−(Ỹ
†∂γX − iY †∂γX̃) + c.c.
+ α↔ β .
(3.14) ku22metric
7The κ-action below has the nice feature of acting as a local fermionic group action by multiplication from
the right. Such a representation was originally suggested in
mcarthur
[26] and was developed more fully for the AdS5×S5
GS action in
glebnotes
[27]; the formulas below are a simple extension of this latter construction to the coset at hand.
Notice that the above variation is consistent with the symmetries and the unimodularity of
√−ggαβ as long as
καi = Π
+ κi , β , κ̃
i = Π
− κ̃i , β . (3.15)
In the above formulas we have decomposed two-component vectors vα as
vα± ≡ Π
± vβ ≡
−ggαβ ± ǫαβ
vβ . (3.16)
3.3 The GS action on U(1|1)2/U(1)2
To obtain the GS action on U(1|1)2/U(1)2 we may simply set
0 = X3 = Y4 = X̃1 = Ỹ2 . (3.17)
in the action (
u22gs
3.11). This is because now the group element g given in equation (
ads2s2param
3.7) belongs to
U(1|1)2 ⊂ U(2|2); this truncation is also consistent with the ZZ4 automorphism (
Z4autu22
3.9). As was
argued at the start of this sub-section these facts imply that setting the above components to
zero is a consistent truncation of the equations of motion for the action (
u22gs
3.11). The GS action
for the truncated theory then is
LGS U(1|1)2/U(1)2 =
(X†∂µX + Y
†∂µY )(X
†∂νX + Y
†∂νY )
−(X̃†∂µX̃ + Ỹ †∂µỸ )(X̃†∂νX̃ + Ỹ †∂ν Ỹ )
−2iǫµν
X†∂µỸ Y
†∂νX̃ + X̃
†∂µY Ỹ
. (3.18) gsu112
It has two U(1) gauge invariances
X → eiθ1X , Y → eiθ1Y , (3.19)
X̃ → eiθ2X̃ , Ỹ → eiθ2 Ỹ , (3.20)
as well as κ-symmetry which is simply the restriction of equations (
ku22coord
3.12) and (
ku22metric
3.14).
If we parametrise the group element g = (X, Y, X̃, Ỹ ) ∈ U(1|1)2 by
X = (eit/2(1 +
ψ2) , 0 , 0 , −e−iα/2ψ̄) , Y = (0 , eit/2(1 + 1
η2) , −e−iα/2η̄ , 0) ,(3.21)
X̃ = (0 , eit/2η , e−iα/2(1− 1
η2) , 0) , Ỹ = (eit/2ψ , 0 , 0 , e−iα/2(1− 1
ψ2)) , (3.22)
where ψ2 ≡ ψ̄ψ and η2 ≡ η̄η, the action (
gsu112
3.18) becomes
LGS U(1|1)2/U(1)2 =
−∂µφ+∂νφ− + i∂µφ+ηi
i − ∂µφ+∂νφ+ηiηi
−ǫµν∂µφ+(η1
∂ν η2 − η1
2) , (3.23) gsu112comp
This action was postulated in
[28] to be a consistent truncation of the full Type IIB GS action
on AdS5 × S5, by checking the absense of certain cubic terms in the latter action, using an ex-
plicit non-unitary representation for PSU(2, 2|4). Here we have shown that on group-theoretic
grounds this action is indeed such a consistent truncation, and have obtained its form using a
unitary representation of the group.
On the local coordinates defined above κ-symmetry acts as
δηi = ǫi , δt = −δα = i
ηiǫi + ηiǫ
. (3.24)
In particular notice that δφ+ = 0. The parameters ǫi are not however free, instead they are
given by
i + i∂αφ− + iη
iηi∂αφ+
καj . (3.25) epskappa
Above καi are complex-valued Grassmann functions of the world-sheet; their complex conjugates
are denoted by κα i. We will also require that the metric vary under κ-symmetry as
−ggαβ) = − i
+ (−η1∂γφ+ − iη2∂γφ+ + 2i∂γη1 + 2∂γη2)
+κ̄(αP
+ (−η1∂γφ+ + iη2∂γφ+ − 2i∂γη1 + 2∂γη2)
+κ̃(αP
− (η1∂γφ+ − iη2∂γφ+ − 2i∂γη1 + 2∂γη2)
1∂γφ+ + iη2∂γφ+ + 2i∂γη
1 + 2∂γη2)
= − i
−ggαγ
κβ i∂γηi + κ
i ∂γη
∂γφ+(κ
β iηi − κβi ηi)
+iǫαγ
1∂γη2 + κ
2∂γη1 − κβ 1∂γη2 − κβ 2∂γη1
ǫαγ∂γφ+
1η2 + κ
2η1 + κ
β 1η2 + κβ 2η1
. (3.26) epskappa
where a(αbβ) = aαbβ + aβbα and
κα1 =
α − κ̄α) , κα2 =
(κ̃α + κα) , (3.27)
with the complex conjugates defined as κ† ≡ κ̄ and κ̃† ≡ ¯̃κ. The above variation of the metric
is symmetric and since
−ggαβ has unit determinant (is uni-modular) we require that
κα = P
+ κβ , κ̃
α = P
− κ̃β . (3.28)
Using the above formulas one can check that the action (
gsu112comp
3.23) is indeed invariant under this
symmetry. However, as we show below this local symmetry is trivial on-shell.
3.4 Fake κ-symmetry
partlim
In this sub-section we show that κ-symmetry acts trivially on-shell on the fermionic GS actions
studied in this section. To see this most easily we will first consider the particle limit (in other
words we remove all σ dependence of fields) for the action LGS U(1|1)2/U(1)2 . This gives
Lparticle = −
dτe−1φ̇+
φ̇− + φ̇+η
iηi − iηiη̇i − iηiη̇i
dτe−1φ̇+a , (3.29) supart
where for convenience we have defined 8
φ̇− + φ̇+η
iηi − iηiη̇i − iηiη̇i
. (3.30)
Setting e = constant, we may solve the the φ+, φ− and ηi equations of motion to get
φ+ = 2κτ , φ− = λτ , ηi = e
−iκτη0 i , (3.31)
where κ, λ (respectively, η0 i) are complex constant Grassmann-even (odd) numbers.
9 Finally,
we turn to the equation for the einbein e which reduces to
κλ = 0 . (3.32)
or in other words forces us to set either κ or λ to zero. As a result the theory consists of two
sectors, one with κ = 0 and the other with λ = 0. The former sector is trivial and uninteresting
as all fields apart from φ− are constant and the energy is zero. The physically more relevant
sector has λ = 0 and κ 6= 0.
Let us now turn to the κ invariance of the action (
supart
3.29). It is easy to see that this action is
invariant under
δφ+ = 0 , δηi = aκi , δφ− = ia(η
iκi + ηiκ
δ(e−1) = 2i(η̇iκi + η̇iκ
i) + φ̇+(η
iκi + ηiκ
i) , (3.33)
where κi are arbitrary Grassmann-odd functions of τ . Since we are free to pick the parameters
κi one might think that we could simply gauge away the femrionic degrees of freedom using
this symmetry; had the κ variations been of the form
δηi = κi ,
8As an aside note that the fermion index i can now run over any number and is not restricted to i = 1, 2 as
is the case for the super-string. This is quite typical of κ-invariant particle actions.
9In the above solution we have, without loss of generality, set the constant parts of φ+ and φ− to zero.
we would have been able to gauge away the fermions. In fact this is not the case: the κ variation
of the fermions instead reads
δηi = aκi , (3.34)
From the equation for the einbein e we see that in fact a = 0 (in the physically important
sector for which κ 6= 0 as discussed above) and so on-shell the above κ symmetry acts trivially
on all fields except the einbein itself. But any κ variation of the einbein e can be compensated
for by a diffeomorphism. We conclude that while the actions (
supart
3.29) and (
gsu112
3.18) formally have a
κ-symmetry, this has a trivial action on-shell and so cannot be used to eliminate any fermions.
The argument in the above paragraph relies on the fact that on fermions κ-symmetry was acting
as δηi = aκi and on-shell a = 0. Returing to the fermionic GS superstring actions discussed in
this section we see from equation (
ku22eps
3.13) that here too κ-symmetry acts as δηi = astringκi, where
astring = (X
†∂αX + Y
†∂αY + X̃
†∂αX̃ + Ỹ
†∂αỸ ) . (3.35)
It is easy to check that because of the Virasoro constrains astring is also zero on-shell. We
conclude that the κ-symmetry of the action (
gsu112comp
3.23) is trivial on-shell and so cannot be used to
eliminate any fermions.
4 Large charge limits of fermionic GS actions
Given a ZZ4 automorphism on some coset G/H we may construct a Green-Schwarz Lagrangian
for it (
3.2). On general grounds the large charge limit of this Lagrangian should be a generalised
Landau Lifshitz sigma model. Further, since we expect the global charges of the two actions
to map onto one another, this LL sigma model should be constructed on a coset G/H̃. In this
section we will attempt to identify H̃.
One step in this direction is to count the number of degrees of freedom that the GS action
has and compare it with that of the LL model. For example in the case of the Type IIB
superstirng on AdS5×S5 there are 10 real bosonic degrees of freedom, and there are 32/2 = 16
fermionic degrees of freedom (where the factor of 1/2 comes from κ symmetry). In the large
charge limit two of the bosonic degrees of freedom are eliminated; the remaining eight are
’doubled’ since the LL Lagrangian should be thought of as a Lagrangian on phase space. The
16 fermions are described by coupled first order equations. When taking the LCL we integrate
out half of the fermions, in order to arrive at second order equations
[19], leaving us with 8 real
fermionic degrees freedom; as in the case of the bosons this should also be ’doubled’, leaving
us with 16 fermionic degrees of freedom. At this point we may simply guess what H̃ is in the
case of G = PSU(2, 2|4), since the only coset of the form G/H̃ with 16 bosonic and fermionic
degrees of freedom each is
H̃ = PS(U(1, 1|2)× U(2|2)) , (4.1)
though of course in this case H̃ is well known from gauge theory.
Let us persue this counting argument further and consider the GS action on
U(1|1)2
U(1)2
. (4.2)
This is a sub-sector of the classical GS string action on AdS5 × S5. It has 2 real bosonic
degrees of freedom and 4 real fermionic degrees of freedom. As was shown in section
partlim
3.4, κ-
symmetry in this case is trivial on-shell, and so, following the counting argument in the previous
paragraph, 10 we expect the LL sigma model corresponding to the LCL of this GS action to
have 4 real fermionic degrees of freedom and no bosonic degrees of freedom. The only such
coset is
U(1|1)2
U(1)4
, (4.3)
in other words H̃ = U(1)4.
Similarily, we may consider the bigger sub-sector of the full classical superstring on AdS5×S5
U(2|2)
SU(2)2
, (4.4)
for which κ-symmetry is also trivial on-shell. This sub-sector has 2 bosonic and 8 fermionic
d.o.f. As a result we expect the LL sigma-model to have no bosonic d.o.f. and 8 fermionic d.o.f.
Again this is enough for us to identify
U(2|2)
U(2)2
, (4.5)
as the coset on which the LL sigma model is constructed. Finally, the largest classical sub-
sector of the GS string action on AdS5 × S5 for which κ-symmetry is trivial is the GS action
PS(U(1, 1|2)× U(2|2))
SU(1, 1)× SU(2)3 . (4.6)
10For the bosons we subtract two real degrees of freedom in the LCL and double the remaining ones. In the
present case this gives 2× (2−2) = 0 d.o.f. For the fermions, the number of d.o.f. in the LL sigma model should
be the same as that of the GS string once κ-symmetry is fixed. This is because, once κ-symmetry is fixed, we
halve the number of d.o.f. since the GS action gives first order differential equations, and the LL action gives
second order differential equations; we then double it because the LL action is an action on phase space. In the
present case, since κ-symmetry is trivial on-shell we end up with 2× 4/2 = 4 fermionic d.o.f.
By our counting argument the corresponding LCL coset should have 16 fermionic and no bosonic
d.o.f. As a result, the LL sigma model which corresponds to the LCL limit of the GS action on
(U(1, 1|2)× U(2|2))/SU(1, 1)× SU(2)3 is constructed over the coset
PS(U(1, 1|2)× U(2|2))
U(1, 1)× U(2)3 . (4.7)
While this counting argument shows how to identify H̃, it is not very clear how the LCL
should be taken in practice and in particular how starting from a GS action one arrives at a LL
action. The rest of this section will address these issues in the three cases ofG = U(1|1)2 , U(2|2)
and U(1, 1|2)×U(2|2). We will restrict our discusion to the leading order term in the LCL and
leave the matching of sub-leading terms to a future publication.
4.1 Matching the U(1|1)2 sub-sectors
In this subsection we will argue that the large charge limit of the Lagrangian given in equa-
tions (
gsu112
3.18) and (
gsu112comp
3.23) which describes the Green-Schwarz string on the coset
U(1|1)2
U(1)2
, (4.8)
is given by the Landau-Lifshitz Lagrangian on the coset 11
U(1|1)2
U(1)4
. (4.9)
We will first arrive at this result in a very pedestrian way. Since general solutions to both
the LL and GS cosets can be given explicitly in full generality we will write them down using
unconstrained coordinates. On the GS side,
φ+ = κτ , (4.10)
the general solution takes the form
eiωnτψ+n + e
−iωnτψ−n
, (4.11)
where ψ±n are constant Grassmann-odd numbers, and
n2 + κ2/4 . (4.12)
11This is somewhat different to the comparison between gauge and string theory done in (
[28]) where it was
argued that on the gauge theory side the coset should be U(1|1)/U(1)2.
η2 is completely determined via the equation of motion
∂ση2 = i∂τη
1 − κ
η1 . (4.13)
In the LCL we take κ→∞ in which case we have
ei(κ/2+n
2/κ)τψ+n + e
−i(κ/2+n2/κ)τψ−n
= eiκτ/2
ψ+0 +
ψ+n e
inσ + ψ+−ne
+e−iκτ/2
ψ−0 +
ψ−n e
inσ + ψ−−ne
≡ eiκτ/2ψ1 LL + e−iκτ/2ψ̄2 LL , (4.14)
where ψ1 LL and ψ2 LL are the 2 complex fermionic d.o.f. for the LL sigma model on (see
equation (
LLu11
2.15))
U(1|1)2
U(1)4
. (4.15)
In particular, after rescaling τ → κτ , they satisfy the equations of motion
∂2σ − i∂τ
ψ1,2 LL . (4.16)
In this way we match, to leading order in the LCL, the classical string Lagrangian with the
corresponding coherent state continuum limit of the gauge theory dilatation operator in the
U(1|1)2 sub-sector.
Notice that physical string solutions have to satisfy the level-matching condition
∂1φ− = 2πm , for m ∈ ZZ . (4.17) levmatch
The winding parameter m does not, however, enter the LCL Lagrangian Rather, it gives a
constraint on its solutions. This matches the spin-chain side where m enters as a constraint
on the Bethe roots, but does not enter the algebraic Bethe equations or the LL sigma-model
action. This feature is very similar to the SL(2) sector discussed in
[29].
4.2 Large Charge Limit of fermionic GS actions
In this section we re-phrase the above discussion in terms of the embedding coordinates
X, Y . . . , and the currents j
µ . This allows for a straightforward generalisation from the
U(1|1)2 sub-sector to the U(2|2) and U(1, 1|2) × U(2|2) sub-sectors. We present the explicit
discussion only for the case of U(2|2), but the other case follows almost trivially.
The first thing to note is that the equation of motion for one of the two bosonic fields, φ+,
is particularily simple in the GS models presently considered. This can be obtained as the
super-trace of equation (
3.3). As a result we may set
X†∂µX + Y
†∂µY − X̃†∂µX̃ − Ỹ †∂µỸ = iκδµ , ,0 . (4.18) phipansatz
Using this, in conformal gauge the equation of motion for the off-diagonal component of the
worldsheet metric implies that
X†∂σX + Y
†∂σY + X̃
†∂σX̃ + Ỹ
†∂σỸ = 0 , (4.19) offdiagVir
while the fermionic equations of motion (
3.4), (
3.5) reduce to 12
0 = κ(j(3)τ − j(3)σ ) + . . . , 0 = κ(j(1)τ + j(1)σ ) + . . . . (4.20) fermrel
As a result of these relations the WZ term does not contribute to the bosonic equation of
motion (
3.3). 13 This fact allows us to check explicitly that the bosonic equations of motion,
together with the ansatz (
phipansatz
4.18), are consistent with the equations of motion for the metric gµν
in conformal gauge. In fact these Virasoro constraints then imply that
Dµt = δµ ,0
, D̃µα = −δµ ,0
. (4.21)
As in the discussion around equation (
levmatch
4.17) above, the level matching condition that follows
from the Virasoro constraints does not enter the LCL action.
Using equations (
phipansatz
4.18), (
offdiagVir
4.19) and (
fermrel
4.20) together with a rescaling τ → κτ we may re-write
12In terms of X, Y, X̃, Ỹ this implies that we have relations of the form
X†∂τ Ỹ = iX̃
†∂σY , X̃
†∂τY = −iX†∂σỸ , etc .
13This is easy to see since the WZ term’s contribution to these equations is proportional to
τ , j
τ , j
. However, since j
τ = −j(1)σ and j(3)τ = j(3)σ each of these commutators vanishes seperately.
the GS Lagrangian in conformal gauge as follows
LGS U(2|2)/SU(2)2 = ηµνStr(j(2)µ j(2)ν ) + ǫµνStr(j(1)µ j(3)ν )
= ηµν
X†∂µX + Y
†∂µY − X̃†∂µX̃ − Ỹ †∂µỸ
X†∂µX + Y
†∂µY + X̃
†∂µX̃ + Ỹ
†∂µỸ
−2Str
j(1)σ j
X†∂τX + Y
†∂τY + X̃
†∂τ X̃ + Ỹ
†∂τ Ỹ
− STr
(j(1)σ + j
σ )(j
σ + j
= LLL U(2|2)/U(2)2 (4.22)
The right-hand side of the above equation is nothing but the LL sigma model Lagrangian
defined on G/H̃, where H̃ is fixed under the ZZ2 automorphism which is the square of the ZZ4
automorphism used in the construction of the GS action. We have thus shown that to leading
order in the LCL the fermionic GS actions constructed in section
3 above reduce to LL sigma
model actions in the manner anticipated by the general argument presented at the start of the
present section. It would be interesting to consider sub-leading corrections to this LCL for
example in a manner similar to
[15].
4.3 A gauge-theory inspired κ gauge
The GS sigma model on AdS5 × S5 has κ-symmetry. This, as well as other symmetries of the
string action, such as world-sheet diffeomorphisms, are not manifest in the corresponding spin-
chain simply because this latter system keeps track only of the physical degrees of freedom. One
of the challenges of defining a LCL is to identify suitable gauges for these stringy symmetries
in which the physical degrees of freedom are written in the most natural coordinates for the
spin-chain: while all gauges should be in principle equivalent it may be much more difficult
to define a LCL between the two theories if we pick an unnatural gauge. In the previous
sub-section we have defined an LCL which matches all 16 fermionic degrees of freedom from
the GS action to the corresponding LL model in a very natural way. This strongly suggests
what κ-gauge should be used in the full AdS5 × S5 string action when comparing to gauge
theory. Specifically it should be the gauge which keeps non-zero the 16 fermions of the coset
PS(U(1, 1|2)×U(2|2))/(SU(1, 1)×SU(2)3). In fact this is the gauge used recently in
[30] and
the above argument can be interpreted as one motivation for their κ-gauge choice.
Acknowledgements
I am grateful to Arkady Tseytlin for many stimulating discussions throughout this project and
to Chris Hull for a number of detailed conversations on κ-symmetry. I would also like to thanks
Charles Young for discussions and Gleb Arutyunov for providing a copy of his notes
glebnotes
[27]. This
research is funded by EPSRC and MCOIF.
A Some examples of Landau-Lifshitz sigma models
In this appendix we collect some expressions for a number of relevant Landau-Lifshitz sigma
models.
A.1 The SU(2|3)/S(U(2|2)× U(1)) model
The SU(2|3) sub-sector sigma model Lagrangian is
LLL SU(2|3) = −iU i∂τUi − iψα∂τψa −
|DσUi|2 −
aDσψa + Λ(UiU
i + ψaψ
a − 1) , (A.1) gtsu23
where
Dµ ≡ ∂µ − iCµ , D̄µ ≡ ∂µ + iCµ , Cµ = −iU i∂µUi − iψa∂µψa , (A.2)
and ψa = ψ∗a and a = 1, 2.
A.2 The SU(4)/S(U(2)× U(2)) model
The SO(6) ∼ SU(4) sub-sector sigma model Lagrangian is
LLL SU(4) = LSU(4) WZ −
Tr(∂1m)
2 − 1
Tr(m∂1m)
2 + Λ(m−m3)
= −iV i∂τVi −
|DσVi|2 + Λ1(V iVi − 1) + Λ2(ViVi − 1) + Λ∗2(V iV i − 1) ,(A.3) gtso6
where mij is a 6× 6 matrix, related to Vi by
mij = ViV
j − VjV i , (A.4)
Dµ ≡ ∂µ − iCµ , Cµ = −iV i∂µVi . (A.5)
Let us define
MAB =
B , mij =
tr(Mρij) , (A.6)
where ρ are the usual SU(4) ρ-matrices. Notice that
TrM = 0 , M † =M , M2 =M . (A.7)
and so we can write it as
M = 2XX† − 1 = −2Y Y † + 1 , (A.8)
where now X and Y are 4× 2 matrices which satisfy
X†X = 12 , Y
†Y = 12 , X
†Y = 0 , Y †X = 0 , (A.9)
XX† + Y Y † = 14 . (A.10)
Further we can write the 4× 2 matrix X as two four-component vectors uA and vA
X = (uA, vA) , (A.11) collvect
in terms of which MAB can be written as
MAB = 2u
AuB + 2v
AvB − δAB , (A.12)
uAuA = 1 , v
AvA = 1 , u
AvA = 0 . (A.13) uvconds
We can relate uA and vA to Vi by
uAρiABv
B , V i =
iABuB . (A.14) natgencoords
It is an easy check to see that these are consistent with
i = 1 , ViVi = 0 , M
B = ViV
jρijAB . (A.15)
In terms of these, the Lagrangian is
LLL SU(4) = −iuA∂0uA − ivA∂0vA −
A∂1uA + ∂1v
A∂1vA
+uA∂1uAu
B∂1uB + v
A∂1vAv
B∂1vB + 2u
A∂1vAv
B∂1uB
= −iTr(X†∂0X)−
Tr(D̄1X
†D1X) , (A.16) goodllso6
As before
X = (uA, vA) , X
, (A.17)
DµX = ∂µX −XX†∂µX . (A.18)
The action (
goodllso6
A.16) has a local U(2) invariance
X → XU(τ, σ) , (A.19)
for U(τ, σ) a general U(2) matrix
U †(τ, σ)U(τ, σ) = U(τ, σ)U †(τ, σ) = 12 . (A.20)
In terms of the uA and vA the action (
goodllso6
A.16) is invariant with respect to the following local
transformations
(uA, vA) → (cos θ(τ, σ) uA + sin θ(τ, σ) vA,− sin θ(τ, σ) uA + cos θ(τ, σ) vA) ,
(uA, vA) → (eiφ1(τ,σ)uA, eiφ1(τ,σ)vA) ,
(uA, vA) → (eiφ2(τ,σ)uA, e−iφ2(τ,σ)vA) ,
(uA, vA) → (eiφ3(τ,σ)vA,−eiφ3(τ,σ)uA) , (A.21)
A.2.1 Subsectors of the SU(4)/S(U(2)× U(2)) model
so6sub
When written in terms of the Vi, the Lagrangian LSU(4) can be reduced to the SU(3) sub-sector
by requiring
V 2a = −iV 2a−1 ≡ 1√
Ua , a = 1, 2, 3 , (A.22)
which can further be restriced to the SU(2) subsector for V 5 = 0 = V 6. In terms of the uA and
vA this restriction is easily enforced by setting for example
uA = (U1, U2, U3, 0) , vA = (0, 0, 0, 1) . (A.23)
Since UaU
a = 1, this choice satisfies the constraints (
uvconds
A.13). Restricting to the SU(2) sector is
achieved by setting u3 = U3 = 0. Upon inserting these ansatze, the Lagrangian (
goodllso6
A.16) reduces
to the Lagrangian (
gtsu3
2.16).
Another interesting sub-sector is obtained by setting
uA = (U1, U2, 0, 0) , vA = (0, 0, V3, V4) , (A.24) su2su2sub
together with the conditions
U1U1 + U
2U2 = 1 , V
1V1 + V
2V2 = 1 . (A.25)
This results in SU(2)×SU(2) subsector consisting of two decoupled SU(2) Landau Lifshitz
Lagrangians.
A.3 The SU(2, 2)/S(U(2)× U(2)) model
For later convenience we present here the SO(2, 4)/S(O(2)×O(4)) ∼ SU(2, 2)/S(U(2)×U(2))
Landau-Lifshitz Lagrangian
LLL SU(2,2) = −iṼ i∂0Ṽi −
|DσṼi|2
= −iũA∂0ũA − iṽA∂0ṽA −
A∂1ũA + ∂1ṽ
A∂1ṽA
−ũA∂1ũAũB∂1ũB − ṽA∂1ṽAṽB∂1tvB − 2ũA∂1ṽAṽB∂1ũB
= iTr(X̃†∂0X̃) +
Tr(D̄1X̃
†D1X̃) , (A.26) goodllso24
where
Ṽ i ≡ Ṽ ∗j ηji , where ηij = diag(−1,−1, 1, 1, 1, 1) , (A.27)
ũA ≡ ũ∗BCBA , ṽA ≡ ṽ∗BCBA , where CAB = (1, 1,−1,−1) . (A.28)
The 4× 2 matrix X̃ has two columns
X̃ = (ũA, ṽA) , (A.29)
and the covariant derivatives are
DµṼi = ∂µṼi + Ṽ
j∂µṼjṼi , (A.30)
DµX̃ = ∂µX̃ − X̃X̃†∂µX̃ . (A.31)
We define
X̃† ≡ −
. (A.32)
This is done for convenience, so that the form of the action in terms of X is independent of the
signature. The fields in the Lagrangian (
goodllso24
A.26) now satisfy the constraints
X̃†X̃ = 12 , (A.33)
Ṽ iṼi = −1 , ṼiṼi = 0 , (A.34)
ũAũA = −1 , ṽAṽA = −1 , ũAṽA = 0 . (A.35) uvlorconst
The action (
goodllso24
A.26) has a local non-compact U(2) invariance
X̃ → X̃U(τ, σ) , (A.36)
for U(τ, σ) a general U(2) matrix
U †(τ, σ)U(τ, σ) = U(τ, σ)U †(τ, σ) = 12 . (A.37)
In terms of the ũA and ṽA the action (
goodllso24
A.26) is invariant with respect to the following local
transformations
(ũA, ṽA) → (cos θ(τ, σ) ũA + sin θ(τ, σ) ṽA,− sin θ(τ, σ) ũA + cos θ(τ, σ) ṽA) ,
(ũA, ṽA) → (eiφ1(τ,σ)ũA, eiφ1(τ,σ)ṽA) ,
(ũA, ṽA) → (eiφ2(τ,σ)ũA, e−iφ2(τ,σ)ṽA) ,
(ũA, ṽA) → (eiφ3(τ,σ)ṽA,−eiφ3(τ,σ)ũA) , (A.38)
To relate the Ṽi coordinates to the ũA, ṽA coordinates recall that the SU(4) ρ matrices could
be combined into 8× 8 γ matrices of SO(6) as follows
0 ρiAB
ρiAB 0
, i = 1, . . . , 6 , (A.39)
with the γi satisfying the SO(6) anti-commutation relations
γi, γj
= 2δij . (A.40) dirso6
The SO(2, 4) γ-matrix algebra is instead
γ̃i, γ̃j
= −2ηij . (A.41) dirso24
Given a set of SO(6) γ matrices we can define
γ̃i =
γi , i = 1, 2 ,
iγi , i = 3, . . . , 6 ,
(A.42)
which satisfy (
dirso24
A.41). Similarily we will define
ρ̃iAB =
ρiAB , i = 1, 2 ,
iρiAB , i = 3, . . . , 6 ,
and ρ̃iAB =
ρiAB , i = 1, 2 ,
iρiAB , i = 3, . . . , 6 ,
(A.43)
which now satisfy
ρ̃iAB ρ̃
jBC + ρ̃iAB ρ̃
jBC = −2δCAηij , (A.44)
as well as
ηij ρ̃
jCD = 2(δCAδ
B − δDA δCB) . (A.45)
Note also that for SU(4) ρ matrices we had
(ρiAB)
∗ = −ρiAB , (A.46)
while for the SU(2,2) ρ̃ matrices we have
(ρ̃iAB)
∗ = ηijρjAB . (A.47)
The relationship between the ṽA, ũA and the Ṽi is
Ṽi =
ũAρ̃iAB ṽ
B , Ṽ i =
ṽAρ̃
iABũB . (A.48)
This can be used to derive the equality between the first and second lines in equation (
goodllso24
A.26).
A.3.1 Subsectors of the SU(2, 2)/S(U(2)× U(2)) model
so24sub
When written in terms of the Vi, the Lagrangian LLL SU(2,2) can be reduced to the SU(1,2)
sub-sector by requiring
Ṽ 2a = −iṼ 2a−1 ≡ 1√
Ũa , a = 1, 2, 3 , (A.49)
which can further be restriced to the SU(2) subsector for Ṽ 5 = 0 = Ṽ 6. In terms of the ũA and
ṽA this restriction is easily enforced by setting for example
ũA = (Ũ1, Ũ2, Ũ3, 0) , ṽA = (0, 0, 0, 1) . (A.50)
We require
ηabŨ∗a Ũb = −1 , (A.51)
so as to satisfy the constraints (
uvlorconst
A.35). Restricting to the SU(1,1) sector is achieved by setting
ũ3 = Ũ3 = 0. Upon inserting these ansatze, the Lagrangian (
goodllso24
A.26) reduces to the standard
SU(1,2) Landau-Lifshitz Lagrangian
LSU(1,2) = −iŨa∂0Ũa −
|DσŨa|2 + Λ(ŨaŨa + 1) , (A.52)
with a = 1, 2, 3 and Ũa ≡ ηabŨ∗b .
A.4 The SU(2|2)/S(U(1|1)× U(1|1)) model
Lets construct the LL model on SU(2|2)/S(U(1|1) × U(1|1)). Starting from equation (
LLsigmamodel
2.4),
with Tr now replaced by STr we may define
g = (X, Y ) , (A.53)
with X and Y super-matrices which satisfy
X†X = 12 , Y
†Y = 12 , XX
† + Y Y † =
. (A.54)
The LL Lagrangian for this model is then
LLL SU(2|2) =
g−1∂0g
(g−1D1g)(g
−1D1g)
= iSTr(X†∂0X)−
, (A.55)
where
D1X ≡ ∂1X −XX†∂1X . (A.56)
The bosonic base of SU(2|2) is SU(2)×SU(2), where in the case of interest to us we write
X = (ũA, vA) , A = 1 . . . , 4 , (A.57)
ũAũA = −1 , vAvA = 1 , (A.58)
ũA ≡ ũ∗BCBA , vA ≡ v∗BCBA , X† ≡ −
, (A.59)
where CBA ≡ diag(−1,−1, 1, 1). Note that the first (last) two components of uA (vA) are
bosonic and the last (first) two components of uA (vA) are fermionic.
LLL SU(2|2) = −iũA∂0ũA − ivA∂0vA −
A∂1ũA + ∂1v
A∂1vA
−ũA∂1ũAũB∂1ũB + vA∂1vAvB∂1vB + 2ũA∂1vAvB∂1ũB
. (A.60) goodllsu22
The action (
goodllsu22
A.60) has a local non-compact U(1|1) invariance
X̃ → X̃U(τ, σ) , (A.61)
for U(τ, σ) a general U(1|1) matrix
U †(τ, σ)U(τ, σ) = U(τ, σ)U †(τ, σ) = 12 . (A.62)
In terms of the ũA and vA the action (
goodllsu22
A.60) is invariant with respect to the following local
transformations
(ũA, vA) → (ũA + vAθ1(τ, σ), vA + ũAθ1(τ, σ)) ,
(ũA, vA) → (ũA − ivAθ2(τ, σ), vA + iũAθ2(τ, σ)) ,
(ũA, vA) → (eiφ1(τ,σ)ũA, eiφ1(τ,σ)vA) ,
(ũA, vA) → (eiφ2(τ,σ)ũA, e−iφ2(τ,σ)vA) , (A.63)
where ψ1, ψ2 (θ1, θ2) are real Grassmann-even (-odd) valued function.
B Quantising the action (
gsu112
3.18) in the t + α = κτ gauge
Given the simple form of the action (
gsu112
3.18), (
gsu112comp
3.23) we present a brief light-cone quantisation of
it here. The main point is that, as expected, the Hamiltonian has a non-zero normal ordering
constant (
normordconst
B.18).
Since the equation of motion for φ+ is
0 = ∂µ (
ggµν∂νφ+) , (B.1)
we may impose conformal gauge (gµν = ηµν) and set
φ+ = 2κτ . (B.2)
The fermionic equations of motion then reduce to
0 = (i∂0 + κ)η
i + ∂1ηj , (B.3) fermeom1
where i 6= j. The fermionic fields have the following periodicity conditions
η1(τ , 2π) = e
iαη1(τ , 0) , η2(τ , 2π) = e
−iαη1(τ , 0) , (B.4)
The fermionic equations of motion then are solved by
i(nσ+ωnτ) + θ̃ne
i(nσ−ωnτ) , (B.5)
−i(nσ+ωnτ) + ξ̃ne
−i(nσ−ωnτ) , (B.6)
where
, (B.7)
and for n 6= 0
ωn + κ
ωn − κ
ξ̃n , (B.8)
while for n = 0
0 = θ0 = ξ̃0 . (B.9)
The Virasoro constraints can be used to find φ− in terms of the other fields
0 = ∂0φ− +
i − κ
ηiηi , (B.10)
0 = ∂1φ− +
i . (B.11)
The Nöther current for time translations t→ t+ ǫ is
jtµ = −∂µφ+ − ∂µφ− + iηi
i − 2∂µφ+ηiηi − ǫµν
∂ν η2 − η1
. (B.12)
We can use the equations of motion to write the Hamiltonian of the system as
Hκ ≡ −
dσjt0 = 2κ+
i . (B.13) Hkappa
The canonical momentum conjugate to ηi is 4iκη
i and so upon quantisation we must have
ηi(τ, σ), ηj(τ, σ
= − 1
δijδ(σ − σ′) . (B.14)
As a consequence the mode oscillators have the following non-zero anti-commutators
ξ̄n, ξm
= −δnm
ωn + κ
16πκωn
ξn, ξ̃m
= −δnm
ωn − κ
16πκωn
, (B.15)
together with
ξ̄0, ξ0
= − 1
θ0, θ̃0
= − 1
, (B.16)
with all other anti-commutators equal to zero. With the convention that ξn,
ξn, ξ0 and θ̃0 are
the annihilaiton operators the normal ordered expression for Hκ in the quantum theory is
Hκ = 2κ(ξ̄0ξ0 +
θ0θ̃0) + 4
n 6=0
ξ̄nξn
ωn + κ
ωn − κ
+ aκ . (B.17)
The normal ordering constant aκ is
ωn . (B.18) normordconst
The remaining non-trivial bosonic Nöther current for the rotations
η1 → eiǫη1 , η2 → e−iǫη2 , (B.19)
jcµ = ∂µφ+(η1η
1 − η2η2) + 2iηµν∂νφ+(η2η1 + η2η1) . (B.20)
The corresponding normal-ordered conserved current is
J = − 1
dσjc0 =
dσ(η2η
2 − η1η1)
θ0θ̃0 − 2κξ̄0ξ0 +
n 6=0
ωn − κ
− ξ̄nξn
ωn + κ
. (B.21)
In this case the normal ordering constant is zero. Since φ− is periodic in σ we require that
dσ∂1φ− = i
i , (B.22)
In the quantum theory this is equivalent to the level matching requirement
n 6=0
ωn − κ
− ξ̄nξn
ωn + κ
| physical 〉 . (B.23)
Finally, we may compute the four non-zero supercharges
Q1 ≡ i
Q10 4 = κe
−iκτ dσ
η1 = κ
θ0 , (B.24)
Q2 ≡ i
Q20 3 = κe
−iκτ dσ
η2 = κξ̄0 , (B.25)
Q̄1 ≡ i
Q40 1 = κe
iκτ dσ
η1 = κθ̃0 , (B.26)
Q̄2 ≡ i
Q30 2 = κe
iκτ dσ
η2 = κξ0 , (B.27)
The above (super-)charges form a U(1|1)2 algebra and in particular we find
[Hκ , J ] = 0 ,
Qi, Q̄j
= − κ
δij . (B.28)
C Comments on conformal invariance of the action (
gsu112
3.18)
In this appendix we entertain the possibility of using the action (
gsu112
3.18), (
gsu112comp
3.23) as a Polyakov
string action. As a warm-up let us integrate out φ− in the action (
gsu112comp
3.23). We arrive at an
effective action for the fermions in which we may set
φ+ = 2κτ . (C.1)
Explicitly we then have
Leff = (−κ)
d2σ iηi
i − 2κηiηi + η1
∂1 η2 − η1
2 . (C.2)
We may represent the worldsheet gamma matrices as
, ρ1 =
, (C.3)
and define a world-sheet Dirac spinor as
, α = 1, 2 . (C.4)
The conjugate spinor is then
ψ̄α = (ψ
†ρ0)α =
−η1, η2
, (C.5)
and the effective action may be written as
Leff = (−κ)
d2σ iδµa ψ̄ρ
a←→∂µψ + 2κψ̄ψ , (C.6)
where
∂µψ ≡ ψ̄ρa∂µψ − ∂µψ̄γaψ . (C.7)
This is simply the Lagrangian for a worldsheet Dirac fermion of mass 2κ. Since such fermions
are not conformal, we get the first indication that the Lagrangian (
gsu112
3.18) is also not conformal.
With the above definitions for ρa, ψα and ψ̄α we can re-write the action (
gsu112
3.18) as
gµν∂µφ+∂νφ− + ie
a ψ̄ρ
a←→∂µψ + 2mψ̄ψ
, (C.8) firstcurved
where
eµa =
gµν∂νφ+ ,
µν∂νφ+
. (C.9)
We have written the above expression in the form of an inverse zwei-bein; we will see shortly
that this is indeed justified. The corresponding zwei-bein is
eaµ = (∂µφ+ , ǫµν∂
νφ+) , (C.10)
and the metric is
Gµν ≡ eaµebνηab = −
gµν . (C.11)
Above
m ≡ gµν∂µφ+∂νφ+ =
−Gµν∂µφ+∂νφ+ , (C.12)
is the norm of φ+, which needs to be non-zero. For completness note that the determinant of
the metric and the zwei-bein are
G ≡ detGµν =
, e ≡ det(eaµ) = −
. (C.13)
Rescaling fermions in the action (
firstcurved
C.8) by
ψ → m−1/2ψ , (C.14)
gives
gµν∂µφ+∂νφ− + im
−1eµa ψ̄ρ
a←→∂µψ + 2ψ̄ψ
. (C.15)
Integrating by parts this can be written as
gµν∂µφ+∂νφ− + 2im
−1eµa ψ̄ρ
a∂µψ +
−gm−1)√
−g ψ̄ρ
aψ + 2ψ̄ψ
gµν∂µφ+∂νφ− + 2im
−1eµa ψ̄ρ
a∂µψ +m
−1∂µ(e
ψ̄ρaψ + 2ψ̄ψ
gµν∂µφ+∂νφ− + 2im
−1eµa ψ̄ρ
a∂µψ +m
−1ω01a ψ̄ρ
aρ01ψ + 2ψ̄ψ
−Gµν∂µφ+∂νφ− + 2ψ̄(iρµDµ +m)ψ
. (C.16) curved
The final form of the action is that of a world-sheet Dirac fermion of mass m together with the
fields φ± moving in a curved metric Gµν . Above we have used the fact that in two dimensions
for any zwei-bein êaµ and corresponding metric ĝµν , the spin connection ω̂
µ can be written as
ω̂abµ = −ǫab
êcµǫc
, (C.17)
where ǫab (ǫc
d) is the flat Minkowski space ǫ-tensor with non-zero components ǫ01 = −ǫ10 = 1
1 = ǫ1
0 = −1). This formula can be derived from the xpressions presentd in Appendix
We may now want to define a string theory path integral for this Lagrangian. To do so we
consider the Polyakov path integral for the Lagrangian (
curved
C.16). Since the path integral integrates
over metrics gµν , and the Lagrangian is a function of the metric Gµν = −m−1gµν we first rescale
gµν → −m1/2gµν , (C.18)
in order to eliminate the metric Gµν . We arrive at a Polyakov-type path-integral with action
gµν∂µφ+∂νφ− + 2ψ̄(iρ
µDµ +
. (C.19) altu11
This Lagrangian is conformally invariant. One way to see this is to generalise the argument
presented in
[31] which considered sigma-models on plane-wave backgrounds. Let us integrate
out the fermions to obtain an effective Lagrangian for φ±
Leff ∼ ηµν∂µφ+∂µφ− +
log det
= ηµν∂µφ+∂µφ− +
log det
∂µ1φ+Π
+ ∂µ2∂ν1φ+Π
− ∂ν2 +
= ηµν∂µφ+∂µφ− +
log(m2)
∼ ηµν∂µφ+∂µφ− + ηµν∂µφ+∂µφ+ ln Λ , (C.20) altu11
where Λ is the cut-off. We can re-absorb this divergent piece by re-defining φ−
φ− → φ− − φ+ ln Λ . (C.21)
This shows that the Lagrangian (
altu11
C.19) is conformal. As it stands however, this Lagrangian is
not Weyl invariant and, just as in
[31], we need to turn on a dilaton
Φ = φ2+ . (C.22)
D Two dimensional spin connection
Let us consider a geenral Lorenzian two dimensional metric gµν which we will parametrise for
convenience as
gµν =
, (D.1)
14I am grateful to A. Tseytlin for a number of discussions and explanations of these issues.
where a, b and d are complex functions of τ and σ the coordinates on the manifold. The
zwei-bein from which this follows is given by
e1µ = (a sinh ρ ,−d sinh ρ) , e2µ = (a cosh ρ , d cosh ρ) , (D.2)
where
. (D.3)
The Christoffel symbols
gµκ (gκν ,λ + gκλ ,ν − gνλ ,κ) (D.4)
are given by
Γ111 = g
aba,1 + ad
2a,0 − bb,0
, (D.5)
Γ112 = Γ
21 = g
d2aa,1 − bdd,0
, (D.6)
Γ122 = g
d2b,1 − bdd,1 − d3d,0
, (D.7)
Γ211 = g
−a3a,1 − aba,0 + a2b,0
, (D.8)
Γ212 = Γ
21 = g
−aba,1 + a2dd,0
, (D.9)
Γ222 = g
a2dd,1 − bb,1 + bdd,0
. (D.10)
where g = det gµν . It is easy to check that these satisfy the defining equation
gµν ,λ − gκνΓκµλ − gκµΓκνλ = 0 . (D.11)
The spin connection ωmnµ can be determined from the following equation
ν = ∂µe
ν + ω
ν − Γκµνemκ = 0 . (D.12)
Since ωmnµ is anti-symmetric in (m,n) the non-zero components are given by
ω010 = −ω100 =
−2a2da,1 − bda,0 + adb,0 + abd,0
−g , (D.13)
ω011 = −ω101 =
−bda,1 − adb,1 + abd,1 + 2ad2d,0
√−g . (D.14)
E T-dual version of the action (
gsu112
3.18)
Performing T-duality for the action (
gsu112
3.18) along α leads to a very simple form for an equivalent
action. In this appendix we breifly present these results. To T-dualise along α we replace ∂µα
by Aα and adding the Lagrange multiplier term ǫ
µνAµ∂ν α̃. The Aµ are then integrated out and
we obtain the action
Ldκ =
1− ηiηi
−(∂µt−
i)(∂νt−
+(∂µα̃− (η1
∂µ η2 − η1
2))(∂να̃− (η1
∂ν η2 − η1
1− ηiηi
(∂µt−
i)(∂να̃− (η1
∂ν η2 − η1
(E.1) tdual
where we have used the fact that up to total derivatives
1− ηiηi
∂µt∂ν α̃η
iηi =
1− ηiηi
∂µt∂ν α̃ . (E.2)
At the level of classical equations of motion we may integrate out the metric to get a Nambu-
Goto type action
LdNGκ =
1− ηiηi
(∂µt− iηi
i)(∂να̃− (η1
∂ν η2 − η1
, (E.3)
where we have rescaled t→ t/2 and multiplied the whole action by a factor of 2. The Nambu-
Goto form of the action is particularily simple due to the ’two-dimensional’ target space form
of the action (
tdual
E.1). The equations of motion for α̃ and t imply that
1− ηiηi
(∂µt−
i) = ∂µχ1 , (E.4)
1− ηiηi
(∂να̃− (η1
∂ν η2 − η1
2) = ∂µχ2 , (E.5)
where χi are arbitrary Grassmann-even functions of τ and σ. The fermion equations of motion
can then be written in form notation as
0 = ηidχ1∧dχ2 − idηi∧dχ2 + idχ1∧dηj , (E.6)
where i 6= j.
Let us combine the (1 + 1 dimensional) spacetime coordinates into a two-vector
xi = (t, α̃)i , i = 1, 2 , (E.7)
and represent the spacetime gamma matrices as
, γ1 =
, (E.8)
a spacetime Dirac spinor as
, α = 1, 2 . (E.9)
The conjugate spinor is then
Ψ̄α = (Ψ
†γ0)α =
−η1, η2
. (E.10)
With these definitions the Nambu-Goto action can be written as
LdNGκ =
1 + Ψ2
ν , (E.11)
where we define
Πiµ ≡ (∂µxi − iΨ̄γi
∂µΨ) , Ψ
2 = Ψ̄Ψ , (E.12)
∂µΨ ≡ Ψ̄γi∂µΨ− ∂µΨ̄γiΨ . (E.13)
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