2. TEMA: LA FAMILIA Y CONSTRUCCIÓN DE VALORES
2.5 Los valores en niños y adolescentes
The description of the vortex dynamics (9.8) is, unfortunately, rather abstract. ForN ≥2 vortices, we have only implicit definitions of the K¨ahler formΩand the angular momen-tum J on the vortex moduli space. It seems plausible that one could make progress using the parametrization of the vortex moduli space introduced in [153]. Here, how-ever, we take a different approach.
An alternative description of the vortex moduli space is provided by D-branes in string theory [155]. This is analogous to the ADHM construction of the instanton
mod-uli space. The vortex modmod-uli spaceMN is parametrized by:
• AnN ×N complex matrixZ
• AN-component complex vectorϕ
These provideN(N + 1)complex degrees of freedom. We will identify configurations related by theU(N)action
Z →U ZU† and ϕ→U ϕ withU ∈U(N). (9.11) We further require thatZ andϕsatisfy the matrix constraint1,
πµ[Z, Z†] +ϕϕ†=k01N. (9.12) This constraint is the moment map for the action (9.11) with levelk0. We define the moduli spaceM˜N through the symplectic quotient
M˜N =
Z, ϕ such that πµ[Z, Z†] +ϕϕ=k0 /U(N).
This space has real dimensiondim( ˜MN) = 2N. The string theory construction of [155]
shows that this space is related to the vortex moduli space M˜N ∼=MN.
These spaces are conjectured to be isomorphic as complex manifolds, and have the same K¨ahler class. The author is not aware, of a direct proof of this conjecture beyond the string theory construction provided in [155].
The matrix description provides a different parametrization of the vortex moduli space. When the vortices are well separated, Z is approximately diagonal. The po-sitions of the vortices are described by theseN diagonal elements. (The normalization ofπµ in (9.12) is associated to the magnetic length which is of the same order as the vortex size.) However, as the vortices approach, Z is no longer approximately diago-nal, reflecting the fact it is better to think of the locations of the vortices as fuzzy, spread out over a disc of radius (9.4). This feature is captured by the matrix description of the vortex moduli space.
The moduli spaceM˜N inherits a natural metric through the quotient construction de-scribed above. This doesnotcoincide with the metric on the vortex moduli spaceMN 1As an aside: for relativistic vortices, the right-hand side of (9.12) is2π/e2, wheree2 is the gauge coupling constant. Comparing the vortex equations (9.1) to their relativistic counterparts shows that this becomesk0in the non-relativistic context. The fact that this is integer valued for vortices in the Chern-Simons theory will prove important below.
described in AppendixB. Nonetheless, there are now a number of examples in which computations of BPS quantities using M˜N coincide with those of computed from the vortex moduli spaceMN because they are insensitive to the details of the metric (see, for example, [156,157,158,159]). Here we will ultimately be interested in holomorphic wavefunctions over the vortex moduli space. Assuming the conjectured equivalence of the spaces as complex manifolds, it will suffice to work with the matrix model descrip-tion of the vortex moduli space.
The Matrix Model Action
It is now a simple matter to write the vortex dynamics in terms of these new fields.
We introduce a U(N)gauge field, α, on the worldline of the vortices. In the absence of a harmonic trap, the low-energy vortex dynamics is governed by theU(N)gauged quantum mechanics
Svortex = ˆ
dt iπµ Tr Z†D0Z
+iϕ†D0ϕ−k0 Tr α (9.13) whereD0Z = ∂0Z −i[α, Z]and D0ϕ = ∂0ϕ−iαϕ. The quantum mechanical Chern-Simons term ensures that Gauss’s law for the matrix model coincides with (9.12). This means that this action describes the same physics as (9.5).
The action (9.13) is thequantum Hall matrix model, previously proposed as a descrip-tion of the fracdescrip-tional quantum Hall effect by Polychronakos [19] and further explored in [134,135, 160, 161,162,163]. The connection to first order vortex dynamics was noted earlier in [137].
We note in passing that we’ve used the D-brane construction of [155] in a fairly indi-rect way to derive the quantum Hall matrix model. A more diindi-rect D-brane derivation of the matrix model was provided previously in [164]. It would be interesting to see how this work, or the string theory construction of [165], is related to the present set-up.
We would also like to add the harmonic trap to the matrix model. This too was ex-plained in [19]. Spatial rotation within the matrix model acts as Z → eiθZ, with the associated chargeJ = πµTr Z†Z. Adding this to the action, we get the matrix model generalization of (9.8),
Svortex = ˆ
dt iπµ Tr Z†DtZ
+iϕ†Dtϕ−k0 Tr α−ωπµ Tr Z†Z
. (9.14) In the rest of this chapter, we describe the properties of this matrix model. Much of this is review of earlier work, in particular [19] and [134, 135]. However, we also make
a number of new observations about the matrix model, most notably the computation of the charge and statistics of quasihole excitations.
The Classical Ground State
In the presence of the harmonic trap, the classical equations of motion comprise the constraint (9.12) and a classical equation of motion forZ:
iDtZ =ωZ. (9.15)
There is a unique time independent solution, withZ˙ = 0, obeying [α, Z] = ωZ. (This equation can also be viewed as the statement that rotating the phase ofZis equivalent to a gauge transformation.) This solution was given in [19], and takes the form
Z0 = s
k0 πµ
0 1
0 √ 2
. ..
0 √ N −1
0
and ϕ0 =√ k0
0 0 ...
0
√N
(9.16)
withα=ωdiag(N−1, N −2, . . . ,2,1,0).
As promised,Z0is not approximately diagonal. This reflects the fact that individual vortices do not have well-defined positions. Nonetheless, we can reconstruct a number of simple properties of the vortex solution from this matrix. The radius-squared of the disc can be thought of as the maximum eigenvalue ofZ0†Z0[19]. To leading order in the vortex numberN, this gives
R2 ≈ k0N πµ
which agrees with our the radius of the classical vortex solution (9.4). Meanwhile, the angular momentum of a given solution isJ = Tr Z†Z. The angular momentum of the ground state is
J0 =πµ Tr Z0†Z0
= k0N(N −1)
2 (9.17)
which, to leading order in 1/N, agrees with the angular momentum of the classical vortex solution (9.9).
The Quantum Ground State
The quantization of the matrix model (9.14) was initiated in [19] and explored in some detail in [134] and [135]. The individual components of the matrixZ and vectorϕare promoted to quantum operators, with commutation relations
πµ[Zab, Zcd† ] =δadδbc and [ϕa, ϕ†b] =δab.
We choose the vacuum state |0isuch thatZab|0i = ϕ|0i = 0. However this does not, in general, correspond to the ground state of the theory because the physical Hilbert space must obey the quantum version of Gauss’s law (9.12). It is useful to view the trace and traceless part of this constraint separately. The trace constraint reads
N
X
a=1
ϕaϕ†a=k0N ⇒
N
X
a=1
ϕ†aϕa = (k0−1)N. (9.18) We now introduce k via k0 ≡ k+ 1; this coincides with the value it will take in non-Abelian theories. This means that physical states must have kN ϕ-excitations. Note that the ordering of the original constraint has resulted in a shift k0 → k. This will prove important below.
Meanwhile, the traceless part of the constraint (9.12) tells us that physical states must beSU(N)⊂U(N)singlets. We can form such singlet operators out ofZ†andϕ†either from baryons or from traces. The baryonic operators are
a1···aN(ϕ†Z†p1)a1· · ·(ϕ†Z†pN)aN
wherep1, . . . , pN are, necessarily distinct, integers. The trace operators are Tr(Z†p).
There can be complicated relations between the baryonic and trace operators; explicit descriptions for low numbers of vortices were given in [166].
The trace constraint (9.18) means that physical states contain exactlyk baryonic op-erators. The harmonic trap endows these with an energy proportional to the number ofZ†excitations,
H =ωJ =ωπµ
N
X
a,b=1
Zab†Zba.
To minimize this energy, we must act withk baryonic operators, each withpi = i−1.
This results in the ground state
|groundik=
a1···aNϕ†a1(ϕ†Z†)a2· · ·(ϕ†Z†N−1)aN
k
|0i. (9.19) The angular momentum of this ground state coincides with that of the classical ground state (9.17) up to a quantum shiftk0 →k.
There is a close resemblance between these ground states and the Laughlin states [9]
forN electrons at filling fractionν = 1/k0,
|Laughlinik0 =Y
a<b
(za−zb)k0e−B4 P|za|2 =
a1···aNza0
1za2· · ·zNa−1
N
k0
e−B4 P|za|2. (9.20) A formal map between the states was suggested in [134]. However, this similarity can be misleading: the operatorsZ†andϕ†are very different objects from the holomorphic position variablesza. To make this connection precise, we need to be more careful about how to relate the two. In fact, there is no canonical map. There are, however, a number of natural ways to make the connection. Two of these, discussed in [135] (see also [161]), are:
• We work with a coherent state representation Zˆ|Z, ϕi = Z|Z, ϕi and ϕˆ|Z, ϕi = ϕ|Z, ϕi where, for once, we’ve used hats to denote the difference between the quantum operatorZˆand the classical matrixZ. We then diagonalizeZ =V DV−1 with D = diag(z1, . . . , zN) and express the resulting wavefunctions as ψ(za) = hza|Ψi. Essentially, we are using the eigenvalues ofZas coordinates on the phase space. (Non-diagonalisable matrices have zero measure.)
• Alternatively, we could decompose the complex operator matrix Zˆ = ˆX + iYˆ and subsequently work in a coherent state representation Xˆ|Xi = X|Xi. This picture has the advantage that the matricesXˆ andYˆ are conjugate, giving us the representation
Zˆab† = 1
√2
Xab− ∂
∂Xba
.
Moreover, calculations in this approach are somewhat easier because the diag-onalization X = U XU† can be achieved by a unitary operator U. The result-ing wavefunctions are written as ψ(xa) = hxa|Ψi. We then analytically continue xa → za to provide holomorphic wavefunctions of the kind appropriate to de-scribe the lowest Landau level.
Both of these approaches were described in [135]. The resulting wavefunctions differ in detail, but share their most important properties.
The first result of [135] is that thek0 = 1ground state, which we have chosen to label as|groundi0 =|0i, is precisely theν = 1Laughlin state describing a filled Landau level.
That is:
|0i=|Laughlini1.
Fork0 > 1, the map to the Laughlin wavefunction is not exact. Instead, the wavefunc-tions agree only at large separation
|groundik→ |Laughlinik0 for |za−zb| 1/πµ.
However, the matrix model states|groundik differ from the Laughlin states as the par-ticle approach: the wavefunctions still vanish as za → zb, but not with the familiar zero-of-order m that is characteristic of the Laughlin wavefunction. Note that these differences only become visible at separations of order the magnetic length. (Indeed, one can obtain the so-called “X-representation” matrix model wavefunction from the Laughlin one by acting with exponentials of derivative operators`B(∂/∂z)on the poly-nomial part.)
As we described above, there is nothing privileged about the choice of coordinates used above – one may try various sets of coordinates and see if there is better short-distance agreement with the Laughlin wavefunction. However, as was found in [135, 161], there seems no obvious way to find an exact match to the Laughlin wavefunctions.
The connection to vortices sheds some light on this. Because vortices are extended objects, there is no “correct” way to specify their positions as they approach. Corre-spondingly, it is not obvious that their physics is captured by a wavefunction describ-ing point particles. Instead, the important questions are those which are independent of the choice of coordinates. The fact that the long-distance correlations in the matrix model ground states (9.19) coincide with those of the Laughlin wavefunction suggests that these states describe the same universality class of quantum Hall fluids. In the rest of Part III, we show that this is indeed correct. We show that excitations of the matrix model describe chiral edge modes and quasiholes. In particular, the latter have charge 1/k0 and fractional statistics, in agreement with the excitations of the Laughlin wavefunction.