The integer quantum Hall plateau transition is a current algebra after all

The scaling behavior near the transition between plateaus of the Integer Quantum Hall Effect (IQHE) has traditionally been interpreted on the basis of a two-parameter renormalization group (RG) flow conjectured from Pruisken's non-linear sigma model. Yet, the conformal field theory (CFT) describing the critical point remained elusive, and only fragments of a quantitative analytical understanding existed up to now. In the present paper we carry out a detailed analysis of the current-current correlation function for the conductivity tensor, initially in the Chalker-Coddington network model for the IQHE plateau transition and then in its exact reformulation as a supersymmetric vertex model. We develop a heuristic argument for the continuum limit of the non-local conductivity response function at criticality and thus identify a non-Abelian current algebra at level n = 4. Based on precise lattice expressions for the CFT primary fields we predict the multifractal scaling exponents of critical wavefunctions to be q(1-q)/4. The Lagrangian of the RG fixed-point theory for r retarded and r advanced replicas is proposed to be the GL(r|r)_4 Wess-Zumino-Witten model deformed by a truly marginal perturbation.


Introduction
Disordered two-dimensional electron gases at low temperatures and in a strong magnetic field exhibit the Integer Quantum Hall Effect (IQHE): they show plateaus in the Hall conductance as a function of the magnetic field strength, with plateau values that are integers in units of the conductance quantum e 2 /h. The present paper revisits the long-standing problem of the transition between adjacent plateaus. From real experiments and numerical simulations one knows that there exists a critical point where the singleelectron localization length diverges (up to a cut-off set by the finite system size or temperature). Thus one is dealing with a critical phenomenon of the type of an Anderson localization-delocalization transition. What exactly happens at the critical point has eluded analytical understanding so far, in spite of considerable efforts that have been expended over the years. Recent developments have rekindled the interest in the plateau transition, as one expects it to be a paradigm for similar transitions that occur between different ground states of topological insulators and superconductors.

Non-linear sigma model
The first attempt to gain an analytical understanding of the IQHE plateau transition was made by Pruisken and collaborators [1]. Building on the theory of weak localization in a weak magnetic field, they wrote down a nonlinear sigma model that has the dissipative conductivity σ xx and the Hall conductivity σ xy for its two coupling constants. Pruisken's key observation was that the breaking of parity symmetry by the magnetic field makes it possible for a so-called topological θ-term (with topological angle θ, and σ xy = θ/2π in units of e 2 /h) to appear in the field-theory Lagrangian.
Pruisken's Lagrangian is usually written in terms of a matrix field where the diagonal matrix Σ 3 accounts for the distinction between the two sectors that originate from the retarded (+) and advanced (−) singleelectron Green's functions. Alternatively, one can express the Lagrangian L NLσM of Pruisken's theory in terms of the (gauge-dependent) matrix-valued one-forms j = u −1 ∂u andj = u −1∂ u, where ∂ = dz ∂ z and∂ = dz ∂z . The precise nature of u depends on whether one handles the disorder by invoking fermionic replicas, or bosonic replicas, or the Wegner-Efetov supersymmetry method [2,3]. In the three cases one has u ∈ U with U = U(2r), or U = U(r, r), or U = U(r, r|2r), respectively, with r the number of replicas. We prefer the last method, in which case Σ 3 = diag(+1 r|r , −1 r|r ) and −iL NLσM = (σ xx + i σ xy ) STr j +− ∧j −+ where STr denotes the supertrace, and the index pairs +− and −+ indicate off-diagonal blocks defined by the decomposition with respect to the eigenspaces of the retarded-advanced signature Σ 3 . It should be stressed that L NLσM is really meant to be a Lagrangian with target space not the group U but rather a Riemannian symmetric (super-)space U/K where K ⊂ U is the centralizer of Σ 3 . While the principal bundle U → U/K is non-trivial (obstructing the existence of a global section u) the lift assumed in Eq. (2) from U/K to U still makes sense, since the offdiagonal blocks of j andj transform as covariant field strengths under local gauge transformations u → uk for k ∈ K. Note that the low-energy degrees of freedom in the U/K-valued field Q = uΣ 3 u −1 are Goldstone modes whose raison d'etre is to restore the symmetry under U which is spontaneously broken to K by the appearance of a non-zero density of states [4,5,6]. Assuming Pruisken's Lagrangian in conjunction with natural expectations of what should happen in limiting cases, a renormalization group (RG) flow for the running couplings σ xx and σ xy was conjectured [1,7]. Its central feature is a fixed point at σ xy = 1/2 (and other half-integers) and an unknown value of σ xx . The fixed point is expected to be isolated and universal; it is stable in the σ xx direction and unstable in the σ xy direction.
While this so-called Pruisken-Khmelnitskii scaling picture of the IQHE plateau transition inspired a lot of further activity and provided an appealing framework in which to think about the transition, it has to be said that Pruisken's Lagrangian (for electrons in the lowest Landau level, or in a strong magnetic field, where the bare coupling σ xx is small) was never derived in a mathematically controlled manner, much less has it led to a quantitative understanding of what is the exact nature of the critical point. Indeed, the obvious and urgent question of what is the conformal field theory describing the scaling limit of the critical point, remained open for 35 years.

The puzzle and its proposed solution
Pruisken's Lagrangian (2) features an invariance under constant transformations Q(r) → u Q(r) u −1 for u ∈ U = U(r, r|2r). This global symmetry is dictated by the supersymmetry method applied to the disordered electron problem at hand. It is also the heart of an apparent paradox, as follows.
A hallmark of conformal field theory in two dimensions is the separation of the energy-momentum tensor into holomorphic and anti-holomorphic parts, giving rise to two Virasoro algebras, one for the left-moving modes and another one for the right-movers; we speak of holomorphic factorization for short. In the presence of Lie group symmetries, one expects the organization by conformal blocks for the holomorphic and anti-holomorphic fields to be refined by an underlying current algebra. For the case of theories with non-Abelian conserved currents, holomorphic factorization is well understood to be realized by Wess-Zumino-Witten models [8,9]. Now since Pruisken's non-linear sigma model enjoys invariance under the global symmetry group U = U(r, r|2r), it has a Noether current: An equivalent formulation of that conservation law is of U as a possible group for gauging in our case. Yet, taking the GKO coset by U(1) fails to match the known properties of the critical point at hand. To summarize, it has been a long-standing puzzle how to reconcile the CFT axiom of holomorphic factorization of conserved currents with the established phenomenology for the critical point of the IQHE plateau transition.
In the present paper, we start from the Chalker-Coddington network model [14] for the IQHE plateau transition and its exact reformulation [15] as a supersymmetric (SUSY) vertex model due to N. Read. Based on formulas taken from the Kubo theory of linear response, we argue that the nonlocal response function, a.k.a. the conductivity tensor, exhibits a form of holomorphic factorization at the critical point. By expressing the critical response function as a current-current correlator in the SUSY vertex model, we are led to propose lattice candidates for the continuum fields of a non-Abelian current algebra. In order to match the known phenomenology, the level of that current algebra has to be n = 4 . Most importantly, our current algebra is gl(r|r) 4 , which has only half the rank of the complexification gl(2r|2r) of Lie U(r, r|2r). The underlying heuristic is that for a suitable choice of basis, gl(2r|2r) splits into four blocks of equal size (r|r) such that the two diagonal blocks give rise to two chiral algebras gl(r|r) for the left-and right-moving modes, while the off-diagonal blocks bosonize to a WZW field M and its inverse M −1 . (The expert reader may recognize some similarity with the free-boson representation of the SU(2) 1 WZW model for the 6vertex model at its SU(2)-invariant point.) The off-diagonal position of M in the global symmetry algebra gl(2r|2r) obliterates the Chamon-Mudry-Wen objection of RG-instability, as it prevents the relevant perturbations STr M q from appearing in the Lagrangian. We should add the remark that the position of our blocks gl(r|r) inside gl(2r|2r) is not unique but depends on a choice of maximal commutative subalgebra in the tangent space T eK (U/K).
The operator product expansions (OPE) for the currents J,J and the field M are those of an integrable deformation of the GL(r|r) n=4 WZW model. The deformation has the important effect of setting the conformal weights of M to zero -a feature required by the physics of Anderson transitions in symmetry class A [12]. Let us stress that the deformation has a direct physical meaning: it reflects the existence of critical current-current correlation functions of two different types, namely J +− J −+ as well as J ++ J −− . It also predicts a universal amplitude ratio for these, which is expected to be verifiable by numerical simulation of the network model.
By the OPE between the currents and the WZW field, powers of the boson-boson block M 00 of M are Kac-Moody primary fields (this is literally true for r = 1 and still true in adapted form for r > 1). It then follows from the Sugawara form of the deformed energy-momentum tensor that the scaling dimension of M q 00 is q(1−q)/n. Since M q 00 represents the q th moment of a critical wavefunction for the network model [15], we are led to predict (with n = 4) that the multifractal scaling exponents of critical wavefunctions are ∆ q = q(1 − q)/4 , in very good agreement with recent computer results.
Let us finish this introduction and overview by offering more perspective on our current algebra scenario. The logic begins with the reminder that we are compelled to regularize the field theory by a perturbation that explicitly breaks the non-compact symmetry U ; we do so with a K-invariant regulator such as µ d 2 r STr Σ 3 Q in the non-linear sigma model. If the physical system were in a metallic phase of delocalized states (realized in space dimensions d ≥ 3) the broken symmetry U would remain spontaneously broken in the limit of a vanishing regulator µ → 0 . On the contrary, in an insulating phase of localized states (as realized, e.g., by the Chalker-Coddington network model off the critical point) the U -symmetry is known to be restored for µ → 0 . The symmetry restoration happens over a scale set by the localization length (or inverse mass scale), making all current-current correlators short-ranged in the infrared limit. Now, at our phase-transition point some of these correlators, J +− J −+ and J ++ J −− , are critical, while some others, J ++ J ++ and J −− J −− , are trivial by first principles and cannot ever become critical. The upshot is that the current-current correlation functions at criticality determine on the Lie superalgebra gl(2r|2r) of symmetries a bilinear form which is degenerate and fails to be U -invariant. In short, the broken U -symmetry is not restored but remains spontaneously broken at the critical point. Hence any attempt at a U -invariant Lagrangian formulation of the RG fixed-point theory leads to novel and exotic CFT mathematics (as communicated by the author in various talks over the past years).
Adopting the formalism of current algebras (or affine Lie superalgebras, to be accurate), the only way to make do with a conventional form thereof is to choose some decomposition gl(r|r) L ⊕ gl(r|r) R ⊂ gl(2r|2r) (explicitly breaking the unbroken K-invariance) such that the current-current bilinear form becomes non-degenerate on restriction to both gl(r|r) L and gl(r|r) R . The latter option is what we develop in the present paper.
The contents are summarized as follows. Section 2 begins with a short introduction to the Chalker-Coddington network model for the IQHE plateau transition. We point out that the model has a Z 4 spectral symmetry, with the consequence that eigenvalues and eigenfunctions of the one-step timeevolution operator come as quadruples modeled after the fourth roots of unity (Sect. 2.1). By introducing a suitable spinor basis, we get a clear view of the Dirac fermion that emerges at long wavelengths in the absence of disorder (Sect. 2.2). In order to handle the strong random phase disorder of the network model, we review Read's method (Sect. 2.3) leading to the exact reformulation as a supersymmetric vertex model (Sect. 2.4). In Section 3 we investigate a toy model of n Dirac species coupled to a random su(n) gauge field. The motivation here is to review some necessary background on current algebra (Sects. 3.1-3.3) and explain what we mean by a GL(r|r) Wess-Zumino-Witten model (Sect. 3.4). The new insights and main results of the paper are presented in Section 4. There, we start from the observa-tion (Sect. 4.1) that the gl(p|q) n current algebra for p = q admits a truly marginal deformation to make the conformal weights of the fundamental field vanish. We then review the current algebra conundrum (Sect. 4.2), before returning to the analysis of the critical point of the IQHE plateau transition. Our method is to go back and forth between the network model and the SUSY vertex model, using identities and results known on one side to complement those on the other side. We begin with a study of the currentcurrent correlation function for the conductivity tensor (Sect. 4.3). Arguing heuristically, we pass to the continuum limit of the non-local conductivity response function at criticality (Sect. 4.4). The outcome is interpreted in the framework of the supersymmetric vertex model (Sect. 4.5), leading to concrete lattice expressions for the holomorphic currents in the continuum limit (Sect. 4.6). We go on to indicate how a Wess-Zumino-Witten field emerges from the lattice theory (Sect. 4.7), and how the WZW model is deformed by a truly marginal perturbation (Sect. 4.8). Finally, we compute the spectrum of multifractal scaling exponents (Sect. 4.9).

Analysis of the network model
The network model, as conceived by Chalker and Coddington in [14], simulates the IQHE single-electron dynamics on a two-dimensional array of directed links connecting the sites of a square lattice; see Fig. 1. The Hilbert space is H = ℓ C ℓ with one copy C ℓ ∼ = C for each link ℓ. Proceeding by discrete steps in time, the quantum dynamics (ψ t+1 = U ψ t ) is generated by a unitary evolution operator U = U r U s composed of a random unitary, U r , and a deterministic unitary, U s . These are defined as follows. Firstly, fixing a basis {|ℓ } for H of unit vectors |ℓ ∈ C ℓ , one takes U r to be diagonal: U r |ℓ = |ℓ e iφ ℓ with random phases φ ℓ that are uniformly distributed and statistically independent of each other. Secondly, to define U s let ℓ be any link and denote by ℓ ± the two links that follow from ℓ by making a left turn (+) or right turn (−). Then, at the critical point to be studied here, (The model becomes non-critical for |a + | = |a − |.) Known as Kac-Ward amplitudes [16], these conventions define an operator U s which is invariant under rotations of the square lattice by π/2 and integer multiples thereof. Most of the discussion below concerns the specific setting of a torus or rectangular network with periodic boundary conditions in both directions.

Z 4 spectral symmetry
The spectrum of U for the network model, on a torus with an even number of links in both directions of the square lattice, is already determined by the spectrum of its fourth power, U 4 . To demonstrate this fact, let the Hilbert space be decomposed as H = 3 l=0 H l where the subspaces H 0 , . . . , H 3 are spanned by the basis vectors |ℓ for links ℓ on even rows, even columns, odd rows and odd columns of the square lattice, respectively; see Fig. 1. Then, taking l modulo 4, one observes that U maps H l to H l+1 , and its fourth power decomposes into four maps U 4 : H l → H l (l = 0, . . . , 3). It directly follows that if ψ ∈ H 0 is an eigenvector of U 4 with eigenvalue e iε , then is an eigenvector of U with eigenvalue e i(ε+2πm)/4 . Thus, eigenvalues and eigenvectors are grouped into quadruplets modeled after the fourth roots e iπm/2 of unity. Note that m in the formula above may be viewed as a discrete angular momentum (or spin), while 2πl/4 is a discrete angle.

Spinor basis
A key step in identifying the critical theory is to pass from the discrete setting to the continuum. To motivate the construction of good variables in which to take the continuum limit, we temporarily turn off the disorder, setting U r ≡ 1. The resulting operator U ≡ U s is translation-invariant and can be diagonalized by hand using standard Bloch theory as follows. l=0 H l , let the 8-dimensional subspace for a given unit cell be spanned by |l int ≡ |l ∈ H l and |l ext ≡ |l ′ ∈ H l (l = 0, . . . , 3). We then introduce an orthonormal spinor basis e ± l as Unit cells are labeled by the position r of their plaquette center. The eight basis vectors for the unit cell r are denoted by e ± l (r), and an orthonormal plane-wave basis for the entire network is provided bỹ In this basis of good momentum k , the matrix of the deterministic factor U s of the network model operator U block-diagonalizes to four 2 × 2 blocks, one for each transition U s : H l → H l+1 . To make the momentum dependence of the blocks explicit, we write k·r = k x x + k y y with r = (x, y) ∈ Z 2 and k = (k x , k y ) ∈ [0, 2π] 2 where k x (resp. k y ) is the wave number along the axis of H 0 and H 2 (resp. H 1 and H 3 ). For the first block of transition (H 0 → H 1 ) we then obtain The three other blocks, namely u 21 , u 32 , and u 03 , follow from the invariance of U s under rotations by integer multiples of π/2. They are where κ = −k y , k x , k y , −k x for l = 1, 2, 3, 0 respectively. For the cyclic products of all four blocks (recall l ∼ l + 4) one easily verifies the result Note that up to linear order in k (i.e. neglecting terms of order k 2 and higher) the cyclic product does not depend on l.
To summarize, in the spinor basis (7) the unitary evolution operator U s : H l → H l+1 at momentum k = 0 simply acts as an l-shift operator The correction terms linear in the momentum act like a Dirac operator; cf. [17]. More precisely, the long wavelength limit of 1 2 (U 4 s − 1) is a massless Dirac operator The Z 4 spectral symmetry principle (5) extends this structure to four Dirac cones, all at k = 0, with quasi-energies at the fourth roots of unity.

Read's method
We now re-instate the disordered factor U r to return to the full operator U = U r U s . The objects of our theoretical analysis are disorder-averaged observables constructed from retarded Green's functions ℓ|(1 − U ) −1 |ℓ ′ where U < 1, and their advanced analogs. These can be computed by Read's variant [15] of the Wegner-Efetov supersymmetry method. To sketch the idea of the method, let u = e iϕ be a sub-unitary number (Im ϕ > 0) and b, b † be a canonical pair of boson annihilation and creation operators acting on the standard Fock space for bosons. Then the quantum statistical trace computes (1−u) −1 . Similarly, for u = e iϕ a super-unitary number (Im ϕ < 0) one has a convergent trace Tr e −iϕbb † = ∞ n=1 u −n = (u − 1) −1 . By adding fermions for the retarded (+) and advanced sectors (−) one arrives at The symbol STr stands for the supertrace (giving a minus sign to the contribution from states with odd fermion number) over the tensor product of four Fock spaces, two of bosonic (b ± ) and fermionic (f ± ) type each. The identity (14) has a direct generalization from numbers u = e ξ to operators U = e X . Let U be given as U = exp |ℓ ℓ|X|ℓ ′ ℓ ′ | . Then we define its second-quantized representation ρ(U ) as where, suppressing the link label ℓ, we introduce the notation Assuming that X is anti-Hermitian, the quantum statistical trace of ρ(U ) over the Fock space, F, is totally oscillatory and can be made absolutely convergent by adding in the exponent an infinitesimal chemical potential µN where µ < 0 and counts the total number of particles. The partition function then exists and is trivial, STr F ρ(U ) = 1, due to the cancelation between bosons and fermions; cf. Eq. (14). A noteworthy property of the second-quantization map U → ρ(U ) is its multiplicativity: ρ(U r U s ) = ρ(U r )ρ(U s ). The Green's function observables of interest are obtained by inserting suitable operators under the trace. To spell out the details, we define the µ-regularized expectation value of an operator A as Since Wick's rule applies in the free-particle setting at hand, all such expectation values are determined by 1 F = 1 and two basic Wick contractions: where we introduced the µ-regularized time-evolution operator T ≡ e µ U and T † = U −1 e µ .

SUSY vertex model
An attractive feature of Read's method is that it makes the step of randomphase averaging very easy and leads to a transparent outcome: the disorderaveraged Fock space model is translation-invariant and has the structure of a supersymmetric (SUSY) vertex model. A quick summary is as follows. By utilizing the multiplicative property ρ(U r U s ) = ρ(U r )ρ(U s ), we see that computing the disorder average E(ρ(U )) amounts to computing E(ρ(U r )). Since U r = |ℓ e iφ ℓ ℓ| is diagonal in the link basis and the random phases φ ℓ for different links are uncorrelated, the disorder average can be carried out for each link separately. For a single link ℓ we have This integral is unity for n + (ℓ) = n − (ℓ) and zero otherwise. Thus the step of taking the disorder average simply projects on the sector of Fock space where for every link ℓ there are as many retarded as advanced particles. We denote by V ℓ the subspace of the Fock space at ℓ which is selected by the constraint n + (ℓ) = n − (ℓ). The total Hilbert space of the model after disorder averaging then is the tensor product V = ℓ V ℓ . Disorder-averaged observables of the network model are now computed by restricting the trace (18) to the subspace V: This re-formulation of our problem is called the SUSY vertex model. The name communicates the fact [18] that ρ(U s ) projected to V factors as a product over vertices, where each factor is made from the Fock operators of the four links connecting to the given vertex. The SUSY vertex model will furnish much of the basis for our heuristic reasoning in Section 4. Note that while most past approaches addressed the transfer matrix (singling out one of two axes of the network model as "imaginary time") and its anisotropic limit (to set up a SUSY spin chain of anti-ferromagnetic type), we will argue with the representation (21) directly.
Let us finish this brief exposition by mentioning that V ℓ is an irreducible highest-weight module for the Lie superalgebra gl(2|2) generated by the set of operators c * α c β (α , β = 0, 1, 2, 3) at the link ℓ. A significant feature of V ℓ is that the vanishing of the first gl(2|2) Casimir invariant (c * α c α = 0) is accompanied by the vanishing of all Casimir invariants [19]: (Here and in the following the summation convention is implied.) The big question then is to what extent these identities survive under renormalization as constraints on the conformal field theory for the critical point.

Toy model: random su(n) gauge field
Our strategy will be to understand the scaling limit of the IQHE plateau transition via the SUSY vertex model (21) at criticality. Now, invoking the principle of universality at generic critical points, it was proposed some time ago [20] that one possible approach to the IQHE plateau transition would be to augment the massless Dirac theory with marginal perturbations as given by a random gauge potential, a random scalar potential, and a random Dirac mass. While this proposal seems reasonable, it is not in concord with the situation for the network model: the random-phase disorder in U r turns out to be strongly relevant [21] at the massless free Dirac limit of U s . Nevertheless, as a preparation for our work ahead, we now invest some time to revisit the continuum approximation by massless Dirac-type fields coupled to a random gauge potential. Recall that α ∈ {0, . . . , 3} is an index labeling retarded bosons (α = 0), retarded fermions (α = 1), advanced bosons (α = 2) and advanced fermions (α = 3). For generality, we assume that n ≥ 1 Dirac species are present in the low-energy effective theory; we label them by l = 0, 1, . . . , n − 1. Since long wavelength implies low frequency for massless Dirac fields, the original nature of our c * αl and c αl as quantum operators on Fock space is expected to give way to low-energy effective behavior as bosonic and fermionic integration variables (or classical fields) in a functional integral that computes the statistical trace. In this vein, we now consider the toy model of a functional integral e −S , S = L , with (graded-)commutative fields c αl , c * αl and continuum-theory Lagrangian By convention, holomorphic fields (with equation of motion ∂zc L = 0) are left-moving, while anti-holomorphic fields (∂ z c R = 0) are right-moving. The coefficients A l, m are complex random fields distributed as Gaussian white noise; they effectively represent some microscopic disorder (which differs from that of the network model). The bar means the complex conjugate. We stipulate that c * L αm = c αm R for α = 0, 2. The numerical (or bosonic) part of S = L for the Dirac Lagrangian (23) then takes values in the imaginary numbers, rendering the functional integral totally oscillatory. Such was the case for the statistical sum STr ρ(U ) in the initial setting of the supersymmetry method, and it is natural to require this property to hold for the toy model as well. As before, the functional integral is made convergent by including an infinitesimal chemical potential term µN (µ < 0). The functional integral expression for the total particle number is The presence of this regularization not only ensures the existence of the functional integral, but it also keeps the partition function trivial: by the symmetry between bosons and fermions. The field theory (23) is a free theory in that it is Gaussian in the Diractype fields c , c * . However, by the rules of the game all observables of the toy model (just like in the network model) are averages over the disorder, i.e., over the random gauge field A. Thus A is meant to be integrated out. Since it is a Gaussian-distributed field, this can be done explicitly and leads to interaction terms that are quartic in the Dirac fields c , c * .
For simplicity of the outcome, we assume that A is traceless: A l, l = 0. In order words, A andĀ (or rather their Cartesian components A x and A y ) take values in su(n). In the strong-coupling limit of broadly distributed A the resulting field theory will turn out (Sects. 3.3, 3.4) to be the GL(r|r) n Wess-Zumino-Witten model for a number r of replicas. The latter (with a reduced value of r) plays a key role in Section 4, where we continue our analysis of the SUSY vertex model formulation of the network model.

A pair of level-rank dual current algebras
To convey the best possible view of our constructions and their meanings, we now make a generalization: instead of the minimal number of replicas assumed so far (one retarded and advanced boson and fermion each), we will consider the general case of any number of replicas. For the toy model with random su(n) gauge field, the distinction between the retarded and advanced sectors actually does not matter; so, we are free to simply speak of r bosonic and s fermionic replicas. Later, when we return to the full problem of the IQHE plateau transition, the distinction will become relevant.
In 2D conformal field theory one has factorization into a holomorphic and an anti-holomorphic sector. In particular, the energy-momentum (or stressenergy) tensor is a sum of two pieces, one for each sector. To construct it, we may focus on the holomorphic part, T (z), of the tensor. With this focus understood, we temporarily simplify the notation: The main basis of the analysis is the operator product expansion (OPE) for the r bosonic and s fermionic replicas of our Dirac-type fields: which follows directly from the free part of the Dirac Lagrangian. The symbol ∼ means that we are writing only the singular part of the OPE. The symbol |α| stands for the (super-)parity: |α| = 0 for even fields (bosonic replicas) and |α| = 1 for odd fields (fermionic replicas). In the following, the normal-ordered product of two local fields J and K will be denoted by (JK). As usual in this context [22], the normal-ordered product is defined by subtracting the singular parts of the operator product and then taking the limit of coinciding points. For example, The normal-ordered products (c * αl c βm ) are called currents. It is known [23] that the totality of all the currents (c * αl c βm ), c * αl c * βm and c αl c βm generates an orthosymplectic current algebra at level one.
Two current sub-algebras will play a role. Still assuming the summation convention for notational convenience, these are generated by To specify their operator product expansions in a concise way, we associate where the bracket [A, B] in the first line is the commutator in gl(n), while [X, Y ] in the second line means the super-commutator in gl(r|s). The first line is referred to as the current algebra of gl(n) at level s − r and is denoted by gl(n) s−r . The second line is a level-n current superalgebra denoted by gl(r|s) n . The two current algebras commute with one another, i.e., mixed operator products do not give rise to any singular terms. In the classical setting of Lie algebras (as opposed to current algebras) one says that gl(n) and gl(r|s) form a so-called Howe pair inside the orthosymplectic Lie superalgebra osp(2nr|2ns). An distinctive feature of this Howe pair is that gl(n) and gl(r|s) have a non-trivial intersection: by their common center, gl(1). In the present setting, this means that the two current algebras share one current, which we denote by E: In conformal field theory one does not usually speak of a Howe pair. Rather, one says that the current algebras gl(n) s−r and gl(r|s) n are levelrank dual to one another. Indeed, the level n of gl(r|s) n is the rank of gl(n) s−r and, conversely, the level s − r of the latter is the rank (i.e. the super-dimension of a Cartan subalgebra) of the former.
Level-rank duality leads to a useful relation between the normal-ordered quadratic Casimir elements of the two current algebras.
. (To appreciate the overall sign of the latter case, note that the theory must become unitary upon restriction to the fermion-fermion sector). Then from the basic operator product expansions (26) one derives the identity where is the energy-momentum tensor of the free theory. In other words, the quartic interaction terms that appear in the normal-ordered products C KK and C JJ are exact negatives of each other and thus cancel in the sum, leaving only a multiple of T free .

Virasoro decomposition of T free
Let us now embark on a brief digression to prepare the coset construction of the next section. Writing C KK = 2(n+s−r)T KK and C JJ = 2(n+s−r)T JJ , one has This decomposition is not Virasoro, which is to say that the individual summands on the right-hand side do not obey the operator product expansion for a Virasoro algebra (with any central charge c). Nonetheless, if r = s the decomposition (35) can be refined to become Virasoro by introducing the traceless currents which generate the current algebras sl(n) s−r and sl(r|s) n , respectively. In this way one gets where the three summands are and each of them individually obeys the OPE (36). The central charges are respectively. These add up to the central charge n(s − r) of T free as required.
Alas, our case of interest is r = s and, there, the decomposition (38) fails. The obstruction is that one cannot arrange for the currents J X at r = s to have vanishing supertrace by subtracting a multiple of the center current E; see (37). What exists irrespective of whether r = s or r = s is a two-summand decomposition, where the second term on the right-hand side is defined to be the sum of T JJ and the traceful part coming from T KK : .
A special case of relevance to our further development is Since both T free and T sl(n) 0 represent the Virasoro algebra (with c = 0 at super-dimension r − s = 0), so does the difference T gl(r|r)n = T free − T sl(n) 0 .

Taking the coset by sl(n) 0
We now inject an observation that could have been made earlier in our text: the terms coupling to the Gaussian random field A in the Dirac Lagrangian (23) can be expressed entirely in terms of the holomorphic gl(n) currents K m l and their anti-holomorphic analogs. Indeed, A l, m couples to (K L ) m l (z) = (c * L αl c αm L )(z) and A m, l does to (K R ) m l (z) = (c * R αl c αm R )(z). As a preparation for Sect. 4, we are now interested in the strong-disorder limit of a widely fluctuating random gauge field A ,Ā. In that limit, the Gaussian A-integral has the effect of a Dirac δ-function, setting the scaling dimensions of the currents K L, R to zero and removing them from the lowenergy effective theory. More precisely, by the assumption of traceless A, it is the currents K in sl(n) ⊂ gl(n) that are killed. One can work through this step of elimination in complete detail, using non-Abelian bosonization [8] and a functional integral version of the Goddard-Kent-Olive coset construction [24], but we will not dwell on that here. In fact, the present model has been analyzed before [10,25,26]; we can therefore be brief.
In short, the theory after A-averaging is a coset conformal field theory with the energy-momentum tensor T gl(r|r)n given in Eq. (45); in the present setting of a Howe pair or level-rank duality, one also speaks of a "conformal embedding" [22]. Since the OPE between the gl(r|r) currents J β α and the sl(n) currents K A − (TrA/n)E is trivial, the latter have vanishing scaling dimension in this CFT. All the currents J β α (including E = J α α ) still have the conformal weights of a holomorphic current, just as they do in the free theory. We turn to the fundamental gl(r|r)-multiplet of fields defined by These are spinless with conformal weight a result that can already be found in [10]. We note that the non-zero value of h(M ) comes solely from the second term (EE)/2n 2 on the right-hand side of (45). Indeed, applying the first term T JJ of (45) to, say c * L αl ≡ c * αl , one has and by one arrives at a free sum over the index β which gives β (−1) |β| = r − s = 0 by supersymmetry. Let us inject a remark here to connect more closely with the literature. For the special case of n = 2 there exists an alternative description that has been publicized in [25,26]. That alternative derives from the accidental isomorphism sl(2) ∼ = sp(2) of complex Lie algebras, which leads to an accidental relation between current algebras. Indeed, sp(n) forms a Howe pair together with osp(2r|2s) ≡ spo(2r|2s) inside osp(2nr|2ns). (The tilde indicates orthogonal bosons and symplectic fermions, while in standard osp the adjectives are interchanged.) The Howe pair property puts sp(n) 2s−2r in level-rank duality with spo(2r|2s) n . Now the current algebras sp(n) 2s−2r and sl(n) s−r coincide for n = 2 and r = s. As a result, the corresponding level-rank duals have the same energy-momentum tensor: The latter theory is discussed in [25,26] under the name of osp(2r|2r) −2 .
We now ask: what is the classical Lagrangian corresponding to T gl(r|r)n (and its analog in the anti-holomorphic sector)? To answer that, we observe that T gl(r|r)n was constructed from the free Dirac theory by gauging with respect to the sl(n) degrees of freedom. Among some other symmetry group actions, that free theory carries a (partly anomalous) action by symmetries G L × G R where G L = G R = GL(r|r). Since gl(r|r) and gl(n) ⊃ sl(n) have the Howe pair property of centralizing each other in the free-particle algebra osp(2nr|2nr), our CFT with energy-momentum tensor T gl(r|r)n still carries the same action by GL(r|r) L × GL(r|r) R . Taken together with the formulas (46, 48) for the scaling dimensions, this means that our putative coset CFT is actually the GL(r|r) Wess-Zumino-Witten (WZW) model at level k = n. Some relevant information about it is collected in the next subsection.

GL(r|s) n WZW model
For the moment, we again relax the condition r = s and consider the general case of any pair r, s (intending to return to r = s when appropriate).
First of all, in view of dissenting proposals in the literature (see [27] for a recent reference), we must offer some basic clarification as to what is meant by a GL(r|s) WZW model. Notwithstanding its misleading name, such a field theory does not have any Lie supergroup GL(r|s) or U(r|s) ⊂ GL(r|s) for its real target space (before complexification). Rather, the target space is what we call a Riemannian symmetric superspace [28]. Indeed, for r = 0 (fermionic replicas only) the target space is well known to be the compact Lie group U(s) ≡ X 1 . On other hand, for s = 0 (bosonic replicas only) it has to be the non-compact dual of U(r), namely the symmetric space X 0 ≡ GL(r, C)/U(r) ≃ Herm + (r) of positive Hermitian matrices M 00 , There are many reasons why the correct identification is X 0 = GL(r, C)/U(r) and not X 0 ? = U(r) as would be the case for the Lie supergroup U(r|s). To give one reason, T gl(r|s)n remembers from T free (through the coset construction) the space of states for the bosonic ghosts in (23). This space is very much larger than the corresponding space for the Dirac fermions in (23), as every mode of excitation can be occupied not just once but an arbitrary number of times (by the absence of the Pauli principle). The choice X 0 ? = U(r) would fail to capture this drastic enlargement of the Hilbert space. To give a second reason, the metric tensors of the symmetric spaces X 0 = GL(r, C)/U(r) and X 1 = U(s) combine by the supertrace form to an invariant metric tensor on X 0 × X 1 which is Riemannian. Here the emphasis is on the adjective "Riemannian", as this property is what is necessary to have a sign-definite action functional and thus a sensible functional integral.
To give yet another reason, the sign in (44) of the JJ terms for the bosonboson sector (|α| = |β| = 0) is opposite to that for the fermion-fermion sector (|α| = |β| = 1). Since the latter translates to a WZW model of compact type, one infers from the functional integral version of the coset construction [24] that the former (with a Sugawara energy-momentum tensor of the opposite sign) must be a WZW model of non-compact type.
Mathematically speaking, the full target space X of the GL(r|s) WZW model is a cs-supermanifold [29] which arises by taking exterior powers of a vector bundle over X 0 × X 1 with standard fiber V , The vector bundle is associated to the direct product of principal bundles GL(r, C) → X 0 and U(s) × U(s) → X 1 by the natural action on V of the direct product of structure groups U(r) × U(s). In physics language, the target space X consists of supermatrices where the left upper block M 00 is a positive Hermitian matrix of size r × r, the right lower block M 11 is a unitary matrix of size s × s, and the offdiagonal blocks M 01 and M 10 are rectangular matrices of sizes r × s and s × r with complex Grassmann variables as matrix entries. The connection to the said vector-bundle picture is made by the re-parametrization which is only determined up to the action of the structure group U(r) × U(s) ∋ (u 0 , u 1 ). The WZW action functional is the usual one. For level k = n one has where d = ∂ +∂ is the exterior derivative, and the domain Σ is a closed Riemann surface. A standard remark is that no space-time metric appears here, as no more than a complex structure for Σ is needed to decompose d = ∂ +∂ = dz ∂ z + dz ∂z . Note that Γ[M ] is real-valued on X 1 but imaginaryvalued on X 0 . Let us also record the observation that the third cohomology group of the odd-odd sector X 1 = U(s) is non-trivial (for s ≥ 2). This has the well-known consequence that the anomalous weight exp(−inΓ[M ]) is well-defined only for integer values of the level number n.
Compared with the Dirac picture, a major change has occurred in that the functional integral is no longer totally oscillatory. In fact, by the Riemannian geometry of X 0 × X 1 the numerical part of the first term on the righthand side of (56) is real and positive. (Note that (i/2) dz ∧ dz = d 2 r > 0.) What remains unchanged from before is that the functional integral needs regularization for the non-compact degrees of freedom. According to the standard rules [8] of non-Abelian bosonization, the total particle number of the infinitesimal chemical potential term −µN takes the form As a quick check, observe that for a positive number m = e ϕ one has m + m −1 = 2 cosh ϕ , so the regulator term S reg does indeed do the required job of cutting off the infinity due to the non-compact zero modes in the WZW field M . We also note that the functional integral regularized by S reg still has all the requisite target-space supersymmetries to make the partition function trivial: Let us finish this subsection with two more remarks on (56). Firstly, no term corresponding to (EE)/2n(n + s − r) in the definition (44) of T gl(r|s)n appears in the WZW Lagrangian. That term becomes negligible in the semiclassical limit of large n and is to be viewed as a quantum correction arising in the Sugawara construction of the energy-momentum tensor. Secondly, the WZW model (56) for level n = 1 is a free theory for any (r, s) and, in particular, for r = s. Indeed, since the current algebra sl(n) s−r for n = 1 is void, the relation (43) gives This means in particular that the level n = 1 theory for r = s, when properly understood, has free-field correlation functions. Different statements exist in the literature, cf. [27], where the questionable choice of a Lie supergroup is made for the target space.

Inferring the IQHE critical theory
Having discussed the toy model (23) in some detail, we now return to our task proper: constructing the CFT for the critical point of the IQHE plateau transition. Two avenues look especially inviting. For one, we could pursue the approach suggested in [20], by adding further random-field perturbations to the Lagrangian (23) to drive the system to the universal fixed point of interest. In this approach we would apply non-Abelian bosonization to the perturbations and analyze them in the WZW model. A second approach is to revert to the SUSY vertex model and advance the analysis there.
In either investigation, it is necessary to pay due attention to the difference between the retarded and advanced degrees of freedom (originating from the retarded and advanced single-electron Green's functions) of the theory. We recall that Σ 3 = +1 in the retarded sector and Σ 3 = −1 in the advanced sector. To avoid the danger of being misled by low-dimensional accidents, we will continue to work with an arbitrary replica number, as long as it does not make the notation overly cumbersome and intransparent. Thus we consider r ≥ 1 replicas of retarded bosons, retarded fermions, advanced bosons, and advanced fermions, r of each kind. For the WZW target space X this means that we double r → 2r and work over the base space

A telling observation
We recall from (48) that the field M of the GL(2r|2r) n WZW model has conformal weights h =h = (2n 2 ) −1 . Now, from the phenomenology of the IQHE plateau transition (as an Anderson transition in symmetry class A) we know with certainty that the fundamental field of the RG fixed-point theory we are seeking must have vanishing scaling dimension [12]. Motivated by this fact, we review in the present section a one-parameter CFT deformation by which the dimension of our field M can be tuned to zero, h =h = 0. This deformation was already described by Chamon, Mudry and Wen [10] a long time ago; while it does not lead us to anything useful per se, it will turn out to be of significance when put into the proper context. By the Chaudhuri-Schwartz criterion [30], a conformal field theory with current algebra can be deformed by adding to the Lagrangian a truly marginal perturbation d 2 r E L E R made from Abelian left-moving and right-moving currents E L and E R . In our case, such an Abelian deformation exists owing to the presence of the central generator E of gl(2r|2r). The pertinent formulas are as follows. Recalling (30), consider the modified current-current operator product expansion where the invariant bilinear form has been deformed by a parameter γ: X, Y n,γ = −n STr(XY ) + γ STr(X) STr(Y ).
(Note that γ is denoted by g A /π in [10].) The deformed energy-momentum tensor, say the holomorphic part T (z), is determined by requiring the OPE expressing the CFT principle that a holomorphic current J X has to be a Virasoro primary field with conformal weights (1, 0). It then follows that which still satisfies the OPE for a Virasoro algebra with central charge c = 0. Now, as was observed earlier, the conformal weights of M stem entirely from the summand (J α α J β β ) of T ; hence they vanish if we set γ = 1. To be sure, the vanishing of the scaling dimension of M at γ = 1 ought to be discarded as coincidental unless we can offer a convincing physical interpretation of the additional term STr(X) STr(Y ) in the current-current OPE (62). Let us therefore anticipate that such an interpretation does in fact exist in the modified scenario developed below. There, the appearance of the second summand in (63) will be explained by the existence of two sets of log-correlated operators in the critical theory. In conventional parlance, they correspond to the two types of basic correlator that can be made from retarded (+) and advanced (−) Green's functions in the microscopic model: Σ(r, r ′ ) = G + (r, r ′ ) G − (r ′ , r) and Υ(r, r ′ ) = G + (r, r) G − (r ′ , r ′ ) .
In the case of our network model, these take the form of where T = QU stands for the time-evolution operator U = U r U s made subunitary by inserting [15] one or more point contacts, Q = 1 − |ℓ c ℓ c |, or by the presence of a homogeneous absorbing background Q = e µ (µ → 0−) as before, or similar. The symbol E(. . .) still denotes the disorder average. At the critical point, both correlators (66) and (67) become logarithms of the distance |ℓ 1 −ℓ 2 |, and the deformed current-current OPE (62) will ultimately predict their universal amplitude ratio.

The current algebra conundrum
If the reader is intrigued and encouraged by the observation that an important scaling dimension can be tuned to the desired value, then we must hasten to caution that the gl(2r|2r) n,γ current algebra (62) is beset, for our purposes, with a serious flaw (for any γ). Using the traditional language of Green's functions G ± , we can phrase the flaw as follows.
Recall that the microscopic foundation of our functional integral or statistical mechanics problem defines a signature Σ 3 that distinguishes between the retarded and advanced sectors of the theory. Now if a correlation function is of mono-type G + G + · · · G + or G − G − · · · G − , i.e., probes only one of the two sectors, then it is well known to be trivial in the infrared limit; cf. [12]. In the network model the triviality is immediate because any product of only, say, retarded Green's functions ℓ|(1 − T ) −1 |ℓ ′ collapses to unity upon random U(1) phase averaging. In our field-theoretical reformulation the collapse is brought about by the supersymmetries in the unprobed sector. Of course the collapse carries over from Green's functions to currentcurrent correlators. Thus all correlation functions J ++ J ++ · · · J ++ V in the SUSY vertex model, and also in the infrared limit of the Dirac Lagrangian (23) with IQHE-generic random perturbations, are trivial, and so are the J −− J −− · · · J −− V , in stark contrast to the OPE (62).
One might now entertain the idea that one should kill these very currents by means of a variant of the Goddard-Kent-Olive coset construction (or by gauging the WZW model). Alas, any such attempt is doomed to fail. For one reason, one would immediately run into a conflict with the global U(r, r|2r) symmetry. For another, correlation functions of mixed type J ++ J −− do not suffer from SUSY collapse but, as we shall see, are actually critical; it is only the mono-cultures J ++ · · · J ++ and J −− · · · J −− that are trivial.
In view of this puzzling state of affairs, it is not clear at all how one might proceed with the assumption of a current algebra for the vertex model, or for the Dirac theory (23) with generic perturbations. We shall therefore revisit our microscopic model for guidance, starting, in the next section, with a physical observable of theoretical and experimental interest: conductivity.
Our strategy from here onwards is to complement known results and physical intuition for the network model with Ward identities in the SUSY network model, and vice versa. As a model of unitary quantum mechanics with conserved probability, the network model has a conserved U(1) charge current. On general grounds one expects the conservation law of that current to be enhanced -at the critical point where conformal invariance emerges in the infrared limit -by a second conservation law to yield a pair of holomorphic and anti-holomorphic U(1) currents. By transferring that Abelian current to the SUSY vertex model, we will identify a conserved current which is non-Abelian, albeit not gl(2r|2r). As a rewarding return, the known constraints on non-Abelian current algebras then improve our understanding of conserved currents in the network model. In a related context, such a scenario of doubling of conservation laws (or symmetry doubling) due to emerging conformal invariance was first pointed out by Affleck [13]. Let us now review that scenario in a language adapted to our purposes.
By way of preparation, we recall some standard facts from differential calculus in the continuum. Let I(C) denote the total charge current flowing across a (d − 1)-dimensional surface C in d space dimensions. To express the total current as an invariantly defined integral I(C) = C j , one equips C with an outer (or transverse) orientation while the current density j is modeled as a twisted differential form [31] of degree d − 1. Assuming the (d.c.) situation of a stationary current flow, the continuity equation of charge conservation says that j is closed: dj = 0. It then follows by Stokes' theorem that the total d.c. current through any boundary C = ∂Σ is zero.
In our two-dimensional setting, the surface C is a curve with outer orientation and j is a twisted 1-form. Now a complex structure in two dimensions is a rotation, R, of tangent vector fields by ±π/2. On 1-forms j, the complex structure R determines a Hodge star operator, ⋆ . Choosing standard Cartesian coordinates x and y for the Euclidean plane, one has ⋆ dx = dy and ⋆ dy = −dx if R is rotation through +π/2. Given ⋆ one associates with j its Hodge dual ⋆j , which is another 1-form. The integral C ⋆j is still invariantly defined; its (un-)physical meaning is that of the current flowing along C. (Using the traditional language of vector calculus, one would say that the current-density vector field is line-integrated along the curve C.) While dj = 0 always holds true for a conserved current, there exists no reason for the Hodge-dual current to obey the stationary continuity equation d ⋆ j ? = 0 in general. Indeed, our network model away from the critical point has circulating currents; in the strong localization regime on one side of the phase transition, currents flow around the elementary plaquettes with one sense of circulation, on the other side they flow around those with the opposite circulation. Yet, at the critical point separating the two phases with opposite circulation, the two opposing tendencies should balance out, and we therefore expect the circulating currents to vanish (around contractible domains) on average over the disorder and after coarse graining to eliminate non-universal behavior on short scales.
If so, the critical network-model current j after disorder averaging and coarse graining satisfies two continuity equations: dj = 0 and d⋆j = 0. Now, ⋆ determines coordinate-free decompositions d = ∂ +∂ and j = j 10 + j 01 by in coordinates: ∂ = dz ∂ z ,∂ = dz ∂z , j 10 = j z dz and j 01 = jz dz . Using these, one re-expresses the two conservation laws for j as Thus, in particular, j 10 = j z dz is holomorphic: ∂z j z = 0. Our plan for the sequel is to uncover a non-Abelian version of this holomorphic networkmodel current in the SUSY vertex model, at criticality.

Bi-local conductivity tensor
According to the Kubo theory of linear response, the electrical conductance is a current-current correlation function. More precisely, the d.c. conductance, G, associates with a pair of homology cycles C 1 and C 2 a quantum statistical expectation value G(C 1 , C 2 ) ∝ I(C 1 )I(C 2 ) where I(C) = C j is a certain operator for the electrical current crossing the hypersurface C. Expressed in suitable physical units, the number G(C 1 , C 2 ) is the linear response current flowing across C 1 when the system is driven by an electrical voltage along a cycle dual (by the intersection pairing) to C 2 . For noninteracting electrons, and in particular for our network model, the currentcurrent correlation function for the d.c. conductance can be reduced to an explicit and simple form, cf. [32], as explained in this section. In a two-dimensional continuum C would be a curve. To formulate a discrete analog of C j we discretize C as a 1-chain on an auxiliary lattice Γ, the so-called medial lattice (of the square lattice of the network model), with 0-cells and 1-cells that are in bijection with a checkerboard of networkmodel plaquettes and connecting nodes, respectively; see Fig. 3a . For each 1-cell of Γ we choose a plus side and a minus side to fix an outer orientation.
Let now the network be in a steady state with stationary current distribution ℓ → j ℓ , so that Kirchhoff's rule holds at every node. Having associated the 1-cells of the medial lattice Γ with nodes of the primary network, we may assign to any such 1-cell, e, the current crossing it. This is done in the obvious way [ Fig. 3b]: the network-model node for e joins four links ℓ of the primary lattice; if these are indexed by i/o for incoming/outgoing and by +/− for the plus/minus side of e, then Kirchhoff's nodal rule states that In this way we re-interpret j as a 1-cochain on the medial lattice Γ. The integral C j is then given by the 1-cochain j paired with the 1-chain C: Note that the 1-cochain e → j(e) is closed (dj = 0), as it was constructed by a scheme of coarse graining that respects the law of current conservation. Put differently, the total current (dj) A flowing into any 2-cell A of Γ is zero. So far, we have left open the details of what current distribution ℓ → j ℓ we have in mind. We might follow [15] and set j ℓ ≡ |ψ ℓ | 2 where ψ = U ψ is a stationary state of quasi-energy zero for the network model with incomingwave boundary conditions at a point contact. However, that is not the choice we want to make here. Instead, we are going to identify the conserved current j with (either one of) the current operators in the current-current correlation function jj for the conductance G(C 1 , C 2 ). To that end, consider the squared Green's function between two links ℓ 1 , ℓ 2 of the primary lattice. It has the property of being doubly closed; in other words, Kirchhoff's rule holds w.r.t. both of its arguments. To check that statement, say for the case of the right argument ℓ 2 (and fixed left argument ℓ 1 ), we sum σ ℓ 1 , ℓ 2 over the two links ℓ 2 = ℓ in that are incoming to any given node of the primary network and compare with the sum over the two links ℓ 2 = ℓ out that are outgoing from the same node. The two sums agree -that's Kirchhoff's rule for ℓ 2 → σ ℓ 1 , ℓ 2 interpreted as a current distribution. Its proof simply utilizes the unitarity of U : Here, assuming that ℓ 1 and ℓ in are distant from each other and from any contact links, we used that In a paragraph above, we described how to convert a current distribution ℓ → j ℓ into a 1-cochain e → j(e). Following that blueprint, we now turn the squared Green's function (ℓ 1 , ℓ 2 ) → σ ℓ 1 , ℓ 2 into a double 1-cochain (e, e ′ ) → σ(e, e ′ ) on pairs e, e ′ of 1-cells, again on the medial lattice Γ. That double 1-cochain σ on Γ is the network-model analog for the Fermi-surface part of the non-local response function of conductivity; cf. Eq. (52) of [32]. On the grounds of that correspondence, the dimensionless conductance associated with two cycles C 1 and C 2 on Γ comes out to be the double sum By Kirchhoff's rule for the squared Green's function, the non-local response function σ is doubly closed. Therefore, the conductance G(C 1 , C 2 ) depends on the cycles C 1 , C 2 only through their homology classes, as required.

Critical conductivity
What are the implications at criticality? As it stands, the non-local response function (e, e ′ ) → σ(e, e ′ ) of conductivity is a random variable due to its dependence on the U(1) random phase factors in U r . By the act of taking the disorder average, the response function becomes translation-invariant provided that a spatially homogeneous regularization scheme is adopted; in formulas we have that holds for any lattice translation t a of (e, e ′ ) to an equivalent pair (t a e, t a e ′ ). Moreover, at the phase-transition critical point we expect conformal invariance to emerge as a symmetry in the infrared limit. To fathom the consequences thereof, we recall from differential calculus in the continuum that (i) a complex structure in two dimensions is a rotation R by ±π/2, which determines (ii) a Hodge star operator ⋆ and (iii) a decomposition of the current density j = j 10 + j 01 by (68). The continuum line integral C j 01 splits as into the current C j flowing across C and the current C ⋆j along C. Recall that if j is closed, then the transverse current C j depends on C only through its homology class. On the other hand, if j is co-closed (d ⋆ j = 0), then it is the longitudinal current that enjoys the property of invariance C 1 ⋆j = C 2 ⋆j for homologous curves (i.e., for C 1 − C 2 = ∂S a boundary).
Let us now discuss a lattice version of the Hodge-dual ⋆j, first for the illustrative example of a 1-cochain e → j(e) and afterwards for the relevant case of our double 1-cochain (e, e ′ ) → σ(e, e ′ ) of conductivity. On the lattice as in the continuum, the complex structure R is rotation by ±π/2. We defined the values of the 1-cochain e → j(e) by the current j(e) across e. In the same vein, we now introduce on Γ another 1-cochain e → (⋆j)(e) by where j(R −1 e) still means the current across the rotated 1-cell R −1 e; by definition it is the same as the current along the unrotated 1-cell e. Referring for notation to Fig. 3c and the text vicinity of Eq. (70), that current is if ℓ + i and ℓ − o are the network-model links on the minus side of the rotated 1-cell R −1 e; otherwise it is the negative thereof.
For a path C on the medial lattice Γ, consider now the sum If C is a cycle, that sum computes (a lattice approximation to) the current circulation around C, cf. Fig. 4; in general, it is not zero. However, at the critical point with conformal invariance, we do expect the current circulation to vanish after disorder averaging and in the scaling limit of large and contractible cycles C. If so, the continuity equation dj = 0 is augmented with a second conservation law d ⋆ j = 0, valid in expectation E(...) and after coarse graining to eliminate lattice effects present for short wavelengths. The lattice currents j 10 = (j + i ⋆ j)/2 and j 01 = (j − i ⋆ j)/2 will then be the parents of holomorphic and anti-holomorphic currents in the continuum. For future reference, we put on record that the lattice currents j 10 and j 01 are given by e → j 10 (e) = j(e) + i ⋆ j(e) /2 and similar for j 01 (with +i → −i). We also note that, using the notation of Figs. 3b and 3c, the expression for j 10 (e) can be rewritten as Figure 4: Illustration of the lattice sum C ⋆j for the current circulation. For ease of drawing, the outer orientation of C is converted into an inner orientation, say by the counterclockwise sense of circulation. (By the same conversion rule, ⋆j is turned into an untwisted 1-cochain.) The lattice sum C ⋆j is then seen to be a weighted sum of primary-lattice currents j ℓ with weights 1 + 1, 1, 0, −1 as indicated by the red symbols +/− at the links ℓ.
An invariant formulation of the same quantity is where the sum is over the four links ℓ joined by the node of e and θ e (ℓ) denotes the angle of positive rotation from the e-perpendicular axis (pointing from minus to plus by the outer orientation of e) to the direction of ℓ. We finally apply the conformal invariance argument (for a secondary conservation law to emerge) to the key object of our endeavor: the non-local response function σ at criticality. To begin, we recall that (e, e ′ ) → σ(e, e ′ ) is a double 1-cochain on Γ and denote its disorder average by Σ(e, e ′ ) = E(σ(e, e ′ )). Earlier, we deduced from Kirchhoff's nodal rule for the squared Green's function that Σ is closed with respect to both of its argumentsthis property already held before taking the disorder average and it still does so afterwards. Now, by virtue of the emerging conformal invariance at the critical point and by the reasoning that led to Eq. (69) above, we expect Σ also to become co-closed with respect to both of its arguments, in the continuum limit. Altogether, this then implies that the tensor component Σ zz ≡ Σ 10,10 becomes holomorphic with respect to both arguments: Here Σ zz (•, • ′ ) means the continuum limit of the linear combination Σ zz (e, e ′ ) = Σ e + iR −1 e , e ′ + iR −1 e ′ /4 of lattice response functions and their ⋆ -duals, and our concise notation exploits the fact that a double 1-cochain is a complex bilinear function of the two chains in its arguments. By the same reasoning, Σzz (defined in the analogous way) becomes anti-holomorphic w.r.t. both arguments.
If we specialize to the continuum of the Euclidean plane with complex coordinate w = x + iy (or the Riemann sphere with complex stereographic coordinate w), then the (anti-)holomorphic part of the response function can be presented in explicit form. Indeed, the rotational invariance of Σ zz = Σ zz (w, w ′ ) dw ⊗ dw ′ in the continuum limit implies for any rotation angle θ. In combination with translational invariance, this determines the holomorphic tensor component to be of the form with a constant n that at the present stage could be any positive real number. The singularity on the diagonal w = w ′ reflects a singularity of Σ that is immanent to its microscopic definition by the squared Green's function (72). The expression for Σzz is similar, with (w − w ′ ) replaced by (w −w ′ ). As a disclaimer, let us stress that the formula (83) assumes translationinvariant regularization of the Green's function ℓ 1 |(1 − T ) −1 |ℓ 2 , T = QU , by an infinitesimal absorbing background Q = e µ (µ → 0−), as stated at the outset of the present subsection. For the real-world purpose of defining and computing a conductance, one must replace the absorbing background by a number n c of terminals (n c ≥ 2). If these are taken to be point contacts, the non-local response function of conductivity becomes an (n c + 2)-point function. The latter has the singularity Σ zz (w, w ′ ) ∼ (w − w ′ ) −2 for w → w ′ but also further singularities when w or w ′ approaches a terminal point.

Response function in the vertex model
From the preceding section we take away the key message that the non-local conductivity response function Σ for the network model at criticality has a (10, 10) tensor component Σ zz which is doubly holomorphic and exhibits the short-distance singularity (83). Guided by this insight we now seek an expression for Σ as a correlator of operators in the SUSY vertex model of Sect. 2.4. Our motivation for doing so is that the resulting operators might be candidates for holomorphic currents of the CFT to be identified.
The first step is to produce operators V (ℓ) and W (ℓ) expressed in terms of the Fock operators at the link ℓ , such that where Σ ℓ 1 , ℓ 2 = E(σ ℓ 1 , ℓ 2 ) is the disorder-averaged squared Green's function introduced in Eq. (66). There exist numerous choices of such operators, and they already exist in the minimal theory with only one replica. For simplicity of notation, let us specialize to that case (r = 1). The good objects to consider then are All operators carry the same link argument ℓ , which has been omitted. The desired correlator Σ ℓ 2 , ℓ 1 is obtained by suitable pairings of these: To verify that claim, e.g. for the pair of V 0 0 with W 0 0 , one starts by using the Wick contraction rule in the free theory (before disorder averaging): together with the formulas (19) for the basic Wick contractions. The desired relation (86) then follows immediately by taking the disorder average on both sides of the equation.
As a direct consequence [15] of the global U(r, r|2r) symmetry of the SUSY vertex model, each of the operators V β α and W β α obeys Kirchhoff's nodal rule in the sense of Sect. 4.3. One might therefore think that, by following the blueprint of Sect. 4.4 to construct from V and W operatorvalued closed 1-cochains which become co-closed at criticality, one could produce the desired holomorphic and anti-holomorphic currents. However, such a direct attempt does not deliver the optimal return. Indeed, the big advantage of the SUSY vertex model, as compared with the network model, is the existence of a boson-fermion multiplet of operators, offering the possibility of a non-Abelian current algebra. Yet, the basic Lie algebra structure needed for a non-Abelian current algebra is absent from the present setup, as the V 's and W 's do not close under commutation (meaning the Lie superbracket). Hence we are going to modify the ansatz (85).

Current algebra from response function
The idea for a modified ansatz that does deliver is very simple: change the basis of Fock operators by mixing the retarded and advanced sectors! Such a change of basis was crucial for the progress made in [15], and it turns out to be key here as well. Thus at every link ℓ of the primary network we now take linear combinations B ± (ℓ) and C ± (ℓ) of the fundamental bosons b ± (ℓ) and b † ± (ℓ): and we make a similar transformation also for the fermions: The unitary factors e iϑ 0 and e iϑ 1 are arbitrary but fixed (independent of ℓ).
Operators denoted by the same letter constitute canonical pairs: and so on (the bracket of two fermionic operators is the anti-commutator).
The relations under Hermitian conjugation in Fock-Hilbert space are diagonal for the fermions but off-diagonal for the bosons: By forming such products as B + B − , taking the left factor from the plus-set and the right factor from the minus-set, one gets 4 2 = 16 quadratic operators. [In the case of r replicas their number would be (4r) 2 = 16r 2 .] Arranged as a matrix, they are  By the canonical bracket relations (89), these quadratic expressions realize the Lie superalgebra gl(2|2) at the given link ℓ, with the matrix position of each operator encoding its behavior w.r.t. the Lie superbracket. The true significance of the matrix arrangement (91) is that it groups the quadratic operators into four blocks, each of which will acquire a distinct meaning. Here comes the working hypothesis that we intend to explain in the sequel [33]: the right upper block bosonizes to a Wess-Zumino-Witten field, M ; the left lower block bosonizes to M −1 . The left upper block gives rise to the holomorphic currents J ↔ ∂M · M −1 , and the right lower block plays the same role on the anti-holomorphic side:J ↔ M −1∂ M .
In the first step, we focus on the operators in the left upper block of the matrix (91), which are made from B ± and F ± . By construction, these generate a subalgebra gl(1|1) ⊂ gl(2|2). [For a number r ≥ 1 of replicas, this would be a subalgebra gl(r|r) ⊂ gl(2r|2r).] For simplicity of notation, we introduce the symbol O β α (α, β = 0, 1) for our gl(1|1) multiplet of operators: Next we express the two-point correlation functions of these operators as disorder averages in the network model. As before, the calculational method is to use the Wick contraction rule for the free-field correlators . . . F and then take the disorder average. Let us state the outcome of this straightforward calculation in a concise manner. For that we recall with X ∈ gl(1|1) the index-free notation O X = X α β O β α introduced for the purpose of writing the operator product expansion (30) of Sect. 3.1. Assuming ℓ 1 = ℓ 2 we then find where Υ ℓ 1 , ℓ 2 was defined in Eq. (67). Our logic now proceeds as follows.
We first look at the special case of odd O X ≡ O 0 1 and O Y ≡ O 1 0 , where the summand in the second line of (93) is absent and we simply have We then recall the procedure that takes the lattice quantity Σ ℓ 1 , ℓ 2 to its holomorphic continuum limit Σ zz in Eqs. (80, 81). (Notice that the symmetrization from Σ ℓ 1 , ℓ 2 to Σ ℓ 1 , ℓ 2 + Σ ℓ 2 , ℓ 1 makes no difference in that limit, thanks to rotational invariance.) In view of the equality (94) we expect that the same procedure applied to the lattice quantities O 0 1 (ℓ 1 ) and O 1 0 (ℓ 2 ) gives rise to holomorphic currents J 0 1 and J 1 0 in the continuum. The expression (83) for the critical response function then implies that the operator product expansion for these currents has the leading singularity shown in (62, 63).
For more general choices of O X,Y the term in the second line of (93) will also be present. To isolate it, we may take to be the operator representing the superparity N = E 0 0 − E 1 1 ∈ gl(1|1) with STrN = 2 and STr(N 2 ) = 0, so that Eq. (93) reduces to (iii) We take the continuum limit at the critical point and denote the limit of the 10-component of j N by J N [34]. The resulting continuum current J N is holomorphic; therefore the singularity of its operator product expansion with itself must be with an undetermined constant γ. Altogether, we see that our reasoning has reproduced the leading singularity of the OPE (62, 63) (for r = 1, but the case r ≥ 1 is no different). Now recall that we want the theory to contain a primary field M with vanishing conformal weights. From the discussion in Sect. 4.1 we know that this will happen if γ = 1; we are thus led to postulate that value for γ. As a direct consequence we can make a prediction: the scaling limit of the network-model correlator Υ ℓ 1 , ℓ 2 projected to its (10, 10)component Υ zz is proportional to Σ zz with a definite amplitude ratio: We offer this as a prediction to be verified by numerical simulation. In order to complete the picture, we ought to check that our lattice currents conform to the next-to-leading singularity in the OPE (62). In the continuum theory, that singularity translates to the statement where C is any integration contour that encloses w (but no points of further operator insertions if such are present). It does not seem easy to check (98) directly for the SUSY vertex model in its present form as a statistical sum (or "path integral"). However, it is textbook knowledge [36] that the path integral relation (98) simply reflects the non-Abelian algebra of conserved charges in the quantum theory. That correspondence suggests to switch to the transfer-matrix (or "quantum") formulation of the SUSY vertex model. The calculation further simplifies if one assumes deformability of the transfer matrix to its anisotropic limit as the Hamiltonian of a quantum (super-)spin chain with Hilbert space . . .
. . . In fact, in the spinchain limit verification of the lattice precursor to the continuum relation (98) becomes a straightforward matter of checking the Lie superalgebra of the operators displayed in the matrix of (91). [The operators (87, 88) act on the dual space V * ℓ by the co-representation.] This algebra is closed for the left upper block (as well as the right lower block), which is why we made the change of basis (87, 88) in the first place.
We finish this subsection with a few remarks. (i) The construction of the anti-holomorphic currentsJ can be done in essentially the same way; for that we take the gl(1|1) algebra generated by the left upper block in (91) and replace it by the gl(1|1) algebra of the right lower block. (ii) One easily checks that the holomorphic currents J have a trivial OPE with the anti-holomorphic currentsJ. (iii) By the relations (90) under Hermitian conjugation, the modes of the holomorphic boson-boson current J 0 0 are adjoint to those of the anti-holomorphic currentJ 0 0 , whereas in the case of the fermion-fermion currents J 1 1 andJ 1 1 they are adjoint to themselves. (iv) While the number n in Eq. (83) could have been any positive number, it is now seen to be the quantized level of a current algebra with compact sector u(r), which is non-Abelian for r > 1; as such it must be a positive integer.

Wess-Zumino-Witten field
Our attention now turns to the right upper block of the matrix array (91); deviating from the standard even-odd ordering we abbreviate its entries as A detailed study of these operators and the observable quantities derived from them was made in [15]; here we summarize the main message. A special role is played by the element B + C − in the right upper corner. This operator is a highest-weight element for the adjoint representation of gl(2|2) (with Cartan subalgebra generated by the diagonal operators B + B − , . . . , C + C − ), but it is also highest-weight for the fundamental action from the left and the anti-fundamental action from the right. Moreover, it has the distinctive property of being a positive operator: B + C − = B + (B + ) † > 0. Building on these properties, we argued in [15] that a coarsegrained form of (B + C − ) q has the infrared behavior of the vertex operator e qϕ for a Gaussian free field ϕ with background charge Q = 1. This proposal had been tested numerically [37] by recognizing (B + C − ) q V as the moment E(|ψ| 2q ) of a critical (hence multifractal) stationary wave function ψ = U ψ.
We now upgrade that argument to the stronger proposal that B + C − corresponds (after coarse graining and non-Abelian bosonization) to the boson-boson matrix element of a Wess-Zumino-Witten field M . Our first remark here is to recall that, for present purposes, an acceptable WZW target space must be a cs-supermanifold [inside the complex Lie supergroup GL(r|r)] based on a Riemannian symmetric space Herm + (r) × U(r). This requirement is satisfied by the proposal (99) as B + C − is positive Hermitian and the fermion-fermion pair F + G − , made from one left-mover (F + ) and one right-mover (G − ), bosonizes to a unitary by standard lore. For a second remark, note that our proposal means that M will be a primary field not just for the Virasoro algebra but even for the underlying Kac-Moody algebra of currents; that will make for a rapid derivation of the multifractality spectrum in Sect. 4.9. Thirdly, note the dependence of M on the unitary constants e iϑ 0 and e iϑ 1 (corresponding to a choice of maximal Abelian subgroup for the non-linear sigma model target manifold U/K).
The operator product expansion between the holomorphic current J and the WZW fundamental field M is [33] and an analogous formula holds for the OPE ofJ with M : As before, this is verified on the lattice by first computing the OPE in the free theory with the Wick contraction rules (19) and then taking the disorder average -with the proviso that insertions of additional operators may be (and in fact must be) present. Let us motivate why the holomorphic current acts on the left while the anti-holomorphic current acts on the right. The basic Wick contractions in the transformed basis (87, 88) are reflecting the U(1, 1|2) invariance of · · · F . All others go to zero by restoration of the global U(1, 1|2) symmetry in the limit of vanishing regularization. For example, using T = e µ U and T † = U −1 e µ one has In order for the last inference (sending µ → 0) to be a rigorous statement, the role of regulator must be taken over by an operator insertion (say, that of a point contact) in the correlation function · · · F to cut off the geometric sum (1 − U ) −1 = n U n . The left-right structure of the OPEs (100) and (101) then follows because the holomorphic current J is constructed from operators B ± , F ± , the anti-holomorphic currentJ from C ± , G ± , while M is made of one from B + , F + (left factor) and one from C − , G − (right factor).

Deformation of WZW model
We now ask what is the Lagrangian of the conformal field theory defined by the operator product expansions (62, 63, 100, 101)? The answer to that question begins with the reminder that the currents of the standard GL(r|r) n WZW model (without deformation) satisfy (62) with γ = 0 . To introduce the deformation parameter γ, we add to the WZW action functional (57)  On general grounds [38] the integrated current J X for a supermatrixvalued holomorphic function X ≡ X(z) is the generator of left translations: (δ X M )(z,z) = X(z)M (z,z).
From the expression (104) one sees immediately that the deformed current responds to infinitesimal left translations as It then follows that the holomorphic current J obeys the operator product expansion (62, 63) with deformation parameter γ. Thus the action functional (103) does the required job of giving the path-integral representation of our deformed conformal field theory [35]. We turn to the energy-momentum tensor, T (z). We recall that T (z) is determined by the requirement (64) that the holomorphic current J be a Virasoro-primary field of conformal weights (1, 0). By SUSY cancelation due to the equal number of bosonic and fermionic fields, the number of replicas r versus 2r does not change the result for T (z). Thus the expression (65) remains valid in the present case of a gl(r|r) n current algebra.
We already know from Sect. 4.1 that the deformation parameter γ must be set to unity in order for M to have conformal weights (0, 0) as required by the phenomenology of Anderson transitions in class A. This leaves the current algebra level n as the only open parameter. In the final section we will demonstrate that the quantized level n ∈ N must be n = 4 to match the multifractality spectrum known from numerical simulations.

Multifractal scaling exponents
The expression (65) for the energy-momentum tensor T (z) simplifies to at γ = 1. Now, we have argued that M is a primary field for the Kac-Moody (or affine) Lie superalgebra gl(r|r) n,γ , and is so, in particular, for γ = 1. We expect this property to carry over to a continuum of infinitedimensional gl(r|r)-irreducible representations. Exactly what those are in our non-compact current algebra setting is a non-trivial question, at least from the mathematics perspective of serious analysis. Here we will content ourselves with a simple physics-style argument. We set r = 1 and look at any power q ∈ N of the boson-boson field M 0 0 . (Standard heuristics indicates that the "good" powers are q = 1/2 + iλ with λ ∈ R.) Building on the assumption that (M 0 0 ) q is an affine primary field, we obtain [38] the OPE where Cas 2 (q) is the quadratic gl(1|1) Casimir element evaluated in the representation of (M 0 0 ) q . To compute the latter, we use the formula with E β α = e α ⊗ e β the standard generators of gl(1|1). Our primary field (M 0 0 ) q transforms as the q th symmetric power of the fundamental vector e 0 , which obeys the relations E 1 1 e q 0 = E 1 0 e q 0 = 0 , E 0 0 e q 0 = q e q 0 , (E 0 0 ) 2 e q 0 = q 2 e q 0 .
Using these we obtain the Casimir eigenvalue Cas 2 (q) = −q 2 + q and hence The total scaling dimension of (M 0 0 ) q then is and we expect this to continue analytically to values of q beyond the discrete set N. In our previous work [15] we identified (M 0 0 ) q as the SUSY vertex model operator for the disorder average E |ψ(ℓ)| 2q of a multifractal wave function ℓ → ψ(ℓ) of the network model at criticality. Thus ∆ q is what is known as the spectrum of multifractal scaling exponents. The conclusion from recent numerics [37] was that a good fit of the numerical data can be had with ∆ q = Xq(1 − q) for X in the range of 0.26-0.28. We now predict with confidence that careful finite-size scaling for large systems will converge to the result (112) with n = 4.