Experimental probes of emergent symmetries in the quantum Hall system

Experiments studying renormalization group flows in the quantum Hall system provide significant evidence for the existence of an emergent holomorphic modular symmetry Γ₀(2)${\Gamma }_0(2)$. We briefly review this evidence and show that, for the lowest temperatures, the experimental determination of the position of the quantum critical points agrees to the parts per mille   level with the prediction from Γ₀(2)${\Gamma }_0(2)$. We present evidence that experiments giving results that deviate substantially from the symmetry predictions are not cold enough to be in the quantum critical domain. We show how the modular symmetry extended by a non-holomorphic particle–hole duality leads to an extensive web of dualities related to those in plateau–insulator transitions, and we derive a formula relating dual pairs (B,Bd)$(B,B_d)$ of magnetic field strengths across any transition. The experimental data obtained for the transition studied so far is in excellent agreement with the duality relations following from this emergent symmetry, and rule out the duality rule derived from the “law of corresponding states”. Comparing these generalized duality predictions with future experiments on other transitions should provide stringent tests of modular duality deep in the non-linear domain far from the quantum critical points.     

OPE for null Wilson loops and open spin chains

Maximal helicity-violating scattering amplitudes in N=4$N=4$ supersymmetric Yang–Mills theory are dual to Wilson loops on closed null polygons. We perform their operator product expansion analysis in two-dimensional kinematics in the soft-collinear approximation which corresponds to the case when some light-cone distances vanish. We construct the expansion in terms of multi-particle “heavy”–light operators, where the “heavy” fields are identified with the Wilson lines defining the OPE channel and the light fields emerge from the curvature of the contour. The correlation function of these define the remainder function. We study the dilatation operator for these operators at one-loop order and find that it corresponds to a non-compact open spin chain. This provides an alternative view on elementary excitations propagating on the GKP string at weak coupling, which now correspond to particles traveling along an open spin chain. The factorized structure of the Wilson loop in the soft limit allows one to represent the two-loop correction to the octagon Wilson loop as a convolution formula and find the corresponding remainder function.     

Are scattering amplitudes dual to super Wilson loops?

The MHV scattering amplitudes in planar N=4 SYM are dual to bosonic light-like Wilson loops. We explore various proposals for extending this duality to generic non-MHV amplitudes. The corresponding dual object should have the same symmetries as the scattering amplitudes and be invariant to all loops under the chiral half of the N=4 superconformal symmetry. We analyze the recently introduced supersymmetric extensions of the light-like Wilson loop (formulated in Minkowski space-time) and demonstrate that they have the required symmetry properties at the classical level only, up to terms proportional to field equations of motion. At the quantum level, due to the specific light-cone singularities of the Wilson loop, the equations of motion produce a nontrivial finite contribution which breaks some of the classical symmetries. As a result, the quantum corrections violate the chiral supersymmetry already at one loop, thus invalidating the conjectured duality between Wilson loops and non-MHV scattering amplitudes. We compute the corresponding anomaly to one loop and solve the supersymmetric Ward identity to find the complete expression for the rectangular Wilson loop at leading order in the coupling constant. We also demonstrate that this result is consistent with conformal Ward identities by independently evaluating corresponding one-loop conformal anomaly.     

Light-like noncommutativity,light-front quantization and new light on UV/IR mixing

We revisit the problem of quantizing field theories on noncommutative Moyal space–time with light-like   noncommutativity. To tackle the issues arising from noncommuting and hence nonlocal time, we argue that for this case light-front quantization procedure should be employed. In this appropriate quantization scheme we perform the non-planar loop analysis for the light-like noncommutative field theories. One of the important and peculiar features of light-front quantization is that the UV cutoff of the light-cone Hamiltonian manifests itself as an IR cutoff for the light-cone momentum, p⁺$p^{+}$. Due to this feature, the naive results of covariant quantization for the light-like case allude to the absence of the UV/IR mixing in the light-front quantization. However, by a careful analysis of non-planar loop integrals we show that this is not the case and the UV/IR mixing persists. In addition, we argue in favour of the perturbative unitarity of light-like noncommutative field theories in the light-front quantization scheme.     

Connectivities of Potts Fortuin–Kasteleyn clusters and time-like Liouville correlator

Recently, two of us argued that the probability that an FK cluster in the Q-state Potts model connects three given points is related to the time-like Liouville three-point correlation function (Delfino and Viti, 2011) [1]. Moreover, they predicted that the FK three-point connectivity has a prefactor which unveils the effects of a discrete symmetry, reminiscent of the _SQ$S_Q$ permutation symmetry of the Q=2,3,4$Q=2,3,4$ Potts model. We revisit the derivation of the time-like Liouville correlator (Zamolodchikov, 2005) [2] and show that this is the only consistent analytic continuation of the minimal model structure constants. We then present strong numerical tests of the relation between the time-like Liouville correlator and percolative properties of the FK clusters for real values of Q.     

B to light tensor meson form factors derived from light-cone sum rules

Using the recent results for the two-parton light-cone distribution amplitudes of the tensor meson, we calculate the form factors for the decays of B_u,d,s$B_{u,d,s}$ into the light JPC=²⁺⁺$J^{\mathit{PC}}=2^{++}$ tensor mesons via the vector/axial–vector/tensor current with the light-cone sum rules. We also obtain the q²$q^2$-dependence of the form factors.     

Composite representation invariants and unoriented topological string amplitudes

Sinha and Vafa [1] had conjectured that the SO   Chern–Simons gauge theory on S³$S^3$ must be dual to the closed A  -model topological string on the orientifold of a resolved conifold. Though the Chern–Simons free energy could be rewritten in terms of the topological string amplitudes providing evidence for the conjecture, we needed a novel idea in the context of Wilson loop observables to extract cross-cap c=0,1,2$c=0,1,2$ topological amplitudes. Recent paper of Marino [2] based on the work of Morton and Ryder [3] has clearly shown that the composite representation placed on the knots and links plays a crucial role to rewrite the topological string cross-cap c=0$c=0$ amplitude. This enables extracting the unoriented cross-cap c=2$c=2$ topological amplitude. In this paper, we have explicitly worked out the composite invariants for some framed knots and links carrying composite representations in U(N)$U(N)$ Chern–Simons theory. We have verified generalised Rudolph's theorem, which relates composite invariants to the invariants in SO(N)$\mathit{SO}(N)$ Chern–Simons theory, and also verified Marino's conjectures on the integrality properties of the topological string amplitudes. For some framed knots and links, we have tabulated the BPS integer invariants for cross-cap c=0$c=0$ and c=2$c=2$ giving the open-string topological amplitude on the orientifold of the resolved conifold.     

Green functions and twist correlators for N branes at angles

We compute the Green functions and correlator functions for N   twist fields for branes at angles on T²$T^2$ and we show that there are N−2$N-2$ different configurations labeled by an integer M   which is roughly associated with the number of reflex angles of the configuration. In order to perform this computation we use an SL(2,R)$\mathit{SL}(2,\mathbb{R})$ invariant formulation and geometric constraints instead of Pochammer contours. In particular the M=1$M=1$ or M=N−1$M=N-1$ amplitude can be expressed without using transcendental functions. We determine the amplitudes normalization from N→N−1$N\to N-1$ reduction without using the factorization into the untwisted sector. Both the amplitudes normalization and the OPE of two twist fields are unique (up to one constant) when the ?↔1−?$?\leftrightarrow 1-?$ symmetry is imposed. For consistency we find also an infinite number of relations among Lauricella hypergeometric functions.     

Properties of CFTs dual to charged BTZ black hole

We study properties of strongly coupled CFT's with non-zero background electric charge in 1+1$1+1$ dimensions by studying the dual gravity theory—which is a charged BTZ black hole. Correlators of operators dual to scalars, gauge fields and fermions are studied at both T=0$T=0$ and T≠0$T\ne 0$. In the T=0$T=0$ case we are also able to compare with analytical results based on AdS₂${\mathit{AdS}}_2$ and find reasonable agreement. In particular the correlation between log periodicity and the presence of finite spectral density of gapless modes is seen. The real part of the conductivity (given by the current–current correlator) also vanishes as ω→0$\omega \to 0$ as expected. The fermion Green's function shows quasiparticle peaks with approximately linear dispersion but the detailed structure is neither Fermi liquid nor Luttinger liquid and bears some similarity to a “Fermi–Luttinger” liquid. This is expected since there is a background charge and the theory is not Lorentz or scale invariant. A boundary action that produces the observed non-Luttinger liquid like behavior (k  -independent non-analyticity at ω=0$\omega =0$) in the Green's function is discussed.