Parisi-Sourlas Supersymmetry Research Articles

Random field ϕ3 model and Parisi-Sourlas supersymmetry

We use the RG framework set up in [1] to explore the ϕ3 theory with a random field interaction. According to the Parisi-Sourlas conjecture this theory admits a fixed point with emergent supersymmetry which is related to the pure Lee-Yang CFT in two less dimensions. We study the model using replica trick and Cardy variables in d = 8 − ϵ where the RG flow is perturbative. Allowed perturbations are singlets under the Sn symmetry that permutes the n replicas. These are decomposed into operators with different scaling dimensions: the lowest dimensional part, ‘leader’, controls the RG flow in the IR; the other operators, ‘followers’, can be neglected. The leaders are classified into: susy-writable, susy-null and non-susy-writable according to their mixing properties. We construct low lying leaders and compute the anomalous dimensions of a number of them. We argue that there is no operator that can destabilize the SUSY RG flow in d ≤ 8. This agrees with the well known numerical result for critical exponents of Branched Polymers (which are in the same universality class as the random field ϕ3 model) that match the ones of the pure Lee-Yang fixed point according to dimensional reduction in all 2 ≤ d ≤ 8. Hence this is a second strong check of the RG framework that was previously shown to correctly predict loss of dimensional reduction in random field Ising model.

Random field Ising model and Parisi-Sourlas supersymmetry. Part II. Renormalization group

We revisit perturbative RG analysis in the replicated Landau-Ginzburg description of the Random Field Ising Model near the upper critical dimension 6. Working in a field basis with manifest vicinity to a weakly-coupled Parisi-Sourlas supersymmetric fixed point (Cardy, 1985), we look for interactions which may destabilize the SUSY RG flow and lead to the loss of dimensional reduction. This problem is reduced to studying the anomalous dimensions of “leaders” — lowest dimension parts of Sn-invariant perturbations in the Cardy basis. Leader operators are classified as non-susy-writable, susy-writable or susy-null depending on their symmetry. Susy-writable leaders are additionally classified as belonging to superprimary multiplets transforming in particular OSp(d|2) representations. We enumerate all leaders up to 6d dimension ∆ = 12, and compute their perturbative anomalous dimensions (up to two loops). We thus identify two perturbations (with susy- null and non-susy-writable leaders) becoming relevant below a critical dimension dc ≈ 4.2 - 4.7. This supports the scenario that the SUSY fixed point exists for all 3 < d ⩽ 6, but becomes unstable for d < dc.

Dimensional reduction of higher-point conformal blocks

Recently, with the help of Parisi-Sourlas supersymmetry an intriguing relation was found expressing the four-point scalar conformal block of a (d − 2)-dimensional CFT in terms of a five-term linear combination of blocks of a d-dimensional CFT, with constant coefficients. We extend this dimensional reduction relation to all higher-point scalar conformal blocks of arbitrary topology restricted to scalar exchanges. We show that the constant coefficients appearing in the finite term higher-point dimensional reduction obey an interesting factorization property allowing them to be determined in terms of certain graphical Feynman-like rules and the associated finite set of vertex and edge factors. Notably, these rules can be fully determined by considering the explicit power-series representation of just three particular conformal blocks: the four-point block, the five-point block and the six-point block of the so-called OPE/snowflake topology. In principle, this method can be applied to obtain the arbitrary-point dimensional reduction of conformal blocks with spinning exchanges as well. We also show how to systematically extend the dimensional reduction relation of conformal partial waves to higher-points.

How to succeed at Witten diagram recursions without really trying

Witten diagrams are basic objects for studying dynamics in AdS space, and also play key roles in the analytic functional bootstrap. However, these diagrams are notoriously hard to evaluate, making it extremely difficult to search for recursion relations among them. In this note, we present simple methods to obtain recursion relations for exchange Witten diagrams from conformal block recursion relations. We discover a variety of new relations, including the dimensional reduction formulae for exchange Witten diagrams. In particular, we find a five-term recursion relation relating exchange Witten diagrams in d and d − 2 dimensions. This gives the holographic analogue of a similar formula for conformal blocks due to Parisi-Sourlas supersymmetry. We also extend the analysis to two-point functions in CFTs with conformal boundaries, and obtain similar results.

Random Field Ising Model and Parisi-Sourlas supersymmetry. Part I. Supersymmetric CFT

Quenched disorder is very important but notoriously hard. In 1979, Parisi and Sourlas proposed an interesting and powerful conjecture about the infrared fixed points with random field type of disorder: such fixed points should possess an unusual supersymmetry, by which they reduce in two less spatial dimensions to usual non-supersymmetric non- disordered fixed points. This conjecture however is known to fail in some simple cases, but there is no consensus on why this happens. In this paper we give new non-perturbative arguments for dimensional reduction. We recast the problem in the language of Conformal Field Theory (CFT). We then exhibit a map of operators and correlation functions from Parisi-Sourlas supersymmetric CFT in d dimensions to a (d − 2)-dimensional ordinary CFT. The reduced theory is local, i.e. it has a local conserved stress tensor operator. As required by reduction, we show a perfect match between superconformal blocks and the usual conformal blocks in two dimensions lower. This also leads to a new relation between conformal blocks across dimensions. This paper concerns the second half of the Parisi-Sourlas conjecture, while the first half (existence of a supersymmetric fixed point) will be examined in a companion work.

On p-adic stochastic dynamics, supersymmetry and the Riemann conjecture

We construct (assuming the quantum inverse scattering problem has a solution) the operator that yields the zeroes of the Riemann zeta function by defining explicitly the supersymmetric quantum mechanical model (SUSY QM) associated with the p-adic stochastic dynamics of a particle undergoing a Brownian random walk. The zig-zagging occurs after collisions with an infinite array of scattering centers that fluctuate randomly. Arguments are given to show that this physical system can be modelled as the scattering of the particle about the infinitely many locations of the prime numbers positions. We are able then to reformulate such a p-adic stochastic process, that has an underlying hidden Parisi–Sourlas supersymmetry, as the effective motion of a particle in a potential which can be expanded in terms of an infinite collection of p-adic harmonic oscillators with fundamental (Wick-rotated imaginary) frequencies ω p= i log p ( p is a prime) and whose harmonics are ω p,n= i log p n . The p-adic harmonic oscillator potential allows us to determine a one-to-one correspondence between the amplitudes of oscillations a n (and phases) with the imaginary parts of the zeroes of the Riemann zeta function, λ n , after solving the inverse scattering problem. We review our recent proof of the Riemann hypothesis that the non-trivial zeroes of zeta are of the form s=1/2+i λ n and the solution to the quantum inverse scattering problem.

On QCD string theory and AdS dynamics

The AdS/CFT correspondence of elementary string theory has been recently suggested as a ``microscopic'' approach to QCD string theory in various dimensions. We use the microscopic theory to show that the ultraviolet regime on the string world-sheet is mapped to the ultraviolet effects in QCD. In the case of QCD_2, a world-sheet path integral representation of QCD strings is known, in terms of a topological rigid string theory whose world-sheet supersymmetry is reminiscent of Parisi-Sourlas supersymmetry. We conjecture that the supersymmetric rigid string theory is dual to the elementary Type IIB string theory in the singular AdS background that corresponds to the large-N limit of QCD_2. We also generalize the rigid string with world-sheet Parisi-Sourlas supersymmetry to dimensions greater than two, and argue that the theory is asymptotically free, a non-zero string tension is generated dynamically through dimensional transmutation, and the theory is topological only asymptotically in the ultraviolet.

Quenched two dimensional supersymmetric Yang-Mills theory

By studying the pure Yang-Mills theory on a circle, as well as an adjoint scalar coupled to the gauge field on a circle, we propose a quenching prescription in which the combination of the spatial component of the gauge field and P is treated as a dynamic variable. Averaging over momentum is not necessary, therefore the usual ultraviolet cut-off is eliminated. We then apply this prescription to study the large N two dimensional supersymmetric gauge theory. An one dimensional supersymmetric matrix model is obtained. It is not known whether this model can be solved exactly. However, an extended model with one more complex fermion is exactly solvable, with N = 1 supersymmetry as Parisi-Sourlas supersymmetry. The exact solvability may have some implications for the N = 1 quenched model.

Geometrical reduction and Parisi-Sourlas supersymmetry

Parisi-Sourlas supersymmetry is shown to be described geometrically as a generalized reduction. This geometrical interpretation uses a group theoretical BRST construction that works beyond canonical phase space and extends ordinary BRST covariant quantization of constrained Hamiltonian theories.

Multicritical phases of the O( n) model on a random lattice

We exhibit the multicritical phase structure of the loop gas model on a random surface. The dense phase is reconsidered, with special attention paid to the topological points g = 1/ p. This phase is complementary to the dilute and higher multicritical phases in the sense that dense models contain the same spectrum of bulk operators (found in the continuum by Lian and Zuckerman) but a different set of boundary operators. This difference illuminates the well-known ( p, q) asymmetry of the matrix chain models. Higher multicritical phases are constructed, generalizing both Kazakov's multicritical models as well as the known dilute phase models. They are quite likely related to multicritical polymer theories recently considered independently by Saleur and Zamolodchikov. Our results may be of help in defining such models on flat honeycomb lattices; an unsolved problem in polymer theory. The phase boundaries correspond again to “topological” points with g = p/1 integer, which we study in some detail. Two qualitatively different types of critical points are discovered for each such g. For the special point g = 2 we demonstrate that the dilute phase O(−2) model does not correspond to the Parisi-Sourlas model, a result likely to hold as well for the flat case. Instead it is proven that the first multicritical O(−2) point possesses the Parisi-Sourlas supersymmetry.

Polymers and percolation in two dimensions and twisted N = 2 supersymmetry

We show how a large class of geometrical critical systems including dilute polymers, polymers at the θ-point, percolation and to some extent brownian motion, are described by a twisted N = 2 supersymmetric theory with k = 1 (it is broken in the dense polymer phase that is described simply by a η, ξ system). This allows us to give for the first time a consistent conformal field theory description of these problems. The fields that were described so far by formally allowing half-integer labels in the Kac table are built and their four-point functions studied. Geometrical operators are organized in a few representations of the twisted N = 2 algebra. A noticeable feature is that in addition to Neveu-Schwarz and Ramond, a sector with quarter twists sometimes has to be introduced. The algebra of geometrical operators is determined. Fermions boundary conditions are geometrically interpreted, and the partition functions that were so far defined formally as generating functions for the critical exponents are naturally understood, sector by sector. Twisted N = 2 provides moreover a very unified description of all these geometrical models, explaining for instance why the exponents of polymers and percolation coincide. It must be stressed that the physical states in these geometrical problems are not the physical states for string theory, which are usually extracted by the BRS cohomology. In polymers for instance, Q BRS is precisely the operator that creates polymers out of the vacuum, such that the topological sector is the sector without any polymers. It seems that twisted N = 2 is the correct continuum limit (in two dimensions) for models with Parisi-Sourlas supersymmetry. Some possible explanation of this fact is advanced. As an example of application of N = 2 supersymmetry we discuss the still unsolved problem of backbone of percolation. We conjecture in particular the value D = 25 16 for the fractal dimension of the backbone, in good agreement with numerical computations. Finally some of these ideas are extended to the off-critical case. It is shown how to give a meaning to the n → 0 limit of the O( n) model S-matrix recently introduced by Zamolodchikov by introducing fermions.

Geometrical lattice models for N = 2 supersymmetric theories in two dimensions

We introduce in this paper two-dimensional lattice models whose continuum limit belongs to the N = 2 series. The first kind of model is integrable and obtained through a geometrical reformulation, generalizing results known in the k = 1 case, of the Γ k models (based on the quantum algebra U q sl(2) and representation of spin j = k/2). We demonstrate in particular that at the N = 2 point, the free energy of the Γ k vertex model can be obtained exactly by counting arguments, without any Bethe ansatz computation, and we exhibit lattice operators that reproduce the chiral ring. The second class of models is more adequately described in the language of twisted N = 2 supersymmetry, and consists of an infinite series of multicritical polymer points, which should lead to experimental realizations. The presence of N = 2 in that case is traced back to the Parisi-Sourlas supersymmetry of the lagrangians usually used to replace n → 0 limits. Boundary conditions as well as fermionic and bosonic variables are geometrically interpreted. Moreover it turns out that the exponents ν = (k + 2) 2(k + 1) for these multicritical polymer points coincide with the old phenomenological formulas of Flory. We therefore confirm that these formulas are exact in two dimensions, and suggest that their unexpected validity due to non-renormalization theorems for the N = 2 underlying theories. We also discuss the status of the much discussed θ-point for polymers in the light of N = 2 renormalization group flows.

Stochastic Quantization Method with Field-Dependent Metric

We investigate a general framework of the stochastic quantization method for theories with field-dependent metrics. The differential of a field has different properties from ordinary (i.e., not stochastic) one, which enforces us to define the Langevin equation, the invariant measure and others in rather complicated forms. The Parisi-Sourlas supersymmetry in this framework is also discussed and shown to be defined in a reference frame.

Parisi-Sourlas fermions from Ramond superstring

We show that the zero energy limit of the Ramond superstring naturally describes Dirac fermions with an OSP( D + 1,1¦2) or Parisi-Sourlas supersymmetry, and establish an equivalence between ordinary Dirac fermions and a massive (0 + 1)-dimensional supergravity theory.

OSP(1,1|2) and Parisi-Sourlas supersymmetry for closed strings

Nonabelian OSP(1,1|2) generators for a closed string are constructed and the hole role of subsidiary constraints in dealing with the zero modes is explained.

Negative dimensional integration. II. Path integrals and fermionic equivalence

We present a working definition of negative dimensional integration which reduces the evaluation of integrals to the algebraic problem of resumming polynomial expressions. We demonstrate its equivalence to fermionic integration. Then we apply this to the calculation of Green's functions for the harmonic and anharmonic oscillators. Finally we comment on the connection between negative dimensional integration and Parisi-Sourlas supersymmetry.