Self-gravitating Brownian particles in two dimensions: the case of N = 2 particles

We study the motion of N = 2 overdamped Brownianparticles in gravitational interaction in a space of dimensiond = 2. This is equivalent to the simplified motion of twobiological entities interacting via chemotaxis when time delay anddegradation of the chemical are ignored. This problem also bearssimilarities with the stochastic motion of two point vorticesin viscous hydrodynamics [O. Agullo, A. Verga, Phys. Rev. E 63,056304 (2001)]. We analytically obtain the probability density offinding the particles at a distance r from each other at timet. We also determine the probability that the particles havecoalesced and formed a Dirac peak at time t(i.e. the probability that the reduced particle has reached r = 0at time t). Finally, we investigate the meansquare separation \(\langle\) r ² \(\rangle\) and discuss the proper formof the virial theorem for this system. The reduced particle has anormal diffusion behavior for small times with a gravity-modifieddiffusion coefficient \(\langle\) r ² \(\rangle\) = r ₀ ² + (4k _B /ξ μ)(T–\(T_{*}\))t, wherek _B \(T_{*}\) = Gm ₁ m ₂/2 is a critical temperature, and an anomalousdiffusion for large times \(\langle\) r ² \(\rangle\) \(\propto\) \(t^{1-T_*/T}\). As a by-product, our solution also describes thegrowth of the Dirac peak (condensate) that forms at large time inthe post collapse regime of the Smoluchowski-Poisson system (orKeller-Segel model in biology) for T < T _c = GMm/(4k _B ). We find thatthe saturation of the mass of the condensate to the total mass isalgebraic in an infinite domain and exponential in a boundeddomain. Finally, we provide the general form of the virial theoremfor Brownian particles with power law interactions.     

Nonlocal current algebra and N = 2 superconformal field theory in two dimensions

We present a new construction of the unitary highest weight representations of the N = 2 superconformal algebras in two dimensions. This construction is based on the non-local current in the Z_N conformal theory and a free scalar field. It provides a physical realization of all the unitary N = 2 superconformal field theories by critical systems. The correlation functions of the theories can be calculated through this construction.     

N = 1 superstring in 2 + 2 dimensions

In this paper we construct a (2,2) dimensional string theory with manifest N = 1 spacetime supersymmetry. We use Berkovits' approach of augmenting the spacetime supercoordinates by the conjugate momenta for the fermionic variables. The worldsheet symmetry algebra is a twisted and truncated “small” N = 4 superconformal algebra. The realisation of the symmetry algebra is reducible with an infinite order of reducibility. We study the physical states of the theory by two different methods. In one of them, we identify a subset of irreducible constraints, which is by itself critical. We construct the BRST operator for the irreducible constraints, and study the cohomology and interactions. This method breaks the SO(2,2) spacetime symmetry of the original reducible theory. In another approach, we study the theory in a fully covariant manner, which involves the introduction of infinitely many ghosts for ghosts.     

N=2 super Yang-Mills theory in projective superspace

We constructN=2 Yang-Mills theory in projective superspace by exploiting the analogy to Ward's twistor construction of self-dual Yang-Mills fields.Work supported in part by NSF grant No. PHY 85-07627     

Massive N = 2a supergravity in ten dimensions

A new version of the non-chiral N = 2a supergravity in ten dimensions is obtained, in which the two-index tensor field of the theory “eats” the single vector field and acquires a mass in a Higgs-type mechanism. The new theory, although it contains no fundamental vectors, bears many formal resemblances to gauged supergravities (in particular, the recently constructed F(4) theory in six dimensions). The scalar potential has no extrema, but nevertheless the classical equations of motion admit a wide variety of spontaneous compactifications, many to four dimensions.     

Study of N = 2 superconformal field theories in 4 dimensions

Making use of the exact solutions of the N = 2 supersymmetric gauge theories, we construct new classes of superconformal field theories (SCFTs) by fine-tuning the moduli parameters and bringing the theories to critical points. SCFTs we have constructed represent universality classes of the 4-dimensional N = 2 SCFTs.     

On the integrable hierarchies associated with the N = 2 super Wn algebra

A new Lax operator is proposed from the viewpoint of constructing the integrable hierarchies related with the N = 2 super W_n algebra. It is shown that the Poisson algebra associated to the second Hamiltonian structure for the resulting hierarchy contains the N = 2 super Virasoro algebra as a proper subalgebra. The simplest cases are discussed in detail. In particular, it is proved that the supersymmetric two-boson hierarchy is one of the N = 2 supersymmetric KdV hierarchies. Also, a Lax operator is supplied for one of the N = 2 supersymmetric Boussinesq hierarchies.     

The (N=1) Supersymmetric Sine-Gordon model in two dimensions. I

In this paper and its companion (II) we prove that the SupersymmetricN=1 massless Sine-Gordon field theory, at finite (space) volume, exists and is analytic in the coupling constant . Moreover at finite (space) volume is Witten index is=1.     

The (N=1) Supersymmetric Sine-Gordon model in two dimensions. II

In this second paper the technical part of the results about the SupersymmetricN=1 massless Sine-Gordon field theory, at finite (space) volume, is given. The proof that the theory exists and is analytic in the coupling constant and that, at finite (space) volume, its Witten index is 1, is, therefore, completed.     

N = 8 supergravity in various dimensions and the implications for four dimensions

The possibilities for constructing new spontaneously broken N = 8 supergravity theories in four dimensions by applying the generalized dimensional reduction method to symmetrical theories in higher dimensions are explored.     

Gauged N = 8 supergravity in five dimensions

We construct gauged N = 8 supergravity theories in five dimensions. Instead of the twenty-seven vector fields of the ungauged theory, the gauged theories contain fifteen vector fields and twelve second-rank antisymmetric tensor fields satisfying self-dual field equations. The fifteen vector fields can be used to gauge any of the fifteen-dimensional semisimple subgroups of SL(6,R), specially SO(p, 6?p) for p = 0, 1, 2, 3. The gauged theories also have a physical global SU(1,1) symmetry which survives from the E₆₍₆₎ symmetry of the ungauged theory. This SU(1,1) for the SO(6) gauging is presumably related to that of the chiral N = 2 theory in ten dimensions. In our formalism we maintain a composite local USp(8) symmetry analogous to SU(8) in four dimensions.     

Broken N=2 super Yang-Mills theories with a finite vacuum energy

It is shown that an explicit symmetry breaking of the N=2 ultra-violet finite super Yang-Mills theories through the addition of soft mass terms does not affect the finiteness of the vacuum energy above one loop. One-loop vacuum energy is finite provided the mass terms satisfy the sum rules Σ_J(2J+1)(−1)^2Jm_J^2K = 0 (K = 0, 1, 2).     

Operator algebra of the N=1 super Wess-Zumino model

The “projective” Ward identities for the super Kac-Moody algebra are solved in order to determine the three- and four-point functions in the N=1 superspace. Linear super-equations are derived for the correlation functions of degenerate operators. These equations are solved for the three- and four-point functions, proving the closure of the operator algebra of the super Wess-Zumino model.     

A new derivation of the Picard-Fuchs equations for effective N = 2 super Yang-Mills theories

A new method to obtain the Picard-Fuchs equations of effective, N = 2 supersymmetric gauge theories in 4 dimensions is developed. It includes both pure super Yang-Mills and supersymmetric gauge theories with massless matter hypermultiplets. It applies to all classical gauge groups, and directly produces a decoupled set of second-order, partial differential equations satisfied by the period integrals of the Seiberg-Witten differential along the 1-cycles of the algebraic curves describing the vacuum structure of the corresponding N = 2 theory.     

The structure ofN=16 supergravity in two dimensions

We extend the previously constructed linear system forN=16 supergravity in two dimensions by including the unphysical gravitino degrees of freedom. This theory has a residualN=16 superconformal invariance that can be bosonized to localE ₉ transformations. The modifications to the linear system described here suggest a further extension to an infinite hierarchy of fields and associated gauge transformations related toE ₉.On leave of absence from: Department of Mathematics, Massachusetts Institute of Technology, Cambridge MA 02139, USAWork supported in part by National Science Foundation Grant #85-07109     

A new finite N = 2 supersymmetric sigma model with higher derivatives in four dimensions

A new finite N = 2 supersymmetric sigma model with higher derivatives in four dimensions is formulated. The action of the model is determined in N = 2, d = 4 superspace in terms of N = 2 chiral real superfields. The Lagrangian of the theory under consideration is evaluated in N = 1 super-space and in components. By the methods of N = 2 superfield perturbation theory, it is shown that the constructed N = 2 model is ultraviolet-finite in all orders of quantum perturbation theory in 4-dimensional space-time.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 4, pp. 85–92, April, 1990.     

N = 1 super quantum chromo dynamics and fractional branes

In this talk we show how to get the one-loop beta function and the chiral anomaly ofN = 1 super QCD from a stack of fractionalN D3-branes localized inside the world-volume of 2M fractionalD7-branes in the orbifoldC ³/(Z ₂ ×Z ₂). They are obtained by analyzing the classical supergravity background generated by such a brane configuration, in the spirit of the gauge/gravity correspondence.