Fermionic Symmetry Research Articles

Calculating the prepotential by localization on the moduli space of instantons

We describe a new technique for calculating instanton effects in supersymmetric gauge theories applicable on the Higgs or Coulomb branches. In these situations the instantons are constrained and a potential is generated on the instanton moduli space. Due to existence of a nilpotent fermionic symmetry the resulting integral over the instanton moduli space localizes on the critical points of the potential. Using this technology we calculate the one- and two-instanton contributions to the prepotential of SU(N) gauge theory with N=2 supersymmetry and show how the localization approach yields the prediction extracted from the Seiberg-Witten curve. The technique appears to extend to arbitrary instanton number in a tractable way.

Toward a Super Yang-Mills Theory on the Lattice

We present an entirely new approach towards a realization of the supersymmetric Yang-Mills theory on the lattice. The action consists of the staggered fermion and the plaquette variables distributed in the Euclidean space with a particular pattern. The system is shown to have fermionic symmetries relating the fermion and the link variables.

Action for supergravity interacting with super-p-brane sources

We describe the coupled system of supergravity and a superbrane source by the sum of the group manifold action for D--dimensional supergravity and the action for a super--$p$--brane. We derive the generalized Einstein equation with the source and discuss the local fermionic symmetries of the coupled action. Our scheme could be especially relevant in $D=11, 10$, in which the superfield actions for supergravity are not known.

Symmetries of fermions in the lowest Landau level

In the first paper we presented a general theory of conformal fields on a complex plane, and in the second paper we wrote down, for the first time, a supersymmetric transformation of the conformal fields. I think I was expected to present a historical talk at the conference. But, since I had already written an essay on the reminiscences of my entire research activities in [3] and furthermore since a part of the article has been printed again in [4], I chose not to talk about my recollections on this matter. Instead, under the title Symmetries of Fermions in the Lowest Landau Level, I discussed the subjects, which may or may not be suited to the theme of this conference, namely (a) the dynamic symmetries of edge excitations and (b) the method of collective coordinates for planar fermions and the bosonization. Since I believe the viewpoint I took on the latter subject is unconventional, in this written report I expanded the discussion and have written it in detail.

The D-instanton partition function

The D-instanton partition function is a fascinating quantity because in the presence of N D3-branes, and in a certain decoupling limit, it reduces to the functional integral of = 4 U(N) supersymmetric gauge theory for multi-instanton solutions. We study this quantity as a function of non-commutativity in the D3-brane theory, VEVs corresponding to separating the D3-branes and α'. Explicit calculations are presented in the one-instanton sector with arbitrary N, and in the large-N limit for all instanton charge. We find that for general instanton charge, the matrix theory admits a nilpotent fermionic symmetry and that the action is -exact. Consequently the partition function localizes on the minima of the matrix theory action. This allows us to prove some general properties of these integrals. In the non-commutative theory, the contributions come from the ``Higgs Branch'' and are equal to the Gauss-Bonnet-Chern integral of the resolved instanton moduli space. Separating the D3-branes leads to additional localizations on products of abelian instanton moduli spaces. In the commutative theory, there are additional contributions from the ``Coulomb Branch'' associated to the small instanton singularities of the instanton moduli space. We also argue that both non-commutativity and α'-corrections play a similar rôle in suppressing the contributions from these singularities. Finally we elucidate the relation between the partition function and the Euler characteristic of the instanton moduli space.

Non-local Conservation Laws and Flow Equations¶ for Supersymmetric Integrable Hierarchies

An infinite series of Grassmann-odd and Grassmann-even flow equations is defined for a class of supersymmetric integrable hierarchies associated with loop superalgebras. All these flows commute with the mutually commuting bosonic ones originally considered to define these hierarchies and, hence, provide extra fermionic and bosonic symmetries that include the built-in N=1 supersymmetry transformation. The corresponding non-local conserved quantities are also constructed. As an example, the particular case of the principal supersymmetric hierarchies associated with the affine superalgebras with a fermionic simple root system is discussed in detail.

Lattice QCD with the overlap fermions at strong gauge coupling (II)

In the previous paper we developed a strong-coupling expansion for the lattice QCD with the overlap fermions and showed that L\"usher's "extended" chiral symmetry is spontaneously broken in some parameter region of the overlap fermions. In this paper, we derive a low-energy effective action of hadrons and show that there exist quasi-Nambu-Goldstone bosons which are identified as the pions. The pion field is a {\em nonlocal} composite field of quark and anti-quark even at the strong-coupling limit because of the nonlocality of the overlap fermion formalism and L\"usher's chiral symmetry. The pions become massless in the limit of the vanishing bare-quark mass as it is desired. We furthermore examine symmetries of the overlap fermions with the hopping expansion and argue that there appear no other massless modes other than the pions.

Two-dimensional lattice Gross-Neveu model with domain-wall fermions

We investigate the two-dimensional lattice Gross-Neveu model in the large flavor number limit using the domain-wall fermion formulation as a toy model of lattice QCD. We study the nonperturbative behavior of the restoration of the chiral symmetry of the domain-wall fermions as the extent of the extra dimension ${(N}_{s})$ is increased to infinity. We find the parity broken phase (Aoki phase) for finite ${N}_{s},$ and study the phase diagram, which is related to the mechanism of chiral restoration in the ${N}_{s}\ensuremath{\rightarrow}\ensuremath{\infty}$ limit. The continuum limit is taken and the $O(a)$ scaling violation of observables vanishes in the ${N}_{s}\ensuremath{\rightarrow}\ensuremath{\infty}$ limit. We also examine the systematic dependencies of observables to the parameters.

Symplectic fermions

We study two-dimensional conformal field theories generated from a ``symplectic fermion'' - a free two-component fermion field of spin one - and construct the maximal local supersymmetric conformal field theory generated from it. This theory has central charge c=-2 and provides the simplest example of a theory with logarithmic operators. Twisted states with respect to the global SL(2,C)-symmetry of the symplectic fermions are introduced and we describe in detail how one obtains a consistent set of twisted amplitudes. We study orbifold models with respect to finite subgroups of SL(2,C) and obtain their modular invariant partition functions. In the case of the cyclic orbifolds explicit expressions are given for all two-, three- and four-point functions of the fundamental fields. The C_2-orbifold is shown to be isomorphic to the bosonic local logarithmic conformal field theory of the triplet algebra encountered previously. We discuss the relation of the C_4-orbifold to critical dense polymers.

Fermionic symmetries: Extension of the two to one relationship between the spectra of even-even and neighboring odd mass nuclei

In the single j shell there is a two to one relationship between the spectra of certain even-even and neighboring odd mass nuclei; e.g., the calculated energy levels of ${J=0}^{+}$ states in ${}^{44}\mathrm{Ti}$ are at twice the energies of corresponding levels in ${}^{43}\mathrm{Ti}{(}^{43}\mathrm{Sc})$ with $J=j=7/2.$ Here an approximate extension of the relationship is made by adopting a truncated seniority scheme; i.e., for ${}^{46}\mathrm{Ti}$ and ${}^{45}\mathrm{Sc}$ we get the relationship if we do not allow the seniority $v=4$ states to mix with the $v=0$ and $v=2$ states. Better than that, we get very close to the two to one relationship if seniority $v=4$ states are admixed perturbatively. In addition, it is shown that for the $J=0 T=3$ state in ${}^{46}\mathrm{Ti}$ and for the $J=j T=5/2$ state in ${}^{45}\mathrm{Sc}$ (i.e., the states of higher isospin) there are no admixtures in which the neutrons have seniority 4.

Fermion masses and symmetry breaking of a U(2) flavour symmetry

We show how a specific sequential breaking pattern of a U(2) flavour symmetry occurs automatically in a broad framework. The relative orientation in U(2) space of the spurion fields that breaks the U(2) symmetry is uniquely fixed, thus determining the form of the fermion mass matrices in a predictive way.

NONGENERIC SUSY IN SPINNING NUT-KERR-NEWMAN SPACE-TIME

The supersymmetric extension of theNUT–Kerr–Newman (NUT–KN)space–time is investigated. Along with fourstandard supersymmetries, this type of space–timeadmits fermionic symmetry generated by the square root of the bosonic constant of motion exceptthe Hamiltonian. Such a new supersymmetry corresponds tothe Killing–Yano tensor, which plays an importantrole in solving various field equations in thisspace–time.

Twisted N=4 SUSY algebra in topological models of Schwarz type

We reinvestigate the twisted N=4 supersymmetry present in Schwarz type topological field models. We show that Chern-Simons theory in three dimensions can be untwisted to a kind of sigma-model with reversed statistics only in the free case. By dimensional reduction we define then the two-dimensional BF-model. We establish an analog result concerning the untwisting. As a consequence of the definition through dimensional reduction we find new fermionic scalar symmetries that have been overlooked so far in the literature.

Quantization of p-branes, D-p-branes and M-branes

Killing spinors of space-time BPS configurations play an important role in quantization of theories with the fermionic worldvolume local symmetry. We show here how it works for the CS superstring, BST supermembrane and M-5-brane. We show that the non-linear generalization of the (2,0) d=6 tensor supermultiplet action is the M-5-brane action in a Killing gauge. For D-p-branes the novel feature of quantization is that they can be quantized Lorentz covariantly, in particular, for D-0-brane a gauge exists where the action is covariant and free. We present a general condition on possible choice of gauges for the κ-symmetric branes.

GENERATING SYMMETRY OPERATORS FOR THE KdV EQUATION AND SEEKING THEIR SUPERSYMMETRIC GENERALIZATIONS

Exploiting the Lax form of the Korteweg-de Vries equation, we determine the whole set of its symmetry operators and relate them to conserved quantities. We also study a KdV supersymmetric version and put in evidence its even (bosonic) and odd (fermionic) symmetries. We then get the Lie superalgebra characterizing this supersymmetric nonlinear equation.

Background independent matrix models

A class of background independent matrix models is made for which the structure of both local gauge symmetries and classical solutions is clarified. These matrix models do not involve a space-time metric and provide the matrix analogs of topological Chern-Simons and BF theories. It is explicitly shown that the BF type of matrix model can be formulated in any space-time dimension and include 3+1 dimensional gravity as a special case. Moreover, we discuss some generalization of the model to include a fermionic BRST-like symmetry whose partition function is related to the Casson invariant.

Solar neutrinos and grand unification

We consider the Grand Unification (GU) scenario for neutrino masses which is based on the see-saw mechanism with the mass of the heaviest right handed (RH) neutrino at the GU-scale: $M_3 \sim \Lambda_{GU}$, and on the quark-lepton symmetry for fermions from the third generation. The scenario predicts for the light neutrinos: $m_3 \sim (2 - 4) \cdot 10^{-3}$ eV and $m_2 \sim (0.3 - 3) \cdot 10^{-5}$ eV (in the case of a linear mass hierarchy of the RH neutrinos or/and in presence of the Planck scale suppressed non-renormalizable operators). It also predicts large $\nu_e - \nu_{\mu}$ mixing: $~\sin^2 2\theta_{e\mu} \stackrel{_>}{_\sim} 0.2$. In this scenario the solar neutrinos ($\nu_{\odot}$) undergo both the \nue $\to$ \nutau resonance conversion in the Sun and substantial \nue $\to$ \numu vacuum oscillations on the way from the Sun to the Earth. The interplay of both effects enlarges the range of neutrino parameters which solve the $\nu_{\odot}$-problem. In particular, $\nu_e - \nu_{\tau}$ mixing angle can be as small as the corresponding quark mixing: $\sin^2 2\theta_{e\tau} \geq (2~-~5) \cdot 10^{-4}$. The scenario predicts peculiar (oscillatory) distortion of the boron neutrino energy spectrum and seasonal variations of signals. Manifestations of these effects in the Super-Kamiokande and SNO experiments are studied.

Superparticles, p-form coordinates and the BPS condition

A model for $n$ superparticles in $(d-n,n)$ dimensions is studied. The target space supersymmetry involves a product of $n$ momentum generators, and the action has $n(n+1)/2$ local bosonic symmetries and $n$ local fermionic symmetries. The precise relation between the symmetries presented here and those existing in the literature is explained. A new model is proposed for superparticles in arbitrary dimensions where coordinates are associated with all the $p$-form charges occuring in the superalgebra. The model naturally gives rise to the BPS condition for the charges.

Superparticles in D>11

Actions for two-superparticle system in (10,2) dimensions and three-superparticle systems in (11,3) dimensions are constructed. These actions have worldline bosonic and fermionic local symmetries, and target space global supersymmety generalizing the reparametrization, kappa-symmetry and Poincare supersymmetry of the usual superparticle. With the second particle, or the second and third particles on-shell, they describe a superparticle propagating in the background of a second superparticle in (10,2) dimensions, or two other superparticles in (11,3) dimensions. Symmetries of the action are shown to exist in presence of super Yang-Mills background as well.

Correlation functions in topological Yang-Mills theory with two ferminionic charges

The solution of the Donaldson cohomology problem for the topological Yang-Mills theory with two fermionic symmetries needs besides the gauge field and its descendants additional fields, hereafter called ascendants of the gauge field. It is shown that the dependence of the ascendants disappears in the all the correlation functions. This property allows one for the usual interpretation of the Donaldson invariants as cocycles of the instanton moduli space.