Andrei Johansen

Infinite conformal algebras in supersymmetric theories on four manifolds

Abstract We study a supersymmetric theory twisted on a Kahler four-manifold M = Σ 1 × Σ 2 , where Σ 1,2 are two-dimensional Riemann surfaces. We demonstrate that it possesses a “left-moving” conformal stress tensor on Σ 1 ( Σ 2 ) in a BRST cohomology, which generates the Virasoro algebra with the conventional commutation relations. The central charge of the Virasoro algebra has a purely geometric origin and is proportional to the Euler characteristic χ of the Σ 2 ( Σ 1 ) surface. It is shown that this construction can be extended to include a realization of a Kac-Moody algebra in BRST cohomology with a level proportional to the Euler characteristic χ. This structure is shown to be invariant under renormalization group. A representation of the algebra W 1+∞ in terms of a free chiral supermultiplet is also given. We discuss the role of instantons and a possible relation between the dynamics of four-dimensional Yang-Mills theories and those of two-dimensional sigma models.

The Level crossing phenomenon with Yukawa interactions

Abstract In the minimal model of electroweak interactions we carefully investigate the spectrum of the massive euclidean Dirac operator in three, four and five dimensions (D) in the presence of topologically non-trivial external fields. More specifically we study the cases of the instanton (D = 4), sphaleron (D = 3) as well as that of the (D = 5) Dirac operator that pertains to the existence of the global SU(2) anomaly. We establish the existence of normalizable massive fermion zero modes in all three cases. We give closed form expressions which relate the massive with the massless zero modes. As a consequence the level crossing phenomenon is shown to be manifest and generic in the presence of Yukawa interactions.

REALIZATION OF W1+∞ AND VIRASORO ALGEBRAS IN SUPERSYMMETRIC THEORIES ON FOUR MANIFOLDS

We demonstrate that a supersymmetric theory twisted on a Kahler four-manifold M= Σ1×Σ2, where Σ1,2 are 2-D Riemann surfaces, possesses a “left-moving” conformal stress tensor on Σ1 (Σ2) in the BRST cohomology. The central charge of the Virasoro algebra has a purely geometric origin and is proportional to the Euler characteristic of the Σ2 (Σ1) surface. This structure is shown to be invariant under renormalization group. We also give a representation of the algebra W1+∞ in terms of a free chiral supermultiplet.

ON SPHALERON DEFORMATIONS INDUCED BY YUKAWA INTERACTIONS

Due to the presence of the chiral anomaly sphalerons, half a number of Chern-Simons (CS=1/2) are the only static configurations that allow a fermion level crossing in the two-dimensional Abelian-Higgs model with massless fermions, i.e. in the absence of Yukawa interactions. In the presence of fermion-Higgs interactions we demonstrate the existence of zero energy solutions to the one-dimensional Dirac equation at deformed sphalerons with CS ≠ 1/2. Induced level crossing due to Yukawa interactions illustrates a non-trivial generalization of the Atiyah-Patodi-Singer index theorem and of the equivalence between parity anomaly in odd and chiral anomaly in even dimensions. We discuss a subtle manifestation of this effect in the standard electroweak theory at finite temperatures.Due to the presence of the chiral anomaly sphalerons with Chern-Simons number a half (CS=1/2) are the only static configurations that allow for a fermion level crossing in the two-dimensional Abelian-Higgs model with massless fermions, i.e. in the absence of Yukawa interactions. In the presence of fermion-Higgs interactions we demonstrate the existence of zero energy solutions to the one-dimensional Dirac equation at deformed sphalerons with CS

A Z2 structure in the configuration space of Yang–Mills theories

\neq 1/2 .

Exact multiparticle amplitudes at threshold in φ4 theories with softly broken O(∞) symmetry

Induced level crossing due to Yukawa interactions illustrates a non-trivial generalization of the Atiyah-Patodi-Singer index theorem and of the equivalence between parity anomaly in odd and the chiral anomaly in even dimensions. We discuss a subtle manifestation of this effect in the standard electroweak theory at finite temperatures.

TOPOLOGICAL CLASSIFICATION OF ODD-PARITY SPHALERON DEFORMATIONS

We argue for the presence of a Z2 topological structure in the space of static gauge–Higgs field configurations of SU(2n) and SO(2n) Yang–Mills theories. We rigorously prove the existence of a Z2 homotopy group of mappings from the two‐dimensional projective sphere RP2 into SU(2n)/Z2 and SO(2n)/Z2 Lie groups, respectively. Consequently the symmetric phase of these theories admits infinite surfaces of odd‐parity static and unstable gauge field configurations which divide into two disconnected sectors with integer Chern–Simons numbers n and n+1/2, respectively. Such a Z2 structure persists in the Higgs phase of the above theories and accounts for the existence of CS=1/2 odd‐parity saddle point solutions to the field equations which correspond to spontaneous symmetry breaking mass scales.

AN O(3) GLOBAL ANOMALY IN 0+1 DIMENSION

Abstract We consider the problem of multiparticle production at threshold in a φ 4 -theory with an O ( N 1 + N 2 ) symmetry softly broken down to O ( N 1 ) × O ( N 2 ) by unequal masses. We derive the set of recurrence relations between the multiparticle amplitudes which sums all relevant diagrams with arbitrary number of loops in the large- N limit with fixed number of produced particles. We transform it into a quantum mechanical problem and show how it can be obtained directly from the operator equations of motion by applying the factorization at large N . We find the exact solutions to the problem by using the Gelfand-Dikiĭ representation of the diagonal resolvent of the Schrodinger operator. The result coincides with the tree amplitudes while the effect of loops is the renormalization of the coupling constant and masses. The form of the solution is due to the fact that the exact amplitude of the process 2 → n vanishes at n > 2 on mass shell when averaged over the O ( N 1,2 ) indices of incoming particles. We discuss what dynamical symmetry is behind this property. We also give an exact solution in the large- N limit for the model of the O ( N ) + singlet scalar particle with the spontaneous breaking of a reflection symmetry.

THE SU(2) GLOBAL ANOMALY THROUGH LEVEL CIRCLING

We discuss topological aspects of the electroweak sphaleron and its odd-parity deformations. We demonstrate that they are uniquely classified in terms of their odd or even-parity pure gauge field behavior at spatial infinity. Fermion level crossing occurs only for odd-parity configurations, which are topologically disconnected from the vacuum. They contribute to the high temperature unsuppressed baryon violating thermal transitions. Deformations with even-parity pure gauge behavior at spatial infinity are topologically trivial and do not mediate baryon number violation in the early universe.

The global SU (2) anomaly for massive fermions

We present a simple exactly solvable quantum mechanical example of the global anomaly in an O(3) model with an odd number of fermionic triplets coupled to a gauge field on a circle. Because the fundamental group is non-trivial, π1(O(3))=Z2, fermionic level crossing—circling occurs in the eigenvalue spectrum of the one-dimensional Dirac operator under continuous external field transformations. They are shown to be related to the presence of an odd number of normalizable zero modes in the spectrum of an appropriate two-dimensional Dirac operator. We argue that fermionic degrees of freedom in the presence of an infinitely large external field violate perturbative decoupling.