Yuji Igarashi

Anomalous gauge theories and subcritical strings based on the Batalin-Fradkin formalism

Abstract On the basis of the generalized hamiltonian formalism of Batalin and Fradkin for systems under second-class constraints, we propose a systematic treatment of anomalous gauge theories. We derive a BRST gauge-fixed covariant action for the four-dimensional chiral U(1) gauge theory, showing that the unphysical fields introduced in the formalism turn into the Wess-Zumino scalar. We discuss within this context how the conservation of BRST current is connected with the anomalous current divergence. The Nambu-Goto string theory is consistently quantized at the subcritical dimensions in various gauges, and it is shown how the conformal degree of freedom appears in the light-cone gauge.

Realization of Symmetry in the ERG Approach to Quantum Field Theory

We review the use of the exact renormalization group for realization of symmetry in renormalizable field theories. The review consists of three parts. In part I (sects. 2,3,4), we start with the perturbative construction of a renormalizable field theory as a solution of the exact renormalization group (ERG) differential equation. We show how to characterize renormalizability by an appropriate asymptotic behavior of the solution for a large momentum cutoff. Renormalized parameters are introduced to control the asymptotic behavior. In part II (sects. 5--9), we introduce two formalisms to incorporate symmetry: one by imposing the Ward-Takahashi identity, and another by imposing the generalized Ward-Takahashi identity via sources that generate symmetry transformations. We apply the two formalisms to concrete models such as QED, YM theories, and the Wess-Zumino model in four dimensions, and the O(N) non-linear sigma model in two dimensions. We end this part with calculations of the abelian axial and chiral anomalies. In part III (sects. 10,11), we overview the Batalin-Vilkovisky formalism adapted to the Wilson action of a bare theory with a UV cutoff. We provide a few appendices to give details and extensions that can be omitted for the understanding of the main text. The last appendix is a quick summary for the readers convenience.

Quantum Master Equation for QED in Exact Renormalization Group

Recently, one of us (H. S.) gave an explicit form of the Ward-Takahashi identity for the Wilson action of QED. We first rederive the identity using a functional method. The identity makes it possible to realize the gauge symmetry even in the presence of a momentum cutoff. In the cutoff dependent realization, the nilpotency of the BRS transformation is lost. Using the Batalin-Vilkovisky formalism, we extend the Wilson action by including the antifield contributions. Then, the Ward-Takahashi identity for the Wilson action is lifted to a quantum master equation, and the modified BRS transformation regains nilpotency. We also obtain a flow equation for the extended Wilson action.

BRS symmetry, the quantum master equation, and the Wilsonian renormalization group

Recently we made a proposal for realization of an effective BRS symmetry along the Wilsonian renormalization group flow. In this paper we show that the idea can be naturally extended to the most general gauge theories. Extensive use of the antifield formalism is made to reveal some remarkable structure of the effective BRS symmetry. The average action, defined with a continuum analog of the block spin transformation, obeys the quantum master equation (QME), provided that a UV action does so. We show that the RG flow described by the exact flow equations is generated by canonical transformations in the field-antifield space. Using the relation between the average action and the Legendre effective action, we establish the equivalence between the QME for the average action and the modified WardTakahashi identity for the Legendre action. The QME remains intact when the regularization is removed.

A new insight into BRST anomalies in string theory

Abstract Using the generalized hamiltonian method of Batalin, Fradkin and Vilkovisky, we investigate the algebraic structure of anomalies in the Polyakov string theory that appear as the Schwinger terms in super-commutation relations between BRST charge and total hamiltonian. We obtain the most general form of anomalies in the extended phase space, without any reference to a two-dimensional metric. This pregeometrical result, referred to as the generalized Virasoro anomaly which is independent of the gauge and the regularization under a minor assumption, is a non-perturbative result, and valid for any space-time dimension. In a configuration space, in which the two-dimensional metric can be identified, we can geometrize the result without assuming the weak gravitational field, showing that the most general anomaly exactly exhibits the Weyl anomaly.

Classical Solutions and Order of Zeros in Open String Field Theory

Earlier an analytic approach is proposed for classical solutions describing tachyon vacuum in open string field theory. Based on the approach, we construct a certain class of classical solutions written in terms of holomorphic functions with higher order zeros. Taking the simplest among the new classical solutions, we study the cohomology of the new BRS charge and make a numerical analysis of the vacuum energy. The results indicate that the new non-trivial solution is another analytic candidate for the tachyon vacuum.

Exact symmetries realized on the renormalization group flow

Abstract We show that symmetries are preserved exactly along the (Wilsonian) renormalization group flow, though the IR cutoff deforms concrete forms of the transformations. For a gauge theory the cutoff dependent Ward–Takahashi identity is written as the master equation in the antifield formalism: one may read off the renormalized BRS transformation from the master equation. The Maxwell theory is studied explicitly to see how it works. The renormalized BRS transformation becomes non-local but keeps off-shell nilpotency. Our formalism is applicable for a generic global symmetry. The master equation considered for the chiral symmetry provides us with the continuum analog of the Ginsparg–Wilson relation and the Luschers symmetry.

Self-duality in super D3-brane action

We establish self-duality of super D3-brane theory as an exact symmetry of the action both in the Lagrangian and Hamiltonian formalisms. In the Lagrangian formalism, the action is shown to satisfy the Gaillard-Zumino condition. This algebraic relation is recognized in our previous paper to be a necessary and sufficient condition for generic action of U(1) gauge field strength coupled with gravity and matters to be self-dual. For the super D3-brane action, SO(22) duality transformation of a world-volume gauge field should be associated with SO(2) rotation of fermionic brane coordinates in the N = 2 SUSY multiplet. This SO(2) duality symmetry is lifted to SL(2, R) symmetry in the presence of a dilaton and axion background fields. In the canonical formalism, we show that the duality rotation is described by a canonical transformation, and the Hamiltonian of the D3-brane action is invariant under the transformation.

Electric-magnetic duality rotations and invariance of actions

Abstract For D = 4 theories of a single U (1) gauge field strength coupled to gravity and matters, we show that the electric-magnetic duality can be formulated as an invariance of the actions. The symmetry is associated with duality rotation acting directly on the gauge field. The rotation is constructed in flat space, and an extension to curved spaces is also given. It is non-local and non-covariant, yet generates off-shell extended transformation of the field strength. The algebraic condition of Gaillard and Zumino turns out to be a necessary and sufficient condition for the invariance of actions. It may be used as a guiding principle in constructing self-dual actions in string and field theories.

Exploring vacuum manifold of open string field theory

The global symmetry generated by Kn is a subgroup of the stringy gauge symmetry. We explore the part of the vacuum manifold related by this symmetry. Strong evidence is presented that the analytic classical solutions to the cubic string field theory found earlier in Refs. 5) [T. Takahashi and S. Tanimoto, J. High Energy Phys. 03 (2002), 033; hepth/0202133] and 6) [I. Kishimoto and T. Takahashi, Prog. Theor. Phys. 108 (2002), 591; hep-th/0205275] are actually related by the symmetry and, therefore, all of them describe the same tachyon vacuum. Some remaining subtlety is pointed out.