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M. B. Halpern

Publications and source records attributed to M. B. Halpern.

At least 19 recordsLinked to original sources
arXiv2011

The orbifold-string theories of permutation-type: II. Cycle dynamics and target space-time dimensions

We continue our discussion of the general bosonic prototype of the new orbifold-string theories of permutation type. Supplementing the extended physical-state conditions of the previous paper, we construct here the extended Virasoro generators with cycle central charge $\hat{c}_j(σ)=26f_j(σ)$, where $f_j(σ)$ is the length of cycle $j$ in twisted sector $σ$. We also find an equivalent, reduced formulation of each physical-state problem at reduced cycle central charge $c_j(σ)=26$. These tools are used to begin the study of the target space-time dimension $\hat{D}_j(σ)$ of cycle $j$ in sector $σ$, which is naturally defined as the number of zero modes (momenta) of each cycle. The general model-dependent formulae derived here will be used extensively in succeeding papers, but are evaluated in this paper only for the simplest case of the "pure" permutation orbifolds.

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arXiv2010

The Lorentzian Space-Times of the Orientation-Orbifold String Systems

To illustrate our recent discussions of the target space-times in general orbifold-string theories of permutation-type, we return here to a detailed analysis of some simple examples of this type, namely an explicit set of orientation-orbifold string systems. These orientation-orbifold string systems provide twisted, multisector generalizations of ordinary critical open-closed bosonic string systems -- each such system exhibiting a unique graviton. Furthermore, each sector $σ$ of these string systems shows the following properties: a) 26 effective degrees of freedom, b) a Lorentzian space-time with space-time dimension $D(σ)\leq 26$, c) an $SO(D(σ)-1,1)$-invariant ordinary string subsystem with quantized intercept less than or equal one, and d) an extra set of $(26-D(σ))$ twisted fields which are $SO(D(σ)-1,1)$ scalars. Subexamples of non-tachyonic strings and four-dimensional strings are noted. Additionally, we discuss certain subsets of physical states of these theories, concluding that these investigations are so far consistent with the no-ghost conjecture for all the Lorentzian orbifold-string theories of permutation-type.

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arXiv2010

The Orbifold-String Theories of Permutation-Type: III. Lorentzian and Euclidean Space-Times in a Large Example

To illustrate the general results of the previous paper, we discuss here a large concrete example of the orbifold-string theories of permutation-type. For each of the many subexamples, we focus on evaluation of the \emph{target space-time dimension} $\hat{D}_j(σ)$, the \emph{target space-time signature} and the \emph{target space-time symmetry} of each cycle $j$ in each twisted sector $σ$. We find in particular a gratifying \emph{space-time symmetry enhancement} which naturally matches the space-time symmetry of each cycle to its space-time dimension. Although the orbifolds of $\Z_{2}$-permutation-type are naturally Lorentzian, we find that the target space-times associated to larger permutation groups can be Lorentzian, Euclidean and even null (\hat{D}_{j}(σ)=0), with varying space-time dimensions, signature and symmetry in a single orbifold.

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arXiv2010

The Orbifold-String Theories of Permutation-Type: I. One Twisted BRST per Cycle per Sector

We resume our discussion of the new orbifold-string theories of permutation-type, focusing in the present series on the algebraic formulation of the general bosonic prototype and especially the target space-times of the theories. In this first paper of the series, we construct one twisted BRST system for each cycle $j$ in each twisted sector $σ$ of the general case, verifying in particular the previously-conjectured algebra $[Q_{i}(σ),Q_{j}(σ)]_{+} =0$ of the BRST charges. The BRST systems then imply a set of extended physical-state conditions for the matter of each cycle at cycle central charge $\hat{c}_{j}(σ)=26f_{j}(σ)$ where $f_{j}(σ)$ is the length of cycle $j$.

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arXiv2009

The Dual Quark Models

We briefly recall the historical environment around our 1971 and 1975 constructions of current-algebraic internal symmetry on the open string. These constructions included the introduction of world sheet fermions, the independent discovery of affine Lie algebra in physics (level one of affine su(3)), the first examples of the affine-Sugawara and coset constructions, and finally - from compactified spatial dimensions on the string - the first vertex-operator constructions of the fermions and level one of affine su(n).

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arXiv2008

The Orbifolds of Permutation-Type as Physical String Systems at Multiples of $\mathbf{c=26}$ V. Cyclic Permutation Orbifolds

I consider the $\mathbb{Z}_λ,$ $λ$ prime free-bosonic permutation orbifolds as interacting physical string systems at $\hat{c} = 26λ$. As a first step, I introduce twisted tree diagrams which confirm at the interacting level that the physical spectrum of each twisted sector is equivalent to that of an ordinary $c=26$ closed string. The untwisted sectors are surprisingly more difficult to understand, and there are subtleties in the sewing of the loops, but I am able to propose provisional forms for the full modular-invariant cosmological constants and one-loop diagrams with insertions.

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arXiv2007

The Orbifolds of Permutation-Type as Physical String Systems at Multiples of c=26 IV. Orientation Orbifolds Include Orientifolds

In this fourth paper of the series, I clarify the somewhat mysterious relation between the large class of {\it orientation orbifolds} (with twisted open-string CFT's at $\hat c=52$) and {\it orientifolds} (with untwisted open strings at $c=26$), both of which have been associated to division by world-sheet orientation-reversing automorphisms. In particular -- following a spectral clue in the previous paper -- I show that, even as an {\it interacting string system}, a certain half-integer-moded orientation orbifold-string system is in fact equivalent to the archetypal orientifold. The subtitle of this paper, that orientation orbifolds include and generalize standard orientifolds, then follows because there are many other orientation orbifold-string systems -- with higher fractional modeing -- which are not equivalent to untwisted string systems.

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arXiv2007

The Orbifolds of Permutation-Type as Physical String Systems at Multiples of c=26 II. The Twisted BRST Systems of \hatc=52 Matter

This is the second in a series of papers which consider the orbifolds of permutation-type as candidates for new physical string systems at higher central charge. In the first paper, I worked out the extended actions of the twisted sectors of these orbifolds -- which exhibit new permutation-twisted world-sheet gravities and correspondingly extended diffeomorphism groups. In this paper I begin the study of these systems as operator string theories, limiting the discussion for simplicity to the strings with ${\hat c} = 52$ matter (which are those governed by ${\mathbb Z}_2$-twisted permutation gravity). In particular, I present here a construction of the twisted reparametrization ghosts and {\em new twisted BRST systems} of all ${\hat c} = 52$ strings. The twisted BRST systems also imply new {\em extended physical state conditions}, whose analysis for individual ${\hat c} = 52$ strings is deferred to the next paper of the series.

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arXiv2007

The Orbifolds of Permutation-Type as Physical String Systems at Multiples of $c=26$ III. The Spectra of $\hat{c}=52$ Strings

In the second paper of this series, I obtained the twisted BRST systems and extended physical-state conditions of all twisted open and closed $\hat{c} = 52$ strings. In this paper, I supplement the extended physical-state conditions with the explicit form of the extended (twisted) Virasoro generators of all $\hat{c} = 52$ strings, which allows us to discuss the physical spectra of these systems. Surprisingly, all the $\hat{c}=52$ spectra admit an equivalent description in terms of generically-unconventional Virasoro generators at $c=26$. This description strongly supports our prior conjecture that the $\hat{c}=52$ strings are free of negative-norm states, and moreover shows that the spectra of some of the simpler cases are equivalent to those of ordinary untwisted open and closed $c=26$ strings.

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arXiv2007

The Orbifolds of Permutation-Type as Physical String Systems at Multiples of c=26. I.Extended Actions and New Twisted World-Sheet Gravities

This is the first in a series of papers in which I investigate the orbifolds of permutation-type as candidates for {\it new physical string systems} at multiples of critical closed-string central charges. Examples include the bosonic orientation orbifolds, the bosonic permutation orbifolds and others, as well as superstring extensions. In this paper I use the extended (twisted) Virasoro algebras of these orbifolds to construct the corresponding extended action formulations of the twisted open- and closed-string sectors of all the bosonic orbifolds at $\hatc=26K$. The extended actions exhibit a large set of {\it new twisted world-sheet gravities}, whose extended diffeomorphism groups clearly indicate that the associated operator-string theories can be free of negative- norm states at higher central charge.

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arXiv2004

The General Twisted Open WZW String

We recently studied two large but disjoint classes of twisted open WZW strings: the open-string sectors of the WZW orientation orbifolds and the so-called basic class of twisted open WZW strings. In this paper, we discuss {\it all T-dualizations} of the basic class to construct the {\it general} twisted open WZW string -- which includes the disjoint classes above as special cases. For the general case, we give the {\it branes} and {\it twisted non-commutative geometry} at the classical level and the {\it twisted open-string KZ equations} at the operator level. Many examples of the general construction are discussed, including in particular the simple case of twisted free-bosonic open strings. We also revisit the open-string sectors of the general WZW orientation orbifold in further detail. For completeness, we finally review the {\it general twisted boundary state equation} which provides a complementary description of the general twisted open WZW string.

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arXiv2004

A Basic Class of Twisted Open WZW Strings

Recently, Giusto and Halpern reported the open-string description of a certain basic class of untwisted open WZW strings, including their associated non-commutative geometry and open-string KZ equations. In this paper, we combine this development with results from the theory of current-algebraic orbifolds to find the open-string description of a corresponding basic class of {\it twisted} open WZW strings, which begin and end on different WZW branes. The basic class of twisted open WZW strings is in 1-to-1 correspondence with the twisted sectors of all closed-string WZW orbifolds, and moreover, the basic class can be decomposed into a large collection of open-string WZW orbifolds. At the classical level, these open-string orbifolds exhibit new {\it twisted non-commutative geometries}, and we also find the relevant {\it twisted open-string KZ equations} which describe these orbifolds at the quantum level. In a related development, we also formulate the closed-string description (in terms of twisted boundary states) of the {\it general} twisted open WZW string.

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arXiv2004

Twisted Open Strings from Closed Strings: The WZW Orientation Orbifolds

Including {\it world-sheet orientation-reversing automorphisms} $\hat{h}_σ \in H_-$ in the orbifold program, we construct the operator algebras and twisted KZ systems of the general WZW {\it orientation orbifold} $A_g (H_-) /H_-$. We find that the orientation-orbifold sectors corresponding to each $\hat{h}_σ \in H_-$ are {\it twisted open} WZW strings, whose properties are quite distinct from conventional open-string orientifold sectors. As simple illustrations, we also discuss the classical (high-level) limit of our construction and free-boson examples on abelian $g$.

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arXiv2003

On the Target-Space Geometry of Open-String Orientation-Orbifold Sectors

Including world-sheet orientation-reversing automorphisms in the orbifold program, we recently reported the twisted operator algebra and twisted KZ equations in each open-string sector of the general WZW orientation orbifold. In this paper we work out the corresponding classical description of these sectors, including the {\it WZW orientation-orbifold action} -- which is naturally defined on the solid half cylinder -- and its associated WZW orientation-orbifold branes. As a generalization, we also obtain the {\it sigma-model orientation-orbifold action}, which describes a much larger class of open-string orientation-orbifold sectors. As special cases, this class includes twisted open-string {\it free boson} examples, the open-string WZW sectors above and the open-string sectors of the {\it general coset orientation orbifold}. Finally, we derive the {\it orientation- orbifold Einstein equations}, in terms of twisted Einstein tensors -- which hold when the twisted open-string sigma-model sectors are 1-loop conformal.

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arXiv2003

Twisted Einstein Tensors and Orbifold Geometry

Following recent advances in the local theory of current-algebraic orbifolds, we study various geometric properties of the general WZW orbifold, the general coset orbifold and a large class of (non-linear) sigma model orbifolds. Phase-space geometry is emphasized for the WZW orbifolds - while for the sigma model orbifolds we construct the corresponding {\it sigma model orbifold action}, which includes the previously-known general WZW orbifold action and general coset orbifold action as special cases. We focus throughout on the {\it twisted Einstein tensors} with diagonal monodromy, including the twisted Einstein metric, the twisted B field and the twisted torsion field of each orbifold sector. Finally, we present strong evidence for a conjectured set of {\it twisted Einstein equations} which should describe those sigma model orbifolds in this class which are also 1-loop conformal.

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arXiv2003

Extended Operator Algebra and Reducibility in the WZW Permutation Orbifolds

Recently the operator algebra, including the twisted affine primary fields, and a set of twisted KZ equations were given for the WZW permutation orbifolds. In the first part of this paper we extend this operator algebra to include the so-called orbifold Virasoro algebra of each WZW permutation orbifold. These algebras generalize the orbifold Virasoro algebras (twisted Virasoro operators) found some years ago in the cyclic permutation orbifolds. In the second part, we discuss the reducibility of the twisted affine primary fields of the WZW permutation orbifolds, obtaining a simpler set of single-cycle twisted KZ equations. Finally we combine the orbifold Virasoro algebra and the single-cycle twisted KZ equations to investigate the spectrum of each orbifold, identifying the analogues of the principal primary states and fields also seen earlier in cyclic permutation orbifolds. Some remarks about general WZW orbifolds are also included.

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arXiv2003

On the Large N Limit of Conformal Field Theory

Following recent advances in large N matrix mechanics, I discuss here the free (Cuntz) algebraic formulation of the large N limit of two-dimensional conformal field theories of chiral adjoint fermions and bosons. One of the central results is a new {\it affine free algebra} which describes a large N limit of su(N) affine Lie algebra. Other results include the associated {\it free-algebraic partition functions and characters}, a free-algebraic coset construction, free- algebraic construction of osp(1|2), {\it free-algebraic vertex operator constructions} in the large N Bose systems and a provocative new free-algebraic factorization of the ordinary Koba-Nielsen factor.

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