Title of the Thesis:

Two-Point Functions and Boundary States
in Boundary Logarithmic Conformal Field Theories

Abstract

     Amongst conformal field theories, there exist logarithmic conformal field theories such as c_p,1 models, various WZNW models, and a large variety of statistical models. It is well known that these theories generally contain a Jordan cell structure, which is a reducible but indecomposable representation. Our main aim in this thesis is to address the results and prospects of boundary logarithmic conformal field theories: theories with boundaries that contain the above Jordan cell structure.

     In this thesis, we briefly review conformal field theory and the appearance of logarithmic conformal field theories in the literature in the chronological order. Thereafter, we introduce the conventions and basic facts of logarithmic conformal field theory, and sketch an essential note on boundary conformal field theory. We have investigated c_p,q boundary theory in search of logarithmic theories and have found logarithmic solutions of two-point functions in the context of the Coulomb gas picture. Other two-point functions have also been studied in the free boson construction of BCFT with SU(2)_k symmetry. In addition, we have analyzed and obtained the boundary Ishibashi state for a rank-2 Jordan cell structure [22]. We have also examined the (generalised) Ishibashi state construction and the symplectic fermion construction at c=-2 for boundary states in the context of the c=-2 triplet model [23, 24]. It is also presented how the differences between two constructions should be interpreted, resolved and extended beyond each case. Some discussions on possible applications are given in the final chapter.

Introduction

     Nearly two decades after the celebrated paper by Belavin, Polyakov and Zamolodchikov [1], it is now clear that two-dimensional conformal field theory (CFT) is an essential mathematical tool and background to explore various models and theories in physics. Among them, there are two main branches of physics where CFT lies deeply. One is particle physics and the other is condensed matter physics. CFT was extended to logarithmic conformal field theory nine years later [2], but let us first focus on brief descriptions of CFT applications.

     In particle physics, symmetry has played a great role in quantum field theory. It revealed the electroweak theory as a simplest example of unification, and QCD as a theory for quarks. They together gave rise to the successful standard model, leaving quantum gravity as a last requirement towards the final unification -- the ultimate theory of everything (TOE). For the consistency of quantum gravity, string theory was reintroduced in the early 1980s, when the basic formulation of CFT was established in [1]. Strings and their interactions form a two-dimensional surface called the `world sheet' and replace many complicated graviton loops in quantum field theory with simpler surfaces. It is two-dimensional conformal symmetry that emerges in the world-sheet physics and is a part of reparametrisation symmetry which in turn requires the whole theory to be in the specific dimensions, 26 (10) for the bosonic (fermionic) case. Here, CFT shows up as a necessity for string theories, towards the TOE.

     On the other hand, in condensed matter physics, consider the (2+1)-dimensional electron systems which have a phase transition at some temperature. Before [1], it was known that a certain class of such systems has some strange behaviours at its critical point. Namely, order parameters of the system obey power laws with constant exponents near the critical point as if scaling symmetry is enhanced. By a symmetry argument, which we will briefly mention in the following section, it turns out that CFT describes the system at criticality, giving rise to a correct set of those critical exponents of the order parameters [1]. It was also shown that, in some statistical models, the corresponding CFT gives not only the exponents but, in principle, all physics of the systems. The Ising model is an explicit example of such statistical models. CFT arises as a requirement that any underlying non-conformal theory of such systems must flow into the CFT as it approaches the critical point.

     Many contributions have been made to the conformal zoo of theories [3,4]. Starting from the Kac determinant and table [5], the unitary minimal models have been found and developed in particle theories and statistical models [1,3-8]. Some other mathematical frameworks have also been invented such as representation theories of various CFTs and partition functions on a torus in terms of character functions. We should also count here the affine Lie algebras, the Sugawara construction and the related Wess-Zumino-Novikov-Witten models (WZNW models) [3-13]. Free field realisations of the minimal models were given, including the Dotsenko-Fateev construction [14,15], conformal ghost systems and other fermionic systems for superstring theories [16,17]. Superconformal field theories and their free field representations have also been developed in relation to strings [18,19]. Especially, boundary conformal field theory (boundary CFT or BCFT) was nicely introduced by Cardy in the context of open strings and statistical models with finite size effects [20]. We will mention this later.

     There is much more research on this subject. However, as stated above, most of the literature is restricted to the unitary CFTs with some extra symmetries. Vast areas of other possibilities have not been greatly discussed or focussed on, especially those away from the unitary cages of the tamed zoo. It is often the case with experiments, that physical events happen and are detected in the absence of unitarity or outside the conformal region. Thus, we need another tool to explore the frontier of unitary CFT and beyond for a breakthrough in the subject. In fact, there is a class of the theories called `logarithmic conformal field theory (LCFT)' which is believed to lie widely at the border of unitary and non-unitary conformal theories. It should be clarified here that what we are going to discuss in this thesis is those LCFTs with one or more boundaries. The Coulomb gas construction of c_p,q models, a c_2,1=-2 model in particular, and the free boson realisation of the SU(2)_k WZNW model will be given in the presence of a boundary. The results on their two-point functions will be shown explicitly. We also investigate boundary states in LCFTs, especially at c=-2, and study a few different constructions such as of (generalised) Ishibashi states and of coherent states [22-24].

     It was first found and discussed by Knizhnik that a four-point function of CFT has a logarithmic singularity in orbifold models [25]. The same sort of singularity was also discussed by Rozansky and Saleur in the GL(1,1) WZNW model [26]. Later, Gurarie revealed that, given such logarithms, logarithmic fields appear in the theory as a degenerate pair of fields [2]. Thus discovered theory was named LCFT, since known CFTs didn't contain such fields. The main feature and a heuristic definition of LCFT is that there appears a set of `logarithmic' operators which form a reducible but indecomposable representation of the L<sub>0</sub> operator, the zero mode of the Virasoro algebra [27]. In [28], this structure was generally defined as the Jordan cell structure, and the pair of `logarithmic' operators in [2] emerged as a rank-2 Jordan cell. One of the pair is purely logarithmic, giving a logarithmic singularity in its correlator, and the other gives a primary state of zero norm [29]. In principle, some minimal models of CFT may possess such fields, although in many cases they are non-unitary or their central charges are irregular. Nevertheless, this class of theories is worth studying for new physics. Note that we can always ignore their non-unitary nature as if they were a subsystem. Thus far, many studies have been devoted to this subject and have found the same sort of logarithmic behaviours in various models. For example, the gravitationally dressed CFT and WZNW models at different levels or on different groups [26,29-42]. c_p,1 and non-minimal c_p,q models, including the c_2,1=-2 model [2, 21,22-24, 27, 28, 30, 43-58]. c=0 models [59-62]. Critical polymers and percolation [30,49,59,63,64], quantum Hall effect, quenched disorder and localization in planar systems [65-70]. 2D-magneto-hydrodynamic and ordinary turbulence [71-76]. LCFT in general [77-85]. In string theory, D-brane recoil, target-space symmetries and AdS/CFT correspondence have been studied and discussed with respect to LCFTs in the literature [86-106].

     On the other hand, there is another class of theories called BCFT first considered by Cardy in [20,107,108] (see also [109-112]). BCFT is CFT with one or more boundaries. It was shown by Cardy that many tools developed in ordinary CFT can be imported into BCFT by the use of his `mirror method', hence n-point functions become manageable [20]. As is often the case with experiments, when critical systems are restricted to finite size, the corresponding theories turn out to be BCFTs. Therefore, this class of theories is as essential as CFT in both particle physics and condensed matter physics. In fact in string theory, theories of open strings are defined on an infinite strip with two boundaries in its simplest cases. A periodicity along its boundaries induces a dual `closed string' picture defined on the same geometry. Modular invariance leads to a one-to-one correspondence between the boundary conditions in the first picture and the boundary states in the other. It was found in [107] that these boundary states are spanned by boundary Ishibashi states [113], by which the Verlinde formula in [114] is proven to hold for unitary minimal models of boundary CFT. Note, from the point of view of brane solutions in string theory, the boundary states prescribe the dynamics between closed-string states on both branes.

     Although the study of LCFT is still in progress, much progress has been made in both areas, LCFT and BCFT. Despite this, only a small number of papers have contributed to LCFT with boundaries (boundary LCFT) and the effects of the presence of the boundaries [22-24, 50, 55, 56, 71, 83]. This is partially because there is a problem of reducible but indecomposable representations which cannot be applied to boundary CFT in a straightforward way. The first systematic attempt to formulate a boundary LCFT was made by Kogan and Wheater in [50]. Several important problems were discussed, including the boundary operators, their correlation functions, the structure of boundary states in LCFT, using a model at c=-2 as an example, and a possibility towards the Verlinde formula. In 2001, our previous paper showed that a `pure' rank-2 Jordan cell has only one Ishibashi state in the cell. It propagates from one boundary to another, behaving as a ghost state [22]. From a basic mathematical framework developed by Rohsiepe [28], the structure of initial and final states of Jordan cells were reintroduced and a brief proof for the above statement was given in a purely algebraic manner. The `pure' Jordan cell means that the Virasoro representations built on the cell do not contain any subrepresentations of lowest weight states nor become subrepresentations of any other representation. From the calculation on the cell [22,83], we state that there is only one Ishibashi state in each `pure' rank-2 Jordan cell and that it is the state built on the primary state of zero norm.

     Soon, along the same lines, another computation was given in a symplectic fermion system at c=-2 by Kawai and Wheater [23]. In this fermionic ghost system appearing in [30], they constructed the coherent states which satisfy Cardy's equation for boundary states, and found another set of boundary states which differs from our set. Logarithmic terms occur in their character functions, which was absent in our previous result. However, recent results found by Bredthauer and Flohr showed that one may have generalised Ishibashi states for boundary states in the c=-2 theory, and that they might give the same logarithms in their character functions [24]. A following paper by Bredthauer revealed that with a choice of (generalised) Ishibashi states one can see the isomorphic relation between their results and other results on fermions in terms of characters [56].

     A question arises, namely what is the difference between the boundary states in [22], those in [23,56], and those in [24, 56]. Our answer for the first two results is that they deal with the same model where our definition didn't count their generalised Ishibashi states but give a smaller set of their solutions. Then, the question is transfered to the difference between two constructions, namely, between the Ishibashi state construction and the coherent state, or the symplectic fermion, construction. By examining each construction, we come to an answer that these two cases should be called different models because they are constructed in mathematically and physically distinct ways. We will show these and additional results in chapter 5.

     Apart from the boundary states, we would like to consider free field realisations in boundary logarithmic theories. While boundary states would give much information w.r.t. the boundary, the actual correlation functions need to be calculated or confirmed that they satisfy a certain differential equations. From such results of correlation functions, one can see the relations between their normalisation factors and those of boundary operators defined only on the boundary. In chapter 4 we will examine a few cases of free boson constructions on the upper half-plane and single out necessary conditions for boundary LCFT in this context. Some of the boundary two-point functions will also be shown, which are in complete agreement with [50].

     The thesis is organised as follows.

     In this chapter, we shall give a pedagogical introduction to the celebrated world of CFTs, chiefly in two dimensions. All referred facts and equations are well-known to those with experience in this subject. In chapter 2, we start reviewing the emergence of logarithms and a brief history of logarithmic CFTs, roughly in chronological order. In the following sections of the same chapter, we briefly review the definitions and formal properties of LCFTs. This includes the definition of Jordan cell structure, a review of sections of [28] and a brief introduction to the models at c=-2, which we will focus on later. In chapter 3, we discuss BCFTs that were introduced by Cardy in his successive papers [20,107,108,110]. Chapters 4 and 5 contain the main results of this thesis. In chapter 4, we examine the Coulomb gas construction of general c_p,q models and describe the c=-2 case in particular, in the presence of a boundary. Using the same techniques, the free boson realisation of the SU(2)_k WZNW model with a boundary is studied. We also present boundary two-point functions of these models. In chapter 5, we prove the existence of the boundary Ishibashi state for the rank-2 Jordan cell structure and show its explicit form [22]. In the following sections, the other two constructions are reviewed and confirmed [23,24]. Then, we describe and clarify the differences between the constructions, referring our results and [56]. Some additional results to compare will also be presented.

     In chapter 6, we present and summarise the results on chapters 4 and 5. We give a remark on the Verlinde formula for LCFT and possible applications to string theory. We also discuss the generalisation of the above results and possibilities of boundary LCFTs and prospects of LCFTs to close. The appendices contain: the first appendix is the explanation of radial quantisation mainly for the introduction to CFT. The second appendix is a proof of relations between hypergeometrical functions appearing in chapter 4.


(References are the same as in the original.)

References

[22] Yukitaka Ishimoto, Boundary states in boundary logarithmic CFT, Nucl. Phys. B 619 (2001) 415 [arXiv:hep-th/0103064].

[83] Yukitaka Ishimoto, Boundary states in boundary logarithmic CFT: an algebraic approach, in the Proceedings of the "School & Workshop on Logarithmic Conformal Field Theory and Its Applications 2001" IPM, Tehran, Iran (2003) [Int. J. Mod. Phys. A 18, No. 25 (2003) 4639].