Boundary States and Symplectic Fermions

12 Pages·0.18 MB·English

by  Andreas Bredthauer

hep-th/0207181
ITP-UH-18/02
Boundary States and
Symplectic Fermions
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n Andreas Bredthauer∗
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Institut fu¨r Theoretische Physik

2 Universit¨at Hannover
v Appelstraße 2, 30167 Hannover, Germany
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0 July 19, 2002
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Abstract
We investigate the set of boundary states in the symplectic fermion
description of the logarithmic conformal field theory with central charge
c = 2. We show that the thus constructed states correspond exactly
−
to those derived under the restrictions of the maximal chiral symmetry
algebra for this model, (2,3,3,3). This connects our previous work to
W
the coherent state approach of Kawai and Wheater.
∗[email protected]
1 Introduction
During the last 20 years, conformal field theory (CFT) in two dimensions [1] has become a
very important tool in theoretical physics. Especially, two different directions are subject
of current interest: The study of critical systems on surfaces involving boundaries led to a
good knowledge of the so-called boundary CFTs (BCFT) [2, 3, 4, 5]. On the other hand,
already in 1991, Saleur showed the existence of density fields with scaling dimension zero
occurring in the treatment of dense polymers [6]. These fields may cause the existence
of operators yielding logarithmically diverging correlation functions. The two subjects,
BCFT and logarithmic conformal field theory (LCFT), enjoy increasing popularity in
both condensed matter physics and string theory.
Even though there has been much progress in the field, LCFTs are not yet completely
understood. However, it has been found that many properties of ordinary rational CFTs
can be generalized to LCFT, such as characters, partition functions and fusion rules,
see, e.g., [7, 8, 9, 10, 11, 12, 13, 14] and [15, 16] for some recent reviews. In ordinary
CFTs, especiallyinunitaryminimalmodels, thepresence ofaboundaryismathematically
and physically described by a standard procedure introduced by Ishibashi [2] and Cardy
[3] that allows to derive boundary states encoding the physical boundary conditions.
Unfortunately, LCFTs involving a boundary happen to be more difficult to treat. There
have been different approaches towards a consistent description of boundary LCFTs in
terms of boundary states emerging first for two years ago [18, 19, 20, 21], see also [22, 23].
LCFT in the vicinity of a boundary is also dealt with in [24, 25]. All those works focus
on the best understood example of a LCFT, the c = 2 realization with the maximally
−
extended chiralsymmetry algebra (2,3,3,3). The earlier results aredifferent andpartly
W
contradictory. Most successful seem the ideas of Kawai and Wheater [20] using symplectic
fermions and coherent states and of ourselves [21]. The concept of symplectic fermions
was first introduced by Kausch [17] in order to describe the rational c = 2 (bulk) LCFT.
−
In our own work, a general, very basic approach towards the derivation of boundary states
in the case of the -algebra is presented that allows to handle complicated structures
W
such as indecomposable representations in LCFTs.
This letter is positioned exactly at this point. We show that the two different symmetries
– the symplectic fermions vs. (2,3,3,3)-algebra – lead to the same set of boundary
W
states. In particular, the former one, though extending the latter, implies no additional
restrictionsontheboundarystates. Bythis, wecanshowthatthecoherent stateapproach
is fully equivalent to ours yielding the same results. This corresponds to the presumption
of Kawai [23] that the coherent states are indeed as good as taking the usual Ishibashi
states.
The paper proceeds as follows: In section 2, a short introduction to the rational c = 2
−
LCFT is given both for the (2,3,3,3)-algebra and in the symplectic fermion picture.
W
Then, section 3 and 4 review the results of Kawai and Wheater and those deduced by
us. In section 5, the boundary states for the symplectic fermion symmetry algebra are
derived using the method of [21] and compared to both of the previous results. Finally,
section 6 concludes the paper with a short discussion.
1
2 The model
The CFT realization at c = 2 is based on the extended chiral symmetry algebra
−
(2,3,3,3) consisting of the energy-momentum tensor L(z) and a triplet of spin-3 fields
W
Wa(z). With the two quasi-primary normal-ordered fields Λ = :L2: 3/10∂2L and
−
Va = :LWa: 3/14∂2Wa the commutation relations for the corresponding modes read:
−
1
L ,L =(m n)L m3 m δ ,
m n m+n m+n,0
− − 6 −
(cid:2)L ,Wa(cid:3) =(2m n)Wa , (cid:0) (cid:1)
m n − m+n
1
(cid:2)Wa,Wb(cid:3) =gˆab 2(m n)Λ + (m n) 2m2 +2n2 mn 8 L
m n − m+n 20 − − − m+n
(cid:18)
(cid:2) (cid:3) 1 (cid:0) (cid:1)
m m2 1 m2 4 δ (1)
m+n,0
−120 − −
(cid:19)
5 (cid:0) (cid:1)(cid:0) (cid:1) 12
+fˆab 2m2 +2n2 3mn 4 Wc + Vc .
c 14 − − m+n 5 m+n
(cid:18) (cid:19)
(cid:0) (cid:1)
Here, gˆab is the metric and fˆab are the structure constants of su(2). It is convenient to
c
arrange the fields in a Cartan-Weyl basis W0, W±. In this framework, we have gˆ00 = 1,
gˆ+− = gˆ−+ = 2, fˆ0± = fˆ±0 = 1, and fˆ+− = fˆ−+ = 2.
± − ± ± 0 − 0
The algebra yields a set of six representations that close under fusion. There are four
ordinary highest weight representations: is based on the vacuum state Ω with weight
0
V
h = 0, emerges from the state µ with weight h = 1/8. Then, one has two doublet
−1/8
V −
representations based on the states φ± and built from ν±. Furthermore, two
1 3/8
V V
indecomposable or generalized highest weight representations and emerge. They
0 1
R R
base on the states ω and ψ±, respectively. These states form rank-2 Jordan blocks in L
0
together with the states Ω and φ±. Thus, and are subrepresentations of and
0 1 0
V V R
, respectively. also contains two subrepresentations of type built on the states
1 0 1
R R V
Ψ+ = W+ ω, Ψ+ = W0 + 1L ω,
1 −1 2 −1 2 −1 (2)
Ψ− = W0 + 1L ω, Ψ− = W− ω.
1 − −1 2 −1 2 (cid:0) −1 (cid:1)
(cid:0) (cid:1)
For the bulk states of the and we use the metric of [21] that reads:
0 1
R R
Ω Ω = 0, Ω ω = 1, ω ω = d,
(3)
(cid:10)φ+(cid:12)φ−(cid:11) = 0, φ(cid:10)+(cid:12)ψ−(cid:11) = 1, ψ(cid:10)+(cid:12)ψ(cid:11)− = t,
(cid:12) (cid:12) − (cid:12) −
where d and t a(cid:10)re (cid:12)in p(cid:11)rinciple ar(cid:10)bitr(cid:12)ary(cid:11)real number(cid:10)s. T(cid:12)he (cid:11)fusion rules for this model
(cid:12) (cid:12) (cid:12)
read:
Ψ = Ψ, = , = ,
0 −1/8 −1/8 0 3/8 1 −1/8
V × V ×V R V ×V V
= , = , = 2 +2 ,
1 1 0 −1/8 3/8 1 −1/8 m −1/8 3/8
V ×V V V ×V R V ×R V V (4)
= , = , = 2 +2 ,
1 0 1 −1/8 1 3/8 3/8 m −1/8 3/8
V ×R R V ×V V V ×R V V
= , = , = 2 +2 .
1 1 0 3/8 3/8 0 m n 0 1
V ×R R V ×V R R ×R R R
2
Here, m and n can take the values 0, 1. From(4) one reads off that , , ,
0 1 −1/8 3/8
R R V V
is a sub-group closed under fusion itself. The characters for the model are given by:
(cid:8) (cid:9)
1 1
χ (q) = Θ (q)+(∂Θ) (q) , χ (q) = Θ (q),
V0 2η(q) 1,2 1,2 V−1/8 η(q) 0,2
1 (cid:0) (cid:1) 1
χ (q) = Θ (q) (∂Θ) (q) , χ (q) = Θ (q), (5)
V1 2η(q) 1,2 − 1,2 V3/8 η(q) 2,2
(cid:0) 2 (cid:1)
χ (q) χ (q) = χ (q) = Θ (q).
R ≡ R0 R1 η(q) 1,2
Note, that the physical characters are only χ , χ and χ forming a three-dimen-
V−1/8 V3/8 R
sional representation of the modular group that corresponds to the above-mentioned sub-
group. Here, η(q) = q1/24 (1 qn) is the Dedekind eta function and Θ (q) and
n∈N − r,2
(∂Θ) (q) are the ordinary and affine Riemann-Jacobi theta functions:
1,2 Q
2 2
Θ (q) = q(2kn+r) /4k and (∂Θ) (q) = (2kn+r)q(2kn+r) /4k. (6)
r,k r,k
n∈ n∈
XZ XZ
In ordinary CFTs, the characters coincide with the torus amplitudes. Here, this is no
longer the case: The torus amplitudes form a slightly larger, five-dimensional representa-
tion of the modular group. It reads:
2
χ , χ , χ , χ , and χ (q) Θ (q)+iαlog(q)(∂Θ) (q) . (7)
V0 V−1/8 V1 V3/8 R ≡ η(q) 1,2 1,2
(cid:16) (cid:17)
This representation was analyzed bey Flohr [7]. There, the -matrix transforming the
S
“characters” under τ 1/τ was constructed and it was shown that it yields the
−→ −
fusion rules (4) only in the limit α 0 under which the logarithmic term in (7) vanishes.
−→
However, in this limit, S became singular.
There exists an explicit Lagrangian formulation for the c = 2 LCFT based on two
−
fermionic fields η and ξ of scaling dimension 1 and 0, respectively:
1
S = d2z η∂¯ξ +η¯∂ξ¯ . (8)
π
Z
(cid:0) (cid:1)
This is the fermionic ghost system at c = 2 with the operator product expansions
−
1
η(z)ξ(w) = ξ(z)η(w) = +... . (9)
z w
−
All other products are regular. Kausch [17] showed, that these two fields combine into a
two-component symplectic fermion
χ+ η and χ− ∂ξ. (10)
≡ ≡
Thechoiceassuresthatχ+ andχ− havethesameconformalweighth = 1. Thisdescription
differs from the ghost system only by the treatment of the zero modes in χ− and ξ. The
fermion modes are defined by the usual power series expansion
χ±(z) = χ±z−m−1, (11)
m
m∈ +λ
XZ
3
where λ = 0 in the untwisted (bosonic) sector and λ = 1 in the twisted (fermionic) sector.
2
The modes satisfy the anticommutation relations
χα, χβ = mεαβδ , (12)
m n m+n,0
with the totally antisymmetric(cid:8)tensor ε(cid:9)±∓ = 1. The symplectic fermion decomposes the
±
Virasoro modes and the modes of the three spin-3 fields W(z) of (2,3,3,3) [8, 9, 15]:
W
1 λ(λ 1)
L = ε : χαχβ : + − δ ,
n 2 αβ j n−j 2 n,0
j∈ +λ
XZ
1
W0 = j : χ+ χ− : + : χ− χ+ : , (13)
n −2 · n−j j n−j j
j∈ +λ
XZ (cid:8) (cid:9)
W± = j χ± χ±.
n · n−j j
j∈ +λ
XZ
The highest weight states become related to each other by introducing the fermion sym-
metry: In the twisted sector, the doublet states of weight h = 3/8 are connected to the
singlet at weight h = 1/8 by να = χα µ. The states of weight 0 in the untwisted
− −1/2
sector are related by ξ± = χ±ω, Ω = χ−χ+ω. Furthermore, one finds φα = χα Ω and
− 0 0 0 −1
ψα = χα ω. Thus, this additional symmetry intertwines the representation with
−1 R0 R1
and with .
−1/8 3/8
V V
3 Approach 1: Coherent boundary states
Starting point for any derivation of boundary conditions is the absence of energy-mo-
mentum flow across the boundary and corresponding gluing conditions for the extended
symmetry fields. On a cylinder, the boundary conditions are identified with an initial
and final state of a propagating closed string: the boundary states B . After radial
ordering and in the framework of symplectic fermions this yields the following consistency
(cid:12) (cid:11)
equations: (cid:12)
L L B = 0, (14)
n −n
−
χ(cid:0)± e±iφχ±(cid:1)(cid:12) (cid:11)B = 0, (15)
n − −n(cid:12)
where φ is a phase that occurs i(cid:0)n the gluing c(cid:1)o(cid:12)ndi(cid:11)tion of χ and χ. The latter equation
(cid:12)
implies the first one due to (13). Kawai and Wheater showed that (15) is solved by the
coherent states [20]
eiφ e−iφ
B = N exp χ− χ+ + χ− χ+ 0 . (16)
0φ k −k −k k −k −k φ
!
k>0
(cid:12) (cid:11) X (cid:12) (cid:11)
(cid:12) (cid:12)
Here, N is a normalization factor and 0 is a non-chiral ground state. The boundary
φ
states were designed in such a way that they are compatible with the -algebra and thus
(cid:12) (cid:11) W
obey (14) and (cid:12)
Wa +Wa B = 0. (17)
n −n
(cid:0) (cid:1)(cid:12) (cid:11)
4(cid:12)
This implies that the phase φ can only take the values φ = 0 and φ = π. Therefore, the
non-chiral ground states are given by the “invariant vacua” Ω Ω , ω ω , µ µ .
⊗ ⊗ ⊗
This yields six possible boundary states, denoted by (+) if φ = 0 and ( ) for φ = π:
(cid:8)(cid:0) (cid:1) (cid:0)− (cid:1) (cid:0) (cid:1)(cid:9)
BΩ+ BΩ,φ=0 , BΩ− , Bω± , and Bµ± . (18)
≡
The corresponding(cid:12)cylin(cid:11)der(cid:12)amplitu(cid:11)de(cid:12)s are(cid:11)gi(cid:12)ven b(cid:11)y the (cid:12)natu(cid:11)ral pairings B qˆC =
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
B qH C = B (q1/2)(L0+L0+1/6) C . For the interesting (untwisted) sector, they are
(cid:10) (cid:12) (cid:12) (cid:11)
(cid:12) (cid:12)
(cid:10) (cid:12)(cid:12) (cid:12)(cid:12) (cid:11) (cid:10) (cid:12)(cid:12) BΩ+ BΩ− (cid:12)(cid:12) (cid:11) Bω+ Bω−
BΩ+ (cid:12)(cid:12) 0 (cid:11) (cid:12)(cid:12) 0 (cid:11) (cid:12)(cid:12)η(q)2(cid:11) Θ(cid:10)1,2(q(cid:12)(cid:12))
(cid:10)BΩ−(cid:12) 0 0 Θ1,2(q) η(q)2 
(cid:12) . (19)
(cid:10)Bω+(cid:12) η(q)2 Θ1,2(q) d(d+ln(q))η(q)2 d(d+ln(q))Θ1,2(q)
(cid:12) 
(cid:10)Bω−(cid:12)(cid:12)Θ1,2(q) η(q)2 d(d+ln(q))Θ1,2(q) d(d+ln(q))η(q)2 
 
(cid:10) (cid:12) 
The different(cid:12)factors and signs in contrast to [20] arise due to our different normalization
of the metric. To get rid of the unphysical terms proportional to log(q)Θ (q), one of the
1,2
states B was discarded and the physical boundary conditions were derived with this
ω±
reduced set. This was possible according to the symmetry φ φ+π mod 2π.
(cid:12) (cid:11) Z2 −→
Candid(cid:12)atesfortheIshibashistateswerededucedbydiagonalizingthecylinder amplitudes,
i.e., i qˆj = δ χ (q). However, it was not possible to express the physical boundary
ij i
states in terms of this basis. Kawai and Wheater proposed the following five states and
(cid:10) (cid:12) (cid:12) (cid:11)
five cor(cid:12)re(cid:12)sponding duals:
V0 = 21 BΩ+ + 12 BΩ− , V0 = −21 Bω+ − 12 Bω− ,
(cid:12)(cid:12)V1(cid:11) = 21(cid:12)(cid:12)BΩ+(cid:11)− 21(cid:12)(cid:12)BΩ−(cid:11), (cid:10)V1(cid:12)(cid:12) = 12 B(cid:10)ω+ −(cid:12)(cid:12) 21 B(cid:10)ω− ,(cid:12)(cid:12)
V(cid:12) (cid:11) = 1(cid:12)B (cid:11)+ 1 (cid:12)B (cid:11), V(cid:10) (cid:12) = 1(cid:10)B (cid:12)+ 1(cid:10)B (cid:12), (20)
−(cid:12)1/8 2(cid:12) µ+ 2 (cid:12) µ− −1/8(cid:12) 2 µ+(cid:12) 2 µ−(cid:12)
(cid:12) V (cid:11) = 1(cid:12)B (cid:11) 1(cid:12)B (cid:11), (cid:10) V (cid:12) = 1(cid:10)B (cid:12) 1(cid:10)B (cid:12),
(cid:12) 3/8 2(cid:12) µ+ − 2(cid:12) µ− 3/8(cid:12) 2 µ+(cid:12)− 2 µ−(cid:12)
(cid:12)(cid:12) R(cid:11) = √(cid:12)(cid:12)2 BΩ(cid:11)+ , (cid:12)(cid:12) (cid:11) (cid:10) R(cid:12)(cid:12) = (cid:10)√2 B(cid:12)(cid:12) ω− .(cid:10) (cid:12)(cid:12)
−
The (ket-)stat(cid:12)es(cid:11)form on(cid:12)ly a f(cid:11)our-dimensional sp(cid:10)ac(cid:12)e. Especia(cid:10)lly, (cid:12)R is associated to the
(cid:12) (cid:12) (cid:12) (cid:12)
indecomposable representations but only built on the subrepresentations. It is evident
(cid:12) (cid:11)
that the states B cannot obey equation (15) without furth(cid:12)er restrictions because
ω±
they are based on the state ω ω which is obviously not a proper ground state:
(cid:12) (cid:11) ⊗
(cid:12)
L L(cid:0) ω ω(cid:1) = Ω ω ω Ω = 0, (21)
0 0
− ⊗ ⊗ − ⊗ 6
unless the right-han(cid:2)d side st(cid:3)a(cid:0)te is d(cid:1)iscar(cid:0)ded as(cid:1)in t(cid:0)he uniq(cid:1)ue local c = 2 LCFT [12].
−
There, a chiral and an anti-chiral version of the rational c = 2 LCFT are glued together
−
to obtain a non-chiral theory. In order to keep locality of the correlators, certain states
had to be divided out, namely the image of (L L ). This was not mentioned by Kawai
0 0
−
and Wheater. It is shown in the following that their considerations are indeed compatible
with the result of [21] and lead to the same results if starting from the “vacua” of the
complete chiral theory.
5
4 Approach 2: Boundary states for the -algebra
W
In [21] the span of boundary states under the constraints of the (2,3,3,3)-algebra was
W
derived. This was done by inventing a straight-forward method that uses only basic
properties of the theory and its representations. Due to that it was possible to keep
especially the inner structure of the indecomposable representations and and their
0 1
R R
subrepresentations visible. This allowed to find relations between the derived states. Ten
boundary states were identified:
The states V and V corresponding to the admissible irreducible representations
−1/8 3/8
and are the usual Ishibashi states for these modules.
V−1/8 V(cid:12)3/8 (cid:11) (cid:12) (cid:11)
For the ind(cid:12)ecomposable(cid:12)representations , to stay close to the usual notions, the defi-
λ
R
nition of the Ishibashi states was generalized. The two states
R = γλlm l,m l,n , λ = 0, 1 (22)
λ n 1⊗U ⊗
l,m,n
(cid:12) (cid:11) X (cid:0) (cid:1)(cid:12) (cid:11) (cid:12) (cid:11)
(cid:12) (cid:12) (cid:12)
are called generalized Ishibashi states. Here, l,m ; l = h,h+1,..., m = 1,... is an
arbitrary basis over the representation where l counts the levels beginning from the
Rλ (cid:8)(cid:12) (cid:11) (cid:9)
top-most, which is h = 0 in our case. The basis(cid:12)states on each level of the representation
arecountedbym. Similarly, l,n isthebasisfortheanti-holomorphicmodule . The
λ
R
matrix γλ was identified to be the inverse metric on . In ordinary CFTs, these bases
(cid:8)(cid:12) (cid:11)(cid:9) Rλ
can be chosen orthonormal an(cid:12)d then the result would coincide with the usual Ishibashi
state. It was argued in [21] that this is not applicable here.
The Ishibashi states corresponding to the two subrepresentations and were derived
0 1
V V
ˆ ˆ ˆ¯ ˆ
with the help of an operator = δ +δ, where δ is the off-diagonal part of L that was
0
considered to be in Jordan forNm. Since there are rank-2 Jordan cells at most, δˆ2 = 0 and
thus, ˆ3 = 0. It was argued that the states
N
1
V = ˆ R (23)
λ λ
2N
(cid:12) (cid:11) (cid:12) (cid:11)
do not vanish and fulfill (14) and ((cid:12)17), i.e., ar(cid:12)e properly defined boundary states. These
are called level-2 Ishibashi states and contain only contributions from the corresponding
subrepresentations.
In addition, two doublets of states were found that glue together the two different inde-
composable representations and at the boundary. They were given in terms of
0 1
operators ˆ and ˆ† that inteRrtwine thRe two representations and have the following action
P P
on the (bulk) states:
ˆ† ω = ξ± , ˆ† Ω = 0, ˆ ψ± = Ψ± ,
P± P± P+ − 2 (24)
ˆ ξ(cid:12)∓(cid:11) = (cid:12) Ω(cid:11) , ˆ φ(cid:12)±(cid:11) = 0, ˆ (cid:12)ψ±(cid:11) = Ψ(cid:12)± .(cid:11)
P± (cid:12) ±(cid:12) P± (cid:12) P−(cid:12) (cid:12)1
This yields the so(cid:12)-ca(cid:11)lled m(cid:12)ixe(cid:11)d Ishibas(cid:12)hi s(cid:11)tates R± an(cid:12)d R(cid:11)± :(cid:12) (cid:11)
(cid:12) (cid:12) (cid:12) 01 (cid:12) 10 (cid:12)
R± = ˆ R = ˆ† R , V (cid:12)(cid:12) = (cid:11)ˆ R±(cid:12)(cid:12) =(cid:11) ˆ R± ,
01 P± 1 P± 0 0 P∓ 01 P∓ 10 (25)
(cid:12)R±(cid:11) = ˆ (cid:12)R (cid:11) = ˆ†(cid:12)R (cid:11), (cid:12)V (cid:11) = ˆ†(cid:12)R±(cid:11) = ˆ†(cid:12)R±(cid:11).
(cid:12) 10 P±(cid:12) 1 P±(cid:12) 0 (cid:12) 1 P∓(cid:12) 01 P∓(cid:12) 10
(cid:12) (cid:11) (cid:12) (cid:11) (cid:12) (cid:11) (cid:12) (cid:11) (cid:12) (cid:11) (cid:12) (cid:11)
(cid:12) (cid:12) (cid:12) 6(cid:12) (cid:12) (cid:12)
These relations can be drawn schematically. It is not quite unexpected that there is a
one-to-one correspondence to the embedding scheme of the local theory [12]: The states
that are divided out there are due to (14) exactly those that do not contribute to the
boundary states.
V R φ ψ
1 (cid:27) 1 (cid:27)
• Qk (cid:17)• • Qk (cid:17)•
(cid:12)(cid:12) (cid:11) J]JQQQ R1±0 (cid:17)(cid:17)(cid:17)(cid:10)(cid:10) (cid:12)(cid:12) (cid:11) J]JQQQ ρ¯± (cid:17)(cid:17)(cid:17)(cid:10)(cid:10)
J Q (cid:17)+ (cid:10) J Q (cid:17)+ (cid:10)
J (cid:12) (cid:11) (cid:10) J (cid:10)
J (cid:10)(cid:12)••J] (cid:10) J (cid:10)••J] (cid:10)
J(cid:10) J(cid:10) J(cid:10) J(cid:10)
(cid:10)J (cid:10)J (cid:10)J (cid:10)J
(cid:10) J (cid:10)(cid:29) J (cid:10) J (cid:10)(cid:29) J
(cid:10) J (cid:10) J
(cid:10)(cid:17)(cid:10)(cid:17)(cid:17)(cid:17)R••0±1QkQQQJJ (cid:10)(cid:17)(cid:10)(cid:17)(cid:17)(cid:17)ρ••±QkQQQJJ
(cid:10)(cid:29)(cid:17)+ QJ (cid:10)(cid:29)(cid:17)+ QJ
(cid:27) (cid:27)
(cid:12) (cid:11)
• (cid:12) • • •
V R Ω ω
0 0
(cid:12) (cid:11) boundary states (cid:12) (cid:11)
(cid:12) (cid:12) R
figure 1: boundary states vs. local theory
The lines in the left picture in figure 1 refer to the action of ˆ, ˆ, ˆ†, ˆ, and ˆ† while
N P P P P
in the right one they denote the action of the (non-chiral) symmetry algebra.
The non-vanishing natural pairings of the boundary states are given by
V qˆV = χ (q), V qˆV = χ (q),
−1/8 −1/8 V−1/8 3/8 3/8 V3/8
(26)
(cid:10) (cid:12) (cid:12) (cid:11) (cid:10) (cid:12) (cid:12) (cid:11)
R0(cid:12)qˆ(cid:12)R0 = χR(q), R1(cid:12)qˆ(cid:12)R1 = χR(q).
These coincide w(cid:10)ith(cid:12)th(cid:12)e p(cid:11)hysical characters form(cid:10)ing(cid:12)th(cid:12)e th(cid:11)ree-dimensional representation
(cid:12) (cid:12) (cid:12) (cid:12)
of the modular group. The torus amplitudes, on the other hand, are first seen with the
help of additional, so-called weak boundary states X and Y , λ = 0,1, that obey
λ λ
R = ˆ X + Y(cid:12) . (cid:11) (cid:12) (cid:11) (27)
λ λ (cid:12)λ (cid:12)
N
These states could be chosen u(cid:12)niqu(cid:11)ely in(cid:12)suc(cid:11)h a(cid:12)way(cid:11) that they serve as the duals to the
(cid:12) (cid:12) (cid:12)
null states V obtaining
λ
(cid:12) (cid:11)X qˆV = χ (q), X qˆR = log(q) χ (q),
(cid:12) λ λ Vλ λ λ · Vλ
(cid:10)X (cid:12)qˆ(cid:12)Y (cid:11) = 0, (cid:10)Y (cid:12)qˆ(cid:12)R (cid:11) = χ (q) 2χ (q), (28)
λ(cid:12) (cid:12) λ λ(cid:12) (cid:12) λ R − Vλ
(cid:10)R (cid:12)qˆ(cid:12)R (cid:11) = χ (q). (cid:10) (cid:12) (cid:12) (cid:11)
λ(cid:12) (cid:12) λ R (cid:12) (cid:12)
Obviously, this(cid:10)doe(cid:12)s(cid:12)not(cid:11)exactly reproduce the elements of the five-dimensional represen-
(cid:12) (cid:12)
tationgiven in(5) but rather linear combinations of them andthe unphysical contribution
log(q)Θ (q). This has to be taken care of when calculating physical relevant boundary
1,2
conditions with the help of Cardy’s consistency equation.
7
5 With the general method
The method presented in [21] provides an efficient tool for the investigation of the bound-
ary states under the restrictions of the symplectic fermion algebra. It bases on the general
ansatz for a boundary state connecting a holomorphic and an anti-holomorphic represen-
tation and at the boundary
Mh Mh
B = cl l,m l,n . (29)
mn 1⊗U ⊗
l,m,n
(cid:12) (cid:11) X (cid:0) (cid:1)(cid:12) (cid:11) (cid:12) (cid:11)
(cid:12) (cid:12) (cid:12)
The task is to directly calculate the matrix c. This is done in an iterative procedure.
Since the sum in (29) is infinite, the coefficients cl can only be derived up to any finite
mn
level l = L. The idea is that this provides the basis for the second step, the identification
of the boundary states.
The boundary state consistency equation for this symmetry algebra is given by (15):
χ± e±iφχ± B = 0, (30)
m − −m
where φ is the spin which can t(cid:0)ake the values φ(cid:1)(cid:12)=(cid:11)0,π at the boundary, since we force B
(cid:12)
to be compatible with the -algebra. It is then clear that (14) and (17) are automatically
W (cid:12) (cid:11)
satisfied once (30) is valid. This implies that the solutions are linear combinations of(cid:12)the
boundary states of section 4. The naturallyarising question is, especially when comparing
the results presented in the two previous sections, whether the fermion symmetry is more
restrictive than the -algebra, i.e., if less states are found here than in the latter theory.
W
The opposite is the case: Using the method of [21] we again find ten proper boundary
states. Denoting the φ = 0 case by the quantum number (+) and φ = π by ( ) as in the
−
previous discussion, these states are:
Ω,Ω; = Ω,Ω φ+,φ− φ−,φ+ +... ,
± ± ∓
Ω,ω; = Ω,ω + ω,Ω ξ+,ξ− ξ−,ξ+ +... ,
(cid:12) (cid:11) (cid:12) (cid:11) (cid:12) (cid:11) (cid:12) (cid:11)
(cid:12) ± (cid:12) (cid:12) ± (cid:12) ∓ (31)
Ω,ξa; = Ω,ξa ξa,Ω +... , a = +, ,
(cid:12) (cid:11) (cid:12) (cid:11) (cid:12) (cid:11) (cid:12) (cid:11) (cid:12) (cid:11)
(cid:12) ± (cid:12) ±(cid:12) (cid:12) (cid:12) −
µ,µ; = µ,µ ν+,ν− ν−,ν+ +... .
(cid:12) (cid:11) (cid:12) (cid:11) (cid:12) (cid:11)
(cid:12) ± (cid:12) ± (cid:12) ∓
(cid:12) (cid:11) (cid:12) (cid:11) (cid:12) (cid:11) (cid:12) (cid:11)
Here, m,n is us(cid:12)ed as a sho(cid:12)rt-hand f(cid:12)or m n(cid:12) . This result may be compared to the
⊗
one for the -algebra. We obtain the following identities
(cid:12) (cid:11)W (cid:12) (cid:11) (cid:12) (cid:11)
(cid:12) (cid:12) (cid:12)
Ω,Ω; = V V , Ω,ω; = R +d V R t V ,
0 1 0 0 1 1
± ± ± ± − (32)
(cid:12)(cid:12)Ω,ξa;±(cid:11) = (cid:12)(cid:12)R0a1(cid:11) ±(cid:12)(cid:12) R1a(cid:11)0 , (cid:12)(cid:12)µ,µ;±(cid:11) = (cid:0)V(cid:12)(cid:12) 3/8(cid:11) ± (cid:12)(cid:12)V−1(cid:11)/(cid:1)8 . (cid:0)(cid:12)(cid:12) (cid:11) (cid:12)(cid:12) (cid:11)(cid:1)
Thi(cid:12)s identifi(cid:11)catio(cid:12)n us(cid:11)es t(cid:12)he fa(cid:11)ct tha(cid:12)t the bo(cid:11)unda(cid:12)ry st(cid:11)ates(cid:12) fulfill(cid:11)(14) and (17). Thus, the
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
first level contributions of (31) can be compared to the results of section 4 to gain the
corresponding linear combinations of the states given there.
To show that the result (20) of Kawai and Wheater is compatible to ours one has to keep
in mind that the coherent states obey the consistency equation (30), and hence (14) and
(17). Therefore, they can be expressed in terms of the states (31). Indeed, we find
BΩ± = Ω,Ω; and Bµ± = µ,µ; , (33)
± ±
(cid:12) (cid:11) (cid:12) (cid:11) (cid:12) (cid:11) (cid:12) (cid:11)
(cid:12) (cid:12) 8 (cid:12) (cid:12)
up to possible additional contributions from null-states and the different normalization.
It seems contradictory that here, no boundary state based on ω ω is found. But
⊗
reviewing [20] as quoted in section 3 these are the states B (or rather B ) which
ω± ω±
(cid:0) (cid:1)
occur only as the duals to BΩ± in the Ishibashi states. This is remarkable, since in our
(cid:12) (cid:11) (cid:10) (cid:12)
framework the only states having such logarithmic contrib(cid:12)utions, i.e., ω ω -lik(cid:12)e terms,
(cid:12) (cid:11) ⊗
are X that we used in p(cid:12)recisely the same manner. This suggests, that the coherent
λ
(cid:0) (cid:1)
states based on ω ω are related to X in the same way as above:
(cid:12) (cid:11) ⊗ λ
(cid:12)
(cid:0) (cid:1) ω,ω; (cid:12)= (cid:11)X X . (34)
± (cid:12) 0 ± 1
Observe the fact that these sta(cid:12)tes do n(cid:11)ot e(cid:12)xac(cid:11)tly c(cid:12)orr(cid:11)espond to B due to the connec-
(cid:12) (cid:12) (cid:12) ω±
tion to the local theory as discussed above.
(cid:12) (cid:11)
The generic procedure of [21] yields a much bigger collection o(cid:12)f states in comparison to
Kawai and Wheater. Especially, the mixed boundary states were not discussed by them
and the Ishibashi boundary state for the module was obtained by the identification
R
2 + 2 . Presumably therefore and by referring to the local theory by setting
0 1
V V ≡ R
Ω ω ω Ω to zero, their physical boundary conditions differ from the set of
⊗ − ⊗
Ishibashi states.
(cid:0) (cid:1) (cid:0) (cid:1)
Indeed, we find that the coherent state method produces exactly the same amount of
states when starting from the “invariant vacua” that we have:
Ω Ω , Ω ω + ω Ω , Ω ξa eiφ ξa Ω , µ µ . (35)
⊗ ⊗ ⊗ ⊗ − ⊗ ⊗
(cid:8)(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)(cid:9)
The symplectic fermions decompose the L operator in such a way that
0
χ±ω = ξ± and χ±χ∓ω = Ω. (36)
0 − 0 0 ∓
With respect to (24) and (25) this suggest that the intertwining operators ˆ and ˆ†
P P
and the corresponding boundary states Ra and Ra might be closely related to the
01 10
fermionic zero modes.
(cid:12) (cid:11) (cid:12) (cid:11)
(cid:12) (cid:12)
6 Discussion
We worked out the space of boundary states in the rational LCFT with central charge
c = 2 under the restrictions of the symplectic fermion symmetry. It turned out that
−
these states coincide with the solutionwe presented in [21]. In particular, this implies that
the symplectic fermion algebra gives no additional constraints on the boundary states in
comparison to the (2,3,3,3)-algebra of the rational c = 2 LCFT. This is interesting
W −
because the latter one is embedded in the former. One might guess that the boundary
state consistency equation for the symplectic fermion symmetry is more restrictive than
the oneforthe -algebra. Onthe other hand, already in[21] we noticed theclose relation
W
between the derivation of boundary states and the construction of a local theory (see fig.
1). At least for c = 2 the latter one is uniquely defined which would suggest, that there
−
exists exactly one consistent solution for the set of boundary states.
To construct the boundary states, we used the same method that we presented in [21]
for the -algebra case. This shows that this method really yields a general prescription
W
9