Boundary action of N
Ä2 super-Liouville theory
Changrim Ahn*and Masayoshi Yamamoto†Department of Physics, Ewha Womans University, Seoul 120-750, Korea 共Received 17 October 2003; published 29 January 2004兲
We derive a boundary action of N⫽2 super-Liouville theory which preserves both N⫽2 supersymmetry and conformal symmetry by imposing explicitly T⫽T¯ and G⫽G¯ . The resulting boundary action shows a new duality symmetry.
DOI: 10.1103/PhysRevD.69.026007 PACS number共s兲: 11.25.Hf, 11.55.Ds
I. INTRODUCTION
Two-dimensional Liouville field theory 共LFT兲 has been studied actively for its relevance to noncritical string theories and two-dimensional quantum gravity关1,2兴. This theory has been extended to the supersymmetric Liouville field theories
共SLFTs兲 which can describe the noncritical superstring
theo-ries. In particular, the N⫽2 SLFT has been studied actively because the world sheet supersymmetry can generate space-time supersymmetry. In addition to applications to string theories, these models provide theoretically challenging problems. The Liouville theory and its supersymmetric gen-eralizations are irrational conformal field theories 共CFTs兲 which have a continuously infinite number of primary fields. Because of this property, most CFT formalisms developed for rational CFTs do not apply to this class of model. An interesting problem is to extend the conventional CFT for-malism to irrational CFTs. There has been a lot of progress in this field. Various methods have been proposed to derive structure constants and reflection amplitudes, which are basic building blocks to complete the conformal bootstrap关3–5兴. These have been extended to the N⫽1 SLFT in 关6,7兴.
A more challenging problem is to extend these formalisms to the CFTs defined in the two-dimensional space-time ge-ometry with a boundary condition共BC兲 which preserves the conformal symmetry. Cardy showed that the conformally in-variant BCs can be associated with the primary fields in terms of modular S-matrix elements for the case of rational CFTs关8兴. It has been an issue whether the Cardy formalism can be extended to the irrational CFTs. There are active ef-forts to understand the conformally invariant boundary states in the context of string theories related to D-branes关9,10兴.
Important progress in this direction was made in 关11兴 where the functional relation method developed in 关4兴 was used in the boundary LFT. With a boundary action which preserves conformal symmetry, a one-point function of a bulk operator in the presence of the boundary interaction and two-point correlation functions of boundary operators have been computed using the functional relation method 关11兴. Here the conformal BC is denoted by a continuous parameter appearing in the boundary action. A similar treatment of the LFT defined in the classical Lobachevskiy plane—namely,
the pseudosphere—has been made in 关12兴. For the N⫽1 SLFT, the one-point functions and the boundary two-point functions have been obtained in关13,14兴 based on the conjec-tured boundary action. It is desirable to show that indeed this action preserves both supersymmetry and conformal symme-try.
In this paper we derive the boundary actions of the N
⫽1,2 SLFTs by imposing the symmetries. This approach to
obtain the boundary actions has been made before. In 关15兴, based on a superfield formulation, the N⫽2 supersymmetric boundary action has been derived for a general N⫽2 super-symmetric quantum field theory. For integrable quantum field theories with infinite conserved charges, the situation becomes much more complicated. As shown in a pioneering work关16兴, the boundary action which preserves the integra-bility can be fixed by imposing a first few conservation laws. For the supersymmetric integrable models, the two conditions—the supersymmetry and integrability—have been successfully imposed to get appropriate boundary ac-tions 关17–19兴. We continue this approach to the N⫽1,2 SLFTs and impose the boundary superconformal invariance conditions to derive the boundary actions. We will show that even at the classical level, the boundary actions are deter-mined uniquely.
This paper is organized as follows. In Sec. II we review a superfield formulation of the N⫽1 SLFT boundary action proposed previously. Then, we show that this action satisfies the superconformal invariance. Our main result—the super-conformally invariant boundary action of the N⫽2 SLFT—is derived in Sec. III. After repeating the superfield formulation, we derive the boundary action by imposing N
⫽2 superconformal symmetry. We conclude in Sec. IV with
a few discussions and provide technical details in the Appen-dixes.
II. BOUNDARY NÄ1 SUPER-LIOUVILLE THEORY In this section, we review a superfield formulation of the boundary action of the N⫽1 SLFT which preserves the boundary N⫽1 supersymmetry. Then, we will show that the same result can be obtained by imposing directly the N⫽1 superconformal symmetry.
A. Superfield formulation of the NÄ1 boundary action The action of the N⫽1 SLFT is given by 关20兴
S⫽
冕
d2zd2冉
12D¯⌽D⌽⫹ie
b⌽
冊
, 共2.1兲*Email address: ahn@ewha.ac.kr
where⌽ is a real scalar superfield:
⌽⫽⫹i⫺i¯¯⫹i¯ F. 共2.2兲
共See Appendix A 1 for our conventions of the N⫽1
super-symmetry.兲 This theory contains a dimensionless Liouville coupling constant b and the cosmological constant . Note that we consider a trivial background and omit a linear dila-ton coupling. We can express the action in terms of the com-ponent fields, S⫽
冕
d2z冋
1 2共¯⫹¯⫹¯¯兲⫹ib 2¯ eb ⫹1 2 2b2e2b册
, 共2.3兲by integrating over the and¯ coordinates in Eq.共2.1兲 and eliminating the auxiliary field F from its equation of motion. To introduce the boundary action, we consider first a gen-eral N⫽1 supersymmetric theory on the lower half-plane:
⫺⬁⬍x⫽Re z⬍⬁, ⫺⬁⬍y⫽Im z⭐0. Following 关10兴, we
can write the action as follows:
S⫽
冕
⫺⬁ ⬁ dx冕
⫺⬁ 0 d y冕
d2L, 共2.4兲whereL is the Lagrangian density in superspace. The super-symmetry variation of the action is
␦S⫽
冕
⫺⬁ ⬁ dx冕
⫺⬁ 0 d y冕
d2共Q⫹¯ Q¯兲L ⫽⫺i 2冕
⫺⬁ ⬁ dx共L兩¯⫹¯L兩兲兩y⫽0. 共2.5兲To cancel the surface term共2.5兲, we add a boundary action SB⫽
i 2
冕
⫺⬁⬁
dxL兩⫽¯⫽0, ⫽⫾1, 共2.6兲 which is defined at y⫽0. When⫽¯ , the supersymmetry variation of the total action vanishes:␦S⫹␦SB⫽0. Only one
supercharge Q⫹Q¯ is preserved. Conservation of this charge imposes the boundary condition on the supercurrent: G⫹G¯⫽0 at y⫽0. The superderivatives in the tangential and normal directions are given by Dt⫽D⫹D¯ and Dn
⫽D⫺D¯ , respectively. Their conjugate coordinates are t
⫽(⫹¯ )/2 andn⫽(⫺¯ )/2.
For the total variation of S⫹SB to vanish, two types of
boundary conditions can be imposed.
共i兲 Dirichlet boundary conditions Dt⌽兩y⫽n⫽0⫽0: For
the N⫽1 SLFT, this corresponds to
⫺¯兩
y⫽0⫽0, x兩y⫽0⫽0. 共2.7兲
These conditions can be identified with the supersymmetric version of the ZZ brane 关12–14兴
共ii兲 Neumann boundary conditions Dn⌽兩y⫽n⫽0⫽0: For
the N⫽1 SLFT, these give
⫹¯兩
y⫽0⫽0, y⫺2be b兩
y⫽0⫽0. 共2.8兲
These boundary conditions correspond to the supersymmet-ric version of the FZZT brane 关11,21兴.
In关10兴, it is shown that one can add an additional term to the boundary action
SB
⬘
⫽⫺ 1 2冕
⫺⬁ ⬁ dx冕
dt冉
⌫Dt⌫⫹ 4 biB⌫e b⌽/2冊
, 共2.9兲with a fermionic boundary superfield ⌫⫽a⫹ith. In fact,
this boundary action is equivalent to that considered previ-ously in 关13,14兴. We will show in the next subsection that this action indeed preserves the boundary superconformal symmetry.
B. Boundary superconformal symmetry
To derive a boundary action which preserves both N⫽1 supersymmetry and conformal symmetry, we start with a general form of boundary action
SB⫽
冕
⫺⬁ ⬁ dx冋
⫺ i 4¯⫹ 1 2axa⫺ f共兲a共⫹¯兲⫹B共兲册
, 共2.10兲where a is a real fermionic boundary degree of freedom which anticommutes with and ¯ . The boundary action
共2.10兲 was first proposed in the boundary N⫽1
supersym-metric sine-Gordon model 关18兴. f () and B() are func-tions of the scalar field to be determined by the boundary conditions which preserve N⫽1 supersymmetry. The fermi-onic boundary degree of freedom a was first introduced in the Ising model in a boundary magnetic field关16兴 and in the N⫽1 SLFT with appropriate kinetic term 关14兴.
The boundary N⫽1 superconformal symmetry imposes the following constraints on the stress tensor and supercur-rent:
T⫽T¯, G⫽G¯ at y⫽0. 共2.11兲
Here we choose ⫽⫺1 and preserve only one supercharge Q⫺Q¯ . Hence, it is called sometimes as N⫽1/2 supersym-metry.
The stress tensor T and the supercurrent G are given by
T⫽⫺1 2关共兲 2⫹兴⫹1 2Qˆ 2, G⫽i共⫺Qˆ兲, 共2.12兲
where Qˆ is the background charge. By using the equations of motion, one can easily show that the conservation laws¯ T
⫽T¯⫽¯ G ⫽G¯⫽0 are satisfied at the classical level with Qˆ ⫽1/b.
¯⫽b3共i¯⫹eb兲eb,
¯⫽⫺ib2¯ eb, ¯⫽ib2eb, 共2.13兲
and the boundary equations of motion,
y⫽4 f a共⫹¯兲⫺4 B , ⫺¯⫽⫺4i f a, xa⫽ f共⫹¯兲, 共2.14兲 we obtain G⫺G¯⫽2
冉
f⫺2 b f 冊
xa ⫹冉
⫺2 f B ⫺ 4 bf⫹be b1 f冊
xa. 共2.15兲 Here we eliminated,¯ assuming f is not zero.The condition G⫺G¯⫽0 can be satisfied by the following f and B: f⫽Beb/2, B⫽
冉
⫺ 2 b2B 2⫹1 2冊
e b, 共2.16兲where B is the boundary cosmological constant. One can
show similarly that T⫺T¯⫽0 can be also satisfied. One can easily check that the boundary action 共2.9兲 with 共2.6兲 in terms of the superfields is indeed the same as Eqs. 共2.10兲 with共2.16兲. Therefore, this action preserves not only bound-ary N⫽1 supersymmetry but also conformal symmetry.
So far, we have considered only the classical equations of motion. Even at this level, the boundary action has been determined uniquely. We can consider quantum corrections in a similar approach. For this, we interpret eb in Eqs.
共2.13兲 as the normal-ordered exponential :eb:. The fields in
the stress tensor and the supercurrent in Eq.共2.12兲 should be also normal ordered. With this change, we obtain
¯ T⫽b2共1⫹b2⫺Qˆb兲关共:eb:兲2
⫹i¯:eb:⫺i¯:eb:兴. 共2.17兲
The conservation law ¯ T ⫽0 共and others兲 can be satisfied when the background charge is renormalized to Qˆ⫽1/b
⫹b. We will show in Appendix B that the boundary
super-conformal symmetry T⫺T¯⫽0 and G⫺G¯⫽0 is also pre-served at the quantum level with this Qˆ .
III. BOUNDARY NÄ2 SUPER-LIOUVILLE THEORY In this section, we use previous method to derive the su-perconformal boundary action of the N⫽2 SLFT.
A. Superfield formulation The action of the N⫽2 SLFT is given by S⫽
冕
d2z冋
1
冕
d4⌽⫹⌽⫺⫹冉
i冕
d2⫹eb⌽⫹
⫹c.c.
冊
册
,共3.1兲
where⌽⫾are the chiral superfields which satisfy
D⫿⌽⫾⫽D¯⫿⌽⫾⫽0. 共3.2兲
Therefore,⌽⫾ can be expanded as
⌽⫾⫽⫾共y⫾, y¯⫾兲⫹i⫾⫿共y⫾,y¯⫾兲⫺i¯⫾¯⫿共y⫾, y¯⫾兲
⫹i⫾¯⫾F⫾共y⫾, y¯⫾兲, 共3.3兲
where y⫾⫽z⫹1
2⫾⫿ and y¯⫾⫽z¯⫹ 1
2¯⫾¯⫿. 共See Appendix
A 2 for conventions.兲 The action can be written in terms of the component fields as
S⫽
冕
d2z冋
1 2共⫺¯⫹⫹⫹¯⫺⫹⫺¯⫹
⫹⫹¯⫺⫹¯⫺¯⫹⫹¯⫹¯⫺兲⫹ib2⫺¯⫺eb⫹ ⫹ib2⫹¯⫹eb⫺⫹2b2eb(⫹⫹⫺)
册
. 共3.4兲 Now we consider boundary conditions in the N⫽2 SLFT on the lower half-plane. The action can be written asS⫽
冕
⫺⬁ ⬁ dx冕
⫺⬁ 0 dy冋
冕
d4K共⌽⫹,⌽⫺兲 ⫹冕
d2⫹W⫹共⌽⫹兲⫺冕
d2⫺W⫺共⌽⫺兲册
⫽SK⫹SW, 共3.5兲where K(⌽⫹,⌽⫺) is a Ka¨hler potential and W⫾(⌽⫾) are superpotentials. Consider first the case where only the Ka¨hler potential term exists 关22兴. The supersymmetric variation of SK is ␦SK⫽
冕
⫺⬁ ⬁ dx冕
⫺⬁ 0 d y冕
d4共⫹Q⫹⫹¯⫹Q¯⫹⫹⫺Q⫺ ⫹¯⫺Q¯⫺兲K共⌽⫹,⌽⫺兲 ⫽4i冕
⫺⬁ ⬁ dx共⫹K兩⫹¯⫹¯⫺⫹¯⫹K兩⫹¯⫹⫺ ⫹⫺K兩 ¯⫹⫺¯⫺⫹¯⫺K兩⫹⫺¯⫺兲兩y⫽0. 共3.6兲We can cancel Eq. 共3.6兲 by adding two types of boundary actions
SBK⫽i 4
冕
⫺⬁⬁
and
SBK⫽
i 4
冕
⫺⬁⬁
dx共eiK兩⫹¯⫺⫹e⫺iK兩⫺¯⫹兲, 共3.8兲 where ei is an arbitrary phase. In the first case, the super-symmetry variation of SK⫹SBK vanishes when ¯⫾
⫽e⫾i⫿. The conserved supercharges are Q
⫹⫹e⫺iQ¯⫺
and Q⫺⫹eiQ¯⫹. This leads to a condition on the supercur-rents: G⫾⫹e⫿iG¯⫿⫽0 at y⫽0. This case is called A-type boundary conditions 关23兴. The second case is¯⫾⫽e⫿i⫾ where conserved supercharges are Q⫹⫹e⫺iQ¯⫹ and Q⫺
⫹eiQ¯
⫺. Associated boundary conditions on the
supercur-rents will be called B-type boundary condition: G⫾
⫹e⫿iG¯⫾⫽0 at y⫽0. In this paper, we will consider ei
⫽⫺1 for simplicity.
With nonvanishing superpotential W⫾, the supersymmet-ric variation becomes
␦SW⫽ 1 2
冕
⫺⬁ ⬁ dx冋
共¯⫺⫺⫺⫺¯⫺兲W ⫹ ⫹ ⫹共⫹¯⫹⫺¯⫹⫹兲W ⫺ ⫺册
. 共3.9兲We classify the boundary conditions into two classes follow-ing关15,22兴.
1. A-type boundary condition
We set¯⫾⫽⫺⫿in Eq.共3.9兲 and assume that the fermi-ons satisfy the condition
⫾⫺¯⫿兩
y⫽0⫽0. 共3.10兲
The boundary conditions for the bosons are given by
x共⫹⫺⫺兲⫽0, y共⫹⫹⫺兲⫽0. 共3.11兲
If the superpotentials W⫾satisfy
W⫹ ⫹⫺ W⫺ ⫺
冏
y⫽0 ⫽0, 共3.12兲 ␦SW⫽0 can be achieved.2. B-type boundary condition If¯⫾⫽⫺⫾, Eq. 共3.9兲 becomes ␦SW⫽ 1 2
冕
⫺⬁ ⬁ dx冋
⫺⫺共⫺⫹¯⫺兲W ⫹ ⫹ ⫹⫹共⫹⫹¯⫹兲W⫺ ⫺册
. 共3.13兲This vanishes for two types of boundary conditions.
共i兲 Dirichlet boundary conditions
⫾⫹¯⫾兩y
⫽0⫽0, x⫾兩y⫽0⫽0. 共3.14兲
共ii兲 Neumann boundary conditions
⫾⫺¯⫾兩y
⫽0⫽0, y⫾兩y⫽0⫽0. 共3.15兲
While no additional condition is needed for共i兲, the additional conditions
W⫾
⫾
冏
y⫽0⫽0 共3.16兲
are necessary for case共ii兲. To avoid this unphysical situation, one must add an additional boundary term
SBW⫽i 2
冕
⫺⬁ ⬁ dx共W⫹⫺W⫺兲冏
⫾⫽¯⫾⫽0 . 共3.17兲The variation of this term cancels ␦SW in Eq. 共3.13兲 if ⫺
⫹⫹⫽0 is satisfied. This leads to the boundary conditions
for ⫾: y⫾⫿2i W⫿ ⫿
冏
y⫽0 ⫽0. 共3.18兲Therefore, only N⫽1 supersymmetry is preserved. B. Boundary action of NÄ2 super-Liouville theory Here we construct the boundary action with a B-type boundary condition which preserves N⫽2 superconformal invariance. We start with
SB⫽
冕
⫺⬁ ⬁ dx冋
⫺ i 4共¯ ⫹⫺⫹¯⫺⫹兲⫹1 2a ⫺ xa⫹ ⫺12共 f⫹共⫹兲a⫹⫹ f˜⫹共⫹兲a⫺兲共⫺⫹¯⫺兲 ⫺12共 f⫺共⫺兲a⫺⫹ f˜⫺共⫺兲a⫹兲共⫹⫹¯⫹兲 ⫹B共⫹,⫺兲册
, 共3.19兲where a⫾ are complex fermionic boundary degrees of free-dom, which anticommute with ⫾ and ¯⫾. The boundary action of the form共3.19兲 was first proposed in the context of the N⫽2 supersymmetric sine-Gordon model 关19兴. f⫾(⫾), f˜⫾(⫾), and B(⫹,⫺) are functions of ⫾ to be deter-mined by the boundary conditions.
The stress tensor T, the supercurrent G⫾, and the U(1) current J are given by
T⫽⫺⫺⫹⫺1 2共
⫺⫹⫹⫹⫺兲
G⫾⫽
冑
2i共⫾⫾⫺Qˆ⫾兲, 共3.21兲J⫽⫺⫺⫹⫹Qˆ共⫹⫺⫺兲, 共3.22兲
where Qˆ is the background charge.
One can show that the conservation laws¯ T ⫽T¯⫽¯ G ⫾
⫽G¯⫾⫽¯ J ⫽¯J⫽0 are satisfied at the classical level when Qˆ⫽1/b in the same way as the N⫽1 case. One major dif-ference for the N⫽2 SLFT is that Qˆ has no quantum correc-tion. The above conservation laws hold at the quantum level with Qˆ⫽1/b due to :eb⫹::eb⫺ª:eb⫺::eb⫹:. This means that a classical level computation is sufficient for our consideration. Also, without the correction, dual symmetry
b→1/b disappears. The lack of dual symmetry makes it
much harder to solve even bulk N⫽2 SLFT 关24兴.
To preserve N⫽2 superconformal symmetry, we impose the following boundary conditions on the conserved currents: T⫽T¯, G⫾⫽G¯⫾, J⫽J¯ at y⫽0. 共3.23兲
Substituting the bulk equations of motion,
¯⫾⫽b3共i⫾¯⫾⫹eb⫾兲eb⫿,
¯⫾⫽⫺ib2¯⫿eb⫾,
¯⫾⫽ib2⫿eb⫾, 共3.24兲
and the boundary equations of motion,
y⫾⫽2
冉
f⫿ ⫿a⫿⫹ f˜⫿ ⫿a⫾冊
共⫾⫹¯⫾兲⫺4 B ⫿, ⫾⫺¯⫾⫽⫺2i共 f⫾a⫾⫹ f˜⫾a⫿兲, xa⫾⫽ f⫿共⫾⫹¯⫾兲⫹ f˜⫾共⫿⫹¯⫿兲, 共3.25兲into G⫾⫺G¯⫾ and eliminating⫾and¯⫾, we obtain
G⫾⫺G¯⫾⫽
冉
f⫾⫺2 b f⫾ ⫾冊
x⫾a⫾⫹冉
f˜⫾⫺ 2 b f˜⫾ ⫾冊
x⫾a⫿⫹冉
⫺ 2 f⫾ f⫾f⫿⫺ f˜⫾f˜⫿ B ⫿⫺ 2 bf ⫾ ⫺ b f˜⫿ f⫾f⫿⫺ f˜⫾f˜⫿e b⫾冊
xa⫾⫹冉
2 f˜⫾ f⫾f⫿⫺ f˜⫾f˜⫿ B ⫿⫺ 2 bf˜ ⫾⫹ b f⫿ f⫾f⫿⫺ f˜⫾f˜⫿e b⫾冊
xa⫿. 共3.26兲The condition G⫾⫺G¯⫾⫽0 determines f⫾, f˜⫾, and B as follows:
f⫾⫽C⫾eb⫾/2, f˜⫾⫽C˜⫾eb⫾/2, 共3.27兲 B⫽⫺ 2
b2共C
⫹C⫺⫹C˜⫹C˜⫺兲eb(⫹⫹⫺)/2, 共3.28兲
where C⫾ and C˜⫾ are complex constants which obey C⫾C˜⫾⫽b2/4.
We next consider the stress tensor. Eliminating a⫾ from Eq. 共3.25兲 and using Eq. 共3.27兲 and 共3.28兲, we obtain
y⫾⫽i b 2共 ⫿⫺¯⫿兲共⫾⫹¯⫾兲 ⫹4b共C⫹C⫺⫹C˜⫹C˜⫺兲eb(⫹⫹⫺)/2, 共3.29兲 x⫾⫺x¯⫾⫽ b 2x ⫾共⫾⫺¯⫾兲 ⫺2i共C⫹C⫺⫹C˜⫹C˜⫺兲eb(⫹⫹⫺)/2 ⫻共⫾⫹¯⫾兲⫺4iC⫾C˜⫾eb⫾共⫿⫹¯⫿兲. 共3.30兲
Substituting the above equations into T⫺T¯ and J⫺J¯, one can show that our solution satisfies both T⫽T¯ and J⫽J¯.
We have obtained the boundary action 共3.19兲 with Eqs.
共3.27兲 and 共3.28兲. Moreover, we impose the invariance of LB
under complex conjugation. This invariance implies C⫹
⫽(C⫺)*and C⫾can be written as C⫾⫽
Be⫾i␣, where␣is
a real parameter. This phase factor can be gauged away by redefining the fermionic zero modes a⫾→e⫿i␣a⫾. There-fore, the final form of the boundary action is
SB⫽
冕
⫺⬁ ⬁ dx冋
⫺ i 4共¯ ⫹⫺⫹¯⫺⫹兲⫹1 2a ⫺ xa⫹ ⫺12eb⫹/2冉
Ba⫹⫹ b2 4B a⫺冊
共⫺⫹¯⫺兲 ⫺1 2e b⫺/2冉
Ba⫺⫹ b2 4B a⫹冊
共⫹⫹¯⫹兲 ⫺ 2 b2冉
B 2⫹ 2b4 16B 2冊
e b(⫹⫹⫺)/2册
. 共3.31兲This is the main result of this paper. This action preserves two conserved supercharges Q⫹⫺Q¯⫹and Q⫺⫺Q¯⫺.
We can rewrite this boundary action in terms of boundary superfields. Defining supertranslation operators Dt⫾⫽D⫾
⫺D¯
⫾, which satisfy
兵Dt⫹,Dt⫺其⫽x, Dt2⫹⫽Dt2⫺⫽0, 共3.32兲 and their conjugate coordinates t⫾⫽(⫾⫺¯⫾)/2, we intro-duce fermionic boundary chiral superfields⌫⫾:
Dt⫿⌫⫾⫽0. 共3.33兲
The boundary superfields⌫⫾ can be expanded as
⌫⫾⫽a⫾共x⫾兲⫹i t
⫾h⫾共x⫾兲, 共3.34兲
where x⫾⫽x⫹1
2t⫾t⫿. In terms of these superfields, the
boundary action can be written as
SB⫽
冕
⫺⬁ ⬁ dx再
⫺ i 4共⌽ ⫹⌽⫺兩 ⫹¯⫺⫹⌽⫹⌽⫺兩⫺¯⫹兲⫺ 1 2共e b⌽⫹⫹eb⌽⫺兲冏
⫾⫽¯⫾⫽0 ⫹1 2冕
dt ⫹d t ⫺⌫⫹⌫⫺⫺冋
i b冕
dt ⫹冉
B⌫⫹eb⌽ ⫹/2⫹b 2 4B共⌫ ⫺eb⌽⫹/2⫺⌫⫹eb⌽⫺/2兲冊
冏
t⫺⫽0⫹c.c.册
. 共3.35兲When the terms including the superfields ⌫⫾ do not exist, Eq. 共3.35兲 reduces to the boundary action which preserves only N⫽1 supersymmetry under Neumann boundary condi-tions. In this case the N⫽2 supersymmetry transformation of the action S⫹SB has a nonvanishing surface term which is
canceled by those of the terms including ⌫⫾. IV. DISCUSSION
Our result contains one boundary parameter B which
generates a continuous family of conformal boundary condi-tions. One remarkable result is that the boundary action has a dual symmetry
B→
b2 4B
. 共4.1兲
This means that two conformal boundary conditions of the N⫽2 SLFT can be identified. To understand the further im-plications of this, we need to derive some exact correlation functions such as boundary one-point functions. Our bound-ary action is a first step toward this. It is possible to derive a functional relation for the one-point functions using the boundary action as a screening boundary operator. The main difficulty arises, as in the bulk case 关24兴, from the lack of coupling constant duality. In a recent paper 关25兴, the one-point functions for the N⫽2 SLFT are conjectured from the modular transformations of the characters for a special value of the coupling constant. It would be interesting to see if these one-point functions are consistent with the functional relations based on our boundary action and to derive them for arbitrary values of the coupling constant.
ACKNOWLEDGMENTS
We thank C. Kim, R. Nepomechie, J. Park, C. Rim, and M. Stanishkov for helpful discussions. This work was sup-ported in part by the Korea Research Foundation, Grant No. 2002-070-C00025.
APPENDIX A: CONVENTIONS
In this appendix we present our conventions for N⫽1,2 supersymmetries.
1. NÄ1 supersymmetry
We use (1,1) superspace with bosonic coordinates z, z¯ and fermionic coordinates ,¯ . Here we define z ⫽x⫹iy, z¯⫽x
⫺iy and ⫽(x⫺iy)/2, ¯⫽(x⫹iy)/2. The integration
measure is 兰d2zd2⫽兰dxdydd¯ . The covariant deriva- tives are given by
D⫽ ⫹, D¯⫽ ¯⫹¯¯ , 共A1兲 which satisfy 兵D,D其⫽2, 兵D¯ ,D¯其⫽2¯ , 兵D,D¯其⫽0. 共A2兲 The supercharges Q⫽ ⫺, Q¯⫽ ¯⫺¯¯ 共A3兲 satisfy 兵Q,Q其⫽⫺2, 兵Q¯ ,Q¯其⫽⫺2¯ , 兵Q,Q¯其⫽0, 共A4兲 and anticommute with D, D¯ .
2. NÄ2 supersymmetry
We use (2,2) superspace with bosonic coordinates z, z¯ and fermionic ones ⫹, ¯⫹, ⫺, ¯⫺. Complex conjugation of fermionic coordinates is defined by (⫾)*⫽¯⫿. The
D⫾⫽ ⫾⫹ 1 2 ⫿, D¯ ⫾⫽ ¯⫾⫹ 1 2¯ ⫿¯ 共A5兲 satisfy 兵D⫹,D⫺其⫽, 兵D¯⫹,D¯⫺其⫽¯ ,
all other共anti兲commutators⫽0. 共A6兲 The supercharges are given by
Q⫾⫽ ⫾⫺ 1 2 ⫿, Q¯⫾⫽ ¯⫾⫺ 1 2¯ ⫿¯ , 共A7兲 which obey 兵Q⫹,Q⫺其⫽⫺, 兵Q¯⫹,Q¯⫺其⫽⫺¯ ,
all other共anti兲commutators⫽0, 共A8兲 and anticommute with D⫾, D¯⫾. Chiral superfields⌽i⫾ sat-isfy D⫿⌽i⫾⫽D¯⫿⌽i⫾⫽0. An N⫽2 supersymmetric action is constructed from D terms and F terms1 and is written as
S⫽
冕
d2zd4K共⌽i⫹,⌽i⫺兲⫹冉
冕
d2zd2⫹W⫹共⌽i⫹兲⫹c.c.冊
,共A9兲
where K(⌽i⫹,⌽i⫺) is an arbitrary differentiable function of superfields and W⫹(⌽i⫹) is a holomorphic function of chiral superfields ⌽i⫹. The integration measures in Eq. 共A9兲 are defined by
冕
d2zd4K共⌽ i ⫹,⌽ i ⫺兲⫽冕
dxdy d⫹d¯⫹d⫺d¯⫺ ⫻K共⌽i⫹,⌽i⫺兲, 共A10兲冕
d2zd2⫹W⫹共⌽ i ⫹兲⫽冕
dxdy d⫹d¯⫹ ⫻W⫹共⌽ i ⫹兲兩 ⫺⫽¯⫺⫽0. 共A11兲APPENDIX B: QUANTUM CORRECTIONS OF THE NÄ1 BOUNDARY SUPERSYMMETRY
In this appendix we show how the classical boundary ac-tion共2.10兲 with Eqs. 共2.16兲 is modified at the quantum level. At the quantum level, we replace e␣ with the normal-ordered exponential :e␣:. Therefore, we cannot eliminate
,¯ from G,G¯ because f⫺1may not be well defined. There-fore, we need to keep the fermionic fields. Then, Eq.共2.15兲 becomes G⫺G¯⫽2f ax⫹2共⫹¯兲 f a共⫹¯兲 ⫺2共⫹¯兲B ⫺4Qxf a⫺4Q f2共⫹¯兲 ⫹Qb2共⫹¯兲⌳⫺2b2共:eb/2:兲2, 共B1兲 where⌳ is a cutoff scale and f ⫽B:eb/2:.
The first and second terms on the right-hand side of Eq.
共B1兲 can be dealt with the point-splitting technique
devel-oped in关26兴. The first term can be calculated as f ax⫽ B 2 xlim 1→x2 关:eb(x1)/2:a共x1兲 x共x2兲⫹共x1↔x2兲兴 ⫽B:eb/2x:a⫹b lim x1→x2 1 x1⫺x2 ⫻关 f共x1兲a共x1兲⫺ f 共x2兲a共x2兲兴 ⫽2 bxf a⫹bxf a⫹b f 2共⫹¯兲, 共B2兲
where we used Eq. 共2.14兲. Similarly the second term be-comes 共⫹¯兲f a共⫹¯兲⫽2xf a⫹2 f2共⫹¯兲. 共B3兲 This leads to G⫺G¯⫽共⫹¯兲
冋冉
⫺4 bB 2⫹Qˆ b2⌳⫺2b2冊
共:eb/2:兲2 ⫺2B册
, 共B4兲where we used Qˆ⫽1/b⫹b. The condition G⫺G¯⫽0 gives B⫽
冉
⫺ 2 b2B 2⫹1 2Qˆ b⌳ ⫺2b2冊
共:eb/2:兲2. 共B5兲Compared with the classical result 共2.16兲, B gets the quan-tum correction.
The condition T⫺T¯⫽0 can be also satisfied in this way. Substituting Eqs.共2.13兲 and 共2.14兲 into T⫺T¯, we obtain
T⫺T¯⫽i关2xf a共⫹¯兲⫺x„f a共⫹¯兲…兴
⫹12共¯
x¯⫺x兲. 共B6兲
Using Eq. 共2.14兲 for and ¯ , one can show that Eq.共B6兲 vanishes. Therefore, the boundary action共2.10兲 along with f and B given above preserves boundary conformal symmetry up to the quantum level.
1One can also consider twisted F terms which we do not mention
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