Duality in N=2 super-Liouville theory

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Duality in N

Ä2 super-Liouville theory

Changrim Ahn,1,*Chanju Kim,1,†Chaiho Rim,2,‡and M. Stanishkov1,§ 1_{Department of Physics, Ewha Womans University, Seoul 120-750, Korea} 2_{Department of Physics, Chonbuk National University, Chonju 561-756, Korea}

共Received 8 February 2004; published 27 May 2004兲

In this paper we consider a strong-weak coupling duality of the N⫽2 super-Liouville field theory 共SLFT兲. Without the self-duality found in other Liouville theories, the N⫽2 SLFT, we claim, is associated with a ‘‘dual’’ action by a transformation b→1/b where b is the coupling constant. To justify our conjecture, we compute the reflection amplitudes共or two-point functions兲 of the Neveu-Schwarz and Ramond operators of the N⫽2 SLFT based on the conjectured dual action and show that the results are consistent with known results. DOI: 10.1103/PhysRevD.69.106011 PACS number共s兲: 11.25.Hf, 11.55.Ds

I. INTRODUCTION

Two-dimensional Liouville field theory 共LFT兲 appears naturally in the context of 2D quantum gravity and string theories 关1,2兴. This theory has been extended to the super-symmetric Liouville theories to accommodate world-sheet supersymmetries appearing in the string theories. These Liouville-type theories are also interesting for quantum field theoretical properties as well. They possess both conformal symmetry and the weak coupling duality. The strong-weak coupling duality has been attracting much attention recently to understand nonperturbative aspects of the various quantum field theories rigorously. For example, the seminal work of Seiberg and Witten on the supersymmetric Yang-Mills theories in 3⫹1 dimensions is based on the intuitive observation that the gauge theories in the strong coupling limit are described by a weak coupling region of some effec-tive action 关3兴. More recently, similar duality arises in the string theory context to understand strings and branes non-perturbatively. This duality also arises in statistical systems such as the 2D Ising model.

For the LFT and the N⫽1 super-LFT 共SLFT兲, the strong-weak coupling duality appears as a quantum symmetry closely connected to 共super兲conformal symmetries. It has been observed that the background charge is renormalized to Q⫽b⫹1/b by quantum corrections and the theories preserve the quantum conformal symmetries 关2,4兴. This means the two LFTs are invariant under b→1/b—i.e., self-dual. These two symmetries are essential to determine exact correlation functions for the LFT关5兴 and the N⫽1 SLFT 关6兴. Now let us consider the duality symmetry of the N⫽2 SLFT. With or without conformal symmetry, two-dimensional models with N⫽2 supersymmetry show an interesting feature: namely, the nonrenormalization. The parameters in the supersymmet-ric action do not change in all orders of perturbative calcu-lations. This means that the N⫽2 SLFT maintains conformal

symmetry without renormalization of the background charge and loses self-duality.

Our main proposal in this paper is that the N⫽2 SLFT still shows an interesting duality behavior. Under the dual transformation b→1/b, the theory maps to a ‘‘dual’’ action which is another N⫽2 super conformal field theory 共CFT兲. The N⫽2 SLFT with a strong coupling can be described by the dual action perturbatively. We compute the reflection am-plitudes 共the two-point functions兲 of the theory using func-tional relations derived from the actions. This procedure pro-vides an exact relation between two parameters␮,␮˜ , which fixes the relations between the two actions completely. To check the self-consistency of our proposal, we compare the reflection amplitudes derived from the conjecture action with some independent result derived in a totally different con-text.

CFTs with N⫽2 supersymmetry have been actively stud-ied mainly due to possible applications to string theories. In particular, the N⫽2 SLFT has been conjectured to be dual to a fermionic black hole based on the coset SL(2)/U(1) 关7兴. This proposal was further explored in关8兴. We will relate our ‘‘dual’’ theory to the fermionic black hole theory by compar-ing the reflection amplitudes and explicit Lagrangians.

II. NÄ2 SUPER-LIOUVILLE THEORY AND ITS DUAL ACTION

The action of the N⫽2 SLFT at the flat background is given by

AI⫽

冕

d2z

冋

冕

d4␪SS†⫹␮

冕

d2␪ebS⫹c.c.

册

, 共1兲 where S is a scalar superfield satisfying

D_⫺S⫽D¯_⫺S⫽0, D_⫹S†⫽D¯_⫹S†⫽0. 共2兲

The Lagrangian of the N⫽2 SLFT can be expressed in terms of the component fields as follows:

*Electronic address: ahn@dante.ewha.ac.kr †_{Electronic address: cjkim@ewha.ac.kr} ‡_{Electronic address: rim@mail.chonbuk.ac.kr}

§_{On leave of absence from Institute of Nuclear Research and} Nuclear Energy, Sofia, Bulgaria. Electronic address: stanishkov@dante.ewha.ac.kr

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L⫽ 1 4␲关2␸⳵⳵¯␸ †_⫹2_␸†_⳵⳵_¯_␸_⫹_␺†_¯_⳵_␺_⫹_␺⳵_¯_␺†_⫹_␺_¯†_⳵␺_¯ ⫹␺¯_⳵␺¯†_兴⫺5 4␲␮ 2_b2_eb␸⫹b␸†_⫹␮b 2 2 ␺ †_␺_¯†_eb␸ ⫹␮b 2 2 ␺␺¯ e b␸†_. _共3兲

As in the LFT and the N⫽1 SLFTs, one should introduce a background charge 1/b so that the second term in Eq. 共1兲 becomes the screening operator of the CFT. However, a fun-damental difference arises where the background charge is unrenormalized due to the N⫽2 supersymmetry. For the LFT and the N⫽1 SLFTs, the background charge is renor-malized to Q⫽1/b⫹b and the theories are invariant under the dual transformation b→1/b. This self-duality plays an essential role in determining various exact correlation func-tions of those Liouville theories. Unrenormalized, the N⫽2 SLFT is not self-dual.

This theory is a CFT with a central charge

c⫽3⫹6/b2. 共4兲

The primary operators of the N⫽2 SLFT are classified into Neveu-Schwarz共NS兲 and the Ramond 共R兲 sectors and can be written in terms of the component fields as follows 关9兴:

N_␣␣¯⫽e␣␸†⫹␣¯␸, R⫾_␣␣¯⫽␴⫾e␣␸†⫹␣¯␸, 共5兲 where␴⫾ are spin operators. The conformal dimensions of these fields are given by

⌬_␣␣¯N ⫽⫺␣␣¯_⫹ 1 2b共␣⫹␣¯兲, ⌬␣␣¯ R ⫽⌬_␣␣¯N ⫹1₈. 共6兲 The U(1) charges are given by

Q_␣␣¯N ⫽⫺ 1

2b共␣⫺␣¯兲, Q␣␣¯

R⫾

⫽Q_␣␣¯N

⫾1₄. 共7兲 From these expressions, one can notice that

␣→1/b⫺␣¯ , ␣¯_→1/b⫺_␣ _共8兲 do not change the same conformal dimension and U(1) charge. From the CFT point of view, this means that N1/b⫺␣¯,1/b⫺␣ should be identified with N␣␣¯and similarly for the 共R兲 operators up to normalization factors. The reflection amplitudes are determined by these normalization factors.

Without self-duality, it is possible that there exists a ‘‘dual’’ action to Eq.共1兲 whose perturbative 共weak coupling兲 behaviors describe the N⫽2 SLFT in the strong coupling region. This action should be another CFT. Our proposal for the dual action is as follows:

AII⫽

冕

d2z

冕

d4␪关SS†⫹␮˜ e(S⫹S

†_)/b

兴, 共9兲

with background charge b. The N⫽2 supersymmetry is pre-served because S⫹S† is a N⫽2 scalar superfield. One can see that this action is conformally invariant because the in-teraction term is a screening operator. Our conjecture is that the two actions AI and AII are equivalent. To justify this conjecture, we will compute the reflection amplitudes based on these actions and will compare with some independent results.

III. REFLECTION AMPLITUDES

As mentioned above, the reflection amplitudes of the Liouville-type CFT is defined by the linear transformations between different exponential fields, corresponding to the same primary field of chiral algebra. In this paper, we apply the method developed in关10兴 to derive the reflection ampli-tudes of the N⫽2 SLFT. For simplicity, we will restrict our-selves to the case of␣⫽␣¯ in Eq.共5兲 where the U(1) charge of the共NS兲 operators becomes 0. We will refer to this case as the ‘‘neutral’’ sector. 共From now on, we will suppress the second indices ␣¯ .) The physical states in this sector are given by

␣⫽ 1

2b⫹iP, 共10兲

where P is a real parameter. This parameter is transformed by P→⫺P under the reflection relation 共8兲 and can be thought of as a ‘‘momentum’’ which is reflected off from a potential wall.

Two-point functions of the same operators can be ex-pressed as

具

N_␣共z,z¯兲N_␣共0,0兲

典

⫽ DN_共_␣_兲 兩z兩4⌬_␣N , 共11兲

具

R_␣⫹共z,z¯兲R_␣⫺共0,0兲

典

⫽ DR共␣兲 兩z兩4⌬_␣R , 共12兲

where⌬_␣N,⌬_␣Rare given by Eq.共6兲. The reflection amplitudes are given by the normalization factors DN(␣),DR(␣) and should satisfy

DN共␣兲DN共1/b⫺␣兲⫽1, DR共␣兲DR共1/b⫺␣兲⫽1.

共13兲

To find these amplitudes explicitly, we consider the operator product expansions 共OPEs兲 with degenerate operators.

The共NS兲 and 共R兲 degenerate operators in the neutral sec-tor are N_␣

nmand R␣nm

⫾ _{with integers n,m and}

␣nm⫽

1⫺n 2b ⫺

mb

2 , n,m⭓0. 共14兲

The OPE of a共NS兲 field with a degenerate operator N_⫺b/2is simply given by

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Here the structure constant can be obtained from the screen-ing integral as follows:

C_⫺N共␣兲⫽␬1␥共1⫺␣b兲␥共1/2⫺␣b⫺b2/2兲␥共⫺1/2 ⫹␣b兲␥共␣b⫹b2/2兲, 共16兲 where ␬1⫽ ␮2_b4_␲2 2 ␥共⫺b 2_⫺1兲_␥

冉

₁_⫹b 2 2

冊

␥

冉

b2 2 ⫹ 3 2

冊

, with␥(x)⫽⌫(x)/⌫(1⫺x) as usual.

To use this OPE, we consider a three-point function

具

N_␣⫹b/2N_␣N_⫺b/2

典

and take the OPE by N_⫺b/2 either on N_␣⫹b/2or on N_␣ using Eq. 共15兲. This leads to a functional equation

C_⫺N共␣兲DN共␣⫹b/2兲⫽DN共␣兲. 共17兲 This functional equation can determine the共NS兲 reflection amplitude in the following form:

DN_共_␣_兲⫽

冉

␬1 b4

冊

⫺2␣/b ␥共2␣/b⫺1/b2_兲␥共b␣⫹1/2兲 ␥共b␣兲 f共␣兲, 共18兲

with an aribitrary function f (␣) satisfying f (␣)⫽ f (␣⫹b). To fix this unknown function, we need an additional func-tional equation. It is natural that this relation is provided by the dual actionAII.

For this purpose, we consider OPEs with another degen-erate operator: namely,

N_␣R_⫺1/2b⫹ ⫽R_␣⫺1/2b⫹ ⫹C˜_⫺N共␣兲R_␣⫹1/2b⫹ , 共19兲 R_␣⫺R_⫺1/2b⫹ ⫽N_␣⫺1/2b⫹C˜_⫺R共␣兲N_␣⫹1/2b. 共20兲 The structure constants can be computed by the screening integrals using the dual action AII, which is equivalent to

AI. The results are

C˜_⫺N共␣兲⫽␬₂共b兲␥共2␣/b⫺1/b 2_兲 ␥共2␣/b兲 , 共21兲 C˜_⫺R共␣兲⫽␬2共b兲 ␥共2␣/b⫺1/b2⫹1兲 ␥共2␣/b⫹1兲 , 共22兲 where ␬2共b兲⫽␮˜␲␥

冉

1 b2⫹1

冊

. 共23兲

These results are consistent with the N⫽2 superminimal CFT results 关9兴.

Now we consider the three-point functions

具

R_␣⫹1/2b⫺ N_␣R_⫺1/2b⫹

典

and

具

N_␣⫹1/2bR_␣⫺R_⫺1/2b⫹

典

. Taking an OPE

with R_⫺1/2b⫹ on one of two other operators in the correlation functions and using the OPE relations共19兲 and 共20兲, we ob-tain an independent set of functional relations as follows:

C ˜ ⫺ N_共_␣_兲DR_共_␣_{⫹1/2b兲⫽D}N_共_␣_兲, _共24兲 C ˜ ⫺ R 共␣兲DN_共_␣_{⫹1/2b兲⫽D}R_共_␣_兲. _共25兲

Solving for the DN(␣), we find that a most general solution of Eq.共24兲 is

DN共␣兲⫽␬₂⫺2␣b⌫

2_共_␣_b_⫹1/2兲

⌫2_共_␣_b_兲 ␥共2␣/b⫺1/b2兲g共␣兲,

共26兲

where g(␣) is another arbitrary function satisfying g(␣)

⫽g(␣⫹1/b). Combining Eqs. 共18兲 and 共26兲, and requiring

the normalization DN(␣⫽1/2b)⫽1, we can determine the

共NS兲 reflection amplitude completely as follows:

DN共␣兲⫽⫺ 2 b2␬2 ⫺2␣b⫹1_␥

冉

2␣ b ⫺ 1 b2

冊

␥

冉

␣b⫹ 1 2

冊

IV. CONSISTENCY CHECK

␥共⫺1⫺b2_兲 ␥共⫺1/2⫺b2_/2_兲. 共32兲

This result agrees with Eq. 共27兲 for ␣→1/2b⫺b/2. Simi-larly, when ␣→0, one can compute the two-point function directly from the actionAIIand get

具

N_␣共0兲N_␣共1兲

典

⫽⫺ ␮ ˜ b2

冕

d2z

具

e␣(␸⫹␸†)共0兲e␣(␸⫹␸†)共1兲 ⫻e(␸⫹␸†)/b_⳵_共_␸_⫹_␸†_兲_¯_⳵_共_␸_⫹_␸†_{兲共z,z¯兲}

_典

⫽⫺␮ ˜ b2␲␣ b⫽0 as ␣→0. 共33兲

Here, the insertion we have considered is the only term in the actionAIIwhich can give nonvanishing contribution. Again, this result agrees with Eq.共27兲 for ␣⫽0.

In the semiclassical limit b→0, the reflection amplitudes can be interpreted as the quantum mechanical reflection am-plitudes of the wave function of the zero modes from the exponential potential wall arising from the action 共1兲. One can easily find that the reflection amplitude corresponding to the 共NS兲 operator is given by

RN共P兲⬃⌫共1⫹2iP/b兲

⌫共1⫺2iP/b兲, 共34兲

which is consistent with Eq. 共31兲 in the limit b→0. V. RELATION TO THE FERMIONIC BLACK HOLES It was proposed in Ref.关7兴 that N⫽2 SLFT is T dual to the theory of the fermionic black hole based on the coset SL(2)/U(1). The purpose was to compute some physical

observables in the so-called little string theory using a holo-graphic projection. This statement was not proved but based on a few observations in 关7兴. The matching of the central charges in both theories requires a connection between the slope of the dilaton and the level k of the coset theory. Simi-larly, the compactification radius of a boson in N⫽2 SLFT is the inverse of that of the asymptotic radius of the black hole. There is a similar dualty between the bosonic black hole and the so-called sine-Liouville theory关13兴. The two-point func-tion of vertex operators Vj;m,m¯ was computed in关7兴 based on

a previous result found in 关14兴 using the coset theory SL(2)/U(1). One can check that this two-point function co-incides exactly with ours in the neutral case m⫽0 provided the spin j⫽⫺␣b and the level k⫽b2.

The above proposal was further explored in 关8兴 where it was argued that these two theories can be viewed as a mirror image of each other. Starting with certain gauged Wess-Zumino-Witten共WZW兲 model and its T dual transformation, the authors of 关8兴 showed that these lead to the fermionic black hole and the N⫽2 SLFT, respectively, in the IR limit. The fermionic black hole is defined as a Kazama-Suzuki coset model described by a gauged supersymmetric WZW theory. After solving for the nondynamical gauge fields and performing some implicit transformations between the SL(2) and N⫽2 superfields, one obtains an action of the form

S⫽

冕

d2zd4␪K共S,S⫹兲, 共35兲 where K(S,S⫹) is a Ka¨hler potential depending on the chiral superfields S,S⫹with the black hole metric:

1 1⫺ ␮ ˜ b2e ␸⫹␸⫹ 共36兲

(␸ and ␸⫹ are the first components of the superfields兲. At first sight this action looks different fromAIIwhich we pro-posed above. But if one expands interacting terms in the action perturbatively in the manner of the screening proce-dure, it is easy to see that the result for the physical observ-ables is the same. This is particularly evident for the calcu-lation of the structure constants computed in Sec. III where only the first-order perturbation共i.e., one screening insertion兲 is needed.

In summary, we have conjectured a strong coupling effec-tive action which is dual to the N⫽2 SLFT. Based on this conjecture, we have computed the reflection amplitudes共the two-point functions兲 of the 共NS兲 and the 共R兲 primary fields in neutral sectors exactly. We have fixed the relation between the two parameters ␮ and ␮˜ . Then, we have checked the validity of these amplitudes by comparing with an indepen-dent result along with some other consistency checks. Also we related our result to the fermionic black hole which has been conjectured to be dual to the N⫽2 SLFT. The reflection

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amplitudes derived in this paper can be applied to compute the effective central charge of the N⫽2 supersymmetric sinh-Gordon共or sine-Gordon兲 model. The comparison of this with the thermodynamic Bethe ansatz can provide another stringent check for our conjecture. We hope to report this progress in a future publication.

ACKNOWLEDGMENTS

We thank J. Teschner for helpful discussions. This work was supported in part by Korea Research Foundation, Grant No. 2002-070-C00025. M.S. is supported by the Brain Pool program from Korean Association Science and Technology.

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