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Matrix Factorizations and Effective Superpotentials
Nuclear Physics B (Proc. Suppl.) 171 (2007) 288–289 www.elsevierphysics.com
Matrix Factorizations and Effective Superpotentials Johanna Knappa a
Department of Physics, Theory Division CERN, Geneva, Switzerland 1. D–Branes in B–type Landau–Ginzburg Models D–branes in the topological B–model are characterized with matrix facotrizations of a Landau– Ginzburg superpotential W (xi ), i.e. one can find pairs of matrices E, J with polynomial entries such that W · 1 = E · J.
(1)
The properties of E, J encode which brane it is. Two matrix factorizations (E, J) and (E ′ , J ′ ) describe the same D–brane if they can be related by a similarity transformation: E ′ = U1 EU2−1
J ′ = U2 JU1−1 ,
(2)
where U1 , U2 ∈ GL(N, R) are invertible matrices with polynomial entries. Having a set of matrix factorizations, one can construct a BRST operator: 0 E Q= , (3) J 0 This operator defines a graded differential d which acts as follows on open string states Ψ: dΨ = QΨ − (−1)|Ψ| ΨQ,
(4)
where |Ψ| is the Z2 –grading of the states. This operator squares to 0 in the quotient ring C/W . Physical staes lie in the cohomology of d which splits into an even and an odd part: H(d) = H e (d) ⊕ H o (d). States in H e are referred to as bosons and denoted by φi , states in H o are called fermions and are denoted by ψi . 2. The Effective Superpotential The effective superpotential Wef f can be interpreted in various ways: 0920-5632/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysbps.2007.06.031
1. Wef f represents the generating functional of open string disk amplitudes. Once all the amplitudes are known, they can be integrated to give the effective superpotential. The values of the amplitudes are constrained by worldsheet consistency conditions. In particular, they have to satisfy the A∞ –relations. Solving these constraints determines the values of the amplitudes and therefore the effective superpotential. 2. In the context of string compactifications, Wef f is the four dimensional space–time superpotential in N = 1 string compactifications with D–branes. 3. Wef f encodes the obstructions to the deformations of D-branes. Since turning on generic deformations usually leads away from the critical pint this problem cannot be approached within a CFT context. 2.1. Massey Products and Deformations of D–Branes We will now present a method to determine Wef f by calculating the full non–linear deformations of a matrix factorization. This method is designed for the minimal models. These models only have massive deformations, however a generalization of this method to models which also have marginal deformations is presently under investigation. Further information concerning the method presented in this section can be found in [1]. We start with an ansatz for a non–linear deformation of the opeator Q: X m ~ Qdef = Q + αm (5) ~u , m ~
J. Knapp / Nuclear Physics B (Proc. Suppl.) 171 (2007) 288–289 ~ mr 1 where um = um with r = dimH o and 1 · · · ur αm ~ are matrices which can be determined recursively. At the level of linear deformations, where PdimH o |m| ~ := i=0 mi = 1, the matrices αm ~ are given by the odd states ψi , which constitute the deformations. To each odd state φa we associate a deformation parameter ua . This is all the input we need to determine the higher αm ~ . As will be explained below, the even states φi ∈ H e give the obstructions. In order to have a valid deformation of the Q– operator the factorization condition should still ! be satisfied. Instead of demanding Q2def = W · 1, which is impossible to achieve, we need to satisfy a slightly more general condition: !
Q2def = W · 1 +
e dimH X
fi (u)φi ,
(6)
i=1
In [1] this method was applied to many examples of branes in minimal models. Comparing the results to those obtained from solving the worldsheet consistecy constraints one finds perfect agreement if bulk fields are turned off. Turning on bulk fields, one finds a descrepancy which implies that some of the worldsheet constraints need to be modified. An interesting observation is that the effective superpotentials for certain branes in minimal models are related to the superpotentials of certain Kazama-Suzuki coset models. REFERENCES 1. J. Knapp, H. Omer, “Matrix Factorizations, Minimal Models and Massey Products”, JHEP 05 (2006) 064, hep-th/0604189 2. A. Siqveland, “The Method of Computing Formal Moduli”, J.Alg 241 (2001) 292.
where fi (u) are the vanishing relations in the polynomial ring k[u1 , . . . , ur ]/(f1 , . . . fr ). Squaring (5), we obtain: X X m ~ m ~ 1 +m ~2 Q2def = W ·1+ {Q, αm αm (7) ~ }u + ~ 1 αm ~2u | {z } m ~
m ~ 1 +m ~2
y(m) ~
The expression y(m) ~ is called matric Massey product[2]. If y(m) ~ 6= 0 we have to distinguish between two cases:
• y(m) ~ is not in H e . Then we can find an αm ~ and two ~ such that {Q, αm ~ } = −y(m) contributions in (7) cancel. • y(m) ~ is in H e . Since this expression is by definition not exact we cannot find an αm ~ to cancel this contibution. Therefore this represents an obstruction and yields a contribution to the vanishing relations fi (u). In this way we build up vanishing relations fi (u) and deformation matrices αm ~ step by step for each level |m|. ~ At higher levels these relations have to be taken into account when comuting the Massey products. The procedure terminates at a certain level. One then finds that certain combinations of the vanishing relations fi (u) can be integrated to give Wef f .
289
Matrix Factorizations and Effective Superpotentials Johanna Knappa a
Department of Physics, Theory Division CERN, Geneva, Switzerland 1. D–Branes in B–type Landau–Ginzburg Models D–branes in the topological B–model are characterized with matrix facotrizations of a Landau– Ginzburg superpotential W (xi ), i.e. one can find pairs of matrices E, J with polynomial entries such that W · 1 = E · J.
(1)
The properties of E, J encode which brane it is. Two matrix factorizations (E, J) and (E ′ , J ′ ) describe the same D–brane if they can be related by a similarity transformation: E ′ = U1 EU2−1
J ′ = U2 JU1−1 ,
(2)
where U1 , U2 ∈ GL(N, R) are invertible matrices with polynomial entries. Having a set of matrix factorizations, one can construct a BRST operator: 0 E Q= , (3) J 0 This operator defines a graded differential d which acts as follows on open string states Ψ: dΨ = QΨ − (−1)|Ψ| ΨQ,
(4)
where |Ψ| is the Z2 –grading of the states. This operator squares to 0 in the quotient ring C/W . Physical staes lie in the cohomology of d which splits into an even and an odd part: H(d) = H e (d) ⊕ H o (d). States in H e are referred to as bosons and denoted by φi , states in H o are called fermions and are denoted by ψi . 2. The Effective Superpotential The effective superpotential Wef f can be interpreted in various ways: 0920-5632/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysbps.2007.06.031
1. Wef f represents the generating functional of open string disk amplitudes. Once all the amplitudes are known, they can be integrated to give the effective superpotential. The values of the amplitudes are constrained by worldsheet consistency conditions. In particular, they have to satisfy the A∞ –relations. Solving these constraints determines the values of the amplitudes and therefore the effective superpotential. 2. In the context of string compactifications, Wef f is the four dimensional space–time superpotential in N = 1 string compactifications with D–branes. 3. Wef f encodes the obstructions to the deformations of D-branes. Since turning on generic deformations usually leads away from the critical pint this problem cannot be approached within a CFT context. 2.1. Massey Products and Deformations of D–Branes We will now present a method to determine Wef f by calculating the full non–linear deformations of a matrix factorization. This method is designed for the minimal models. These models only have massive deformations, however a generalization of this method to models which also have marginal deformations is presently under investigation. Further information concerning the method presented in this section can be found in [1]. We start with an ansatz for a non–linear deformation of the opeator Q: X m ~ Qdef = Q + αm (5) ~u , m ~
J. Knapp / Nuclear Physics B (Proc. Suppl.) 171 (2007) 288–289 ~ mr 1 where um = um with r = dimH o and 1 · · · ur αm ~ are matrices which can be determined recursively. At the level of linear deformations, where PdimH o |m| ~ := i=0 mi = 1, the matrices αm ~ are given by the odd states ψi , which constitute the deformations. To each odd state φa we associate a deformation parameter ua . This is all the input we need to determine the higher αm ~ . As will be explained below, the even states φi ∈ H e give the obstructions. In order to have a valid deformation of the Q– operator the factorization condition should still ! be satisfied. Instead of demanding Q2def = W · 1, which is impossible to achieve, we need to satisfy a slightly more general condition: !
Q2def = W · 1 +
e dimH X
fi (u)φi ,
(6)
i=1
In [1] this method was applied to many examples of branes in minimal models. Comparing the results to those obtained from solving the worldsheet consistecy constraints one finds perfect agreement if bulk fields are turned off. Turning on bulk fields, one finds a descrepancy which implies that some of the worldsheet constraints need to be modified. An interesting observation is that the effective superpotentials for certain branes in minimal models are related to the superpotentials of certain Kazama-Suzuki coset models. REFERENCES 1. J. Knapp, H. Omer, “Matrix Factorizations, Minimal Models and Massey Products”, JHEP 05 (2006) 064, hep-th/0604189 2. A. Siqveland, “The Method of Computing Formal Moduli”, J.Alg 241 (2001) 292.
where fi (u) are the vanishing relations in the polynomial ring k[u1 , . . . , ur ]/(f1 , . . . fr ). Squaring (5), we obtain: X X m ~ m ~ 1 +m ~2 Q2def = W ·1+ {Q, αm αm (7) ~ }u + ~ 1 αm ~2u | {z } m ~
m ~ 1 +m ~2
y(m) ~
The expression y(m) ~ is called matric Massey product[2]. If y(m) ~ 6= 0 we have to distinguish between two cases:
• y(m) ~ is not in H e . Then we can find an αm ~ and two ~ such that {Q, αm ~ } = −y(m) contributions in (7) cancel. • y(m) ~ is in H e . Since this expression is by definition not exact we cannot find an αm ~ to cancel this contibution. Therefore this represents an obstruction and yields a contribution to the vanishing relations fi (u). In this way we build up vanishing relations fi (u) and deformation matrices αm ~ step by step for each level |m|. ~ At higher levels these relations have to be taken into account when comuting the Massey products. The procedure terminates at a certain level. One then finds that certain combinations of the vanishing relations fi (u) can be integrated to give Wef f .
289