Representations of the Exceptional Lie Superalgebra E(3,6) II: Four Series of Degenerate Modules

Kač, Victor G.; Rudakov, A N

doi:10.1007/pl00005581

Cited by 21 publications

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“…In the present paper we continue the study of irreducible representations of E(3, 6), the Lie superalgebra which has apparent relations to the Standard Model (see [4]). …”

Section: Introductionmentioning

confidence: 90%

“…We are to consider a linear combination w = Aw (1) + Bw (4) where w (1) is given by (5.1) with n = 1 and w (4) is the vector of the case (4) of Theorem 3.15, thus…”

Section: Cases (1)and(4)mentioning

confidence: 99%

“…Let us fix and keep for the following the above the notations a, b, c, d for the elements of U (L − ) (these notations were also used in [4]). …”

Section: From Semi-singular To Singular Vectorsmentioning

confidence: 99%

“…4. The non-trivial g 0 -highest singular vectors v are the following ones (the value of θ is written in the square brackets):…”

Section: From Semi-singular To Singular Vectorsmentioning

confidence: 99%

“…The article is a sequel to our papers [3,4] and together they provide the classification and description of all irreducible L 0 -locally finite representations of E(3, 6) (see [3] for the definition).…”

Section: Introductionmentioning

confidence: 99%

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Representations of the Exceptional Lie Superalgebra E(3,6) Iii: Classification of Singular Vectors

Kač

Rudakov

2005

J. Algebra Appl.

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Abstract. We continue the study of irreducible representations of the exceptional Lie superalgebra E(3, 6). This is one of the two simple infinite-dimensional Lie superalgebras of vector fields which have a Lie algebra sℓ(3) × sℓ(2) × gℓ(1) as the zero degree component of its consistent Z-grading. We provide the classification of the singular vectors in the degenerate Verma modules over E(3, 6), completing thereby the classification and construction of all irreducible E(3, 6)-modules that are L0-locally finite. Introduction.There are only two infinite-dimensional simple linearly compact Lie superalgebras, E(3, 6) and E(3, 8), that have the Lie algebra sℓ(3) × sℓ(2) × gℓ(1) as the zero degree component g 0 in their consistent Z-grading [2]. In the present paper we continue the study of irreducible representations of E(3, 6), the Lie superalgebra which has apparent relations to the Standard Model (see [4]).The article is a sequel to our papers [3, 4] and together they provide the classification and description of all irreducible L 0 -locally finite representations of E(3, 6) (see [3] for the definition).As was shown in [3] the problem can be solved in two steps. First one obtains the classification of the so called degenerate (generalized) Verma modules (the Verma modules that have non-trivial singular vectors), and then one describes the irreducible factors of these degenerate Verma modules.In [4], both probelms were solved modulo the classification of singular vectors. In this article we provide the proof of this classification (the result was announced in [4]). Thus we complete the classification of the irreducible E(3, 6)-modules that are L 0 -locally finite.As before all vector spaces, linear maps and tensor products are considered over the field C of complex numbers.

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“…In the present paper we continue the study of irreducible representations of E(3, 6), the Lie superalgebra which has apparent relations to the Standard Model (see [4]). …”

Section: Introductionmentioning

confidence: 90%

“…We are to consider a linear combination w = Aw (1) + Bw (4) where w (1) is given by (5.1) with n = 1 and w (4) is the vector of the case (4) of Theorem 3.15, thus…”

Section: Cases (1)and(4)mentioning

confidence: 99%

“…Let us fix and keep for the following the above the notations a, b, c, d for the elements of U (L − ) (these notations were also used in [4]). …”

Section: From Semi-singular To Singular Vectorsmentioning

confidence: 99%

“…4. The non-trivial g 0 -highest singular vectors v are the following ones (the value of θ is written in the square brackets):…”

Section: From Semi-singular To Singular Vectorsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Representations of the Exceptional Lie Superalgebra E(3,6) Iii: Classification of Singular Vectors

Kač

Rudakov

2005

J. Algebra Appl.

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Effective Lagrangian, general supersymmetric system

Siegel¹,

Zachos²,

Spallucci³

et al. 2004

Concise Encyclopedia of Supersymmetry

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A noncritical string in six dimensions with ð0; 1Þ spacetime supersymmetry [1] which is dual to the heterotic string with an E 8 instanton of zero size [2], and which in F-theory corresponds to a particular P 1 shrinking to zero size [3]. The ð0; 1Þ noncritical string can be viewed as an M5 brane wrapped around 1 2 K 3. The study of the BPS states of E-strings was made in [4].The characters of the n-string bound state are captured by N ¼ 4 U ðnÞ topological Yang-Mills theory on 1 2 K 3, and this relation can also be used to shed light on the properties of N ¼ 4 topological Yang-Mills theories on manifolds with b þ 2 ¼ 1 [5]. In particular the E-string partition functions, which can be computed using local mirror symmetry on a Calabi-Yau threefold, give the Euler characteristics of the Yang-Mills instanton moduli space on 1 2 K 3. The partition functions are determined by a gap condition combined with a simple recurrence relation which has its origins in a holomorphic anomaly that has been conjectured to exist for N ¼ 4 topological Yang-Mills on manifolds with b þ 2 ¼ 1 and is also related to the holomorphic anomaly for higher genus topological strings on Calabi-Yau threefolds [5]. The case with n < 9 is important for counting BPS states for fivedimensional versions of E-strings [6]. Bibliography Steven DuplijECTOPLASM A superspace approach to component expansion of supergravity actions, based on recognizing component actions as differential forms on bosonic subspaces of the full superspace [1]. The basic idea is that any integrand is a differential form, and any integral over a subspace is independent of the choice of surface only if that form is closed. The most familiar example is a conserved charge: The integral of the time component of a current over space is conserved if the current is divergenceless; it is completely independent of spacelike hypersurface (locally conserved) only if the current is so. The current is then actually a ðD À 1Þ-form in a Ddimensional space, and the current conservation law is actually the statement that it is curl free, which does not rely on a metric:In superspace this condition is solved after converting the curved indices to flat: For any p-form J , dJ ¼ 0 ) 1 p r ½A J BC :::DÞ ¼ 1 2 T ½ABj E J EjC :::DÞ in terms of the covariant derivative r A and torsion T AB C . Spinor indices carry a lower engineering dimension than vector ones, so the constraints with more spinor indices are solved first. Many of the higherdimension components then follow by using the constant part of the torsion. Similar remarks apply to solving more general constraints, such as for the field strengths of physical forms like the vector and tensor multiplets. In fact, the D-form that describes component Lagrangians can be derived inductively in p for general p-forms, since they satisfy generic sets of constraints, independently of what type of object they are being used to describe: For any system where the gauge parameters are p-forms, the gauge fields are ðp þ 1Þ-forms, the field strengths are ðp þ 2...

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Exceptional Superalgebra

Miškinis¹,

Duplij²,

Parks³

et al. 2004

Concise Encyclopedia of Supersymmetry

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A noncritical string in six dimensions with ð0; 1Þ spacetime supersymmetry [1] which is dual to the heterotic string with an E 8 instanton of zero size [2], and which in F-theory corresponds to a particular P 1 shrinking to zero size [3]. The ð0; 1Þ noncritical string can be viewed as an M5 brane wrapped around 1 2 K 3. The study of the BPS states of E-strings was made in [4].The characters of the n-string bound state are captured by N ¼ 4 U ðnÞ topological Yang-Mills theory on 1 2 K 3, and this relation can also be used to shed light on the properties of N ¼ 4 topological Yang-Mills theories on manifolds with bIn particular the E-string partition functions, which can be computed using local mirror symmetry on a Calabi-Yau threefold, give the Euler characteristics of the Yang-Mills instanton moduli space on 1 2 K 3. The partition functions are determined by a gap condition combined with a simple recurrence relation which has its origins in a holomorphic anomaly that has been conjectured to exist for N ¼ 4 topological Yang-Mills on manifolds with b þ 2 ¼ 1 and is also related to the holomorphic anomaly for higher genus topological strings on Calabi-Yau threefolds [5]. The case with n < 9 is important for counting BPS states for fivedimensional versions of E-strings [6]. Bibliography Steven DuplijECTOPLASM A superspace approach to component expansion of supergravity actions, based on recognizing component actions as differential forms on bosonic subspaces of the full superspace [1]. The basic idea is that any integrand is a differential form, and any integral over a subspace is independent of the choice of surface only if that form is closed. The most familiar example is a conserved charge: The integral of the time component of a current over space is conserved if the current is divergenceless; it is completely independent of spacelike hypersurface (locally conserved) only if the current is so. The current is then actually a ðD À 1Þ-form in a Ddimensional space, and the current conservation law is actually the statement that it is curl free, which does not rely on a metric:In superspace this condition is solved after converting the curved indices to flat: For any p-form J , in terms of the covariant derivative r A and torsion T AB C . Spinor indices carry a lower engineering dimension than vector ones, so the constraints with more spinor indices are solved first. Many of the higherdimension components then follow by using the constant part of the torsion. Similar remarks apply to solving more general constraints, such as for the field strengths of physical forms like the vector and tensor multiplets. In fact, the D-form that describes component Lagrangians can be derived inductively in p for general p-forms, since they satisfy generic sets of constraints, independently of what type of object they are being used to describe: For any system where the gauge parameters are p-forms, the gauge fields are ðp þ 1Þ-forms, the field strengths are ðp þ 2Þ-forms, and the Bianchi identities are ðp þ 3Þ-forms, dA ¼ dK ; F ¼ d...

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Representations of the Exceptional Lie Superalgebra E(3,6) II: Four Series of Degenerate Modules

Cited by 21 publications

References 4 publications

Representations of the Exceptional Lie Superalgebra E(3,6) Iii: Classification of Singular Vectors

Representations of the Exceptional Lie Superalgebra E(3,6) Iii: Classification of Singular Vectors

Effective Lagrangian, general supersymmetric system

Exceptional Superalgebra

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