Extended Galilean conformal algebras in two dimensions

Raymond, Christopher

doi:10.14264/uql.2015.914

Cited by 4 publications

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“…In our continued exploration [13,14] of Galilean contractions, we have presented a generalisation of the contraction prescription to allow for inputs of any finite number of OPAs or vertex algebras. This has resulted in hierarchies of higher-order Galilean conformal algebras, including Virasoro, affine Kac-Moody and W 3 algebras.…”

Section: Discussionmentioning

confidence: 99%

“…Vir 2 G is the familiar Galilean Virasoro algebra [2,3,4,5,6,13,14]. For small N , the Galilean Virasoro algebras Vir N G have recently appeared in [29].…”

Section: Galilean Virasoro and Affine Kac-moody Algebrasmentioning

confidence: 99%

“…It can be constructed [2,3,4,5,6] as an Inönü-Wigner contraction [7,8,9,10] of a commuting pair of Virasoro algebras. The Galilean W 3 algebra [11,12,13,14] likewise follows by contracting a pair of W 3 algebras [15]. Many other Galilean conformal algebras with extended symmetries have been worked out [16,13,14], including contractions of higher-rank W N algebras [17,18,19,20,21].…”

Section: Introductionmentioning

confidence: 99%

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Higher-order Galilean contractions

Rasmussen

Raymond

2019

Nuclear Physics B

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A Galilean contraction is a way to construct Galilean conformal algebras from a pair of infinitedimensional conformal algebras, or equivalently, a method for contracting tensor products of vertex algebras. Here, we present a generalisation of the Galilean contraction prescription to allow for inputs of any finite number of conformal algebras, resulting in new classes of higher-order Galilean conformal algebras. We provide several detailed examples, including infinite hierarchies of higher-order Galilean Virasoro algebras, affine Kac-Moody algebras and the associated Sugawara constructions, and W 3 algebras.

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Section: Discussionmentioning

confidence: 99%

“…Vir 2 G is the familiar Galilean Virasoro algebra [2,3,4,5,6,13,14]. For small N , the Galilean Virasoro algebras Vir N G have recently appeared in [29].…”

Section: Galilean Virasoro and Affine Kac-moody Algebrasmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Higher-order Galilean contractions

Rasmussen

Raymond

2019

Nuclear Physics B

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“…Their Galilean contractions are therefore readily obtained, although not presented explicitly here. Details can be found in [82].…”

Section: N = 4 Scas and Inönü-wigner Contractionsmentioning

confidence: 99%

“…CR was funded by an Australian Postgraduate Award from the Australian Government and by a University of Queensland Research Scholarship. This work is largely based on CR's thesis [82] submitted in July 2015 for the degree of MPhil at the University of Queensland. JR thanks Costas Zoubos for stimulating discussions during the formative stages of the project, and the Niels Bohr Institute for making them possible with their hospitality in July 2013.…”

Section: Acknowledgementsmentioning

confidence: 99%

Galilean contractions of W -algebras

Rasmussen

Raymond

2017

Nuclear Physics B

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Infinite-dimensional Galilean conformal algebras can be constructed by contracting pairs of symmetry algebras in conformal field theory, such as W -algebras. Known examples include contractions of pairs of the Virasoro algebra, its N = 1 superconformal extension, or the W 3 algebra. Here, we introduce a contraction prescription of the corresponding operator-product algebras, or equivalently, a prescription for contracting tensor products of vertex algebras. With this, we work out the Galilean conformal algebras arising from contractions of N = 2 and N = 4 superconformal algebras as well as of the W -algebras W (2, 4), W (2, 6), W 4 , and W 5 . The latter results provide evidence for the existence of a whole new class of W -algebras which we call Galilean W -algebras. We also apply the contraction prescription to affine Lie algebras and find that the ensuing Galilean affine algebras admit a Sugawara construction. The corresponding central charge is level-independent and given by twice the dimension of the underlying finite-dimensional Lie algebra. Finally, applications of our results to the characterisation of structure constants in W -algebras are proposed.

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Algebraic aspects of conformal field theory

Raymond¹

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This thesis presents an algebraic study of two areas of conformal field theory. The first part extends the theory of Galilean conformal algebras. These algebras are extended conformal symmetry algebras for two-dimensional quantum field theories, formed through a process known as Galilean contraction. Analogous to an Inonu-Wigner contraction, the Galilean contraction procedure takes two conformal symmetry algebras, equivalent up to central charge, as input and produces a new conformal symmetry algebra. We extend the theory of Galilean contractions in several directions. First, we develop the theory to allow input of any number of symmetry algebras. These generalised algebras have a truncated graded structure. We develop a theory for multi-graded Galilean algebras, whereby we can extend structures which are graded by sequences. Finally, we present a comprehensive analysis of the possible algebras which arise when the input algebras are no longer required to be equivalent. We refer to this as the asymmetric Galilean contraction. For each stage of generalisation, we present several pertinent examples, discuss the Sugawara construction of a Galilean Virasoro algebra given an affine Lie algebra, and apply our results to the W-algebra W 3. The second part of this thesis presents an exploratory study of reducible but indecomposable modules of the N = 2 Superconformal algebras, known as staggered modules. These modules are characterised by a non-diagonalisable action of L 0 , the Virasoro zero-mode. Using recent results on the coset construction of N = 2 minimal models, we are able to construct the first examples of staggered modules for the N = 2 algebras. We determine the structure of a family of such modules for all admissible values of the central charge. Furthermore, we investigate the action of symmetries such as spectral flow on these modules. We present the results along with a range of examples, and discuss possible paths towards a classification of such modules for the Neveu-Schwarz and Ramond N = 2 superconformal algebras.

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Extended Galilean conformal algebras in two dimensions

Cited by 4 publications

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Higher-order Galilean contractions

Higher-order Galilean contractions

Galilean contractions of W -algebras

Algebraic aspects of conformal field theory

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