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Counting gauge invariants: The Plethystic program
 JHEP 0703 (2007) 090 hepth/0701063
"... We propose a programme for systematically counting the single and multitrace gauge invariant operators of a gauge theory. Key to this is the plethystic function. We expound in detail the power of this plethystic programme for worldvolume quiver gauge theories of Dbranes probing CalabiYau singulari ..."
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Cited by 49 (17 self)
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We propose a programme for systematically counting the single and multitrace gauge invariant operators of a gauge theory. Key to this is the plethystic function. We expound in detail the power of this plethystic programme for worldvolume quiver gauge theories of Dbranes probing CalabiYau singularities, an illustrative case to which the programme is not limited, though in which a full intimate web of relations between the geometry and the gauge theory manifests herself. We can also use generalisations of HardyRamanujan to compute the entropy of gauge theories from the plethystic exponential. In due course, we also touch upon fascinating connections to Young Tableaux, Hilbert schemes and the
Resolution of stringy singularities by noncommutative algebras
 JHEP 0106
"... Preprint typeset in JHEP style. PAPER VERSION ..."
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Lectures on Dbranes, gauge theories and CalabiYau singularities
, 2004
"... These lectures, given at the Chinese Academy of Sciences for the BeiJing/HangZhou International Summer School in Mathematical Physics, are intended to introduce, to the beginning student in string theory and mathematical physics, aspects of the rich and beautiful subject of Dbrane gauge theories co ..."
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Cited by 11 (6 self)
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These lectures, given at the Chinese Academy of Sciences for the BeiJing/HangZhou International Summer School in Mathematical Physics, are intended to introduce, to the beginning student in string theory and mathematical physics, aspects of the rich and beautiful subject of Dbrane gauge theories constructed from local CalabiYau spaces. Topics such as orbifolds, toric singularities, del Pezzo surfaces as well as chaotic duality will be covered.
Generalised Clifford groups and simulation of associated quantum circuits
, 2007
"... Quantum computations that involve only Clifford operations are classically simulable despite the fact that they generate highly entangled states; this is the content of the GottesmanKnill theorem. Here we isolate the ingredients of the theorem and provide generalisations of some of them with the ai ..."
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Cited by 7 (3 self)
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Quantum computations that involve only Clifford operations are classically simulable despite the fact that they generate highly entangled states; this is the content of the GottesmanKnill theorem. Here we isolate the ingredients of the theorem and provide generalisations of some of them with the aim of identifying new classes of simulable quantum computations. In the usual construction, Clifford operations arise as projective normalisers of the first and second tensor powers of the Pauli group. We consider replacing the Pauli group by an arbitrary finite subgroup G of U(d). In particular we seek G such that G ⊗ G has an entangling normaliser. Via a generalisation of the GottesmanKnill theorem the resulting normalisers lead to classes of quantum circuits that can be classically efficiently simulated. For the qubit case d = 2 we exhaustively treat all finite subgroups of U(2) and find that the only ones (up to unitary equivalence and trivial phase extensions) with entangling normalisers are the groups generated by X and the n th root of Z for n ∈ N. 1
Finite Heisenberg groups from nonabelian orbifold quiver gauge theories
, 2007
"... A large class of orbifold quiver gauge theories admits the action of finite Heisenberg groups of the form ∏ i Heis(Zq i × Zq i). For an Abelian orbifold generated by Γ, the Zq i shift generator in each Heisenberg group is one cyclic factor of the Abelian group Γ. For general nonAbelian Γ, however, ..."
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Cited by 4 (1 self)
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A large class of orbifold quiver gauge theories admits the action of finite Heisenberg groups of the form ∏ i Heis(Zq i × Zq i). For an Abelian orbifold generated by Γ, the Zq i shift generator in each Heisenberg group is one cyclic factor of the Abelian group Γ. For general nonAbelian Γ, however, we find that the shift generators are the cyclic factors in the Abelianization of Γ. We explicitly show this for the case Γ = ∆(27), where we construct the finite Heisenberg group symmetries of the field theory. These symmetries are dual to brane number operators counting branes on homological torsion cycles, which therefore do not commute. We compare our field theory results with string
Discrete torsion orbifolds and Dbranes. ii,” hepth/0101143
"... The consistency of the orbifold action on open strings between Dbranes in orbifold theories with and without discrete torsion is analysed carefully. For the example of the C 3 / Z2 × Z2 theory, it is found that the consistency of the orbifold action requires that the Dbrane spectrum contains brane ..."
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Cited by 1 (0 self)
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The consistency of the orbifold action on open strings between Dbranes in orbifold theories with and without discrete torsion is analysed carefully. For the example of the C 3 / Z2 × Z2 theory, it is found that the consistency of the orbifold action requires that the Dbrane spectrum contains branes that give rise to a conventional representation of the orbifold group as well as branes for which the representation is projective. It is also shown how the results generalise to the orbifolds C 3 / ZN × ZN, for which a number of novel features arise. In particular, the N> 2 theories with minimal discrete torsion have nonBPS branes charged under twisted RR potentials that couple to none of the (known) BPS branes.
unknown title
, 2007
"... Copyright & reuse City University London has developed City Research Online so that its users may access the research outputs of City University London's staff. Copyright © and Moral Rights for this paper are retained by the individual author(s) and / or other copyright holders. All materia ..."
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Copyright & reuse City University London has developed City Research Online so that its users may access the research outputs of City University London's staff. Copyright © and Moral Rights for this paper are retained by the individual author(s) and / or other copyright holders. All material in City Research Online is checked for eligibility for copyright before being made available in the live archive. URLs from City Research Online may be freely distributed and linked to from other web pages. Versions of research The version in City Research Online may differ from the final published version. Users are advised to check the Permanent City Research Online URL above for the status of the paper. Enquiries If you have any enquiries about any aspect of City Research Online, or if you wish to make contact with the author(s) of this paper, please email the team at publications@city.ac.uk.Counting Gauge Invariants: the Plethystic Program
unknown title
, 2003
"... Preprint typeset in JHEP style. PAPER VERSION hepth/0210168 ..."
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