Sanjaye Ramgoolam - Academia.edu (original) (raw)
Papers by Sanjaye Ramgoolam
Journal of High Energy Physics
The structure of half-BPS representations of psu(2, 2|4) leads to the definition of a super-polyn... more The structure of half-BPS representations of psu(2, 2|4) leads to the definition of a super-polynomial ring mathcalR\mathcal{R}mathcalR R (8|8) which admits a realisation of psu(2, 2|4) in terms of differential operators on the super-ring. The character of the half-BPS fundamental field representation encodes the resolution of the representation in terms of an exact sequence of modules of mathcalR\mathcal{R}mathcalR R (8|8). The half-BPS representation is realized by quotienting the super-ring by a quadratic ideal, equivalently by setting to zero certain quadratic polynomials in the generators of the super-ring. This description of the half-BPS fundamental field irreducible representation of psu(2, 2|4) in terms of a super-polynomial ring is an example of a more general construction of lowest-weight representations of Lie (super-) algebras using polynomial rings generated by a commuting subspace of the standard raising operators, corresponding to positive roots of the Lie (super-) algebra. We illustrat...
arXiv (Cornell University), Feb 18, 2022
Two and three-point functions of primary fields in four dimensional CFT have a simple space-time ... more Two and three-point functions of primary fields in four dimensional CFT have a simple space-time dependences factored out from the combinatoric structure which enumerates the fields and gives their couplings. This has led to the formulation of two dimensional topological field theories with SO(4, 2) equivariance which are conjectured to be equivalent to higher dimensional conformal field theories. We review this CFT4/TFT2 construction in the simplest possible setting of a free scalar field, which gives an algebraic construction of the correlators in terms of an infinite dimensional so(4, 2) equivariant algebra with finite dimensional subspaces at fixed scaling dimension. Crossing symmetry of the CFT4 is related to associativity of the algebra. This construction is then extended to describe perturbative CFT4, by making use of deformed co-products. Motivated by the Wilson-Fisher CFT we outline the construction of U(so(d,2)) equivariant TFT2 for non-integer d, in terms of diagram algebras and their representations.
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Eighth-BPS local operators in N = 4 SYM are dual to quantum states arising from the quantization ... more Eighth-BPS local operators in N = 4 SYM are dual to quantum states arising from the quantization of a moduli space of giant gravitons in AdS5 × S5. Earlier results on the quantization of this moduli space give a Hilbert space of multiple harmonic oscillators in 3 dimensions. We use these results, along with techniques from fuzzy geometry, to develop a map between quantum states and brane geometries. In particular there is a map between the oscillator states and points in a discretization of the base space in the toric fibration of the moduli space. We obtain a geometrical decomposition of the space of BPS states with labels consisting of U(3) representations along with U(N) Young diagrams and associated group theoretic multiplicities. Factorization properties in the counting of BPS states lead to predictions for BPS world-volume excitations of specific brane geometries. Some of our results suggest an intriguing complementarity between localisation in the moduli space of branes and l...
Scattering of zero branes off the fixed point in R8 /Z2, as described by a super-quantum mechanic... more Scattering of zero branes off the fixed point in R8 /Z2, as described by a super-quantum mechanics with eight supercharges, displays some novel effects relevant to Matrix theory in non-compact backgrounds. The leading long distance behaviour of the moduli space metric receives no correction at one loop in Matrix theory, but does receive a correction at two loops. There are no contributions at higher loops. We explicitly calculate the two-loop term, finding a non-zero result. It has the right dependence on v and b for the scattering of zero branes off the (fractional) two-brane charge, expected in M-theory, at the fixed point. We discuss this result in the light of the Matrix theory conjecture, taking into account possible N dependent rescalings of velocity and impact parameter. We also discuss scattering in the orbifolds, R 5 /Z2 and R 9 /Z2 where we find the predicted fractional charges. May
Physical Review Letters, 2017
We develop general counting formulae for primary fields in free four dimensional (4D) scalar conf... more We develop general counting formulae for primary fields in free four dimensional (4D) scalar conformal field theory (CFT). Using a duality map between primary operators in scalar field theory and multi-variable polynomial functions subject to differential constraints, we identify a sector of holomorphic primary fields corresponding to polynomial functions on a class of permutation orbifolds. These orbifolds have palindromic Hilbert series, which indicates they are Calabi-Yau. We construct the top-dimensional holomorphic form expected from the Calabi-Yau property. This sector includes and extends previous constructions of infinite families of primary fields. We sketch the generalization of these results to free 4D vector and matrix CFTs. I.
Journal of High Energy Physics, 2020
The correlators of free four dimensional conformal field theories (CFT4) have been shown to be gi... more The correlators of free four dimensional conformal field theories (CFT4) have been shown to be given by amplitudes in two-dimensional so(4, 2) equivariant topological field theories (TFT2), by using a vertex operator formalism for the correlators. We show that this can be extended to perturbative interacting conformal field theories, using two representation theoretic constructions. A co-product deformation for the conformal algebra accommodates the equivariant construction of composite operators in the presence of nonadditive anomalous dimensions. Explicit expressions for the co-product deformation are given within a sector of N = 4 SYM and for the Wilson-Fischer fixed point near four dimensions. The extension of conformal equivariance beyond integer dimensions (relevant for the Wilson-Fischer fixed point) leads to the definition of an associative diagram algebra U , abstracted from Uso(d) in the limit of large integer d, which admits extension of Uso(d) representation theory to general real (or complex) d. The algebra is related, via oscillator realisations, to so(d) equivariant maps and Brauer category diagrams. Tensor representations are constructed where the diagram algebra acts on tensor products of a fundamental diagram representation. A similar diagrammatic algebra U ,2 , related to a general d extension for Uso(d, 2) is defined, and some of its lowest weight representations relevant to the Wilson-Fischer fixed point are described.
Journal of High Energy Physics, 2012
Eighth-BPS local operators in N = 4 SYM are dual to quantum states arising from the quantization ... more Eighth-BPS local operators in N = 4 SYM are dual to quantum states arising from the quantization of a moduli space of giant gravitons in AdS 5 × S 5. Earlier results on the quantization of this moduli space give a Hilbert space of multiple harmonic oscillators in 3 dimensions. We use these results, along with techniques from fuzzy geometry, to develop a map between quantum states and brane geometries. In particular there is a map between the oscillator states and points in a discretization of the base space in the toric fibration of the moduli space. We obtain a geometrical decomposition of the space of BPS states with labels consisting of U (3) representations along with U (N) Young diagrams and associated group theoretic multiplicities. Factorization properties in the counting of BPS states lead to predictions for BPS world-volume excitations of specific brane geometries. Some of our results suggest an intriguing complementarity between localisation in the moduli space of branes and localisation in space-time.
Nuclear Physics B, 1998
Scattering of zero branes off the fixed point in R 8 /Z 2 , as described by a super-quantum mecha... more Scattering of zero branes off the fixed point in R 8 /Z 2 , as described by a super-quantum mechanics with eight supercharges, displays some novel effects relevant to Matrix theory in non-compact backgrounds. The leading long distance behaviour of the moduli space metric receives no correction at one loop in Matrix theory, but does receive a correction at two loops. There are no contributions at higher loops. We explicitly calculate the two-loop term, finding a non-zero result. It has the right dependence on v and b for the scattering of zero branes off the (fractional) two-brane charge, expected in M-theory, at the fixed point. We discuss this result in the light of the Matrix theory conjecture, taking into account possible N dependent rescalings of velocity and impact parameter. We also discuss scattering in the orbifolds, R 5 /Z 2 and R 9 /Z 2 where we find the predicted fractional charges.
Journal of High Energy Physics
Permutation group algebras, and their generalizations called permutation centralizer algebras (PC... more Permutation group algebras, and their generalizations called permutation centralizer algebras (PCAs), play a central role as hidden symmetries in the combinatorics of large N gauge theories and matrix models with manifest continuous gauge symmetries. Polynomial functions invariant under the manifest symmetries are the observables of interest and have applications in AdS/CFT. We compute such correlators in the presence of matrix or tensor witnesses, which by definition, can include a matrix or tensor field appearing as a coupling in the action (i.e a spurion) or as a classical (un-integrated) field in the observables, appearing alongside quantum (integrated) fields. In both matrix and tensor cases we find that two-point correlators of general gauge-invariant observables can be written in terms of gauge invariant functions of the witness fields, with coefficients given by structure constants of the associated PCAs. Fourier transformation on the relevant PCAs, relates combinatorial bas...
Cornell University - arXiv, Apr 5, 2022
A number of finite algorithms for constructing representation theoretic data from group multiplic... more A number of finite algorithms for constructing representation theoretic data from group multiplications in a finite group G have recently been shown to be related to amplitudes for combinatoric topological strings (G-CTST) based on Dijkgraaf-Witten theory of flat G-bundles on surfaces. We extend this result to projective representations of G using twisted Dijkgraaf-Witten theory. New algorithms for characters are described, based on handle creation operators and minimal multiplicative generating subspaces for the centers of group algebras and twisted group algebras. Such minimal generating subspaces are of interest in connection with information theoretic aspects of the AdS/CFT correspondence. For the untwisted case, we describe the integrality properties of certain character sums and character power sums which follow from these constructive G-CTST algorithms. These integer sums appear as residues of singularities in G-CTST generating functions. S-duality of the combinatoric topological strings motivates the definition of an inverse handle creation operator in the centers of group algebras and twisted group algebras.
Permutations and associated algebras allow the construction of half and quarter BPS operators in ... more Permutations and associated algebras allow the construction of half and quarter BPS operators in maximally supersymmetric Yang Mills theory with U(N), SO(N) and Sp(N) gauge groups. The construction leads to bases for the operators, labelled by Young diagrams and associated group theory data, which have been shown to be orthogonal under the inner product defined by the free field two-point functions. In this paper, we study in detail the orientifold projection map between the Young diagram basis for U(N) theories and the Young diagram basis for SO(N) (and Sp(N)) half-BPS operators. We find a simple connection between this map and the plethystic refinement of the Littlewood Richardson coefficients which couple triples of Young diagrams where two of them are identical. This plethystic refinement is known to be computable using an algorithm based on domino tilings of Young diagrams. We discuss the domino combinatorics of the orientifold projection map in terms of giant graviton branes. ...
Journal of High Energy Physics, 2021
We give a construction of general holomorphic quarter BPS operators in mathcalN\mathcal{N}mathcalN N = 4 SY... more We give a construction of general holomorphic quarter BPS operators in mathcalN\mathcal{N}mathcalN N = 4 SYM at weak coupling with U(N) gauge group at finite N. The construction employs the Möbius inversion formula for set partitions, applied to multi-symmetric functions, alongside computations in the group algebras of symmetric groups. We present a computational algorithm which produces an orthogonal basis for the physical inner product on the space of holomorphic operators. The basis is labelled by a U(2) Young diagram, a U(N) Young diagram and an additional plethystic multiplicity label. We describe precision counting results of quarter BPS states which are expected to be reproducible from dual computations with giant gravitons in the bulk, including a symmetry relating sphere and AdS giants within the quarter BPS sector. In the case n ≤ N (n being the dimension of the composite operator) the construction is analytic, using multi-symmetric functions and U(2) Clebsch-Gordan coefficients. Co...
arXiv: High Energy Physics - Theory, 2017
Counting formulae for general primary fields in free four dimensional conformal field theories of... more Counting formulae for general primary fields in free four dimensional conformal field theories of scalars, vectors and matrices are derived. These are specialised to count primaries which obey extremality conditions defined in terms of the dimensions and left or right spins (i.e. in terms of relations between the charges under the Cartan subgroup of SO(4,2)SO(4,2)SO(4,2)). The construction of primary fields for scalar field theory is mapped to a problem of determining multi-variable polynomials subject to a system of symmetry and differential constraints. For the extremal primaries, we give a construction in terms of holomorphic polynomial functions on permutation orbifolds, which are shown to be Calabi-Yau spaces.
Journal of High Energy Physics, 2018
We define lowest weight polynomials (LWPs), motivated by so(d, 2) representation theory, as eleme... more We define lowest weight polynomials (LWPs), motivated by so(d, 2) representation theory, as elements of the polynomial ring over d × n variables obeying a system of first and second order partial differential equations. LWPs invariant under S n correspond to primary fields in free scalar field theory in d dimensions, constructed from n fields. The LWPs are in one-to-one correspondence with a quotient of the polynomial ring in d × (n − 1) variables by an ideal generated by n quadratic polynomials. The implications of this description for the counting and construction of primary fields are described: an interesting binomial identity underlies one of the construction algorithms. The product on the ring of LWPs can be described as a commutative star product. The quadratic algebra of lowest weight polynomials has a dual quadratic algebra which is non-commutative. We discuss the possible physical implications of this dual algebra.
Journal of High Energy Physics, 2019
Some beautiful identities involving hook contents of Young diagrams have been found in the field ... more Some beautiful identities involving hook contents of Young diagrams have been found in the field of quantum information processing, along with a combinatorial proof. We here give a representation theoretic proof of these identities and a number of generalizations. Our proof is based on trace identities for elements belonging to a class of permutation centralizer algebras. These algebras have been found to underlie the combinatorics of composite gauge invariant operators in quantum field theory, with applications in the AdS/CFT correspondence. Based on these algebras, we discuss some analogies between quantum information processing tasks and the combinatorics of composite quantum fields and argue that this can be fruitful interface between quantum information and quantum field theory, with implications for AdS/CFT.
Journal of High Energy Physics, 2017
Counting formulae for general primary fields in free four dimensional conformal field theories of... more Counting formulae for general primary fields in free four dimensional conformal field theories of scalars, vectors and matrices are derived. These are specialised to count primaries which obey extremality conditions defined in terms of the dimensions and left or right spins (i.e. in terms of relations between the charges under the Cartan subgroup of SO(4, 2)). The construction of primary fields for scalar field theory is mapped to a problem of determining multi-variable polynomials subject to a system of symmetry and differential constraints. For the extremal primaries, we give a construction in terms of holomorphic polynomial functions on permutation orbifolds, which are shown to be Calabi-Yau spaces.
Nuclear Physics B, 2016
In this paper we study the construction of holomorphic gauge invariant operators for general quiv... more In this paper we study the construction of holomorphic gauge invariant operators for general quiver gauge theories with flavour symmetries. Using a characterisation of the gauge invariants in terms of equivalence classes generated by permutation actions, along with representation theory results in symmetric groups and unitary groups, we give a diagonal basis for the 2-point functions of holomorphic and anti-holomorphic operators. This involves a generalisation of the previously constructed Quiver Restricted Schur operators to the flavoured case. The 3-point functions are derived and shown to be given in terms of networks of symmetric group branching coefficients. The networks are constructed through cutting and gluing operations on the quivers.
Journal of High Energy Physics, 2016
In a recent paper we showed that the correlators of free scalar field theory in four dimensions c... more In a recent paper we showed that the correlators of free scalar field theory in four dimensions can be constructed from a two dimensional topological field theory based on so(4, 2) equivariant maps (intertwiners). The free field result, along with results of Frenkel and Libine on equivariance properties of Feynman integrals, are developed further in this paper. We show that the coefficient of the log term in the 1-loop 4-point conformal integral is a projector in the tensor product of so(4, 2) representations. We also show that the 1-loop 4-point integral can be written as a sum of four terms, each associated with the quantum equation of motion for one of the four external legs. The quantum equation of motion is shown to be related to equivariant maps involving indecomposable representations of so(4, 2), a phenomenon which illuminates multiplet recombination. The harmonic expansion method for Feynman integrals is a powerful tool for arriving at these results. The generalization to other interactions and higher loops is discussed.
Journal of High Energy Physics, 2015
A systematic study of holomorphic gauge invariant operators in general N = 1 quiver gauge theorie... more A systematic study of holomorphic gauge invariant operators in general N = 1 quiver gauge theories, with unitary gauge groups and bifundamental matter fields, was recently presented in [1]. For large ranks a simple counting formula in terms of an infinite product was given. We extend this study to quiver gauge theories with fundamental matter fields, deriving an infinite product form for the refined counting in these cases. The infinite products are found to be obtained from substitutions in a simple building block expressed in terms of the weighted adjacency matrix of the quiver. In the case without fundamentals, it is a determinant which itself is found to have a counting interpretation in terms of words formed from partially commuting letters associated with simple closed loops in the quiver. This is a new relation between counting problems in gauge theory and the Cartier-Foata monoid. For finite ranks of the unitary gauge groups, the refined counting is given in terms of expressions involving Littlewood-Richardson coefficients.
Journal of High Energy Physics
The structure of half-BPS representations of psu(2, 2|4) leads to the definition of a super-polyn... more The structure of half-BPS representations of psu(2, 2|4) leads to the definition of a super-polynomial ring mathcalR\mathcal{R}mathcalR R (8|8) which admits a realisation of psu(2, 2|4) in terms of differential operators on the super-ring. The character of the half-BPS fundamental field representation encodes the resolution of the representation in terms of an exact sequence of modules of mathcalR\mathcal{R}mathcalR R (8|8). The half-BPS representation is realized by quotienting the super-ring by a quadratic ideal, equivalently by setting to zero certain quadratic polynomials in the generators of the super-ring. This description of the half-BPS fundamental field irreducible representation of psu(2, 2|4) in terms of a super-polynomial ring is an example of a more general construction of lowest-weight representations of Lie (super-) algebras using polynomial rings generated by a commuting subspace of the standard raising operators, corresponding to positive roots of the Lie (super-) algebra. We illustrat...
arXiv (Cornell University), Feb 18, 2022
Two and three-point functions of primary fields in four dimensional CFT have a simple space-time ... more Two and three-point functions of primary fields in four dimensional CFT have a simple space-time dependences factored out from the combinatoric structure which enumerates the fields and gives their couplings. This has led to the formulation of two dimensional topological field theories with SO(4, 2) equivariance which are conjectured to be equivalent to higher dimensional conformal field theories. We review this CFT4/TFT2 construction in the simplest possible setting of a free scalar field, which gives an algebraic construction of the correlators in terms of an infinite dimensional so(4, 2) equivariant algebra with finite dimensional subspaces at fixed scaling dimension. Crossing symmetry of the CFT4 is related to associativity of the algebra. This construction is then extended to describe perturbative CFT4, by making use of deformed co-products. Motivated by the Wilson-Fisher CFT we outline the construction of U(so(d,2)) equivariant TFT2 for non-integer d, in terms of diagram algebras and their representations.
[
Eighth-BPS local operators in N = 4 SYM are dual to quantum states arising from the quantization ... more Eighth-BPS local operators in N = 4 SYM are dual to quantum states arising from the quantization of a moduli space of giant gravitons in AdS5 × S5. Earlier results on the quantization of this moduli space give a Hilbert space of multiple harmonic oscillators in 3 dimensions. We use these results, along with techniques from fuzzy geometry, to develop a map between quantum states and brane geometries. In particular there is a map between the oscillator states and points in a discretization of the base space in the toric fibration of the moduli space. We obtain a geometrical decomposition of the space of BPS states with labels consisting of U(3) representations along with U(N) Young diagrams and associated group theoretic multiplicities. Factorization properties in the counting of BPS states lead to predictions for BPS world-volume excitations of specific brane geometries. Some of our results suggest an intriguing complementarity between localisation in the moduli space of branes and l...
Scattering of zero branes off the fixed point in R8 /Z2, as described by a super-quantum mechanic... more Scattering of zero branes off the fixed point in R8 /Z2, as described by a super-quantum mechanics with eight supercharges, displays some novel effects relevant to Matrix theory in non-compact backgrounds. The leading long distance behaviour of the moduli space metric receives no correction at one loop in Matrix theory, but does receive a correction at two loops. There are no contributions at higher loops. We explicitly calculate the two-loop term, finding a non-zero result. It has the right dependence on v and b for the scattering of zero branes off the (fractional) two-brane charge, expected in M-theory, at the fixed point. We discuss this result in the light of the Matrix theory conjecture, taking into account possible N dependent rescalings of velocity and impact parameter. We also discuss scattering in the orbifolds, R 5 /Z2 and R 9 /Z2 where we find the predicted fractional charges. May
Physical Review Letters, 2017
We develop general counting formulae for primary fields in free four dimensional (4D) scalar conf... more We develop general counting formulae for primary fields in free four dimensional (4D) scalar conformal field theory (CFT). Using a duality map between primary operators in scalar field theory and multi-variable polynomial functions subject to differential constraints, we identify a sector of holomorphic primary fields corresponding to polynomial functions on a class of permutation orbifolds. These orbifolds have palindromic Hilbert series, which indicates they are Calabi-Yau. We construct the top-dimensional holomorphic form expected from the Calabi-Yau property. This sector includes and extends previous constructions of infinite families of primary fields. We sketch the generalization of these results to free 4D vector and matrix CFTs. I.
Journal of High Energy Physics, 2020
The correlators of free four dimensional conformal field theories (CFT4) have been shown to be gi... more The correlators of free four dimensional conformal field theories (CFT4) have been shown to be given by amplitudes in two-dimensional so(4, 2) equivariant topological field theories (TFT2), by using a vertex operator formalism for the correlators. We show that this can be extended to perturbative interacting conformal field theories, using two representation theoretic constructions. A co-product deformation for the conformal algebra accommodates the equivariant construction of composite operators in the presence of nonadditive anomalous dimensions. Explicit expressions for the co-product deformation are given within a sector of N = 4 SYM and for the Wilson-Fischer fixed point near four dimensions. The extension of conformal equivariance beyond integer dimensions (relevant for the Wilson-Fischer fixed point) leads to the definition of an associative diagram algebra U , abstracted from Uso(d) in the limit of large integer d, which admits extension of Uso(d) representation theory to general real (or complex) d. The algebra is related, via oscillator realisations, to so(d) equivariant maps and Brauer category diagrams. Tensor representations are constructed where the diagram algebra acts on tensor products of a fundamental diagram representation. A similar diagrammatic algebra U ,2 , related to a general d extension for Uso(d, 2) is defined, and some of its lowest weight representations relevant to the Wilson-Fischer fixed point are described.
Journal of High Energy Physics, 2012
Eighth-BPS local operators in N = 4 SYM are dual to quantum states arising from the quantization ... more Eighth-BPS local operators in N = 4 SYM are dual to quantum states arising from the quantization of a moduli space of giant gravitons in AdS 5 × S 5. Earlier results on the quantization of this moduli space give a Hilbert space of multiple harmonic oscillators in 3 dimensions. We use these results, along with techniques from fuzzy geometry, to develop a map between quantum states and brane geometries. In particular there is a map between the oscillator states and points in a discretization of the base space in the toric fibration of the moduli space. We obtain a geometrical decomposition of the space of BPS states with labels consisting of U (3) representations along with U (N) Young diagrams and associated group theoretic multiplicities. Factorization properties in the counting of BPS states lead to predictions for BPS world-volume excitations of specific brane geometries. Some of our results suggest an intriguing complementarity between localisation in the moduli space of branes and localisation in space-time.
Nuclear Physics B, 1998
Scattering of zero branes off the fixed point in R 8 /Z 2 , as described by a super-quantum mecha... more Scattering of zero branes off the fixed point in R 8 /Z 2 , as described by a super-quantum mechanics with eight supercharges, displays some novel effects relevant to Matrix theory in non-compact backgrounds. The leading long distance behaviour of the moduli space metric receives no correction at one loop in Matrix theory, but does receive a correction at two loops. There are no contributions at higher loops. We explicitly calculate the two-loop term, finding a non-zero result. It has the right dependence on v and b for the scattering of zero branes off the (fractional) two-brane charge, expected in M-theory, at the fixed point. We discuss this result in the light of the Matrix theory conjecture, taking into account possible N dependent rescalings of velocity and impact parameter. We also discuss scattering in the orbifolds, R 5 /Z 2 and R 9 /Z 2 where we find the predicted fractional charges.
Journal of High Energy Physics
Permutation group algebras, and their generalizations called permutation centralizer algebras (PC... more Permutation group algebras, and their generalizations called permutation centralizer algebras (PCAs), play a central role as hidden symmetries in the combinatorics of large N gauge theories and matrix models with manifest continuous gauge symmetries. Polynomial functions invariant under the manifest symmetries are the observables of interest and have applications in AdS/CFT. We compute such correlators in the presence of matrix or tensor witnesses, which by definition, can include a matrix or tensor field appearing as a coupling in the action (i.e a spurion) or as a classical (un-integrated) field in the observables, appearing alongside quantum (integrated) fields. In both matrix and tensor cases we find that two-point correlators of general gauge-invariant observables can be written in terms of gauge invariant functions of the witness fields, with coefficients given by structure constants of the associated PCAs. Fourier transformation on the relevant PCAs, relates combinatorial bas...
Cornell University - arXiv, Apr 5, 2022
A number of finite algorithms for constructing representation theoretic data from group multiplic... more A number of finite algorithms for constructing representation theoretic data from group multiplications in a finite group G have recently been shown to be related to amplitudes for combinatoric topological strings (G-CTST) based on Dijkgraaf-Witten theory of flat G-bundles on surfaces. We extend this result to projective representations of G using twisted Dijkgraaf-Witten theory. New algorithms for characters are described, based on handle creation operators and minimal multiplicative generating subspaces for the centers of group algebras and twisted group algebras. Such minimal generating subspaces are of interest in connection with information theoretic aspects of the AdS/CFT correspondence. For the untwisted case, we describe the integrality properties of certain character sums and character power sums which follow from these constructive G-CTST algorithms. These integer sums appear as residues of singularities in G-CTST generating functions. S-duality of the combinatoric topological strings motivates the definition of an inverse handle creation operator in the centers of group algebras and twisted group algebras.
Permutations and associated algebras allow the construction of half and quarter BPS operators in ... more Permutations and associated algebras allow the construction of half and quarter BPS operators in maximally supersymmetric Yang Mills theory with U(N), SO(N) and Sp(N) gauge groups. The construction leads to bases for the operators, labelled by Young diagrams and associated group theory data, which have been shown to be orthogonal under the inner product defined by the free field two-point functions. In this paper, we study in detail the orientifold projection map between the Young diagram basis for U(N) theories and the Young diagram basis for SO(N) (and Sp(N)) half-BPS operators. We find a simple connection between this map and the plethystic refinement of the Littlewood Richardson coefficients which couple triples of Young diagrams where two of them are identical. This plethystic refinement is known to be computable using an algorithm based on domino tilings of Young diagrams. We discuss the domino combinatorics of the orientifold projection map in terms of giant graviton branes. ...
Journal of High Energy Physics, 2021
We give a construction of general holomorphic quarter BPS operators in mathcalN\mathcal{N}mathcalN N = 4 SY... more We give a construction of general holomorphic quarter BPS operators in mathcalN\mathcal{N}mathcalN N = 4 SYM at weak coupling with U(N) gauge group at finite N. The construction employs the Möbius inversion formula for set partitions, applied to multi-symmetric functions, alongside computations in the group algebras of symmetric groups. We present a computational algorithm which produces an orthogonal basis for the physical inner product on the space of holomorphic operators. The basis is labelled by a U(2) Young diagram, a U(N) Young diagram and an additional plethystic multiplicity label. We describe precision counting results of quarter BPS states which are expected to be reproducible from dual computations with giant gravitons in the bulk, including a symmetry relating sphere and AdS giants within the quarter BPS sector. In the case n ≤ N (n being the dimension of the composite operator) the construction is analytic, using multi-symmetric functions and U(2) Clebsch-Gordan coefficients. Co...
arXiv: High Energy Physics - Theory, 2017
Counting formulae for general primary fields in free four dimensional conformal field theories of... more Counting formulae for general primary fields in free four dimensional conformal field theories of scalars, vectors and matrices are derived. These are specialised to count primaries which obey extremality conditions defined in terms of the dimensions and left or right spins (i.e. in terms of relations between the charges under the Cartan subgroup of SO(4,2)SO(4,2)SO(4,2)). The construction of primary fields for scalar field theory is mapped to a problem of determining multi-variable polynomials subject to a system of symmetry and differential constraints. For the extremal primaries, we give a construction in terms of holomorphic polynomial functions on permutation orbifolds, which are shown to be Calabi-Yau spaces.
Journal of High Energy Physics, 2018
We define lowest weight polynomials (LWPs), motivated by so(d, 2) representation theory, as eleme... more We define lowest weight polynomials (LWPs), motivated by so(d, 2) representation theory, as elements of the polynomial ring over d × n variables obeying a system of first and second order partial differential equations. LWPs invariant under S n correspond to primary fields in free scalar field theory in d dimensions, constructed from n fields. The LWPs are in one-to-one correspondence with a quotient of the polynomial ring in d × (n − 1) variables by an ideal generated by n quadratic polynomials. The implications of this description for the counting and construction of primary fields are described: an interesting binomial identity underlies one of the construction algorithms. The product on the ring of LWPs can be described as a commutative star product. The quadratic algebra of lowest weight polynomials has a dual quadratic algebra which is non-commutative. We discuss the possible physical implications of this dual algebra.
Journal of High Energy Physics, 2019
Some beautiful identities involving hook contents of Young diagrams have been found in the field ... more Some beautiful identities involving hook contents of Young diagrams have been found in the field of quantum information processing, along with a combinatorial proof. We here give a representation theoretic proof of these identities and a number of generalizations. Our proof is based on trace identities for elements belonging to a class of permutation centralizer algebras. These algebras have been found to underlie the combinatorics of composite gauge invariant operators in quantum field theory, with applications in the AdS/CFT correspondence. Based on these algebras, we discuss some analogies between quantum information processing tasks and the combinatorics of composite quantum fields and argue that this can be fruitful interface between quantum information and quantum field theory, with implications for AdS/CFT.
Journal of High Energy Physics, 2017
Counting formulae for general primary fields in free four dimensional conformal field theories of... more Counting formulae for general primary fields in free four dimensional conformal field theories of scalars, vectors and matrices are derived. These are specialised to count primaries which obey extremality conditions defined in terms of the dimensions and left or right spins (i.e. in terms of relations between the charges under the Cartan subgroup of SO(4, 2)). The construction of primary fields for scalar field theory is mapped to a problem of determining multi-variable polynomials subject to a system of symmetry and differential constraints. For the extremal primaries, we give a construction in terms of holomorphic polynomial functions on permutation orbifolds, which are shown to be Calabi-Yau spaces.
Nuclear Physics B, 2016
In this paper we study the construction of holomorphic gauge invariant operators for general quiv... more In this paper we study the construction of holomorphic gauge invariant operators for general quiver gauge theories with flavour symmetries. Using a characterisation of the gauge invariants in terms of equivalence classes generated by permutation actions, along with representation theory results in symmetric groups and unitary groups, we give a diagonal basis for the 2-point functions of holomorphic and anti-holomorphic operators. This involves a generalisation of the previously constructed Quiver Restricted Schur operators to the flavoured case. The 3-point functions are derived and shown to be given in terms of networks of symmetric group branching coefficients. The networks are constructed through cutting and gluing operations on the quivers.
Journal of High Energy Physics, 2016
In a recent paper we showed that the correlators of free scalar field theory in four dimensions c... more In a recent paper we showed that the correlators of free scalar field theory in four dimensions can be constructed from a two dimensional topological field theory based on so(4, 2) equivariant maps (intertwiners). The free field result, along with results of Frenkel and Libine on equivariance properties of Feynman integrals, are developed further in this paper. We show that the coefficient of the log term in the 1-loop 4-point conformal integral is a projector in the tensor product of so(4, 2) representations. We also show that the 1-loop 4-point integral can be written as a sum of four terms, each associated with the quantum equation of motion for one of the four external legs. The quantum equation of motion is shown to be related to equivariant maps involving indecomposable representations of so(4, 2), a phenomenon which illuminates multiplet recombination. The harmonic expansion method for Feynman integrals is a powerful tool for arriving at these results. The generalization to other interactions and higher loops is discussed.
Journal of High Energy Physics, 2015
A systematic study of holomorphic gauge invariant operators in general N = 1 quiver gauge theorie... more A systematic study of holomorphic gauge invariant operators in general N = 1 quiver gauge theories, with unitary gauge groups and bifundamental matter fields, was recently presented in [1]. For large ranks a simple counting formula in terms of an infinite product was given. We extend this study to quiver gauge theories with fundamental matter fields, deriving an infinite product form for the refined counting in these cases. The infinite products are found to be obtained from substitutions in a simple building block expressed in terms of the weighted adjacency matrix of the quiver. In the case without fundamentals, it is a determinant which itself is found to have a counting interpretation in terms of words formed from partially commuting letters associated with simple closed loops in the quiver. This is a new relation between counting problems in gauge theory and the Cartier-Foata monoid. For finite ranks of the unitary gauge groups, the refined counting is given in terms of expressions involving Littlewood-Richardson coefficients.