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H Saleur - One of the best experts on this subject based on the ideXlab platform.
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a fusion for the periodic temperley lieb algebra and its Continuum Limit
Journal of High Energy Physics, 2018Co-Authors: H Saleur, Azat M Gainutdinov, Jesper Lykke JacobsenAbstract:The equivalent of fusion in boundary conformal field theory (CFT) can be realized quite simply in the context of lattice models by essentially glueing two open spin chains. This has led to many developments, in particular in the context of chiral logarithmic CFT. We consider in this paper a possible generalization of the idea to the case of bulk conformal field theory. This is of course considerably more difficult, since there is no obvious way of merging two closed spin chains into a big one. In an earlier paper, two of us had proposed a " topological " way of performing this operation in the case of models based on the affine Temperley-Lieb (ATL) algebra, by exploiting the associated braid group representation and skein relations. In the present work, we establish—using, in particular, Frobenius reciprocity—the resulting fusion rules for standard modules of ATL in the generic as well as partially degenerate cases. These fusion rules have a simple interpretation in the Continuum Limit, where they correspond to the glueing of the right moving component of one conformal field with the left moving component of the other.
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a fusion for the periodic temperley lieb algebra and its Continuum Limit
arXiv: High Energy Physics - Theory, 2017Co-Authors: H Saleur, Azat M Gainutdinov, Jesper Lykke JacobsenAbstract:The equivalent of fusion in boundary conformal field theory (CFT) can be realized quite simply in the context of lattice models by essentially glueing two open spin chains. This has led to many developments, in particular in the context of chiral logarithmic CFT. We consider in this paper a possible generalization of the idea to the case of bulk conformal field theory. This is of course considerably more difficult, since there is no obvious way of merging two closed spin chains into a big one. In an earlier paper, two of us had proposed a "topological" way of performing this operation in the case of models based on the affine Temperley-Lieb (ATL) algebra, by exploiting the associated braid group representation and skein relations. In the present work, we establish - using, in particular, Frobenius reciprocity - the resulting fusion rules for standard modules of ATL in the generic as well as partially degenerate cases. These fusion rules have a simple interpretation in the Continuum Limit. However, unlike in the chiral case this interpretation does not match the usual fusion in non-chiral CFTs. Rather, it corresponds to the glueing of the right moving component of one conformal field with the left moving component of the other.
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associative algebraic approach to logarithmic cft in the bulk the Continuum Limit of the mathfrak gl 1 1 periodic spin chain howe duality and the interchiral algebra
Communications in Mathematical Physics, 2016Co-Authors: Azat M Gainutdinov, N Read, H SaleurAbstract:We develop in this paper the principles of an associative algebraic approach to bulk logarithmic conformal field theories (LCFTs). We concentrate on the closed \({\mathfrak{gl}(1|1)}\) spin-chain and its Continuum Limit—the \({c=-2}\) symplectic fermions theory—and rely on two technical companion papers, Gainutdinov et al. (Nucl Phys B 871:245–288, 2013) and Gainutdinov et al. (Nucl Phys B 871:289–329, 2013). Our main result is that the algebra of local Hamiltonians, the Jones–Temperley–Lieb algebra JTL N , goes over in the Continuum Limit to a bigger algebra than \({\boldsymbol{\mathcal{V}}}\), the product of the left and right Virasoro algebras. This algebra, \({\mathcal{S}}\)—which we call interchiral, mixes the left and right moving sectors, and is generated, in the symplectic fermions case, by the additional field \({S(z,\bar{z})\equiv S_{\alpha\beta} \psi^\alpha(z)\bar{\psi}^\beta(\bar{z})}\), with a symmetric form \({S_{\alpha\beta}}\) and conformal weights (1,1). We discuss in detail how the space of states of the LCFT (technically, a Krein space) decomposes onto representations of this algebra, and how this decomposition is related with properties of the finite spin-chain. We show that there is a complete correspondence between algebraic properties of finite periodic spin chains and the Continuum Limit. An important technical aspect of our analysis involves the fundamental new observation that the action of JTL N in the \({\mathfrak{gl}(1|1)}\) spin chain is in fact isomorphic to an enveloping algebra of a certain Lie algebra, itself a non semi-simple version of \({\mathfrak{sp}_{N-2}}\). The semi-simple part of JTL N is represented by \({U \mathfrak{sp}_{N-2}}\), providing a beautiful example of a classical Howe duality, for which we have a non semi-simple version in the full JTL N image represented in the spin-chain. On the Continuum side, simple modules over \({\mathcal{S}}\) are identified with “fundamental” representations of \({\mathfrak{sp}_\infty}\).
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the periodic sl 2 1 alternating spin chain and its Continuum Limit as a bulk logarithmic conformal field theory at c 0
Journal of High Energy Physics, 2015Co-Authors: Azat M Gainutdinov, H Saleur, N Read, Romain VasseurAbstract:The periodic sl(2|1) alternating spin chain encodes (some of) the properties of hulls of percolation clusters, and is described in the Continuum Limit by a logarithmic conformal field theory (LCFT) at central charge c = 0. This theory corresponds to the strong coupling regime of a sigma model on the complex projective superspace CP 1|1 = U(2|1)/(U(1)×U(1|1)), and the spectrum of critical exponents can be obtained exactly. In this paper we push the analysis further, and determine the main representation theoretic (logarithmic) features of this Continuum Limit by extending to the periodic case the approach of [1] [N. Read and H. Saleur, Nucl. Phys. B 777 316 (2007)]. We first focus on determining the representation theory of the finite size spin chain with respect to the algebra of local energy densities provided by a representation of the affine Temperley-Lieb algebra at fugacity one. We then analyze how these algebraic properties carry over to the Continuum Limit to deduce the structure of the space of states as a representation over the product of left and right Virasoro algebras. Our main result is the full structure of the vacuum module of the theory, which exhibits Jordan cells of arbitrary rank for the Hamiltonian.
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the periodic sl 2 1 alternating spin chain and its Continuum Limit as a bulk logarithmic conformal field theory at c 0
arXiv: High Energy Physics - Theory, 2014Co-Authors: Azat M Gainutdinov, H Saleur, N Read, Romain VasseurAbstract:The periodic sl(2|1) alternating spin chain encodes (some of) the properties of hulls of percolation clusters, and is described in the Continuum Limit by a logarithmic conformal field theory (LCFT) at central charge c=0. This theory corresponds to the strong coupling regime of a sigma model on the complex projective superspace $\mathbb{CP}^{1|1} = \mathrm{U}(2|1) / (\mathrm{U}(1) \times \mathrm{U}(1|1))$, and the spectrum of critical exponents can be obtained exactly. In this paper we push the analysis further, and determine the main representation theoretic (logarithmic) features of this Continuum Limit by extending to the periodic case the approach of [N. Read and H. Saleur, Nucl. Phys. B 777 316 (2007)]. We first focus on determining the representation theory of the finite size spin chain with respect to the algebra of local energy densities provided by a representation of the affine Temperley-Lieb algebra at fugacity one. We then analyze how these algebraic properties carry over to the Continuum Limit to deduce the structure of the space of states as a representation over the product of left and right Virasoro algebras. Our main result is the full structure of the vacuum module of the theory, which exhibits Jordan cells of arbitrary rank for the Hamiltonian.
Azat M Gainutdinov - One of the best experts on this subject based on the ideXlab platform.
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a fusion for the periodic temperley lieb algebra and its Continuum Limit
Journal of High Energy Physics, 2018Co-Authors: H Saleur, Azat M Gainutdinov, Jesper Lykke JacobsenAbstract:The equivalent of fusion in boundary conformal field theory (CFT) can be realized quite simply in the context of lattice models by essentially glueing two open spin chains. This has led to many developments, in particular in the context of chiral logarithmic CFT. We consider in this paper a possible generalization of the idea to the case of bulk conformal field theory. This is of course considerably more difficult, since there is no obvious way of merging two closed spin chains into a big one. In an earlier paper, two of us had proposed a " topological " way of performing this operation in the case of models based on the affine Temperley-Lieb (ATL) algebra, by exploiting the associated braid group representation and skein relations. In the present work, we establish—using, in particular, Frobenius reciprocity—the resulting fusion rules for standard modules of ATL in the generic as well as partially degenerate cases. These fusion rules have a simple interpretation in the Continuum Limit, where they correspond to the glueing of the right moving component of one conformal field with the left moving component of the other.
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a fusion for the periodic temperley lieb algebra and its Continuum Limit
arXiv: High Energy Physics - Theory, 2017Co-Authors: H Saleur, Azat M Gainutdinov, Jesper Lykke JacobsenAbstract:The equivalent of fusion in boundary conformal field theory (CFT) can be realized quite simply in the context of lattice models by essentially glueing two open spin chains. This has led to many developments, in particular in the context of chiral logarithmic CFT. We consider in this paper a possible generalization of the idea to the case of bulk conformal field theory. This is of course considerably more difficult, since there is no obvious way of merging two closed spin chains into a big one. In an earlier paper, two of us had proposed a "topological" way of performing this operation in the case of models based on the affine Temperley-Lieb (ATL) algebra, by exploiting the associated braid group representation and skein relations. In the present work, we establish - using, in particular, Frobenius reciprocity - the resulting fusion rules for standard modules of ATL in the generic as well as partially degenerate cases. These fusion rules have a simple interpretation in the Continuum Limit. However, unlike in the chiral case this interpretation does not match the usual fusion in non-chiral CFTs. Rather, it corresponds to the glueing of the right moving component of one conformal field with the left moving component of the other.
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associative algebraic approach to logarithmic cft in the bulk the Continuum Limit of the mathfrak gl 1 1 periodic spin chain howe duality and the interchiral algebra
Communications in Mathematical Physics, 2016Co-Authors: Azat M Gainutdinov, N Read, H SaleurAbstract:We develop in this paper the principles of an associative algebraic approach to bulk logarithmic conformal field theories (LCFTs). We concentrate on the closed \({\mathfrak{gl}(1|1)}\) spin-chain and its Continuum Limit—the \({c=-2}\) symplectic fermions theory—and rely on two technical companion papers, Gainutdinov et al. (Nucl Phys B 871:245–288, 2013) and Gainutdinov et al. (Nucl Phys B 871:289–329, 2013). Our main result is that the algebra of local Hamiltonians, the Jones–Temperley–Lieb algebra JTL N , goes over in the Continuum Limit to a bigger algebra than \({\boldsymbol{\mathcal{V}}}\), the product of the left and right Virasoro algebras. This algebra, \({\mathcal{S}}\)—which we call interchiral, mixes the left and right moving sectors, and is generated, in the symplectic fermions case, by the additional field \({S(z,\bar{z})\equiv S_{\alpha\beta} \psi^\alpha(z)\bar{\psi}^\beta(\bar{z})}\), with a symmetric form \({S_{\alpha\beta}}\) and conformal weights (1,1). We discuss in detail how the space of states of the LCFT (technically, a Krein space) decomposes onto representations of this algebra, and how this decomposition is related with properties of the finite spin-chain. We show that there is a complete correspondence between algebraic properties of finite periodic spin chains and the Continuum Limit. An important technical aspect of our analysis involves the fundamental new observation that the action of JTL N in the \({\mathfrak{gl}(1|1)}\) spin chain is in fact isomorphic to an enveloping algebra of a certain Lie algebra, itself a non semi-simple version of \({\mathfrak{sp}_{N-2}}\). The semi-simple part of JTL N is represented by \({U \mathfrak{sp}_{N-2}}\), providing a beautiful example of a classical Howe duality, for which we have a non semi-simple version in the full JTL N image represented in the spin-chain. On the Continuum side, simple modules over \({\mathcal{S}}\) are identified with “fundamental” representations of \({\mathfrak{sp}_\infty}\).
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the periodic sl 2 1 alternating spin chain and its Continuum Limit as a bulk logarithmic conformal field theory at c 0
Journal of High Energy Physics, 2015Co-Authors: Azat M Gainutdinov, H Saleur, N Read, Romain VasseurAbstract:The periodic sl(2|1) alternating spin chain encodes (some of) the properties of hulls of percolation clusters, and is described in the Continuum Limit by a logarithmic conformal field theory (LCFT) at central charge c = 0. This theory corresponds to the strong coupling regime of a sigma model on the complex projective superspace CP 1|1 = U(2|1)/(U(1)×U(1|1)), and the spectrum of critical exponents can be obtained exactly. In this paper we push the analysis further, and determine the main representation theoretic (logarithmic) features of this Continuum Limit by extending to the periodic case the approach of [1] [N. Read and H. Saleur, Nucl. Phys. B 777 316 (2007)]. We first focus on determining the representation theory of the finite size spin chain with respect to the algebra of local energy densities provided by a representation of the affine Temperley-Lieb algebra at fugacity one. We then analyze how these algebraic properties carry over to the Continuum Limit to deduce the structure of the space of states as a representation over the product of left and right Virasoro algebras. Our main result is the full structure of the vacuum module of the theory, which exhibits Jordan cells of arbitrary rank for the Hamiltonian.
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the periodic sl 2 1 alternating spin chain and its Continuum Limit as a bulk logarithmic conformal field theory at c 0
arXiv: High Energy Physics - Theory, 2014Co-Authors: Azat M Gainutdinov, H Saleur, N Read, Romain VasseurAbstract:The periodic sl(2|1) alternating spin chain encodes (some of) the properties of hulls of percolation clusters, and is described in the Continuum Limit by a logarithmic conformal field theory (LCFT) at central charge c=0. This theory corresponds to the strong coupling regime of a sigma model on the complex projective superspace $\mathbb{CP}^{1|1} = \mathrm{U}(2|1) / (\mathrm{U}(1) \times \mathrm{U}(1|1))$, and the spectrum of critical exponents can be obtained exactly. In this paper we push the analysis further, and determine the main representation theoretic (logarithmic) features of this Continuum Limit by extending to the periodic case the approach of [N. Read and H. Saleur, Nucl. Phys. B 777 316 (2007)]. We first focus on determining the representation theory of the finite size spin chain with respect to the algebra of local energy densities provided by a representation of the affine Temperley-Lieb algebra at fugacity one. We then analyze how these algebraic properties carry over to the Continuum Limit to deduce the structure of the space of states as a representation over the product of left and right Virasoro algebras. Our main result is the full structure of the vacuum module of the theory, which exhibits Jordan cells of arbitrary rank for the Hamiltonian.
Romain Vasseur - One of the best experts on this subject based on the ideXlab platform.
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the periodic sl 2 1 alternating spin chain and its Continuum Limit as a bulk logarithmic conformal field theory at c 0
Journal of High Energy Physics, 2015Co-Authors: Azat M Gainutdinov, H Saleur, N Read, Romain VasseurAbstract:The periodic sl(2|1) alternating spin chain encodes (some of) the properties of hulls of percolation clusters, and is described in the Continuum Limit by a logarithmic conformal field theory (LCFT) at central charge c = 0. This theory corresponds to the strong coupling regime of a sigma model on the complex projective superspace CP 1|1 = U(2|1)/(U(1)×U(1|1)), and the spectrum of critical exponents can be obtained exactly. In this paper we push the analysis further, and determine the main representation theoretic (logarithmic) features of this Continuum Limit by extending to the periodic case the approach of [1] [N. Read and H. Saleur, Nucl. Phys. B 777 316 (2007)]. We first focus on determining the representation theory of the finite size spin chain with respect to the algebra of local energy densities provided by a representation of the affine Temperley-Lieb algebra at fugacity one. We then analyze how these algebraic properties carry over to the Continuum Limit to deduce the structure of the space of states as a representation over the product of left and right Virasoro algebras. Our main result is the full structure of the vacuum module of the theory, which exhibits Jordan cells of arbitrary rank for the Hamiltonian.
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the periodic sl 2 1 alternating spin chain and its Continuum Limit as a bulk logarithmic conformal field theory at c 0
arXiv: High Energy Physics - Theory, 2014Co-Authors: Azat M Gainutdinov, H Saleur, N Read, Romain VasseurAbstract:The periodic sl(2|1) alternating spin chain encodes (some of) the properties of hulls of percolation clusters, and is described in the Continuum Limit by a logarithmic conformal field theory (LCFT) at central charge c=0. This theory corresponds to the strong coupling regime of a sigma model on the complex projective superspace $\mathbb{CP}^{1|1} = \mathrm{U}(2|1) / (\mathrm{U}(1) \times \mathrm{U}(1|1))$, and the spectrum of critical exponents can be obtained exactly. In this paper we push the analysis further, and determine the main representation theoretic (logarithmic) features of this Continuum Limit by extending to the periodic case the approach of [N. Read and H. Saleur, Nucl. Phys. B 777 316 (2007)]. We first focus on determining the representation theory of the finite size spin chain with respect to the algebra of local energy densities provided by a representation of the affine Temperley-Lieb algebra at fugacity one. We then analyze how these algebraic properties carry over to the Continuum Limit to deduce the structure of the space of states as a representation over the product of left and right Virasoro algebras. Our main result is the full structure of the vacuum module of the theory, which exhibits Jordan cells of arbitrary rank for the Hamiltonian.
N Read - One of the best experts on this subject based on the ideXlab platform.
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associative algebraic approach to logarithmic cft in the bulk the Continuum Limit of the mathfrak gl 1 1 periodic spin chain howe duality and the interchiral algebra
Communications in Mathematical Physics, 2016Co-Authors: Azat M Gainutdinov, N Read, H SaleurAbstract:We develop in this paper the principles of an associative algebraic approach to bulk logarithmic conformal field theories (LCFTs). We concentrate on the closed \({\mathfrak{gl}(1|1)}\) spin-chain and its Continuum Limit—the \({c=-2}\) symplectic fermions theory—and rely on two technical companion papers, Gainutdinov et al. (Nucl Phys B 871:245–288, 2013) and Gainutdinov et al. (Nucl Phys B 871:289–329, 2013). Our main result is that the algebra of local Hamiltonians, the Jones–Temperley–Lieb algebra JTL N , goes over in the Continuum Limit to a bigger algebra than \({\boldsymbol{\mathcal{V}}}\), the product of the left and right Virasoro algebras. This algebra, \({\mathcal{S}}\)—which we call interchiral, mixes the left and right moving sectors, and is generated, in the symplectic fermions case, by the additional field \({S(z,\bar{z})\equiv S_{\alpha\beta} \psi^\alpha(z)\bar{\psi}^\beta(\bar{z})}\), with a symmetric form \({S_{\alpha\beta}}\) and conformal weights (1,1). We discuss in detail how the space of states of the LCFT (technically, a Krein space) decomposes onto representations of this algebra, and how this decomposition is related with properties of the finite spin-chain. We show that there is a complete correspondence between algebraic properties of finite periodic spin chains and the Continuum Limit. An important technical aspect of our analysis involves the fundamental new observation that the action of JTL N in the \({\mathfrak{gl}(1|1)}\) spin chain is in fact isomorphic to an enveloping algebra of a certain Lie algebra, itself a non semi-simple version of \({\mathfrak{sp}_{N-2}}\). The semi-simple part of JTL N is represented by \({U \mathfrak{sp}_{N-2}}\), providing a beautiful example of a classical Howe duality, for which we have a non semi-simple version in the full JTL N image represented in the spin-chain. On the Continuum side, simple modules over \({\mathcal{S}}\) are identified with “fundamental” representations of \({\mathfrak{sp}_\infty}\).
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the periodic sl 2 1 alternating spin chain and its Continuum Limit as a bulk logarithmic conformal field theory at c 0
Journal of High Energy Physics, 2015Co-Authors: Azat M Gainutdinov, H Saleur, N Read, Romain VasseurAbstract:The periodic sl(2|1) alternating spin chain encodes (some of) the properties of hulls of percolation clusters, and is described in the Continuum Limit by a logarithmic conformal field theory (LCFT) at central charge c = 0. This theory corresponds to the strong coupling regime of a sigma model on the complex projective superspace CP 1|1 = U(2|1)/(U(1)×U(1|1)), and the spectrum of critical exponents can be obtained exactly. In this paper we push the analysis further, and determine the main representation theoretic (logarithmic) features of this Continuum Limit by extending to the periodic case the approach of [1] [N. Read and H. Saleur, Nucl. Phys. B 777 316 (2007)]. We first focus on determining the representation theory of the finite size spin chain with respect to the algebra of local energy densities provided by a representation of the affine Temperley-Lieb algebra at fugacity one. We then analyze how these algebraic properties carry over to the Continuum Limit to deduce the structure of the space of states as a representation over the product of left and right Virasoro algebras. Our main result is the full structure of the vacuum module of the theory, which exhibits Jordan cells of arbitrary rank for the Hamiltonian.
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the periodic sl 2 1 alternating spin chain and its Continuum Limit as a bulk logarithmic conformal field theory at c 0
arXiv: High Energy Physics - Theory, 2014Co-Authors: Azat M Gainutdinov, H Saleur, N Read, Romain VasseurAbstract:The periodic sl(2|1) alternating spin chain encodes (some of) the properties of hulls of percolation clusters, and is described in the Continuum Limit by a logarithmic conformal field theory (LCFT) at central charge c=0. This theory corresponds to the strong coupling regime of a sigma model on the complex projective superspace $\mathbb{CP}^{1|1} = \mathrm{U}(2|1) / (\mathrm{U}(1) \times \mathrm{U}(1|1))$, and the spectrum of critical exponents can be obtained exactly. In this paper we push the analysis further, and determine the main representation theoretic (logarithmic) features of this Continuum Limit by extending to the periodic case the approach of [N. Read and H. Saleur, Nucl. Phys. B 777 316 (2007)]. We first focus on determining the representation theory of the finite size spin chain with respect to the algebra of local energy densities provided by a representation of the affine Temperley-Lieb algebra at fugacity one. We then analyze how these algebraic properties carry over to the Continuum Limit to deduce the structure of the space of states as a representation over the product of left and right Virasoro algebras. Our main result is the full structure of the vacuum module of the theory, which exhibits Jordan cells of arbitrary rank for the Hamiltonian.
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Continuum Limit and symmetries of the periodic gl 1 1 spin chain
Nuclear Physics, 2013Co-Authors: Azat M Gainutdinov, N Read, H SaleurAbstract:Abstract This paper is the first in a series devoted to the study of logarithmic conformal field theories (LCFT) in the bulk. Building on earlier work in the boundary case, our general strategy consists in analyzing the algebraic properties of lattice regularizations (quantum spin chains) of these theories. In the boundary case, a crucial step was the identification of the space of states as a bimodule over the Temperley–Lieb (TL) algebra and the quantum group U q s l ( 2 ) . The extension of this analysis in the bulk case involves considerable difficulties, since the U q s l ( 2 ) symmetry is partly lost, while the TL algebra is replaced by a much richer version (the Jones–Temperley–Lieb — JTL — algebra). Even the simplest case of the g l ( 1 | 1 ) spin chain — corresponding to the c = − 2 symplectic fermions theory in the Continuum Limit — presents very rich aspects, which we will discuss in several papers. In this first work, we focus on the symmetries of the spin chain, that is, the centralizer of the JTL algebra in the alternating tensor product of the g l ( 1 | 1 ) fundamental representation and its dual. We prove that this centralizer is only a subalgebra of U q s l ( 2 ) at q = i that we dub U q odd s l ( 2 ) . We then begin the analysis of the Continuum Limit of the JTL algebra: using general arguments about the regularization of the stress–energy tensor, we identify families of JTL elements going over to the Virasoro generators L n , L ¯ n in the Continuum Limit. We then discuss the s l ( 2 ) symmetry of the (Continuum Limit) symplectic fermions theory from the lattice and JTL point of view. The analysis of the spin chain as a bimodule over U q odd s l ( 2 ) and JTL N is discussed in the second paper of this series.
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associative algebraic approach to logarithmic cft in the bulk the Continuum Limit of the gl 1 1 periodic spin chain howe duality and the interchiral algebra
arXiv: High Energy Physics - Theory, 2012Co-Authors: Azat M Gainutdinov, N Read, H SaleurAbstract:We develop in this paper the principles of an associative algebraic approach to bulk logarithmic conformal field theories (LCFTs). We concentrate on the closed $gl(1|1)$ spin-chain and its Continuum Limit - the $c=-2$ symplectic fermions theory - and rely on two technical companion papers, "Continuum Limit and symmetries of the periodic gl(1|1) spin chain" [Nucl. Phys. B 871 (2013) 245-288] and "Bimodule structure in the periodic gl(1|1) spin chain" [Nucl. Phys. B 871 (2013) 289-329]. Our main result is that the algebra of local Hamiltonians, the Jones-Temperley-Lieb algebra JTL_N, goes over in the Continuum Limit to a bigger algebra than the product of the left and right Virasoro algebras. This algebra, S - which we call interchiral, mixes the left and right moving sectors, and is generated, in the symplectic fermions case, by the additional field $S(z,\bar{z})=S_{ab}\psi^a(z)\bar{\psi}^b(\bar{z})$, with a symmetric form $S_{ab}$ and conformal weights (1,1). We discuss in details how the Hilbert space of the LCFT decomposes onto representations of this algebra, and how this decomposition is related with properties of the finite spin-chain. We show that there is a complete correspondence between algebraic properties of finite periodic spin chains and the Continuum Limit. An important technical aspect of our analysis involves the fundamental new observation that the action of JTL_N in the $gl(1|1)$ spin chain is in fact isomorphic to an enveloping algebra of a certain Lie algebra, itself a non semi-simple version of $sp(N-2)$. The semi-simple part of JTL_N is represented by $Usp(N-2)$, providing a beautiful example of a classical Howe duality, for which we have a non semi-simple version in the full JTL image represented in the spin-chain. On the Continuum side, simple modules over the interchiral algebra S are identified with "fundamental" representations of $sp(\infty)$.
P A Boyle - One of the best experts on this subject based on the ideXlab platform.
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the decay constants f d and f _ d_s in the Continuum Limit of n f 2 1 domain wall lattice qcd
Journal of High Energy Physics, 2017Co-Authors: P A Boyle, L Del Debbio, Andreas Juttner, Ava Khamseh, Francesco Sanfilippo, Justus Tobias TsangAbstract:We present results for the decay constants of the D and D s mesons computed in lattice QCD with N f = 2 + 1 dynamical flavours. The simulations are based on RBC/UKQCD’s domain wall ensembles with both physical and unphysical light-quark masses and lattice spacings in the range 0.11-0.07 fm. We employ the domain wall discretisation for all valence quarks. The results in the Continuum Limit are f D = 208.7(2.8)stat( − 1.8 + 2.1 )sysMeV and $$ {f}_{D_s}=246.4{(1.3)}_{\mathrm{stat}}{\left({}_{-1.9}^{+1.3}\right)}_{\mathrm{sys}}\mathrm{M}\mathrm{e}\mathrm{V} $$ and $$ {f}_{D_s}={f}_D=1.667{(77)}_{\mathrm{stat}}{\left({}_{-43}^{+57}\right)}_{\mathrm{sys}} $$ . Using these results in a Standard Model analysis we compute the predictions |V cd | = 0.2185(50)exp( − 37 + 35 )lat and |V cs | = 1.011(16)exp( − 9 + 4 )lat for the CKM matrix elements.
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Continuum Limit of b_k from 2 1 flavor domain wall qcd
Physical Review D, 2011Co-Authors: Yasumichi Aoki, Rudy Arthur, T Blum, P A Boyle, D Brommel, N H Christ, C Dawson, Taku Izubuchi, Chulwoo Jung, C KellyAbstract:We determine the neutral kaon mixing matrix element BK in the Continuum Limit with 2+1 flavors of domain wall fermions, using the Iwasaki gauge action at two different lattice spacings. These lattice fermions have near exact chiral symmetry and therefore avoid artificial lattice operator mixing. We introduce a significant improvement to the conventional nonperturbative renormalization (NPR) method in which the bare matrix elements are renormalized nonperturbatively in the regularization invariant momentum scheme (RI-MOM) and are then converted into the MS? scheme using Continuum perturbation theory. In addition to RI-MOM, we introduce and implement four nonexceptional intermediate momentum schemes that suppress infrared nonperturbative uncertainties in the renormalization procedure. We compute the conversion factors relating the matrix elements in this family of regularization invariant symmetric momentum schemes (RI-SMOM) and MS? at one-loop order. Comparison of the results obtained using these different intermediate schemes allows for a more reliable estimate of the unknown higher-order contributions and hence for a correspondingly more robust estimate of the systematic error. We also apply a recently proposed approach in which twisted boundary conditions are used to control the Symanzik expansion for off-shell vertex functions leading to a better control of the renormalization in the Continuum Limit. We control chiral extrapolation errors by considering both the next-to-leading order SU(2) chiral effective theory, and an analytic mass expansion. We obtain BKMS? (3??GeV)=0.529(5)stat(15)?(2)FV(11)NPR. This corresponds to B?KRGI? =0.749(7)stat(21)?(3)FV(15)NPR. Adding all sources of error in quadrature, we obtain B?KRGI? =0.749(27)combined, with an overall combined error of 3.6%.
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Continuum Limit physics from 2 1 flavor domain wall qcd
Physical Review D, 2011Co-Authors: Yasumichi Aoki, Rudy Arthur, T Blum, P A Boyle, D Brommel, N H Christ, C Dawson, Taku Izubuchi, J M Flynn, Xiaoyong JinAbstract:We present physical results obtained from simulations using 2+1 flavors of domain wall quarks and the Iwasaki gauge action at two values of the lattice spacing a, (a −1 = 1.73 (3) GeV and a −1 = 2.28 (3) GeV). On the coarser lattice, with 24 3 × 64× 16 points (where the 16 corresponds to Ls, the extent of the 5 th dimension inherent in the domain wall fermion (DWF) formulation