Latest articles of Advances in Theoretical and Mathematical Physics
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https://content.intlpress.com/https://content.intlpress.com/Latest articles of Advances in Theoretical and Mathematical Physics
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enSat, 02 Dec 2023 23:27:02 +0200<![CDATA[A note on the canonical formalism for gravity]]>
https://content.intlpress.com/journal/ATMP/article/11669
https://content.intlpress.com/journal/ATMP/article/11669We describe a simple gauge-fixing that leads to a construction of a quantum Hilbert space for quantum gravity in an asymptotically Anti de Sitter spacetime, valid to all orders of perturbation theory. The construction is motivated by a relationship of the phase space of gravity in asymptotically Anti de Sitter spacetime to a cotangent bundle. We describe what is known about this relationship and some extensions that might plausibly be true. A key fact is that, under certain conditions, the Einstein Hamiltonian constraint equation can be viewed as a way to gauge fix the group of conformal rescalings of the metric of a Cauchy hypersurface. An analog of the procedure that we follow for Anti de Sitter gravity leads to standard results for a Klein–Gordon particle.
PDFXML]]>We describe a simple gauge-fixing that leads to a construction of a quantum Hilbert space for quantum gravity in an asymptotically Anti de Sitter spacetime, valid to all orders of perturbation theory. The construction is motivated by a relationship of the phase space of gravity in asymptotically Anti de Sitter spacetime to a cotangent bundle. We describe what is known about this relationship and some extensions that might plausibly be true. A key fact is that, under certain conditions, the Einstein Hamiltonian constraint equation can be viewed as a way to gauge fix the group of conformal rescalings of the metric of a Cauchy hypersurface. An analog of the procedure that we follow for Anti de Sitter gravity leads to standard results for a Klein–Gordon particle.
PDFXML]]>Edward WittenThu, 13 Jul 2023 00:00:00 +0300<![CDATA[$T \bar{T}$ deformations in general dimensions]]>
https://content.intlpress.com/journal/ATMP/article/11670
https://content.intlpress.com/journal/ATMP/article/11670It has recently been proposed that Zamoldchikov’s $T \bar{T}$ deformation of two-dimensional CFTs describes the holographic theory dual to $\mathrm{AdS}_3$ at finite radius. In this note we use the Gauss–Codazzi form of the Einstein equations to derive a relationship in general dimensions between the trace of the quasi-local stress tensor and a specific quadratic combination of this stress tensor, on constant radius slices of $\mathrm{AdS}$. We use this relation to propose a generalization of Zamoldchikov’s $T \bar{T}$ deformation to conformal field theories in general dimensions. This operator is quadratic in the stress tensor and retains many but not all of the features of $T \bar{T}$. To describe gravity with gauge or scalar fields, the deforming operator needs to be modified to include appropriate terms involving the corresponding $\mathrm{R}$ currents and scalar operators and we can again use the Gauss–Codazzi form of the Einstein equations to deduce the forms of the deforming operators. We conclude by discussing the relation of the quadratic stress tensor deformation to the stress energy tensor trace constraint in holographic theories dual to vacuum Einstein gravity.
PDFXML]]>It has recently been proposed that Zamoldchikov’s $T \bar{T}$ deformation of two-dimensional CFTs describes the holographic theory dual to $\mathrm{AdS}_3$ at finite radius. In this note we use the Gauss–Codazzi form of the Einstein equations to derive a relationship in general dimensions between the trace of the quasi-local stress tensor and a specific quadratic combination of this stress tensor, on constant radius slices of $\mathrm{AdS}$. We use this relation to propose a generalization of Zamoldchikov’s $T \bar{T}$ deformation to conformal field theories in general dimensions. This operator is quadratic in the stress tensor and retains many but not all of the features of $T \bar{T}$. To describe gravity with gauge or scalar fields, the deforming operator needs to be modified to include appropriate terms involving the corresponding $\mathrm{R}$ currents and scalar operators and we can again use the Gauss–Codazzi form of the Einstein equations to deduce the forms of the deforming operators. We conclude by discussing the relation of the quadratic stress tensor deformation to the stress energy tensor trace constraint in holographic theories dual to vacuum Einstein gravity.
PDFXML]]>Marika TaylorThu, 13 Jul 2023 00:00:00 +0300<![CDATA[Crossing symmetry in matter Chern–Simons theories at finite $N$ and $k$]]>
https://content.intlpress.com/journal/ATMP/article/11671
https://content.intlpress.com/journal/ATMP/article/11671We present a conjecture for the crossing symmetry rules for Chern–Simons gauge theories interacting with massive matter in $2 + 1$ dimensions. Our crossing rules are given in terms of the expectation values of particular tangles of Wilson lines, and reduce to the standard rules at large Chern–Simons level. We present completely explicit results for the special case of two fundamental and two antifundamental insertions in $SU(N)_k$ and $U(N)_k$ theories. These formulae are consistent with the conjectured level-rank, Bose–Fermi duality between these theories and take the form of a $q = e^{\frac{2 \pi i}{\kappa}}$ deformation of their large $k$ counterparts. In the ’t Hooft large $N$ limit our results reduce to standard rules with one twist: the Smatrix in the singlet channel is reduced by the factor $\frac{\operatorname{sin} \pi \lambda}{\pi \lambda}$ (where $\lambda$ is the ’t Hooft coupling), explaining ‘anomalous’ crossing properties observed in earlier direct large $N$ computations.
PDFXML]]>We present a conjecture for the crossing symmetry rules for Chern–Simons gauge theories interacting with massive matter in $2 + 1$ dimensions. Our crossing rules are given in terms of the expectation values of particular tangles of Wilson lines, and reduce to the standard rules at large Chern–Simons level. We present completely explicit results for the special case of two fundamental and two antifundamental insertions in $SU(N)_k$ and $U(N)_k$ theories. These formulae are consistent with the conjectured level-rank, Bose–Fermi duality between these theories and take the form of a $q = e^{\frac{2 \pi i}{\kappa}}$ deformation of their large $k$ counterparts. In the ’t Hooft large $N$ limit our results reduce to standard rules with one twist: the Smatrix in the singlet channel is reduced by the factor $\frac{\operatorname{sin} \pi \lambda}{\pi \lambda}$ (where $\lambda$ is the ’t Hooft coupling), explaining ‘anomalous’ crossing properties observed in earlier direct large $N$ computations.
PDFXML]]>Umang Mehta,Shiraz Minwalla,Chintan Patel,Shiroman Prakash,Kartik SharmaThu, 13 Jul 2023 00:00:00 +0300<![CDATA[Fractional quantum Hall effect and $M$-theory]]>
https://content.intlpress.com/journal/ATMP/article/11672
https://content.intlpress.com/journal/ATMP/article/11672We propose a unifying model for FQHE which on the one hand connects it to recent developments in string theory and on the other hand leads to new predictions for the principal series of experimentally observed FQH systems with filling fraction $\nu=\frac{n}{2n \pm 1}$ as well as those with $\nu=\frac{m}{m+2}$. Our model relates these series to minimal unitary models of the Virasoro and super-Virasoro algebra and is based on $SL(2,\mathbf{C})$ Chern–Simons theory in Euclidean space or $SL(2,\mathbf{R}) \times SL(2, \mathbf{R})$ Chern–Simons theory in Minkowski space. This theory, which has also been proposed as a soluble model for $2+1$ dimensional quantum gravity, and its $\mathrm{N}=1$ supersymmetric cousin, provide effective descriptions of FQHE. The principal series corresponds to quantized levels for the two $SL(2,\mathbf{R})$’s such that the diagonal $SL(2,\mathbf{R})$ has level $1$. The model predicts, contrary to standard lore, that for principal series of FQH systems the quasiholes possess non-abelian statistics. For the multi-layer case we propose that complex ADE Chern–Simons theories provide effective descriptions, where the rank of the ADE is mapped to the number of layers. Six dimensional $(2,0)$ ADE theories on the Riemann surface $\Sigma$ provides a realization of FQH systems in $\mathrm{M}$-theory. Moreover we propose that the $\mathrm{q}$-deformed version of Chern–Simons theories are related to the anisotropic limit of FQH systems which splits the zeroes of the Laughlin wave function. Extensions of the model to $3+1$ dimensions, which realize topological insulators with non-abelian topologically twisted Yang–Mills theory is pointed out.
PDFXML]]>We propose a unifying model for FQHE which on the one hand connects it to recent developments in string theory and on the other hand leads to new predictions for the principal series of experimentally observed FQH systems with filling fraction $\nu=\frac{n}{2n \pm 1}$ as well as those with $\nu=\frac{m}{m+2}$. Our model relates these series to minimal unitary models of the Virasoro and super-Virasoro algebra and is based on $SL(2,\mathbf{C})$ Chern–Simons theory in Euclidean space or $SL(2,\mathbf{R}) \times SL(2, \mathbf{R})$ Chern–Simons theory in Minkowski space. This theory, which has also been proposed as a soluble model for $2+1$ dimensional quantum gravity, and its $\mathrm{N}=1$ supersymmetric cousin, provide effective descriptions of FQHE. The principal series corresponds to quantized levels for the two $SL(2,\mathbf{R})$’s such that the diagonal $SL(2,\mathbf{R})$ has level $1$. The model predicts, contrary to standard lore, that for principal series of FQH systems the quasiholes possess non-abelian statistics. For the multi-layer case we propose that complex ADE Chern–Simons theories provide effective descriptions, where the rank of the ADE is mapped to the number of layers. Six dimensional $(2,0)$ ADE theories on the Riemann surface $\Sigma$ provides a realization of FQH systems in $\mathrm{M}$-theory. Moreover we propose that the $\mathrm{q}$-deformed version of Chern–Simons theories are related to the anisotropic limit of FQH systems which splits the zeroes of the Laughlin wave function. Extensions of the model to $3+1$ dimensions, which realize topological insulators with non-abelian topologically twisted Yang–Mills theory is pointed out.
PDFXML]]>Cumrun VafaThu, 13 Jul 2023 00:00:00 +0300<![CDATA[Holographic space-time, Newton’s law, and the dynamics of horizons]]>
https://content.intlpress.com/journal/ATMP/article/11673
https://content.intlpress.com/journal/ATMP/article/11673We revisit the construction of models of quantum gravity in $d$ dimensional Minkowski space in terms of random tensor models, and correct some mistakes in our previous treatment of the subject. We find a large class of models in which the large impact parameter scattering scales with energy and impact parameter like Newton’s law. The scattering amplitudes in these models describe scattering of jets of particles, and also include amplitudes for the production of highly meta-stable states with all the parametric properties of black holes. These models have emergent energy, momentum and angular conservation laws, despite being based on time dependent Hamiltonians. The scattering amplitudes in which no intermediate black holes are produced have a time-ordered Feynman diagram space-time structure: local interaction vertices connected by propagation of free particles (really Sterman–Weinberg jets of particles). However, there are also amplitudes where jets collide to form large meta-stable objects, with all the scaling properties of black holes: energy, entropy and temperature, as well as the characteristic time scale for the decay of perturbations. We generalize the conjecture of Sekino and Susskind, to claim that all of these models are fast scramblers. The rationale for this claim is that the interactions are invariant under fuzzy subgroups of the group of volume preserving diffeomorphisms, so that they are highly non-local on the holographic screen. We review how this formalism resolves the Firewall Paradox.
PDFXML]]>We revisit the construction of models of quantum gravity in $d$ dimensional Minkowski space in terms of random tensor models, and correct some mistakes in our previous treatment of the subject. We find a large class of models in which the large impact parameter scattering scales with energy and impact parameter like Newton’s law. The scattering amplitudes in these models describe scattering of jets of particles, and also include amplitudes for the production of highly meta-stable states with all the parametric properties of black holes. These models have emergent energy, momentum and angular conservation laws, despite being based on time dependent Hamiltonians. The scattering amplitudes in which no intermediate black holes are produced have a time-ordered Feynman diagram space-time structure: local interaction vertices connected by propagation of free particles (really Sterman–Weinberg jets of particles). However, there are also amplitudes where jets collide to form large meta-stable objects, with all the scaling properties of black holes: energy, entropy and temperature, as well as the characteristic time scale for the decay of perturbations. We generalize the conjecture of Sekino and Susskind, to claim that all of these models are fast scramblers. The rationale for this claim is that the interactions are invariant under fuzzy subgroups of the group of volume preserving diffeomorphisms, so that they are highly non-local on the holographic screen. We review how this formalism resolves the Firewall Paradox.
PDFXML]]>Tom Banks,Willy FischlerThu, 13 Jul 2023 00:00:00 +0300<![CDATA[Algebraic interplay between renormalization and monodromy]]>
https://content.intlpress.com/journal/ATMP/article/11674
https://content.intlpress.com/journal/ATMP/article/11674We investigate combinatorial and algebraic aspects of the interplay between renormalization and monodromies for Feynman amplitudes. We clarify how extraction of subgraphs from a Feynman graph interacts with putting edges onshell or with contracting them to obtain reduced graphs. Graph by graph this leads to a study of cointeracting bialgebras. One bialgebra comes from extraction of subgraphs and hence is needed for renormalization. The other bialgebra is an incidence bialgebra for edges put either on- or offshell. It is hence related to the monodromies of the multivalued function to which a renormalized graph evaluates. Summing over infinite series of graphs, consequences for Green functions are derived using combinatorial Dyson–Schwinger equations.
PDFXML]]>We investigate combinatorial and algebraic aspects of the interplay between renormalization and monodromies for Feynman amplitudes. We clarify how extraction of subgraphs from a Feynman graph interacts with putting edges onshell or with contracting them to obtain reduced graphs. Graph by graph this leads to a study of cointeracting bialgebras. One bialgebra comes from extraction of subgraphs and hence is needed for renormalization. The other bialgebra is an incidence bialgebra for edges put either on- or offshell. It is hence related to the monodromies of the multivalued function to which a renormalized graph evaluates. Summing over infinite series of graphs, consequences for Green functions are derived using combinatorial Dyson–Schwinger equations.
PDFXML]]>Dirk Kreimer,Karen YeatsThu, 13 Jul 2023 00:00:00 +0300<![CDATA[Integral and differential structures for quantum field theory]]>
https://content.intlpress.com/journal/ATMP/article/11661
https://content.intlpress.com/journal/ATMP/article/11661The aim of this work is to firstly demonstrate the efficacy of the recently proposed Orlicz space formalism for Quantum theory [44], and secondly to show how noncommutative differential structures may naturally be incorporated into this framework. To start off with we specifically propose regularity conditions which in the context of local algebras corresponding to Minkowski space, ensure good behaviour of field operators as observables, and then show that fields obtained by the Osterwalder–Schrader reconstruction theorem are regular in this sense. This complements earlier work by Buchholz, Driessler, Summers and Wichman, etc, on generalized $H$-bounds. The pair of Orlicz spaces we explicitly use for this purpose, are respectively built on the exponential function (for the description of regular field operators) and on an entropic type function (for the description of the corresponding states). This formalism has been shown to be well suited to a description of quantum statistical mechanics, and in the present work we show that it is also a very useful and elegant tool for Quantum Field Theory. We then introduce the class of tangentially conditioned algebras, which is a large class of local algebras corresponding to globally hyperbolic Lorentzian manifolds that locally “look like” the local algebras of Minkowski space. On the one hand this ensures that at a local level, the Orlicz space formalism discussed above is also relevant for a much more general class of local algebras. On the other hand, the structure of this class of algebras, allows for the development of a non-commutative differential geometric structure along the lines of the du Bois–Violette approach to such a theory. In this way we obtain a complete depiction: integrable structures based on local algebras provide a static setting for an analysis of Quantum Field Theory and an effective tool for describing regular behaviour of field operators, whereas differentiable structures posit indispensable tools for a description of equations of motion.
PDFXML]]>The aim of this work is to firstly demonstrate the efficacy of the recently proposed Orlicz space formalism for Quantum theory [44], and secondly to show how noncommutative differential structures may naturally be incorporated into this framework. To start off with we specifically propose regularity conditions which in the context of local algebras corresponding to Minkowski space, ensure good behaviour of field operators as observables, and then show that fields obtained by the Osterwalder–Schrader reconstruction theorem are regular in this sense. This complements earlier work by Buchholz, Driessler, Summers and Wichman, etc, on generalized $H$-bounds. The pair of Orlicz spaces we explicitly use for this purpose, are respectively built on the exponential function (for the description of regular field operators) and on an entropic type function (for the description of the corresponding states). This formalism has been shown to be well suited to a description of quantum statistical mechanics, and in the present work we show that it is also a very useful and elegant tool for Quantum Field Theory. We then introduce the class of tangentially conditioned algebras, which is a large class of local algebras corresponding to globally hyperbolic Lorentzian manifolds that locally “look like” the local algebras of Minkowski space. On the one hand this ensures that at a local level, the Orlicz space formalism discussed above is also relevant for a much more general class of local algebras. On the other hand, the structure of this class of algebras, allows for the development of a non-commutative differential geometric structure along the lines of the du Bois–Violette approach to such a theory. In this way we obtain a complete depiction: integrable structures based on local algebras provide a static setting for an analysis of Quantum Field Theory and an effective tool for describing regular behaviour of field operators, whereas differentiable structures posit indispensable tools for a description of equations of motion.
PDFXML]]>Louis Labuschagne,Adam MajewskiFri, 30 Jun 2023 00:00:00 +0300<![CDATA[Shifted symplectic reduction of derived critical loci]]>
https://content.intlpress.com/journal/ATMP/article/11662
https://content.intlpress.com/journal/ATMP/article/11662We prove that the derived critical locus of a $G$-invariant function $S : X \to \mathbb{A}^1$ carries a shifted moment map, and that its derived symplectic reduction is the derived critical locus of the induced function $S_{red} : X/G \to \mathbb{A}^1$ on the orbit stack. We also provide a relative version of this result, and show that derived symplectic reduction commutes with derived lagrangian intersections.
PDFXML]]>We prove that the derived critical locus of a $G$-invariant function $S : X \to \mathbb{A}^1$ carries a shifted moment map, and that its derived symplectic reduction is the derived critical locus of the induced function $S_{red} : X/G \to \mathbb{A}^1$ on the orbit stack. We also provide a relative version of this result, and show that derived symplectic reduction commutes with derived lagrangian intersections.
PDFXML]]>Mathieu Anel,Damien CalaqueFri, 30 Jun 2023 00:00:00 +0300<![CDATA[$T$-dual solutions and infinitesimal moduli of the $G_2$-Strominger system]]>
https://content.intlpress.com/journal/ATMP/article/11663
https://content.intlpress.com/journal/ATMP/article/11663We consider $G_2$-structures with torsion coupled with $G_2$-instantons, on a compact $7$-dimensional manifold. The coupling is via an equation for $4$-forms which appears in supergravity and generalized geometry, known as the Bianchi identity. First studied by Friedrich and Ivanov, the resulting system of partial differential equations describes compactifications of the heterotic string to three dimensions, and is often referred to as the $G_2$-Strominger system. We study the moduli space of solutions and prove that the space of infinitesimal deformations, modulo automorphisms, is finite dimensional. We also provide a new family of solutions to this system, on $T^3$-bundles over $K3$ surfaces and for infinitely many different instanton bundles, adapting a construction of Fu–Yau and the second named author. In particular, we exhibit the first examples of $T$-dual solutions for this system of equations.
PDFXML]]>We consider $G_2$-structures with torsion coupled with $G_2$-instantons, on a compact $7$-dimensional manifold. The coupling is via an equation for $4$-forms which appears in supergravity and generalized geometry, known as the Bianchi identity. First studied by Friedrich and Ivanov, the resulting system of partial differential equations describes compactifications of the heterotic string to three dimensions, and is often referred to as the $G_2$-Strominger system. We study the moduli space of solutions and prove that the space of infinitesimal deformations, modulo automorphisms, is finite dimensional. We also provide a new family of solutions to this system, on $T^3$-bundles over $K3$ surfaces and for infinitely many different instanton bundles, adapting a construction of Fu–Yau and the second named author. In particular, we exhibit the first examples of $T$-dual solutions for this system of equations.
PDFXML]]>Andrew Clarke,Mario Garcia-Fernandez,Carl TiplerFri, 30 Jun 2023 00:00:00 +0300<![CDATA[Convergence of eigenstate expectation values with system size]]>
https://content.intlpress.com/journal/ATMP/article/11664
https://content.intlpress.com/journal/ATMP/article/11664Understanding the asymptotic behavior of physical quantities in the thermodynamic limit is a fundamental problem in statistical mechanics. In this paper, we study how fast the eigenstate expectation values of a local operator converge to a smooth function of energy density as the system size diverges. In translation-invariant quantum lattice systems in any spatial dimension, we prove that for all but a measure zero set of local operators, the deviations of finite-size eigenstate expectation values from the aforementioned smooth function are lower bounded by $1/\mathit{O}(N)$, where $N$ is the system size. The lower bound holds regardless of the integrability or chaoticity of the model, and is saturated in systems satisfying the eigenstate thermalization hypothesis.
PDFXML]]>Understanding the asymptotic behavior of physical quantities in the thermodynamic limit is a fundamental problem in statistical mechanics. In this paper, we study how fast the eigenstate expectation values of a local operator converge to a smooth function of energy density as the system size diverges. In translation-invariant quantum lattice systems in any spatial dimension, we prove that for all but a measure zero set of local operators, the deviations of finite-size eigenstate expectation values from the aforementioned smooth function are lower bounded by $1/\mathit{O}(N)$, where $N$ is the system size. The lower bound holds regardless of the integrability or chaoticity of the model, and is saturated in systems satisfying the eigenstate thermalization hypothesis.
PDFXML]]>Yichen HuangFri, 30 Jun 2023 00:00:00 +0300