Recently an action based on Lie 3-algebras was proposed to describe M2-branes. We study the case ... more Recently an action based on Lie 3-algebras was proposed to describe M2-branes. We study the case of infinite dimensional Lie 3-algebras based on the Nambu-Poisson structure of three dimensional manifolds. We show that the model contains self-dual 2-form gauge fields in 6 dimensions, and the result may be interpreted as the M5-brane world-volume action.
Recently an action based on Lie 3-algebras was proposed to describe M2-branes. We study the case ... more Recently an action based on Lie 3-algebras was proposed to describe M2-branes. We study the case of infinite dimensional Lie 3-algebras based on the Nambu-Poisson structure of three dimensional manifolds. We show that the model contains self-dual 2-form gauge fields in 6 dimensions, and the result may be interpreted as the M5-brane world-volume action.
Progress of Theoretical and Experimental Physics, 2021
We explore a formulation of a perfect fluid in $3+1$ dimensions in terms of the Kalb–Ramond field... more We explore a formulation of a perfect fluid in $3+1$ dimensions in terms of the Kalb–Ramond field. This was proposed long ago by Nambu and one of the present authors. In this note, we refine the statements in a more explicit form. We also comment on the duality with the Gross–Pitaevsky formulation written by a complex scalar field.
Recently, Gaiotto and Rapcak proposed a generalization of WN algebra by considering the symmetry ... more Recently, Gaiotto and Rapcak proposed a generalization of WN algebra by considering the symmetry at the corner of the brane intersection (corner vertex operator algebra). The algebra, denoted as YL,M,N, is characterized by three non-negative integers L, M, N. It has a manifest triality automorphism which interchanges L, M, N, and can be obtained as a reduction of W1+∞ algebra with a “pit” in the plane partition representation. Later, Prochazka and Rapcak proposed a representation of YL,M,N in terms of L + M + N free bosons by a generalization of Miura transformation, where they use the fractional power differential operators.In this paper, we derive a q-deformation of the Miura transformation. It gives a free field representation for q-deformed YL,M,N, which is obtained as a reduction of the quantum toroidal algebra. We find that the q-deformed version has a “simpler” structure than the original one because of the Miki duality in the quantum toroidal algebra. For instance, one can f...
We develop some basic properties of the open string on the symmetric product which is supposed to... more We develop some basic properties of the open string on the symmetric product which is supposed to describe the open string field theory in discrete lightcone quantization (DLCQ). After preparing the consistency conditions of the twisted boundary conditions for Annulus/Möbius/Klein Bottle amplitudes in generic nonabelian orbifold, we classify the most general solutions of the constraints when the discrete group is SN . We calculate the corresponding orbifold amplitudes from two viewpoints – from the boundary state formalism and from the trace over the open string Hilbert space. It is shown that the topology of the world sheet for the short string and that of the long string in general do not coincide. For example the annulus sector for the short string contains all the sectors (torus, annulus, Klein bottle, Möbius strip) of the long strings. The boundary/cross-cap states of the short strings are classified into three categories in terms of the long string, the ordinary boundary and t...
Motivated by the recent proposal of an N = 8 supersymmetric action for multiple M2-branes, we stu... more Motivated by the recent proposal of an N = 8 supersymmetric action for multiple M2-branes, we study the Lie 3-algebra in detail. In particular, we focus on the fundamental identity and the relation with Nambu-Poisson bracket. Some new algebras not known in the literature are found. Next we consider cubic matrix representations of Lie 3-algebras. We show how to obtain higher dimensional representations by tensor products for a generic 3-algebra. A criterion of reducibility is presented. We also discuss the application of Lie 3-algebra to the membrane physics, including the Basu-Harvey equation and the Bagger-Lambert model. 1 e-mail address: pmho@phys.ntu.edu.tw 2 e-mail address: matsuo@phys.s.u-tokyo.ac.jp
Recently, Maulik and Okounkov proposed an integrable lattice model where the degree of freedom at... more Recently, Maulik and Okounkov proposed an integrable lattice model where the degree of freedom at each site is identical to the Hilbert space of free boson in two dimensions. We give a brief review of their construction and explain the relation with Wn algebra and CalogeroSutherland model. As a generalization, we examine the Yangian associated with N = 1 superconformal algebra which describes a supersymmetric extension of Calogero-Sutherland model and compare it with the literature.
Recently, Gaiotto and Rapčák (GR) proposed a new family of the vertex operator algebra (VOA) as t... more Recently, Gaiotto and Rapčák (GR) proposed a new family of the vertex operator algebra (VOA) as the symmetry appearing at an intersection of five-branes to which they refer as Y algebra. Procházka and Rapčák, then proposed to interpret Y algebra as a truncation of affine Yangian whose module is directly connected to plane partitions (PP). They also developed GR’s idea to generate a new VOA by connecting plane partitions through an infinite leg shared by them and referred it as the web of W-algebra (WoW). In this paper, we demonstrate that double truncation of PP gives the minimal models of such VOAs. For a single PP, it generates all the minimal model irreducible representations of W-algebra. We find that the rule connecting two PPs is more involved than those in the literature when the U(1) charge connecting two PPs is negative. For the simplest nontrivial WoW, $$ \mathcal{N} $$ N = 2 superconformal algebra, we demonstrate that the improved rule precisely reproduces the known chara...
Instanton partition functions of $$ \mathcal{N}=1 $$ N = 1 5d Super Yang-Mills reduced on S 1 can... more Instanton partition functions of $$ \mathcal{N}=1 $$ N = 1 5d Super Yang-Mills reduced on S 1 can be engineered in type IIB string theory from the (p, q)-branes web diagram. To this diagram is superimposed a web of representations of the Ding-Iohara-Miki (DIM) algebra that acts on the partition function. In this correspondence, each segment is associated to a representation, and the (topological string) vertex is identified with the intertwiner operator constructed by Awata, Feigin and Shiraishi. We define a new intertwiner acting on the representation spaces of levels (1, n) ⊗ (0, m) → (1, n + m), thereby generalizing to higher rank m the original construction. It allows us to use a folded version of the usual (p, q)-web diagram, bringing great simplifications to actual computations. As a result, the characterization of Gaiotto states and vertical intertwiners, previously obtained by some of the authors, is uplifted to operator relations acting in the Fock space of horizontal repre...
In the matrix model approaches of string/M theories, one starts from a generic symmetry gl(∞) to ... more In the matrix model approaches of string/M theories, one starts from a generic symmetry gl(∞) to reproduce the space-time manifold. In this paper, we consider the generalization in which the space-time manifold emerges from a gauge symmetry algebra which is not necessarily gl(∞). We focus on the second nontrivial example after the toroidal compactification, the coset space G/H, and propose a specific infinite-dimensional symmetry which realizes the geometry. It consists of the gauge-algebra valued functions on the coset and Lorentzian generator pairs associated with the isometry. We show that the 0-dimensional gauge theory with the mass and Chern-Simons terms gives the gauge theory on the coset with scalar fields associated with H.
Recently an action based on Lie 3-algebras was proposed to describe M2-branes. We study the case ... more Recently an action based on Lie 3-algebras was proposed to describe M2-branes. We study the case of infinite dimensional Lie 3-algebras based on the Nambu-Poisson structure of three dimensional manifolds. We show that the model contains self-dual 2-form gauge fields in 6 dimensions, and the result may be interpreted as the M5-brane world-volume action.
Recently an action based on Lie 3-algebras was proposed to describe M2-branes. We study the case ... more Recently an action based on Lie 3-algebras was proposed to describe M2-branes. We study the case of infinite dimensional Lie 3-algebras based on the Nambu-Poisson structure of three dimensional manifolds. We show that the model contains self-dual 2-form gauge fields in 6 dimensions, and the result may be interpreted as the M5-brane world-volume action.
Progress of Theoretical and Experimental Physics, 2021
We explore a formulation of a perfect fluid in $3+1$ dimensions in terms of the Kalb–Ramond field... more We explore a formulation of a perfect fluid in $3+1$ dimensions in terms of the Kalb–Ramond field. This was proposed long ago by Nambu and one of the present authors. In this note, we refine the statements in a more explicit form. We also comment on the duality with the Gross–Pitaevsky formulation written by a complex scalar field.
Recently, Gaiotto and Rapcak proposed a generalization of WN algebra by considering the symmetry ... more Recently, Gaiotto and Rapcak proposed a generalization of WN algebra by considering the symmetry at the corner of the brane intersection (corner vertex operator algebra). The algebra, denoted as YL,M,N, is characterized by three non-negative integers L, M, N. It has a manifest triality automorphism which interchanges L, M, N, and can be obtained as a reduction of W1+∞ algebra with a “pit” in the plane partition representation. Later, Prochazka and Rapcak proposed a representation of YL,M,N in terms of L + M + N free bosons by a generalization of Miura transformation, where they use the fractional power differential operators.In this paper, we derive a q-deformation of the Miura transformation. It gives a free field representation for q-deformed YL,M,N, which is obtained as a reduction of the quantum toroidal algebra. We find that the q-deformed version has a “simpler” structure than the original one because of the Miki duality in the quantum toroidal algebra. For instance, one can f...
We develop some basic properties of the open string on the symmetric product which is supposed to... more We develop some basic properties of the open string on the symmetric product which is supposed to describe the open string field theory in discrete lightcone quantization (DLCQ). After preparing the consistency conditions of the twisted boundary conditions for Annulus/Möbius/Klein Bottle amplitudes in generic nonabelian orbifold, we classify the most general solutions of the constraints when the discrete group is SN . We calculate the corresponding orbifold amplitudes from two viewpoints – from the boundary state formalism and from the trace over the open string Hilbert space. It is shown that the topology of the world sheet for the short string and that of the long string in general do not coincide. For example the annulus sector for the short string contains all the sectors (torus, annulus, Klein bottle, Möbius strip) of the long strings. The boundary/cross-cap states of the short strings are classified into three categories in terms of the long string, the ordinary boundary and t...
Motivated by the recent proposal of an N = 8 supersymmetric action for multiple M2-branes, we stu... more Motivated by the recent proposal of an N = 8 supersymmetric action for multiple M2-branes, we study the Lie 3-algebra in detail. In particular, we focus on the fundamental identity and the relation with Nambu-Poisson bracket. Some new algebras not known in the literature are found. Next we consider cubic matrix representations of Lie 3-algebras. We show how to obtain higher dimensional representations by tensor products for a generic 3-algebra. A criterion of reducibility is presented. We also discuss the application of Lie 3-algebra to the membrane physics, including the Basu-Harvey equation and the Bagger-Lambert model. 1 e-mail address: pmho@phys.ntu.edu.tw 2 e-mail address: matsuo@phys.s.u-tokyo.ac.jp
Recently, Maulik and Okounkov proposed an integrable lattice model where the degree of freedom at... more Recently, Maulik and Okounkov proposed an integrable lattice model where the degree of freedom at each site is identical to the Hilbert space of free boson in two dimensions. We give a brief review of their construction and explain the relation with Wn algebra and CalogeroSutherland model. As a generalization, we examine the Yangian associated with N = 1 superconformal algebra which describes a supersymmetric extension of Calogero-Sutherland model and compare it with the literature.
Recently, Gaiotto and Rapčák (GR) proposed a new family of the vertex operator algebra (VOA) as t... more Recently, Gaiotto and Rapčák (GR) proposed a new family of the vertex operator algebra (VOA) as the symmetry appearing at an intersection of five-branes to which they refer as Y algebra. Procházka and Rapčák, then proposed to interpret Y algebra as a truncation of affine Yangian whose module is directly connected to plane partitions (PP). They also developed GR’s idea to generate a new VOA by connecting plane partitions through an infinite leg shared by them and referred it as the web of W-algebra (WoW). In this paper, we demonstrate that double truncation of PP gives the minimal models of such VOAs. For a single PP, it generates all the minimal model irreducible representations of W-algebra. We find that the rule connecting two PPs is more involved than those in the literature when the U(1) charge connecting two PPs is negative. For the simplest nontrivial WoW, $$ \mathcal{N} $$ N = 2 superconformal algebra, we demonstrate that the improved rule precisely reproduces the known chara...
Instanton partition functions of $$ \mathcal{N}=1 $$ N = 1 5d Super Yang-Mills reduced on S 1 can... more Instanton partition functions of $$ \mathcal{N}=1 $$ N = 1 5d Super Yang-Mills reduced on S 1 can be engineered in type IIB string theory from the (p, q)-branes web diagram. To this diagram is superimposed a web of representations of the Ding-Iohara-Miki (DIM) algebra that acts on the partition function. In this correspondence, each segment is associated to a representation, and the (topological string) vertex is identified with the intertwiner operator constructed by Awata, Feigin and Shiraishi. We define a new intertwiner acting on the representation spaces of levels (1, n) ⊗ (0, m) → (1, n + m), thereby generalizing to higher rank m the original construction. It allows us to use a folded version of the usual (p, q)-web diagram, bringing great simplifications to actual computations. As a result, the characterization of Gaiotto states and vertical intertwiners, previously obtained by some of the authors, is uplifted to operator relations acting in the Fock space of horizontal repre...
In the matrix model approaches of string/M theories, one starts from a generic symmetry gl(∞) to ... more In the matrix model approaches of string/M theories, one starts from a generic symmetry gl(∞) to reproduce the space-time manifold. In this paper, we consider the generalization in which the space-time manifold emerges from a gauge symmetry algebra which is not necessarily gl(∞). We focus on the second nontrivial example after the toroidal compactification, the coset space G/H, and propose a specific infinite-dimensional symmetry which realizes the geometry. It consists of the gauge-algebra valued functions on the coset and Lorentzian generator pairs associated with the isometry. We show that the 0-dimensional gauge theory with the mass and Chern-Simons terms gives the gauge theory on the coset with scalar fields associated with H.
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Papers by Yutaka Matsuo