arXiv subjects

Man Hea Kim

Publications and source records attributed to Man Hea Kim.

11 recordsLinked to original sources

arXiv2025

Toward A Celestial Soft Symmetry Algebra in the ${\cal N}=8$ Supergravity

From the classical $SO({\cal N}=8)$ extended superconformal algebra between the lowest ${\cal N}=8$ multiplet in two dimensions obtained by Ademollo et al. (1976), we generalize it for the arbitrary ${\cal N}=8$ multiplet with manifest $SU(8)$ symmetry containing the bosonic $w_{1+\infty}$ algebra. By modifying this ${\cal N}=8$ supersymmetric $w_{1+\infty}$ algebra in two dimensions, we propose a consistent celestial soft current algebra between the graviton, the gravitinos, the graviphotons, the graviphotinos, and the scalars in the ${\cal N}=8$ supergravity theory with $SO(8)$ (or $SU(8)$) global symmetry in four dimensions initiated by de Wit and Freedman (at Stony Brook in 1977). The twenty five couplings in this celestial algebra can be written in terms of eight arbitrary couplings via the Jacobi identity.

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arXiv2025

A Supersymmetric $w_{1+\infty}$ Symmetry, the Extended Supergravity and the Celestial Holography

We determine the ${\cal N}=4$ supersymmetric $W_{1+\infty}^{2,2}[\lambda=\frac{1}{4}]$ algebra which is an extension of ${\cal N}=4$ $SO(4)$ superconformal algebra with vanishing central charge. We identify the soft current algebra between the graviton, the gravitinos, the vectors, the Majorana fermions, the scalar or the pseudoscalar, from the ${\cal N}=4$ supersymmetric $w_{1+\infty}^{2,2}[\lambda=\frac{1}{4}]$ algebra, in two dimensions with the ${\cal N}=4$ supergravity theory with $SO(4)$ global symmetry in four dimensions found by Das (at Stony Brook in 1977), via celestial holography. Furthermore, the truncations of ${\cal N}=4$ supersymmetric soft current algebra provide the soft current algebras for the ${\cal N}=2,3$ supergravity theories, the ${\cal N}=2$ supergravity coupled to its Abelian vector multiplet and the ${\cal N}=1$ supersymmetric Maxwell Einstein theory. For the ${\cal N}=2$ supergravity theory, the soft current algebra can be also realized from the ${\cal N}=2$ supersymmetric $w_{1+\infty}^{K,K}[\lambda=0]$ algebra.

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arXiv2024

A Supersymmetric Extension of $w_{1+\infty}$ Algebra in the Celestial Holography

We determine the ${\cal N}=1$ supersymmetric topological $W_{\infty} $ algebra by using the $λ$ deformed bosons $(β,γ)$ and fermions $(b,c)$ ghost system. By considering the real bosons and the real fermions at $λ=0$ (or $λ=\frac{1}{2}$), the ${\cal N}=1$ supersymmetric $W_{\frac{\infty}{2}}$ algebra is obtained. At $λ=\frac{1}{4}$, other ${\cal N}=1$ supersymmetric $W_{1+\infty}[λ=\frac{1}{4}]$ algebra is determined. We also obtain the extension of Lie superalgebra $PSU(2,2|{\cal N}=4)$ appearing in the worldsheet theory by using the symplectic bosons and the fermions. We identify the soft current algebra between the graviton, the gravitino, the photon (the gluon), the photino (the gluino) or the scalars, equivalent to ${\cal N}=1$ supersymmetric $W_{1+\infty}[λ]$ algebra, in two dimensions with the ${\cal N}=1$ supergravity theory in four dimensions discovered by Freedman, van Nieuwenhuizen and Ferrara in 1976 and its matter coupled theories, via celestial holography.

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arXiv2024

The ${\cal N}=2,4$ Supersymmetric Linear $W_{\infty}[λ]$ Algebras for Generic $λ$ Parameter

The four different kinds of currents are given by the multiple $(β,γ)$ and $(b,c)$ ghost systems with a multiple product of derivatives. We determine their complete algebra where the structure constants depend on the deformation parameter $λ$ appearing in the conformal weights of above fields nontrivially and depend on the generic spins $h_1$ and $h_2$ appearing on the left hand sides in the (anti)commutators. By taking the linear combinations of these currents, the ${\cal N}=4$ supersymmetric linear $W_{\infty}[λ]$ algebra (and its ${\cal N}=4$ superspace description) for generic $λ$ is obtained explicitly. Moreover, we determine the ${\cal N}=2$ supersymmetric linear $W_{\infty}[λ]$ algebra for arbitrary $λ$. As a by product, the $λ$ deformed bosonic $W_{1+\infty}[λ] \times W_{1+\infty}[λ+\frac{1}{2}]$ subalgebra (a generalization of Pope, Romans and Shen's work in $1990$) is obtained. The first factor is realized by $(b,c)$ fermionic fields while the second factor is realized by $(β,γ)$ bosonic fields. The degrees of the polynomials in $λ$ for the structure constants are given by $(h_1+h_2-2)$. Each $w_{1+\infty}$ algebra from the celestial holography is reproduced by taking the vanishing limit of other deformation prameter $q$ at $λ=0$ with the contractions of the currents.

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arXiv2021

Fermionic Construction in the Supersymmetric Coset Model

It is known previously that the operator product expansion (OPE) between the first ${\cal N}=3 $ multiplet and itself contains the second ${\cal N}=3$ multiplet in the supersymmetric coset model. In this paper, by using their realizations in terms of various fermions, we compute the four kinds of OPEs between the first and the second ${\cal N}=3$ multiplets for fixed $N$ and $M$ where the group of the coset contains $SU(N+M)$. By supersymmetrizing the above OPEs in ${\cal N}=3$ superspace and using the various Jacobi identities between the currents, we determine the ${\cal N}=3$ supersymmetric OPE between the first and the second ${\cal N}=3$ multiplets completely. The right hand side of this OPE contains the various ${\cal N}=3$ multiplets: the $SO(3)$ singlet ${\cal N}=3$ multiplets of superspin-$\frac{3}{2},2,3,4$ and the $SO(3)$ triplet ${\cal N}=3$ multiplets of superspin-$\frac{5}{2},3,\frac{7}{2}$. The ${\cal N}=2$ superspace description and the decoupling of the spin-$\frac{1}{2}$ current of the ${\cal N}=3$ superconformal algebra are also described.

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arXiv2021

The ${\cal N}=4$ Higher Spin Algebra for Generic $μ$ Parameter

The ${\cal N}=4$ higher spin generators for general superspin $s$ in terms of oscillators in the matrix generalization of $AdS_3$ Vasiliev higher spin theory at nonzero $μ$ (which is equivalent to the 't Hooft-like coupling constant $λ$) were found previously. In this paper, by computing the (anti)commutators between these ${\cal N}=4$ higher spin generators for low spins $s_1$ and $s_2$ ($s_1+s_2 \leq 11$) explicitly, we determine the complete ${\cal N}=4$ higher spin algebra for generic $μ$. The three kinds of structure constants contain the linear combination of two different generalized hypergeometric functions. These structure constants remain the same under the transformation $μ\leftrightarrow (1-μ)$ up to signs. We have checked that the above ${\cal N}=4$ higher spin algebra contains the ${\cal N}=2$ higher spin algebra, as a subalgebra, found by Fradkin and Linetsky some time ago.

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arXiv2020

The Small ${\cal N}=4$ Superconformal ${\cal W}_{\infty}$ Algebra

The symmetric orbifold of $\mathbb{T}^4$ is the CFT dual of string theory on AdS$_3\times {\rm S}^3 \times \mathbb{T}^4$ with minimal NS-NS flux. We study its symmetry algebra and provide evidence that it does not have any deformation parameter. This suggests that the symmetric orbifold is (at least locally) the most symmetrical CFT in its moduli space.

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arXiv2019

The ${\cal N}=4$ Coset Model and the Higher Spin Algebra

By computing the operator product expansions between the first two ${\cal N}=4$ higher spin multiplets in the unitary coset model, the (anti)commutators of higher spin currents are obtained under the large $(N,k)$ 't Hooft-like limit. The free field realization with complex bosons and fermions is presented. The (anti)commutators for generic spins $s_1$ and $s_2$ with manifest $SO(4)$ symmetry at vanishing 't Hooft-like coupling constant are completely determined. The structure constants can be written in terms of the ones in the ${\cal N}=2$ ${\cal W}_{\infty}$ algebra found by Bergshoeff, Pope, Romans, Sezgin and Shen previously, in addition to the spin-dependent fractional coefficients and two $SO(4)$ invariant tensors. We also describe the ${\cal N}=4$ higher spin generators, by using the above coset construction results, for general super spin $s$ in terms of oscillators in the matrix generalization of $AdS_3$ Vasiliev higher spin theory at nonzero 't Hooft-like coupling constant. We obtain the ${\cal N}=4$ higher spin algebra for low spins and present how to determine the structure constants, which depend on the higher spin algebra parameter, in general, for fixed spins $s_1$ and $s_2$.

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arXiv2019

The Operator Product Expansions in the ${\cal N}=4$ Orthogonal Wolf Space Coset Model

Some of the operator product expansions (OPEs) between the lowest $SO(4)$ singlet higher spin-$2$ multiplet of spins $(2, \frac{5}{2}, \frac{5}{2}, \frac{5}{2}, \frac{5}{2}, 3, 3, 3, 3, 3, 3, \frac{7}{2}, \frac{7}{2}, \frac{7}{2}, \frac{7}{2}, 4)$ in an extension of the large ${\cal N}=4$ (non)linear superconformal algebra were constructed in the ${\cal N}=4$ superconformal coset $\frac{SO(N+4)}{SO(N) \times SO(4)}$ theory with $N=4$ previously. In this paper, by rewriting the above OPEs with $N=5$, the remaining undetermined OPEs are completely determined. There exist additional $SO(4)$ singlet higher spin-$2$ multiplet, six $SO(4)$ adjoint higher spin-$3$ multiplets, four $SO(4)$ vector higher spin-$\frac{7}{2}$ multiplets, $SO(4)$ singlet higher spin-$4$ multiplet and four $SO(4)$ vector higher spin-$\frac{9}{2}$ multiplets in the right hand side of these OPEs. Furthermore, by introducing the arbitrary coefficients in front of the composite fields in the right hand sides of the above complete 136 OPEs, the complete structures of the above OPEs are obtained by using various Jacobi identities for generic $N$. Finally, we describe them as one single ${\cal N}=4$ super OPE between the above lowest $SO(4)$ singlet higher spin-$2$ multiplet in the ${\cal N}=4$ superspace.

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arXiv2017

The Next $16$ Higher Spin Currents and Three-Point Functions in the Large ${\cal N}=4$ Holography

By using the known operator product expansions (OPEs) between the lowest $16$ higher spin currents of spins $(1, \frac{3}{2}, \frac{3}{2}, \frac{3}{2}, \frac{3}{2}, 2,2,2,2,2,2, \frac{5}{2}, \frac{5}{2}, \frac{5}{2}, \frac{5}{2}, 3)$ in an extension of the large ${\cal N}=4$ linear superconformal algebra, one determines the OPEs between the lowest $16$ higher spin currents in an extension of the large ${\cal N}=4$ nonlinear superconformal algebra for generic $N$ and $k$. The Wolf space coset contains the group $G =SU(N+2)$ and the affine Kac-Moody spin $1$ current has the level $k$. The next $16$ higher spin currents of spins $(2,\frac{5}{2}, \frac{5}{2}, \frac{5}{2}, \frac{5}{2}, 3,3,3,3,3,3, \frac{7}{2}, \frac{7}{2}, \frac{7}{2}, \frac{7}{2},4)$ arise in the above OPEs. The most general lowest higher spin $2$ current in this multiplet can be determined in terms of affine Kac-Moody spin $\frac{1}{2}, 1$ currents. By careful analysis of the zero mode (higher spin) eigenvalue equations, the three-point functions of bosonic higher spin $2, 3, 4$ currents with two scalars are obtained for finite $N$ and $k$. Furthermore, we also analyze the three-point functions of bosonic higher spin $2, 3, 4$ currents in the extension of the large ${\cal N}=4$ linear superconformal algebra. It turns out that the three-point functions of higher spin $2,3$ currents in the two cases are equal to each other at finite $N$ and $k$. Under the large $(N,k)$ 't Hooft limit, the two descriptions for the three-point functions of higher spin $4$ current coincide with each other. The higher spin extension of $SO(4)$ Knizhnik Bershadsky algebra is described.

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arXiv2015

The Operator Product Expansion between the 16 Lowest Higher Spin Currents in the N=4 Superspace

Some of the operator product expansions (OPEs) between the lowest 16 higher spin currents of spins (1, 3/2, 3/2, 3/2, 3/2, 2, 2, 2, 2, 2, 2, 5/2, 5/2, 5/2, 5/2, 3) in an extension of the large N=4 linear superconformal algebra were constructed in the N=4 superconformal coset SU(5)/SU(3) theory previously. In this paper, by rewriting the above OPEs in the N=4 superspace developed by Schoutens (and other groups), the remaining undetermined OPEs where the corresponding singular terms possess the composite fields with spins s =7/2, 4, 9/2, 5 are completely determined. Furthermore, by introducing the arbitrary coefficients in front of the composite fields in the right hand sides of the above complete 136 OPEs, reexpressing them in the N=2 superspace and using the N=2 OPEs mathematica package by Krivonos and Thielemans, the complete structures of the above OPEs with fixed coefficient functions are obtained with the help of various Jacobi identities. Then one obtains ten N=2 super OPEs between the four N=2 higher spin currents denoted by (1, 3/2, 3/2, 2), (3/2, 2, 2, 5/2), (3/2, 2, 2, 5/2) and (2, 5/2, 5/2, 3) (corresponding 136 OPEs in the component approach) in the N=4 superconformal coset SU(N+2)/SU(N) theory. Finally, one describes them as one single N=4 super OPE between the above sixteen higher spin currents in the N=4 superspace. The fusion rule for this OPE contains the next 16 higher spin currents of spins of (2, 5/2, 5/2, 5/2, 5/2, 3, 3, 3, 3, 3, 3, 7/2, 7/2, 7/2, 7/2, 4) in addition to the quadratic N=4 lowest higher spin multiplet and the large N=4 linear superconformal family of the identity operator. The various structure constants (fixed coefficient functions) appearing in the right hand side of this OPE depend on N and the level k of the bosonic spin-1 affine Kac-Moody current.

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