arXiv subjects

Shamik Banerjee

Publications and source records attributed to Shamik Banerjee.

At least 19 recordsLinked to original sources

arXiv2026

Note on higher spins and holographic symmetry algebra

In this paper we discuss a higher spin extension of the holographic symmetry algebra for graviton and gluon. Our primary observation is that in the presence of higher spin particles the soft symmetry algebra has a subalgebra isomorphic to $w_{\infty}$ which is generated by the \textit{conformally soft higher spin particles}. This $w_{\infty}$ subalgebra does not commute with the $w_{1+\infty}$ subalegbra generated by the conformally soft gravitons. The same thing holds for the colored higher spin particles. One gets a subalgebra isomorphic to the $S$-algebra which is generated by the conformally soft colored higher spin particles. We further verify the soft algebra for colored higher spin particles using the (tree-level) $4$-point MHV amplitude of the higher spin Yang-Mills theory constructed in arXiv:2210.07130. At the end we also discuss the higher spin extension of the deformed holographic symmetry algebra for non-zero cosmological constant as constructed in arXiv:2312.00876.

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arXiv2026

All loop soft photon theorems and higher spin currents on the celestial sphere

Soft factorization theorems can be reinterpreted as Ward identities for (asymptotic) symmetries of scattering amplitudes in asymptotically flat space-time. In this paper we study the symmetries implied by the all loop soft photon theorems when all the charged particles are highly energetic and the relation $\omega << m << E$ holds where $E$ is the typical energy of a charged particle, $m$ is the typical mass and $\omega$ is the soft photon energy. Loop level soft theorems are qualitatively different from the tree level soft theorems because loop level soft factors contain multi-particle sums. If we want to interpret them as Ward identities or define celestial OPE between between soft and hard operators then we need to introduce additional fields which live on the celestial sphere but do not appear as asymptotic states in any scattering experiment. For example, if we want to interpret the one-loop exact $\mathcal{O}(\ln\omega)$ soft theorem for a positive helicity soft photon (with energy $\omega$) as a Ward identity then we need to introduce a pair of antiholomorphic currents on the celestial sphere which transform as a doublet under the $SL(2,\mathbb{R})_{R}$. We call them dipole currents because the corresponding charges measure the monopole and the dipole moment of an electrically charged particle on the celestial sphere. More generally, the soft photon theorem at $\mathcal{O}(\omega^{2j-1}(\ln\omega)^{2j})$ for every $j\in \frac{1}{2}\mathbb{Z}_+$ gives rise to $(2j+1)$ antiholomorphic currents which transform in the spin-$j$ representation of the $SL(2,\mathbb{R})_{R}$. These currents exist in the quantum theory because they follow from loop level soft theorems. We argue that under certain circumstances the (classical) algebra of the higher spin currents is the wedge subalgebra of the $w_{1+\infty}$.

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arXiv2025

Holographic symmetry algebra for the MHV sector revisited

We revisit the holographic symmetry algebra in the MHV sector. We find an infinite dimensional Abelian symmetry algebra whose generators are the conformally soft negative helicity gravitons and gluons. So the complete symmetry algebra in the MHV graviton sector is a semideirect product of the $w_{1+\infty}$ algebra and the infinite dimensional Abelian algebra. Similarly in the MHV gluon sector the symmetry algebra is a semidirect product of the $S$ algebra and the infinite dimensional Abelian algebra. The extended symmetry algebra has some use. For example, it is known for sometime that an $n$ point MHV amplitude satisfies $(n-2)$ Knizhnik-Zamolodchikov (KZ) type equations. So two equations are missing. We show that the extended symmetry algebra has additional null states whose decoupling give rise to the two missing equations.

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arXiv2025

Worldsheet CFT$_2$ and Celestial CFT$_2$ : An AdS$_3$-CFT$_2$ perspective

Celestial CFT$_d$ is the putative dual of quantum gravity in asymptotically flat $(d+2)$ dimensional space time. We argue that a class of Celestial CFT$_d$ can be engineered via AdS$_{d+1}$-CFT$_d$ correspondence. Our argument is based on the observation that if we zoom in near the boundary of (Euclidean) AdS$_{d+1}$ then the conformal isometry group of EAdS$_{d+1}$, which is SO$(d+2,1)$, contracts to the Poincare group ISO$(d+1,1)$. This suggests that the near boundary scaling limit of a theory of \textit{conformal} gravity on EAdS$_{d+1}$ should be dual to a boundary CFT$_d$ with ISO$(d+1,1)$ symmetry. This dual CFT$_d$, since the symmetries match, is an example of a Celestial CFT$_d$. Similarly, if we have a \textit{non-conformal} theory of gravity on EAdS$_{d+1}$ then the near boundary scaling limit of such a theory is dual to a (boundary) Celestial CFT$_d$ with \textit{only} (SO$(d+1,1)$) Lorentz invariance. Celestial CFTs with only Lorentz invariance have been recently studied in the literature. Now following this logic we discuss, among other things, the near boundary scaling limit of the bosonic string theory on Euclidean AdS$_3$ in the presence of the NS-NS B field. The AdS$_3$ part of the worldsheet theory is free in this limit and has been studied in the literature in different contexts. This limit describes a ``long string'' which wraps the (Euclidean) AdS$_3$ boundary and it has been argued that the space-time CFT$_2$ which describes the radial fluctuations of a long string is a Liouville CFT. According to our proposal, the dual CFT$_2$ which describes the \textit{long string sector} is an example of a \textit{Celestial} CFT$_2$ with \textit{only} (SO$(3,1)$)Lorentz invariance. We do not get a full ISO$(3,1)$ invariant Celestial CFT$_2$ in this way because the string theory does not have target space conformal invariance.

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arXiv2024

Boundary operators in asymptotically flat space-time

In \cite{Jain:2023fxc} the authors have proposed an interesting framework for studying holography in flat space-time. In this note we explore the relationship between their proposal and the Celestial Holography. In particular, we find that in both the massive and in the massless cases the asymptotic boundary limit of the bulk time-ordered Green's function $G$ is related to the Celestial amplitudes by an integral transformation. In the massless case the integral transformation reduces to the well known \textit{shadow transformation} of the celestial amplitude. Now the relation between the asymptotic limit of $G$ and the celestial amplitudes suggests that in asymptotically flat space-time if the scattering states are described by the conformal primary basis then the boundary operators defined by the extrapolate dictionary of \cite{Jain:2023fxc} are given by the \underline{shadow transformation} of the conformal primary operators living on the celestial sphere. This result refers to the non-contact part of the extrapolated Green's function. There are important contact term contributions which we also discuss in the paper.

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arXiv2023

All $S$ invariant gluon OPEs on the celestial sphere

$S$ algebra is an infinite dimensional Lie algebra which is known to be the symmetry algebra of some gauge theories. It is a "coloured version" of the $w_{1+\infty}$. In this paper we write down all possible $S$ invariant (celestial) OPEs between two positive helicity outgoing gluons and also find the Knizhnik-Zamolodchikov type null states for these theories. Our analysis hints at the existence of an infinite number of $S$ invariant gauge theories which include the (tree-level) MHV-sector and the self-dual Yang-Mills theory.

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arXiv2023

Celestial OPE in Self Dual Gravity

In this paper we compute the celestial operator product expansion between two outgoing positive helicity gravitons in the self dual gravity. It has been shown that the self dual gravity is a $ w_{1+\infty} $-invariant theory whose scattering amplitudes are one loop exact with all positive helicity gravitons. Celestial $w_{1+\infty}$ symmetry is generated by an infinite tower of (conformally soft) gravitons which are holomorphic conserved currents. We find that at any given order only a \textit{finite} number of $w_{1+\infty}$ descendants contribute to the OPE. This is somewhat surprising because the spectrum of conformal dimensions in celestial CFT is not bounded from below. However, this is consistent with our earlier analysis based on the representation theory of $w_{1+\infty}$. The phenomenon of truncation suggests that in some (unknown) formulation the spectrum of conformal dimensions in the dual two dimensional theory can be bounded from below.

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arXiv2023

MHV Gluon Scattering in the Massive Scalar Background and Celestial OPE

In this paper we study the OPE between two positive helicity outgoing gluons in the celestial CFT for the Yang-Mills theory chirally coupled to a massive scalar background. This theory breaks the translation as well as scale invariance. We compute the subleading terms in the OPE expansion and show that they are same as the subleading terms of the OPE expansions in the MHV sector. As a result the amplitudes of this theory also satisfy the set of differential equations obtained previously for MHV amplitudes in pure YM theory. This is not surprising because the symmetries coming from the leading and subleading soft gluon theorems do not change in the presence of a massive scalar background.

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arXiv2023

An infinite family of $w_{1+\infty}$ invariant theories on the celestial sphere

In this note we determine the graviton-graviton OPE and the null states in any $w_{1+\infty}$ symmetric theory on the celestial sphere. Our analysis shows that there exists a discrete \textit{infinite} family of such theories. The MHV-sector and the quantum self dual gravity are two members of this infinite family. Although the Bulk Lagrangian description of this family of theories is not currently known to us, the graviton scattering amplitudes in these theories are heavily constrained due to the existence of null states. Presumably they are exactly solvable in the same way as the minimal models of $2$-D CFT.

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arXiv2022

Revisiting the Shadow Stress Tensor in Celestial CFT

We revisit the standard construction of the celestial stress tensor as a shadow of the subleading conformally soft graviton. In its original formulation there is an obstruction to reproducing the expected TT OPE in the double soft limit. We propose a modification to the definition which circumvents this obstruction and then extend this change of basis beyond the conformally soft and single helicity sectors. In the process we investigate how (non)-commutativity of double soft limits is tied to the decoupling of primary descendants, and how our choice of celestial basis determines which symmetries are manifest at the level of the OPE beyond the MHV sector.

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arXiv2022

Scattering Amplitudes: Celestial and Carrollian

Recent attempts at the construction of holography for asymptotically flat spacetimes have taken two different routes. Celestial holography, involving a two dimensional (2d) CFT dual to 4d Minkowski spacetime, has generated novel results in asymptotic symmetry and scattering amplitudes. A different formulation, using Carrollian CFTs, has been principally used to provide some evidence for flat holography in lower dimensions. Understanding of flatspace scattering has been lacking in the Carroll framework. In this work, using ideas from Celestial holography, we show that 3d Carrollian CFTs living on the null boundary of 4d flatspace can potentially compute bulk scattering amplitudes. 3d Carrollian conformal correlators have two different branches, one depending on the null time direction and one independent of it. We propose that it is the time-dependent branch that is related to bulk scattering. We construct an explicit field theoretic example of a free massless Carrollian scalar that realises some desired properties.

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arXiv2021

(Chiral) Virasoro invariance of the tree-level MHV graviton scattering amplitudes

In this paper we continue our study of the tree level MHV graviton scattering amplitudes from the point of view of celestial holography. In arXiv:2008.04330 we showed that the celestial OPE of two gravitons in the MHV sector can be written as a linear combination of $\overline{SL(2,\mathbb C)}$ current algebra and supertranslation descendants. In this note we show that the OPE is in fact manifestly invariant under the infinite dimensional Virasoro algebra as is expected for a $2$-D CFT. This is consistent with the conjecture that the holographic dual in $4$-D asymptotically flat space time is a $2$-D CFT. Since we get only one copy of the Virasoro algebra we can conclude that the holographic dual theory which computes the MHV amplitudes is a chiral CFT with a host of other infinite dimensional global symmetries including $\overline{SL(2,\mathbb C)}$ current algebra, supertranslations and subsubleading soft graviton symmetry. We also discuss some puzzles related to the appearance of the Virasoro symmetry.

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arXiv2021

Subsubleading soft graviton symmetry and MHV graviton scattering amplitudes

In arXiv:2008.04330 it was shown that supertranslation and $\overline{SL(2,\mathbb C)}$ current algebra symmetries, corresponding to leading and subleading soft graviton theorems, are enough to determine the tree level MHV graviton scattering amplitudes. In this note we clarify the role of subsubleading soft graviton theorem in this context.

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arXiv2020

MHV Gluon Scattering Amplitudes from Celestial Current Algebras

We show that the Mellin transform of an $n$-point tree level MHV gluon scattering amplitude, also known as the celestial amplitude in pure Yang-Mills theory, satisfies a system of $(n-2)$ linear first order partial differential equations corresponding to $(n-2)$ positive helicity gluons. Although these equations closely resemble Knizhnik-Zamolodchikov equations for $SU(N)$ current algebra there is also an additional "correction" term coming from the subleading soft gluon current algebra. These equations can be used to compute the leading term in the gluon-gluon OPE on the celestial sphere. Similar equations can also be written down for the momentum space tree level MHV scattering amplitudes. We also propose a way to deal with the non closure of subleading current algebra generators under commutation. This is then used to compute some subleading terms in the mixed helicity gluon OPE and our results match with those obtained from an explicit calculation using the Mellin MHV amplitude.

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arXiv2020

MHV Graviton Scattering Amplitudes and Current Algebra on the Celestial Sphere

The Cachazo-Strominger subleading soft graviton theorem for a positive helicity soft graviton is equivalent to the Ward identities for $\overline{SL(2,\mathbb C)}$ currents. This naturally gives rise to a $\overline{SL(2,\mathbb C)}$ current algebra living on the celestial sphere. The generators of the $\overline{SL(2,\mathbb C)}$ current algebra and the supertranslations, coming from a positive helicity leading soft graviton, form a closed algebra. We find that the OPE of two graviton primaries in the Celestial CFT, extracted from MHV amplitudes, is completely determined in terms of this algebra. To be more precise, 1) The subleading terms in the OPE are determined in terms of the leading OPE coefficient if we demand that both sides of the OPE transform in the same way under this local symmetry algebra. 2) Positive helicity gravitons have null states under this local algebra whose decoupling leads to differential equations for MHV amplitudes. An $n$ point MHV amplitude satisfies two systems of $(n-2)$ linear first order PDEs corresponding to $(n-2)$ positive helicity gravitons. We have checked, using Hodges' formula, that one system of differential equations is satisfied by any MHV amplitude, whereas the other system has been checked up to six graviton MHV amplitude. 3) One can determine the leading OPE coefficients from these differential equations. This points to the existence of an autonomous sector of the Celestial CFT which holographically computes the MHV graviton scattering amplitudes and is completely defined by this local symmetry algebra. The MHV-sector of the Celestial CFT is like a minimal model of $2$-D CFT.

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arXiv2020

BMS Symmetry of Celestial OPE

In this paper we study the BMS symmetry of the celestial OPE of two positive helicity gravitons in Einstein theory in four dimensions. The celestial OPE is obtained by Mellin transforming the scattering amplitude in the (holomorphic) collinear limit. The collinear limit at leading order gives the singular term of the celestial OPE. We compute the first subleading correction to the OPE by analysing the four graviton scattering amplitude directly in Mellin space. The subleading term can be written as a linear combination of BMS descendants with the OPE coefficients determined by BMS algebra and the coefficient of the leading term in the OPE. This can be done by defining a suitable BMS primary state. We find that among the descendants, which appear at the first subleading order, there is one which is created by holomorphic supertranslation with simple pole on the celestial sphere.

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arXiv2019

Modified celestial amplitude in Einstein gravity

In this paper we evaluate the modified celestial amplitude for gravitons and gluons, as defined in arXiv:1801.10171[hep-th]. We find that the modified (tree) amplitude is finite for gravitons in Einstein gravity. The modified amplitude behaves like correlation function of operators inserted at various points of null-infinity in the Minkowski space-time. Therefore, unlike the standard celestial amplitudes, these are three dimensional objects. We also show that this amplitude admits conformal soft factorization recently studied in the literature.

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arXiv2019

Conformal properties of soft operators -- 2 : Use of null-states

Representations of the (Lorentz) conformal group with the soft operators as highest weight vectors have two universal properties, which we clearly state in this paper. Given a soft operator with a certain dimension and spin, the first property is about the existence of "(large) gauge transformation" that acts on the soft operator. The second property is the decoupling of (large) gauge-invariant null-states of the soft operators from the $S$-matrix elements. In each case, the decoupling equation has the form of zero field-strength condition with the soft operator as the (gauge) potential. Null-state decoupling effectively reduces the number of polarisation states of the soft particle and is crucial in deriving soft-theorems from the Ward identities of asymptotic symmetries. To the best of our understanding, these properties are not directly related to the Lorentz invariance of the $S$-matrix or the existence of asymptotic symmetries. We also verify that the results obtained from the decoupling of null-states are consistent with the leading and subleading soft-theorems with finite energy massive and massless particles in the external legs.

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