arXiv subjects

Kausik Ghosh

Publications and source records attributed to Kausik Ghosh.

18 recordsLinked to original sources

arXiv2026

Bootstrapping 3D Conformal Field Theories with Product Analytic Functionals

We introduce a bootstrap method for three-dimensional conformal field theories based on product analytic functionals. The construction combines one-dimensional analytic functionals with dimensional reduction and provides an efficient alternative to the standard derivative basis. We benchmark the method in single correlator gap maximization problems for scalar and spin-two operators. At comparable basis size, the product functional basis gives stronger bounds and converges more rapidly, with the improvement becoming especially pronounced at large external dimension. Setting the external dimension to that of the three-dimensional Ising spin operator, we substantially sharpen the single correlator upper bound on the leading scalar dimension. We also sharpen the Nakayama-Ohtsuki necessary condition for critical points accessible by tuning a single parameter. In the spin-two problem, we find a new kink near $\Delta_\phi \simeq 4.16$, accompanied by a reorganization of the extremal spectrum. We also observe plateau structures at larger external dimension in both gap-maximization problems. Our results make product analytic functionals a promising tool for conformal gauge theories with heavy external operators and for probing the flat-space limit of holographic correlators; accessing both of these scenarios is challenging for conventional methods.

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arXiv2026

Neural Spectral Bias and Conformal Correlators II: Modular and Annulus Bootstrap

We develop a neural network bootstrap framework for reconstructing partition functions of two-dimensional conformal field theories (CFTs) based on modular invariance and the Cardy condition, which are recast as crossing equations for four-point correlators. For torus partition functions, we use the twist-field representation in the symmetric-orbifold description to map modular S-invariance to four-point crossing and focus on the diagonal kinematics of four insertions on a line. For annulus partition functions, we formulate open/closed channel duality as crossing symmetry for mixed four-point functions of defect-changing operators in interface CFT. In both cases, the reconstruction problem is formulated in the anchored-bootstrap form, where the crossing constraints are supplemented by minimal spectral input (a gap) and anchor data. We solve this under-determined problem by using lightweight feed-forward neural networks to parametrise the correlators and their corresponding partition functions. A key ingredient of this approach is the spectral bias of the neural networks in the lazy training regime, which selects specific crossing-symmetric configurations. This reformulation unifies standard modular and annulus constraints in two dimensions with the anchored neural approach for CFT correlators, providing a new way to reconstruct full partition functions from sparse data with remarkable accuracy.

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arXiv2026

The Geometry of Quantum Complexity in Open Systems

We extend Nielsen's geometric approach for quantum complexity from closed to open quantum systems, whose dynamics is governed by Lindbladian evolution. In this framework, complexity is defined through an optimal-control problem on the space of mixed states, with a cost assigned to both unitary and non-unitary generators. We show that the resulting geometric structure differs fundamentally from the Riemannian geometry that emerges in the case of unitary evolution. In the open-system setting, the natural geometry is typically sub-Finslerian. Dissipation makes the geodesics non-reversible, while the admissible tangent directions are restricted by the physically allowed controls. We analyze several physically motivated examples, including a single qubit subject to depolarizing and amplitude-damping channels, as well as the damped harmonic oscillator. We show that, similarly to the unitary case, varying the penalty factors in the cost functional modifies the geometric properties through changes in the flag curvature, the Finslerian analog of sectional curvature. Our results provide a geometric framework for quantifying the abstract notion of complexity in dissipative quantum systems, with potential connections to experimentally realizable setups.

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arXiv2026

Neural Networks Reveal a Universal Bias in Conformal Correlators

We propose that simple neural networks (NNs) trained on crossing symmetry can reconstruct conformal correlators restricted to a line to remarkable accuracy. The input is minimal: an external scaling dimension, a spectral gap, and the value of the correlator at a single point. We present evidence across a wide range of conformal theories and dimensions, for both four-point and thermal two-point functions. We attribute these observations to the spectral bias of gradient-based NN training, which appears to align with an intrinsic smoothness property of conformal field theory. This suggests a novel variational principle for conformal correlators and opens a path towards a powerful new computational framework for non-perturbative quantum field theory.

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arXiv2026

Neural Spectral Bias and Conformal Correlators I: Introduction and Applications

We demonstrate that simple feed-forward neural networks (NNs) can accurately compute correlation functions of conformal field theories (CFTs) on a line. Strikingly, by optimising a NN solely on crossing symmetry and providing only the scaling dimension of the leading non-trivial operator and the correlator's value at a single "anchor point", we can reconstruct target physical correlators to within a few percent. We establish the robustness of this minimal-data approach across a broad class of theories and dimensions, including generalised free fields, contact and one-loop Witten diagrams in AdS$_2$, unitary and non-unitary 2d minimal models, the 3d Ising model, and half-BPS correlators in 4d $\mathcal{N}=4$ super-Yang-Mills theory, together with several thermal two-point functions, notably including those of the 3d Ising model. We argue that this remarkable alignment between NNs and CFTs stems from the spectral bias of gradient-based training, which heavily favours smooth functions. To ground this connection, we analyse the smoothness of conformal correlators using fractional Sobolev semi-norms, Chebyshev spectral decompositions, and a measure based on curvature. Finally, we establish the broader reconstructive power of this technique by extending it beyond the diagonal kinematics of the line.

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arXiv2026

Super Sum rules for Long-Range Models

We study sum rules that control the Regge limit of one-dimensional conformal field theory (CFT) correlators and relate them to dual bulk scattering processes at high energies in $\mathrm{AdS}_2$. By imposing the condition that no scattering takes place in the bulk, these sum rules single out special solutions to crossing symmetry that describe long-range models, which can be understood as free fields in AdS with boundary interactions tuned to criticality. We test these sum rules perturbatively in several distinct theories, namely the 1d long-range versions of the Ising, $O(N)$ and Lee--Yang models, and find that they correctly predict the CFT data characterising these theories. Along the way we compute for the first time the leading contributions of quadruple-twist operators to the long range Ising correlator and analyse their role in the new sum rules. Finally, we explore the consequences of imposing these sum rules in a numerical bootstrap framework and find that they lead to substantial reductions in the allowed parameter space.

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arXiv2025

A Tale of Two Uplifts: Parisi-Sourlas with Defects

Defects in conformal field theories (CFTs) play a key role in critical phenomena by modifying scaling behaviors and generating new universality classes. We introduce Parisi-Sourlas (PS) supersymmetry in the presence of extended operators and demonstrate that any $p$-dimensional defect in a CFT$_d$ can be uplifted to a defect in a PS-supersymmetric CFT$_{d+2}$. Surprisingly, there are actually two distinct uplifted defects--of dimensions $p$ and $p+2$--which reduce to the original one. We show how this reduction works for correlators with insertions both in the bulk and on the defect. As a byproduct, we find new relations between defect conformal blocks in dimensions $d$ and $d+2$. We further show that the reduction of the $p$-dimensional defect implies and extend a "global symmetry reduction" previously considered in the literature. Finally, we provide various examples of these uplifts, including perturbative computations in epsilon expansion of the uplift of the Ising magnetic line defect, as well as exact computations of observables in the four-dimensional uplift of minimal models with boundaries.

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arXiv2025

Solving 1D crossing and QFT$_2$/CFT$_1$

We provide an effective solution of the 1D crossing equation. We begin by arguing that crossing constraints can be recast in terms of bases of sum rules associated to special sets of CFT data -- extremal solutions -- which solve these constraints in a minimal way and naturally saturate positivity bounds on the space of CFTs. We conjecture, argue and check extensively that any extremal solution behaves as a generalized free field in the UV. This allows us to reconstruct the entirety of their CFT data using a rapidly convergent ``hybrid bootstrap'' method, which combines numerics and analytics. Strikingly, as we approach special corners in the space of extremal solutions we find that their CFT data can present non-trivial structure up to arbitrarily large energies. We interpret these corners as flat space limits of QFTs in AdS$_2$, which extremal solutions naturally describe. This picture allows us to bootstrap their CFT data in these limits in terms of 2d S-matrices, and conversely provide a microscopic CFT construction of the latter. Further evidence for this QFT in AdS description of extremal solutions comes from an explicit construction of bulk QFT operators solving an AdS locality problem. Concretly we show that it is possible to canonically associate one or more such operators to any extremal solution by explicitly solving for their BOE data. In the special case where this operator is the bulk stress-tensor we combine crossing and bulk locality constraints to derive stronger bounds on the OPE and BOE data, including an exact bootstrap lower bound on the central charge $C_T\geq 1/2$.

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arXiv2024

Bootstrapping conformal defect operators on a line

We study a conformal field theory with cubic anisotropic symmetry in presence of a line defect. We compute the correlators of the low lying defect operators using Feynman diagrams and derive explicit expressions for the two, three and four point defect correlators at the cubic fixed point in $4-\epsilon$ dimensions to $O(\epsilon)$. We also compute the defect $g$-function for this setup and demonstrate that this is in agreement with the $g$-theorem, which states that the $g$-function is monotonic under the renormalisation group flow along the defect. Next, we focus on conformal bootstrap techniques to determine the CFT data associated with the defect operators, which is the main objective of the paper. We utilize the framework of crossing symmetric Polyakov bootstrap and compute the averaged CFT data to $O(\epsilon)$ up to a finite number of ambiguities. We comment on unmixing the CFT data for the double trace operators at $O(\epsilon)$ and use this to compute the $O(\epsilon^2)$ data. Finally, we study these defect correlators non-perturbatively using numerical methods and isolate them near the free theory limit close to four dimensions.

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arXiv2024

Polyakov blocks for the 1D CFT mixed correlator bootstrap

We introduce manifestly crossing-symmetric expansions for arbitrary systems of 1D CFT correlators. These expansions are given in terms of certain Polyakov blocks which we define and show how to compute efficiently. Equality of OPE and Polyakov block expansions leads to sets of sum rules that any mixed correlator system must satisfy. The sum rules are diagonalized by correlators in tensor product theories of generalized free fields. We show that it is possible to do a change of a basis that diagonalizes instead mixed correlator systems involving elementary and composite operators in a single field theory. As an example, we find the first non-trivial examples of optimal bounds, saturated by the mixed correlator system $ϕ,ϕ^2$ in the theory of a single generalized free field.

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arXiv2023

Numerical Conformal bootstrap with Analytic Functionals and Outer Approximation

This paper explores the numerical conformal bootstrap in general spacetime dimensions through the lens of a distinct category of analytic functionals, previously employed in two-dimensional studies. We extend the application of these functionals to a more comprehensive backdrop, demonstrating their adaptability and efficacy in general spacetime dimensions above two. The bootstrap is implemented using the outer approximation methodology, with computations conducted in double precision. The crux of our study lies in comparing the performance of this category of analytic functionals with conventional derivatives at crossing symmetric points. It is worth highlighting that in our study, we identified some novel kinks in the scalar channel during the maximization of the gap in two-dimensional conformal field theory. Our numerical analysis indicates that these analytic functionals offer a superior performance, thereby revealing a potential alternative paradigm in the application of conformal bootstrap.

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arXiv2022

Crossing Symmetric Spinning S-matrix Bootstrap: EFT bounds

We develop crossing symmetric dispersion relations for describing 2-2 scattering of identical external particles carrying spin. This enables us to import techniques from Geometric Function Theory and study two sided bounds on low energy Wilson coefficients. We consider scattering of photons, gravitons in weakly coupled effective field theories. We provide general expressions for the locality/null constraints. Consideration of the positivity of the absorptive part leads to an interesting connection with the recently conjectured weak low spin dominance. We also construct the crossing symmetric amplitudes and locality constraints for the massive neutral Majorana fermions and parity violating photon and graviton theories. The techniques developed in this paper will be useful for considering numerical S-matrix bootstrap in the future.

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arXiv2021

Dirichlet Baths and the Not-so-Fine-Grained Page Curve

We present a doubly holographic prescription for computing entanglement entropy on a gravitating brane. It involves a Ryu-Takayanagi surface with a Dirichlet anchoring condition. In braneworld cosmology, a related approach was used previously in arXiv:2007.06551. There, the prescription naturally computed a co-moving entanglement entropy, and was argued to resolve the information paradox for a black hole living in the cosmology. In this paper, we show that the Dirichlet prescription leads to reasonable results, when applied to a recently studied wedge holography set up with a gravitating bath. The nature of the information paradox and its resolution in our Dirichlet problem have a natural understanding in terms of the strength of gravity on the two branes and at the anchoring location. By sliding the anchor to the defect, we demonstrate that the limit where gravity decouples from the anchor is continuous -- in other words, as far as island physics is considered, weak gravity on the anchor is identical to no gravity. The weak and (moderately) strong gravity regions on the brane are separated by a "Dirichlet wall". We find an intricate interplay between various extremal surfaces, with an island coming to the rescue whenever there is an information paradox. This is despite the presence of massless gravitons in the spectrum. The overall physics is consistent with the slogan that gravity becomes "more holographic", as it gets stronger. Our observations strengthen the case that the conventional Page curve is indeed of significance, when discussing the information paradox in flat space. We work in high enough dimensions so that the graviton is non-trivial, and our results are in line with the previous discussions on gravitating baths in arXiv:2005.02993 and arXiv:2007.06551.

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arXiv2021

Bulk locality for scalars and fermions with global symmetry

We count the number of independent solutions to crossing constraints of four point functions involving charged scalars and charged fermions in a CFT with large gap in the spectrum. To find the CFT data we employ recently developed analytical functionals to charged fields. We compute the corresponding higher dimensional flat space S matrices in an independent group theoretic manner and obtain agreement with our CFT counting of ambiguities. We also write down the local lagrangians explicitly. Our work lends further evidence to \cite{Heemskerk:2009pn} that any CFT with a large charge expansion and a gap in the spectrum has an AdS bulk dual.

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arXiv2021

Charging Up the Functional Bootstrap

We revisit the problem of bootstrapping CFT correlators of charged fields. After discussing in detail how bounds for uncharged fields can be recycled to the charged case, we introduce two sets of analytic functional bases for correlators on the line. The first, which we call "simple", is essentially a direct sum of analytic functionals for the uncharged case. We use it to establish very general bounds on the OPE density appearing in charged correlators. The second basis is dual to generalized free fields and we explain how it is related to a charged version of the Polyakov bootstrap. We apply these functionals to map out the space of correlators and obtain new improved bounds on the 3d Ising twist defect.

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arXiv2020

Crossing symmetry, transcendentality and the Regge behaviour of 1d CFTs

We develop the technology for Polyakov-Mellin (PM) bootstrap in one-dimensional conformal field theories (CFT$_1$). By adding appropriate contact terms, we bootstrap various effective field theories in AdS$_2$ and analytically compute the CFT data to one loop. The computation can be extended to higher orders in perturbation theory, if we ignore mixing, for any external dimension. We develop PM bootstrap for $O(N)$ theories and derive the necessary contact terms for such theories (which also involves a new higher gradient contact term absent for $N=1$). We perform cross-checks which include considering the diagonal limit of the $2d$ Ising model in terms of the $1d$ PM blocks. As an independent check of the validity of the results obtained with PM bootstrap, we propose a suitable basis of transcendental functions, which allows to fix the four-point correlators of identical scalar primaries completely, up to a finite number of ambiguities related to the number of contact terms in the PM basis. We perform this analysis both at tree level (with and without exchanges) and at one loop. We also derive expressions for the corresponding CFT data in terms of harmonic sums. Finally, we consider the Regge limit of one-dimensional correlators and derive a precise connection between the latter and the large-twist limit of CFT data. Exploiting this result, we study the crossing equation in the three OPE limits and derive some universal constraints for the large-twist limit of CFT data in Regge-bounded theories with a finite number of exchanges.

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arXiv2020

Polyakov-Mellin Bootstrap for AdS loops

We consider holographic CFTs and study their large $N$ expansion. We use Polyakov-Mellin bootstrap to extract the CFT data of all operators, including scalars, till $O(1/N^4)$. We add a contact term in Mellin space, which corresponds to an effective $ϕ^4$ theory in AdS and leads to anomalous dimensions for scalars at $O(1/N^2)$. Using this we fix $O(1/N^4)$ anomalous dimensions for double trace operators finding perfect agreement with \cite{loopal} (for $Δ_ϕ=2$). Our approach generalizes this to any dimensions and any value of conformal dimensions of external scalar field. In the second part of the paper, we compute the loop amplitude in AdS which corresponds to non-planar correlators of in CFT. More precisely, using CFT data at $O(1/N^4)$ we fix the AdS bubble diagram and the triangle diagram for the general case.

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arXiv2018

Simplifying large spin bootstrap in Mellin space

We set up the conventional conformal bootstrap equations in Mellin space and analyse the anomalous dimensions and OPE coefficients of large spin double trace operators. By decomposing the equations in terms of continuous Hahn polynomials, we derive explicit expressions as an asymptotic expansion in inverse conformal spin to any order, reproducing the contribution of any primary operator and its descendants in the crossed channel. The expressions are in terms of known mathematical functions and involve generalized Bernoulli (Norlund) polynomials and the Mack polynomials and enable us to derive certain universal properties. Comparing with the recently introduced reformulated equations in terms of crossing symmetric tree level exchange Witten diagrams, we show that to leading order in anomalous dimension but to all orders in inverse conformal spin, the equations are the same as in the conventional formulation. At the next order, the polynomial ambiguity in the Witten diagram basis is needed for the equivalence and we derive the necessary constraints for the same.

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