arXiv subjects

Sridip Pal

Publications and source records attributed to Sridip Pal.

At least 19 recordsLinked to original sources

arXiv2026

From Z to all: cyclic sum rules for thermal CFT partition functions

The high-temperature expansion of twisted $S^{d-1}\times S^1$ partition functions of CFT$_d$, $d = 2κ$, has recently been shown to obey an infinite tower of universal constraints. We determine these constraints explicitly and demonstrate that they can be packaged into a single equation involving a large conformal diffeomorphism isomorphic to $\mathbb{Z}_{κ+1}$, which cyclically permutes the thermal circle and the rotation angles. Our equation asserts that the $\mathbb{Z}_{κ+1}$-invariant part of $\ln Z/β$ is a constant fixed by the a-anomaly. Our derivation assumes CP-invariance and locality of the thermal effective action, whose structure we constrain using an algebraic formalism starting from embedding space.

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arXiv2026

Resource-Theoretic Quantifiers of Weak and Strong Symmetry Breaking: Strong Entanglement Asymmetry and Beyond

Quantifying how much a quantum state breaks a symmetry is essential for characterizing phases, nonequilibrium dynamics, and open-system behavior. For mixed states, however, conventional diagnostics of weak symmetry breaking can miss a stronger form of symmetry breaking associated with the possibility of exchanging conserved charges with an environment. We develop a resource theory of strong symmetry breaking by identifying the appropriate free states and free operations, and, for channels on a fixed system, by showing that the latter are precisely the operations realizable without exchanging the conserved charge with the environment. We systematically construct measures of strong symmetry breaking, including strong entanglement asymmetry and covariance matrices of symmetry generators, for a broad class of symmetry groups. We further completely characterize i.i.d. convertibility between arbitrary states under any compact Lie group strong symmetry. This identifies the combinations of resource-theoretically valid measures that quantify strong symmetry breaking for arbitrary states, including all mixed states. In particular, for $U(1)$ symmetry, if the states have positive variances of the conserved quantity and equal strong-symmetry periods, the conversion rate is completely determined by the variance ratio whenever the input is pure or the output is weak symmetric. Thus, for these state conversions, the variance of the conserved quantity plays an operational role analogous to that of entanglement entropy in entanglement theory or quantum Fisher information in the resource theory of weak asymmetry. We further show how weak symmetry breaking is irreversibly converted into strong symmetry breaking in open-system dynamics. We illustrate the framework with examples from quantum field theory, strong-to-weak spontaneous symmetry breaking, and a strong-symmetry analogue of Mpemba-type dynamics.

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arXiv2026

From $Z$ to $a$: High-temperature relations, subleading semi-universality, and conformal anomalies

The free energy of any CFT, $ \ln Z(β; ω_i)$, admits two expansions: high temperature ($β\rightarrow 0$) and fast rotation ($ω_i \rightarrow 1$). We demonstrate that in $4d$, locality of the thermal effective action forces $\ln Z$ to take a simple analytic form at all orders in the high temperature expansion, and further imposes an infinite number of sharp relations on the coefficients in this expansion. All are homogeneous, except at order $β^1$ due to the Weyl anomaly. From this, the $a$-anomaly can be extracted from the partition function. The relations resum in the fast-spinning expansion into differential equations in $β$ obeyed by the semi-universal limit and its corrections. We verify the relations in a variety of CFTs. We generalize to any even $d$, but find no similar relations at odd $d$.

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arXiv2026

Semi-universality of CFT$_d$ entropy at large spin

The thermal partition function, $Z$, of a $CFT_d$ on $S^{d-1}$ is parameterized by the inverse temperature $β$ along with $\lfloor d/2\rfloor$ angular velocities $ω_i$. In this paper, we investigate the behaviour of this partition function when $n$ of the $ω_i$ are scaled to unity (the largest allowed value) at fixed values of the other $(\lfloor d/2\rfloor-n)$ angular velocities. We argue that $\ln Z$ develops a simple pole in $(1-ω_i)$ for each $ω_i$ that is scaled to unity. The residue of this product of poles is a theory dependent (so non-universal) function of $β$ and the fixed angular velocities. The inverse Laplace transformation of this partition function constrains the functional form of the field theory entropy as a function of charges in a limit in which angular momenta and the twist are scaled as follows. While $n$ special angular momenta $J_1\ldots J_n$ are scaled to infinity, the twist and the other angular momenta - collectively denoted $x_i$ - are also taken to infinity but at the slower rate that ensures that the scaled charges $x_i/(J_1 J_2 \ldots J_n)^{\frac{1}{n+1}}$ are held fixed. In this limit, we demonstrate that the scaled entropy $S/(J_1 J_2 \ldots J_n)^{\frac{1}{n+1}}$ depends only on the $\lfloor d/2\rfloor-n+1$ scaled charges defined above (the precise form of this dependence is non-universal). We verify our predictions (and compute all non-universal functions) in the case of free scalar theories (which show surprisingly rich behaviour) as well as large $N$, strongly coupled ${\cal N}=4$ Yang Mills theory. The last theory is analyzed in the bulk via the AdS/CFT correspondence. In the scaling limit described above, its phase diagram displays sharp phase transitions between black hole, grey galaxy, and thermal gas phases.

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arXiv2025

Weyl bound for trilinear periods via conformal bootstrap

Let $f_1,f_2$ be holomorphic modular forms of the same weight for a cocompact lattice $Γ< \mathrm{PSL}_2(\mathbf{R})$. We estimate the rate of decay of the coefficients in the expansion of $f_1\overline{f_2}$ in a Laplace eigenbasis. By specializing our main theorem to the case where $Γ$ is arithmetic, we obtain new instances of the Weyl bound for triple product $L$-functions in the spectral aspect. Our method builds on the conformal bootstrap in physics.

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arXiv2025

Universality of the microcanonical entropy at large spin

We consider rigorous consequences of modular invariance for two-dimensional unitary non-rational CFTs with $c > 1$. Simple estimates for the torus partition function can lead to remarkably strong results. We show in particular that the spectral density of spin-$J$ operators must grow like $\exp\left( π\sqrt{\frac{2}{3}(c-1) J} \right)/\sqrt{2J}$ in any twist interval at or above $(c-1)/12$, with a known twist-dependent prefactor. This proves that the large $J$ spectrum becomes dense even without averaging over spins. For twists below $(c-1)/12$ we establish that the growth must be strictly slower. Finally, we estimate how fast the maximal gap between two spin-$J$ operators must go to zero as $J$ becomes large.

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arXiv2025

A universal inequality on the unitary 2D CFT partition function

We prove the conjecture proposed by Hartman, Keller and Stoica [HKS14]: the grand-canonical free energy of a unitary 2D CFT with a sparse spectrum below the scaling dimension $\frac{c}{12}+ε$ and below the twist $\frac{c}{12}$ is universal in the large $c$ limit for all $β_Lβ_R \neq 4π^2$. The technique of the proof allows us to derive a one-parameter (with parameter $α\in(0,1]$) family of universal inequalities on the unitary 2D CFT partition function with general central charge $c\geqslant 0$, using analytical modular bootstrap. We derive an iterative equation for the domain of validity of the inequality on the $(β_L,β_R)$ plane. The infinite iteration of this equation gives the boundary of maximal-validity domain, which depends on the parameter $α$ in the inequality. In the $c \to \infty$ limit, with the additional assumption of a sparse spectrum below the scaling dimension $\frac{c}{12} + ε$ and the twist $\frac{αc}{12}$ (with $α\in (0,1]$ fixed), our inequality shows that the grand-canonical free energy exhibits a universal large $c$ behavior in the maximal-validity domain. This domain, however, does not cover the entire $(β_L, β_R)$ plane, except in the case of $α= 1$. For $α= 1$, this proves the conjecture proposed by [HKS14], and for $α< 1$, it quantifies how sparseness in twist affects the regime of universality. Furthermore, this implies a precise lower bound on the temperature of near-extremal BTZ black holes, above which we can trust the black hole thermodynamics.

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arXiv2025

Universality of Rényi Entropy in Conformal Field Theory

We use the thermal effective theory to prove that, for the vacuum state in any conformal field theory in $d$ dimensions, the $n$-th Rényi entropy $S_A^{(n)}$ behaves as $S_A^{(n)} = \frac{f}{(2πn)^{d-1}} \frac{ {\rm Area}(\partial A)}{(d-2)ε^{d-2}}\left(1+O(n)\right)$ in the $n \rightarrow 0$ limit when the boundary of the entanglement domain $A$ is spherical with the UV cutoff $ε$.The theory dependence is encapsulated in the cosmological constant $f$ in the thermal effective action. Using this result, we estimate the density of states for large eigenvalues of the modular Hamiltonian for the domain $A$. In two dimensions, we can use the hot spot idea to derive more powerful formulas valid for arbitrary positive $n$. We discuss the difference between two and higher dimensions and clarify the applicability of the hot spot idea. We also use the thermal effective theory to derive an analog of the Cardy formula for boundary operators in higher dimensions.

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arXiv2024

Entanglement asymmetry and symmetry defects in boundary conformal field theory

A state in a quantum system with a given global symmetry, $G$, can be sensitive to the presence of boundaries, which may either preserve or break this symmetry. In this work, we investigate how conformal invariant boundary conditions influence the $G-$symmetry breaking through the lens of the entanglement asymmetry, a quantifier of the "distance" between a symmetry-broken state and its symmetrized counterpart. By leveraging 2D boundary conformal field theory (BCFT), we investigate the symmetry breaking for both finite and compact Lie groups. Beyond the leading order term, we also compute the subleading corrections in the subsystem size, highlighting their dependence on the symmetry group $G$ and the BCFT operator content. We further explore the entanglement asymmetry following a global quantum quench, where a symmetry-broken state evolves under a symmetry-restoring Hamiltonian. In this dynamical setting, we compute the entanglement asymmetry by extending the method of images to a BCFT with non-local objects such as invertible symmetry defects.

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arXiv2024

Angular fractals in thermal QFT

We show that thermal effective field theory controls the long-distance expansion of the partition function of a $d$-dimensional QFT, with an insertion of any finite-order spatial isometry. Consequently, the thermal partition function on a sphere displays a fractal-like structure as a function of angular twist, reminiscent of the behavior of a modular form near the real line. As an example application, we find that for CFTs, the effective free energy of even-spin minus odd-spin operators at high temperature is smaller than the usual free energy by a factor of $1/2^d$. Near certain rational angles, the partition function receives subleading contributions from "Kaluza-Klein vortex defects" in the thermal EFT, which we classify. We illustrate our results with examples in free and holographic theories, and also discuss nonperturbative corrections from worldline instantons.

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arXiv2024

Automorphic Spectra and the Conformal Bootstrap

We describe a new method for constraining Laplacian spectra of hyperbolic surfaces and 2-orbifolds. The main ingredient is consistency of the spectral decomposition of integrals of products of four automorphic forms. Using a combination of representation theory of $\mathrm{PSL}_2(\mathbb{R})$ and semi-definite programming, the method yields rigorous upper bounds on the Laplacian spectral gap. In several examples, the bound is nearly sharp. For instance, our bound on all genus-2 surfaces is $λ_1\leq 3.8388976481$, while the Bolza surface has $λ_1\approx 3.838887258$. The bounds also allow us to determine the set of spectral gaps attained by all hyperbolic 2-orbifolds. Our methods can be generalized to higher-dimensional hyperbolic manifolds and to yield stronger bounds in the two-dimensional case. The ideas were closely inspired by modern conformal bootstrap.

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arXiv2023

Holographic Weyl anomaly in string theory

We compute the worldsheet sphere partition function of string theory on global AdS$_3$ with pure NS-NS flux. Because of an unfixed Möbius symmetry on the worldsheet, there is a cancellation of infinities and only a part of the answer is unambiguous. We show that it precisely reproduces the holographic Weyl anomaly and the ambiguous terms correspond to the possible counterterms of the boundary CFT.

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arXiv2023

Bounds on spectral gaps of Hyperbolic spin surfaces

We describe a method for constraining Laplacian and Dirac spectra of two dimensional compact orientable hyperbolic spin manifolds and orbifolds. The key ingredient is an infinite family of identities satisfied by the spectra. These spectral identities follow from the consistency between 1) the spectral decomposition of functions on the spin bundle into irreducible representations of $\mathrm{SL}(2,\mathbb{R})$ and 2) associativity of pointwise multiplication of functions. Applying semidefinite programming methods to our identities produces rigorous upper bounds on the Laplacian spectral gap as well as on the Dirac spectral gap conditioned on the former. In several examples, our bounds are nearly sharp; a numerical algorithm based on the Selberg trace formula shows that the $[0;3,3,5]$ orbifold, a particular surface with signature $[1;3]$, and the Bolza surface nearly saturate the bounds at genus $0$, $1$ and $2$ respectively. Under additional assumptions on the number of harmonic spinors carried by the spin-surface, we obtain more restrictive bounds on the Laplacian spectral gap. In particular, these bounds apply to hyperelliptic surfaces. We also determine the set of Laplacian spectral gaps attained by all compact orientable two-dimensional hyperbolic spin orbifolds. We show that this set is upper bounded by $12.13798$; this bound is nearly saturated by the $[0;3,3,5]$ orbifold, whose first non-zero Laplacian eigenvalue is $λ^{(0)}_1\approx 12.13623$.

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arXiv2023

Symmetry-resolved Entanglement Entropy, Spectra & Boundary Conformal Field Theory

We perform a comprehensive analysis of the symmetry-resolved (SR) entanglement entropy (EE) for one single interval in the ground state of a $1+1$D conformal field theory (CFT), that is invariant under an arbitrary finite or compact Lie group, $G$. We utilize the boundary CFT approach to study the total EE, which enables us to find the universal leading order behavior of the SREE and its first correction, which explicitly depends on the irreducible representation under consideration and breaks the equipartition of entanglement. We present two distinct schemes to carry out these computations. The first relies on the evaluation of the charged moments of the reduced density matrix. This involves studying the action of the defect-line, that generates the symmetry, on the boundary states of the theory. This perspective also paves the way for discussing the infeasibility of studying symmetry resolution when an anomalous symmetry is present. The second scheme draws a parallel between the SREE and the partition function of an orbifold CFT. This approach allows for the direct computation of the SREE without the need to use charged moments. From this standpoint, the infeasibility of defining the symmetry-resolved EE for an anomalous symmetry arises from the obstruction to gauging. Finally, we derive the symmetry-resolved entanglement spectra for a CFT invariant under a finite symmetry group. We revisit a similar problem for CFT with compact Lie group, explicitly deriving an improved formula for $U(1)$ resolved entanglement spectra. Using the Tauberian formalism, we can estimate the aforementioned EE spectra rigorously by proving an optimal lower and upper bound on the same. In the abelian case, we perform numerical checks on the bound and find perfect agreement.

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arXiv2023

Universal fine grained asymptotics of free and weakly coupled Quantum Field Theory

We give a rigorous proof that in any free quantum field theory with a finite group global symmetry $\mathrm{G}$, on a compact spatial manifold, at sufficiently high energy, the density of states $ρ_α(E)$ for each irreducible representation $α$ of $\mathrm{G}$ obeys a universal formula as conjectured by Harlow and Ooguri. We further prove that this continues to hold in a weakly coupled quantum field theory, given an appropriate scaling of the coupling with temperature. This generalizes similar results that were previously obtained in $(1+1)$-D to higher spacetime dimension. We discuss the role of averaging in the density of states, and we compare and contrast with the case of continuous group $\mathrm{G}$, where we prove a universal, albeit different, behavior.

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arXiv2023

Spectral Bounds on Hyperbolic 3-Manifolds: Associativity and the Trace Formula

We constrain the low-energy spectra of Laplace operators on closed hyperbolic manifolds and orbifolds in three dimensions, including the standard Laplace-Beltrami operator on functions and the Laplacian on powers of the cotangent bundle. Our approach employs linear programming techniques to derive rigorous bounds by leveraging two types of spectral identities. The first type, inspired by the conformal bootstrap, arises from the consistency of the spectral decomposition of the product of Laplace eigensections, and involves the Laplacian spectra as well as integrals of triple products of eigensections. We formulate these conditions in the language of representation theory of $\mathrm{PSL}_2(\mathbb{C})$ and use them to prove upper bounds on the first and second Laplacian eigenvalues. The second type of spectral identities follows from the Selberg trace formula. We use them to find upper bounds on the spectral gap of the Laplace-Beltrami operator on hyperbolic 3-orbifolds, as well as on the systole length of hyperbolic 3-manifolds, as a function of the volume. Further, we prove that the spectral gap $λ_1$ of the Laplace-Beltrami operator on all closed hyperbolic 3-manifolds satisfies $λ_1 < 47.32$. Along the way, we use the trace formula to estimate the low-energy spectra of a large set of example orbifolds and compare them with our general bounds, finding that the bounds are nearly sharp in several cases.

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arXiv2023

Lightcone Modular Bootstrap and Tauberian Theory: A Cardy-like Formula for Near-extremal Black Holes

We show that for a unitary modular invariant 2D CFT with central charge $c>1$ and having a nonzero twist gap in the spectrum of Virasoro primaries, for sufficiently large spin $J$, there always exist spin-$J$ operators with twist falling in the interval $(\frac{c-1}{12}-\varepsilon,\frac{c-1}{12}+\varepsilon)$ with $\varepsilon=O(J^{-1/2}\log J)$. We establish that the number of Virasoro primary operators in such a window has a Cardy-like i.e. $\exp\left(2π\sqrt{\frac{(c-1)J}{6}}\right)$ growth. We make further conjectures on potential generalization to CFTs with conserved currents. A similar result is proven for a family of holographic CFTs with the twist gap growing linearly in $c$ and a uniform boundedness condition, in the regime $J\gg c^3\gg1$. From the perspective of near-extremal rotating BTZ black holes (without electric charge), our result is valid when the Hawking temperature is much lower than the "gap temperature".

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arXiv2023

Twist accumulation in conformal field theory. A rigorous approach to the lightcone bootstrap

We prove that in any unitary CFT, a twist gap in the spectrum of operator product expansion (OPE) of identical scalar primary operators (i.e. $ϕ\times ϕ$) implies the existence of a family of primary operators $\mathcal{O}_{τ, \ell}$ with spins $\ell \rightarrow \infty$ and twists $τ\rightarrow 2 Δ_ϕ$ in the same OPE spectrum. A similar twist-accumulation result is proven for any two-dimensional Virasoro-invariant, modular-invariant, unitary CFT with a normalizable vacuum and central charge $c > 1$, where we show that a twist gap in the spectrum of Virasoro primaries implies the existence of a family of Virasoro primaries $\mathcal{O}_{h, \bar{h}}$ with $h \rightarrow \infty$ and $\bar{h} \rightarrow \frac{c - 1}{24}$ (the same is true with $h$ and $\bar{h}$ interchanged). We summarize the similarity of the two problems and propose a general formulation of the lightcone bootstrap.

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