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Bounds for State Degeneracies in 2D Conformal Field Theory
High Energy Physics - Theory · Submitted 5 Jul 2010
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We show that the entropy at inverse temperature 2 pi satisfies a universal lower bound, and we enumerate the principal obstacles to deriving upper bounds on entropies or quantum mechanical degeneracies for fully general CFTs.
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The abstract, sentence by sentence
In this note we explore the application of modular invariance in 2-dimensional CFT to derive universal bounds for quantities describing certain state degeneracies, such as the thermodynamic entropy, or the number of marginal operators. We show that the entropy at inverse temperature 2 pi satisfies a universal lower bound, and we enumerate the principal obstacles to deriving upper bounds on entropies or quantum mechanical degeneracies for fully general CFTs. We then restrict our attention to infrared stable CFT with moderately low central charge, in addition to the usual assumptions of modular invariance, unitarity and discrete operator spectrum. For CFT in the range c_left + c_right < 48 with no relevant operators, we are able to prove an upper bound on the thermodynamic entropy at inverse temperature 2 pi. Under the same conditions we also prove that a CFT can have a number of marginal deformations no greater than ((c_left + c_right) / (48 - c_left - c_right)) e^(4 Pi) - 2.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | In this note we explore the application of modular invariance in 2-dimensional CFT to derive universal bounds for quantities describing certain state degeneracies, such as the thermodynamic entropy, or the number of marginal operators. | contribution 0.96 | 0.80 | 0.14 | asserted 0.67 |
| s1 | We show that the entropy at inverse temperature 2 pi satisfies a universal lower bound, and we enumerate the principal obstacles to deriving upper bounds on entropies or quantum mechanical degeneracies for fully general CFTs. | result 0.86 | 0.96 | 0.38 | asserted 1.00 |
| s2 | We then restrict our attention to infrared stable CFT with moderately low central charge, in addition to the usual assumptions of modular invariance, unitarity and discrete operator spectrum. | limitation 0.44 | 0.70 | 0.04 | asserted 0.65 |
| s3 | For CFT in the range c_left + c_right < 48 with no relevant operators, we are able to prove an upper bound on the thermodynamic entropy at inverse temperature 2 pi. | result 0.99 | 0.94 | 0.27 | asserted 1.00 |
| s4 | Under the same conditions we also prove that a CFT can have a number of marginal deformations no greater than ((c_left + c_right) / (48 - c_left - c_right)) e^(4 Pi) - 2. | result 1.00 | 0.95 | 0.27 | asserted 1.00 |
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The paper
Original abstract
In this note we explore the application of modular invariance in 2-dimensional CFT to derive universal bounds for quantities describing certain state degeneracies, such as the thermodynamic entropy, or the number of marginal operators. We show that the entropy at inverse temperature 2 pi satisfies a universal lower bound, and we enumerate the principal obstacles to deriving upper bounds on entropies or quantum mechanical degeneracies for fully general CFTs. We then restrict our attention to infrared stable CFT with moderately low central charge, in addition to the usual assumptions of modular invariance, unitarity and discrete operator spectrum. For CFT in the range c_left + c_right < 48 with no relevant operators, we are able to prove an upper bound on the thermodynamic entropy at inverse temperature 2 pi. Under the same conditions we also prove that a CFT can have a number of marginal deformations no greater than ((c_left + c_right) / (48 - c_left - c_right)) e^(4 Pi) - 2.
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References
How the literature uses it
23 arXiv papers cite this in 28 sentences
Papers citing it · 14 sentences without a confident label
Builds on it 1 paper
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In section 4, to obtain constraints also for smaller Hodge numbers, we generalize the medium temperature expansion introduced in [ref] and explore consequences of the modular invariance using differential equations on the partition function.
Modular Constraints on Calabi-Yau Compactifications · Introduction · 0.85
Uses its method or tool 2 papers
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To explore consequences of this equality, we use the medium temperature expansion developed in [ref] .
Modular Constraints on Calabi-Yau Compactifications · Differential operators and the medium temperature expansion · 1.00
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The idea is to apply methods used in the modern bootstrap to this amplitude, as has been done in [ref] .
Constraints on 2d CFT partition functions · Introduction · 0.87
Compares with it 1 paper
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The bound ([ref]) was derived [ref] by exploiting the modular invariance of the fundamental string partition function of Type II compactification (with $X_IIA$ as the target space), while the bound ([ref]) is due to the modular property of $Φ$, the generating function of the helicity supertraces of vertical D4–D2–D0 BPS bound states on $X_IIA$.
Modular Forms as Classification Invariants of 4D N=2 Heterotic--IIA Dual Vacua · Lower Bounds on Euler Numbers · 0.93
Cites it as background 9 papers
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Consequences of invariance of the partition function $Z(τ,τ̅)$ under the modular $S$-transformation have been studied in depth (e.g., [ref] ).
Universal Bounds in Even-Spin CFTs · Modular invariance · 1.00
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The modern numerical modular bootstrap was initiated in [ref] and has since been generalized in many different ways in 1+1d [ref] .[Modular consistency outside the 1+1d CFT context has also been effectively utilized in, for example, [ref] .]
Bootstrapping Non-Invertible Symmetries · Introduction · 0.90
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Modular invariance was used in [ref] to bound the number of operators.
Universal Bounds on Operator Dimensions in General 2D Conformal Field Theories · Introduction · 0.80
What it cites · 14 sentences without a confident label
Cites as background 15 papers
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We refer the reader to [ref] for details.
A Universal Inequality for CFT and Quantum Gravity · Review of previous work · 0.89
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This question is explored under various simplifying assumptions in different articles: in [ref] , the two-dimensional case is explored under the assumption of holomorphic factorization of the Hilbert space; in [ref] the two-dimensional case is explored again and extended superconformal symmetry is assumed; in [ref] the authors examine the maximum possible dimension of the lowest-dimension operator appearing in the operator product expansion of two scalar operators whose dimensions are taken as given; in [ref] the two-dimensional case is probed once again and the dimension of the lowest primary operator is bounded in a completely general CFT satisfying the minimal properties of unitarity and discrete operator spectrum.
Three-Dimensional Gravity Revisited · Introduction · 0.89
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This question is explored under various simplifying assumptions in different articles: in [ref] , the two-dimensional case is explored under the assumption of holomorphic factorization of the Hilbert space; in [ref] the two-dimensional case is explored again and extended superconformal symmetry is assumed; in [ref] the authors examine the maximum possible dimension of the lowest-dimension operator appearing in the operator product expansion of two scalar operators whose dimensions are taken as given; in [ref] the two-dimensional case is probed once again and the dimension of the lowest primary operator is bounded in a completely general CFT satisfying the minimal properties of unitarity and discrete operator spectrum.
Extremal N=(2,2) 2D Conformal Field Theories and Constraints of Modularity · Introduction · 0.89
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