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2D BPS Rings from Sphere Partition Functions
High Energy Physics - Theory · Submitted 7 Dec 2017
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We present a way of computing these correlators from the sphere partition function of a deformed theory using localization.
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The abstract, sentence by sentence
We consider extremal correlation functions, involving arbitrary number of BPS (chiral or twisted chiral) operators and exactly one anti-BPS operator in 2D N=(2,2) theories. These correlators define the structure constants in the rings generated by the BPS operators with their operator product expansions. We present a way of computing these correlators from the sphere partition function of a deformed theory using localization. Relating flat space and sphere correlators is nontrivial due to operator mixing on the sphere induced by conformal anomaly. We discuss the supergravitational source of this complication and a resolution thereof. Finally, we demonstrate the process for the Quintic GLSM and the Landau-Ginzburg minimal models.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We consider extremal correlation functions, involving arbitrary number of BPS (chiral or twisted chiral) operators and exactly one anti-BPS operator in 2D N=(2,2) theories. | contribution 0.62 | 0.45 | 0.06 | asserted 0.81 |
| s1 | These correlators define the structure constants in the rings generated by the BPS operators with their operator product expansions. | background 0.47 | 0.29 | 0.18 | asserted 1.00 |
| s2 | We present a way of computing these correlators from the sphere partition function of a deformed theory using localization. | method 0.83 | 0.93 | 0.32 | asserted 1.00 |
| s3 | Relating flat space and sphere correlators is nontrivial due to operator mixing on the sphere induced by conformal anomaly. | background 0.83 | 0.25 | 0.04 | asserted 1.00 |
| s4 | We discuss the supergravitational source of this complication and a resolution thereof. | contribution 0.92 | 0.70 | 0.07 | asserted 0.79 |
| s5 | Finally, we demonstrate the process for the Quintic GLSM and the Landau-Ginzburg minimal models. | contribution 0.88 | 0.92 | 0.10 | asserted 1.00 |
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The paper
Original abstract
We consider extremal correlation functions, involving arbitrary number of BPS (chiral or twisted chiral) operators and exactly one anti-BPS operator in 2D N=(2,2) theories. These correlators define the structure constants in the rings generated by the BPS operators with their operator product expansions. We present a way of computing these correlators from the sphere partition function of a deformed theory using localization. Relating flat space and sphere correlators is nontrivial due to operator mixing on the sphere induced by conformal anomaly. We discuss the supergravitational source of this complication and a resolution thereof. Finally, we demonstrate the process for the Quintic GLSM and the Landau-Ginzburg minimal models.
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References
How the literature uses it
6 arXiv papers cite this in 11 sentences
Papers citing it · 8 sentences without a confident label
Builds on it 1 paper
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The $*$ structure on the algebra and the positivity property of the inner product are obtained as a generalization of certain properties [ref] of for protected sphere two-point functions for $(2,2)$ SCFTs [ref] .
Schur Quantization and Complex Chern-Simons theory · A Chern-Simons motivation · 0.86
Uses its method or tool 1 paper
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The operator mixing can be disentangled with the help of a Gram-Schmidt procedure, described in an analogous four-dimensional setup in [ref] and in two dimensions by [ref] .
Sphere quantization of Higgs and Coulomb branches and Analytic Symplectic Duality · Two-sphere partition functions and positivity · 0.85
Cites it as background 1 paper
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In this paper we limit ourselves to a detailed analysis of the $=2$ supersymmetric rank-1 theories.[Certain 2d and 3d supersymmetric theories also admit some similar classes of protected operators and extremal correlation functions can be defined [ref] .
Extremal Correlators and Random Matrix Theory · Introduction · 0.86
What it cites · 37 sentences without a confident label
Uses the method or tool of 4 papers
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Complex conjugation acts on the contours as follows: C_a^* = _φ_a, φ_a+1^* = _-φ_a+1, -φ_a^- = _φ_-a-1, φ_-a^- = C_-a-1^- Now, the generalization of Riemann bilinear identity [ref] gives us: [equation] The integrals are independent of $a$ and $b$, so we can evaluate the sum separately: ∑_a=1^k+1 ∑_b=1^a e^2πi/k+2[a(m+1)-b(n+1)] = e^-2πi/k+2(n+1)/1-e^-2πi/k+2(n+1) [∑_a=1^k+1 e^2πia/k+2(m+1) - ∑_a=1^k+1 e^2πia/k+2(m-n) ] Note that for $m+1 ≡0 ((k+2))$ int1 vanishes since the Gamma function in the denominator acquires a pole, and therefore the expression e1 will vanish as well.
Irregular Singularities in the H3+ WZW Model · Contour integrals · 0.98
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This implies that insertions of twisted chiral and twisted anti-chiral primaries at the North and the South pole respectively are invariant under $_A$ and the corresponding correlators can be computed by supersymmetric localization using $_A$.[Such a supercharge was used in [ref] to compute the $su(2|1)_A$-invariant partition function using localization.]
Exact Results in D=2 Supersymmetric Gauge Theories · Choice of a localizing supercharge · 0.82
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This procedure will use a generalization of the Riemann bilinear identity as explained in Appendix C of [ref] .[The general idea behind evaluating certain integrals by decomposing them over some cycles comes from Picard–Lefschetz theory, which has been used in the past to compute integrals similar to ours [ref] .
Analytic Continuation Of Chern-Simons Theory · Contour integrals · 0.88
Uses data or values from 1 paper
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In [ref] the integral of the derivative was computed to be $-4πr Y(N)$.
Sphere Partition Functions and the Zamolodchikov Metric · Supersymmetric deformations of the action · 0.95
Compares with 5 papers
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Note: After this paper was finished the paper [ref] came out with which this paper has a large overlap.
On exact correlation functions of chiral ring operators in $2d$ $\mathcal{N}=(2, 2)$ SCFTs via localization · Prologue · 0.96
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This field strength multiplet defines the twisted superpotential action:[Our normalization of this term has an extra factor of $1/2π$ compared to that of [ref] .] -t/4π ∫_^2 ^2x ∫^+ ^- - c.c. = -i/2π ∫_S^2 ^2x (ξD - /2π F_12 ) , where $ξ$ and $$ are the real FI parameter and the topological theta angle respectively: t = i ξ+ /2π .
Partition functions of N=(2,2) gauge theories on S^2 and vortices · Twisted Chiral Ring of the Quintic GLSM · 0.80
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This field strength multiplet defines the twisted superpotential action:[Our normalization of this term has an extra factor of $1/2π$ compared to that of [ref] .] -t/4π ∫_^2 ^2x ∫^+ ^- - c.c. = -i/2π ∫_S^2 ^2x (ξD - /2π F_12 ) , where $ξ$ and $$ are the real FI parameter and the topological theta angle respectively: t = i ξ+ /2π .
Exact Results in D=2 Supersymmetric Gauge Theories · Twisted Chiral Ring of the Quintic GLSM · 0.80
Cites as background 10 papers
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The Killing spinor equation of the supersymmetric $S^2$ background preserving the vector R-symmetry is found by imposing on $η$ and $η$ the follwoing constraints [ref] : η= i/2r , η= i/2r .
New Supersymmetric Localizations from Topological Gravity · Background-A · 0.86
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The Killing spinor equation of the supersymmetric $S^2$ background preserving the vector R-symmetry is found by imposing on $η$ and $η$ the follwoing constraints [ref] : η= i/2r , η= i/2r .
Partition functions of N=(2,2) gauge theories on S^2 and vortices · Background-A · 0.86
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The Killing spinor equation of the supersymmetric $S^2$ background preserving the vector R-symmetry is found by imposing on $η$ and $η$ the follwoing constraints [ref] : η= i/2r , η= i/2r .
Comments on N=(2,2) Supersymmetry on Two-Manifolds · Background-A · 0.86
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