Sphere Partition Function Research Articles

Sphere partition function and tt∗ geometry in abelian GLSM

Sphere partition function integral for abelian Calabi–Yau GLSM is equal to the pairing of the I-function with its complex conjugate, where the real structure is essentially constructed from Iritani’s integral structure on the cohomology of the target.

A double scaling for the 4d/3d reduction of N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 1 dualities

In this paper we revisit the S1 reduction of 4d N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 1 gauge theories, considering a double scaling on the radius of the circle and on the real masses arising from the global symmetries in the compactification. We discuss the implication of this double scaling for SQCD with gauge algebra of ABCD type. We then show how our prescription translates in the reduction of the 4d superconformal index to the 3d squashed three sphere partition function. This allows us to derive the expected integral identities for the 3d dualities directly from the four dimensional ones. This is relevant for the study of orthogonal SQCD, where the derivation from the 4d index is not possible in absence of the double scaling, because of a divergence due to a flat direction in the Coulomb branch of the effective theory on the circle. Furthermore, we obtain, for the even orthogonal case, a 3d duality with a quadratic fundamental monopole superpotential already discussed in the literature, that receives in this way an explanation from 4d.

S-duality in the Cardy-like limit of the superconformal index

We evaluate the superconformal index of 4d N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 SYM with gauge algebra so(2Nc + 1) in the Cardy-like limit. We then study the relation with the results obtained for the S-dual usp(2Nc), discussing the fate of S-duality in different regions of charges. We find that S-duality is preserved thanks to a non-trivial integral identity that relates the three sphere partition functions of pure 3d Chern-Simons gauge theories.

Holographic Weyl anomaly in string theory

We compute the worldsheet sphere partition function of string theory on global AdS_33 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.

Large N partition functions of 3d holographic SCFTs

We study the S1 × {Sigma}_{mathfrak{g}} topologically twisted index and the squashed sphere partition function of various 3d mathcal{N} ≥ 2 holographic superconformal field theories arising from M2-branes. Employing numerical techniques in combination with well-motivated conjectures we provide compact closed-form expressions valid to all orders in the perturbative 1/N expansion for these observables. We also discuss the holographic implications of our results for the topologically twisted index for the dual M-theory Euclidean path integral around asymptotically AdS4 solutions of 11d supergravity. In Lorentzian signature this leads to a prediction for the corrections to the Bekenstein-Hawking entropy of a class of static asymptotically AdS4 BPS black holes.

Large N partition functions of the ABJM theory

We study the large N limit of some supersymmetric partition functions of the U(N)k × U(N)−k ABJM theory computed by supersymmetric localization. We conjecture an explicit expression, valid to all orders in the large N limit, for the partition function on the U(1) × U(1) invariant squashed sphere in the presence of real masses in terms of an Airy function. Several non-trivial tests of this conjecture are presented. In addition, we derive an explicit compact expression for the topologically twisted index of the ABJM theory valid at fixed k to all orders in the 1/N expansion. We use these results to derive the topologically twisted index and the sphere partition function in the ’t Hooft limit which correspond to genus g type IIA string theory free energies to all orders in the α′ expansion. We discuss the implications of our results for holography and the physics of AdS4 black holes.

Webs of 3d \U0001d4a9 = 2 dualities with D-type superpotentials

We study 3d \U0001d4a9 = 2 dualities arising from the compactification of 4d \U0001d4a9 = 1 Usp(2n) SQCD with two antisymmetric rank-two tensors and Dk+2-type superpotential, with odd k. The analysis is carried out by using field theory methods and by checking the various steps on the three sphere partition function. Most of the results are based on a conjectural confining duality that we do not prove but that fits consistently with the web of dualities that we obtain. Along the analysis we recover dualities already claimed in the literature and we propose new ones. The final picture that emerges fits with the general scheme worked out for ordinary SQCD and for adjoint SQCD with Ak-type superpotentials.

Sphere and disk partition functions in Liouville and in matrix integrals

We compute the sphere and disk partition functions in semiclassical Liouville and analogous quantities in double-scaled matrix integrals. The quantity sphere/disk2 is unambiguous and we find a precise numerical match between the Liouville answer and the matrix integral answer. An application is to show that the sphere partition function in JT gravity is infinite.

Odd dimensional nonlocal Liouville conformal field theories

We construct Euclidean Liouville conformal field theories in odd number of dimensions. The theories are nonlocal and non-unitary with a log-correlated Liouville field, a mathcal{Q} -curvature background, and an exponential Liouville-type potential. We study the classical and quantum properties of these theories including the finite entanglement entropy part of the sphere partition function F, the boundary conformal anomaly and vertex operators’ correlation functions. We derive the analogue of the even-dimensional DOZZ formula and its semi-classical approximation.

3d dualities with decoupled sectors and brane transitions

We find exact relations among the sphere partition functions of three-dimensional mathcal{N} = 4 superconformal Chern-Simons theories with circular quiver diagrams. These relations suggest new dualities in gauge theories which are the products of circular quiver gauge theories and decoupled linear quiver gauge theories. The dualities can be interpreted as simple brane transitions in the Hanany-Witten brane construction in type IIB string theory. Interestingly, the brane transitions cannot be generated by the Hanany-Witten transition or the SL (2, ℤ) transformation. Our results can be regarded as generalization and clarification of dualities implied from Weyl group symmetries of quantum curves. To obtain the exact results, we employ the supersymmetric localization technique and the Fermi gas approach.

The two-sphere partition function from timelike Liouville theory at three-loop order

While the Euclidean two-dimensional gravitational path integral is in general highly fluctuating, it admits a semiclassical two-sphere saddle if coupled to a matter CFT with large and positive central charge. In Weyl gauge this gravity theory is known as timelike Liouville theory, and is conjectured to be a non-unitary two-dimensional CFT. We explore the semiclassical limit of timelike Liouville theory by calculating the two-sphere partition function from the perspective of the path integral to three-loop order, extending the work in [6]. We also compare our result to the conjectured all-loop sphere partition function obtained from the DOZZ formula. Since the two-sphere is the geometry of Euclidean two-dimensional de Sitter space our discussion is tied to the conjecture of Gibbons-Hawking, according to which the dS entropy is encoded in the Euclidean gravitational path integral over compact manifolds.

Sphere Partition Function of Calabi–Yau GLSMs

The sphere partition function of Calabi–Yau gauged linear sigma models (GLSMs) has been shown to compute the exact Kähler potential of the Kähler moduli space of a Calabi–Yau. We propose a universal expression for the sphere partition function evaluated in hybrid phases of Calabi–Yau GLSMs that are fibrations of Landau–Ginzburg orbifolds over some base manifold. Special cases include Calabi–Yau complete intersections in toric ambient spaces and Landau–Ginzburg orbifolds. The key ingredients that enter the expression are Givental’s I/J-functions, the Gamma class and further data associated to the hybrid model. We test the proposal for one- and two-parameter abelian GLSMs, making connections, where possible, to known results from mirror symmetry and FJRW theory.

Generating functions for Higgs/Coulomb branch operators from 1d–3d cohomological equivalence

We provide a proof for the conjectured equality of the generating function of integrated Higgs and Coulomb branch topological operators in 3d mathcal{N} ≥ 4 gauge theories and the three sphere partition function deformed by mass or FI parameters. The equality is the result of cohomological equivalence and applies to all theories in this class, including ABJM and other generalized Gaiotto-Witten models, and those without an explicit supersymmetric Lagrangian.

3d [formula omitted] dualities for SU(Nc)×U(1) Chern-Simons gauge theories

We systematically study 3d N=2 dualities for U(Nc) gauge theories with different CS levels for the abelian and the non-abelian factors. We derive such dualities by a gauging/ungauging procedure on other known dualities and by adding an opportune CS contact term for the baryonic symmetry. This allows us to follow the various steps on the three sphere partition function, obtaining exact identities among the new dual phases proposed here. We apply the procedure to various cases, namely SQCD without and with adjoint, chiral dualities and some sporadic cases arising from the real mass flow of models with an E7×U(1) enhancement of the global symmetry.

Magnetic quivers and line defects — On a duality between 3d mathcal{N} = 4 unitary and orthosymplectic quivers

Supersymmetric Sp(k) quantum chromodynamics with 8 supercharges in space-time dimensions 3 to 6 can be realised by two different Type II brane configurations in the presence of orientifolds. Consequently, two types of magnetic quivers describe the Higgs branch of the Sp(k) SQCD theory. This is a salient example of a general phenomenon: a given hyper-Kähler Higgs branch may admit several magnetic quiver constructions. It is then natural to wonder if these different magnetic quivers, which are described by 3d mathcal{N} = 4 theories, are dual theories. In this work, the unitary and orthosymplectic magnetic quiver theories are subjected to a variety of tests, providing evidence that they are IR dual to each other. For this, sphere partition function and supersymmetric indices are compared. Also, we study half BPS line defects and find interesting regularities from the viewpoints of exact results, brane configurations, and 1-form symmetry.

Quantum de Sitter horizon entropy from quasicanonical bulk, edge, sphere and topological string partition functions

Motivated by the prospect of constraining microscopic models, we calculate the exact one-loop corrected de Sitter entropy (the logarithm of the sphere partition function) for every effective field theory of quantum gravity, with particles in arbitrary spin representations. In doing so, we universally relate the sphere partition function to the quotient of a quasi-canonical bulk and a Euclidean edge partition function, given by integrals of characters encoding the bulk and edge spectrum of the observable universe. Expanding the bulk character splits the bulk (entanglement) entropy into quasinormal mode (quasiqubit) contributions. For 3D higher-spin gravity formulated as an sl(n) Chern-Simons theory, we obtain all-loop exact results. Further to this, we show that the theory has an exponentially large landscape of de Sitter vacua with quantum entropy given by the absolute value squared of a topological string partition function. For generic higher-spin gravity, the formalism succinctly relates dS, AdS± and conformal results. Holography is exhibited in quasi-exact bulk-edge cancelation.

Mirror symmetry and mixed Chern-Simons levels for Abelian 3D N=2 theories

We study the mirror symmetry of Abelian three-dimensional (3D) $N=2$ theories with mixed Chern-Simons (CS) levels by turning them into ${\mathcal{T}}_{A,N}$ theories that are defined as $N$ copies of $U(1)\ensuremath{-}[1]$ theory coupled together by mixed CS levels ${k}_{ij}$. We find that ${\mathcal{T}}_{A,N}$ theories have many mirror dual theories with different mixed CS levels and Fayet-Iliopoulos parameter. As an example, we analyze $U(1{)}_{k}+{N}_{C}\mathbf{C}+{N}_{AC}\mathbf{AC}$ theories by transforming these theories into certain ${\mathcal{T}}_{A,N}$ theories and find many equivalent effective CS levels. Finally, we analyze mirror symmetry for theories corresponding to knots. In this work we use sphere partition functions and vortex partition functions to derive dual theories.

Three dimensional mirror symmetry beyond ADE quivers and Argyres-Douglas theories

Mirror symmetry, a three dimensional mathcal{N} = 4 IR duality, has been studied in detail for quiver gauge theories of the ADE-type (as well as their affine versions) with unitary gauge groups. The A-type quivers (also known as linear quivers) and the associated mirror dualities have a particularly simple realization in terms of a Type IIB system of D3-D5-NS5-branes. In this paper, we present a systematic field theory prescription for constructing 3d mirror pairs beyond the ADE quiver gauge theories, starting from a dual pair of A-type quivers with unitary gauge groups. The construction involves a certain generalization of the S and the T operations, which arise in the context of the SL(2, ℤ) action on a 3d CFT with a U(1) 0-form global symmetry. We implement this construction in terms of two supersymmetric observables — the round sphere partition function and the superconformal index on S2 × S1. We discuss explicit examples of various (non-ADE) infinite families of mirror pairs that can be obtained in this fashion. In addition, we use the above construction to conjecture explicit 3d mathcal{N} = 4 Lagrangians for 3d SCFTs, which arise in the deep IR limit of certain Argyres-Douglas theories compactified on a circle.

New modular invariants in mathcal{N} = 4 Super-Yang-Mills theory

We study modular invariants arising in the four-point functions of the stress tensor multiplet operators of the mathcal{N} = 4 SU(N) super-Yang-Mills theory, in the limit where N is taken to be large while the complexified Yang-Mills coupling τ is held fixed. The specific four-point functions we consider are integrated correlators obtained by taking various combinations of four derivatives of the squashed sphere partition function of the mathcal{N} = 2∗ theory with respect to the squashing parameter b and mass parameter m, evaluated at the values b = 1 and m = 0 that correspond to the mathcal{N} = 4 theory on a round sphere. At each order in the 1/N expansion, these fourth derivatives are modular invariant functions of (τ, overline{tau} ). We present evidence that at half-integer orders in 1/N , these modular invariants are linear combinations of non-holomorphic Eisenstein series, while at integer orders in 1/N, they are certain “generalized Eisenstein series” which satisfy inhomogeneous Laplace eigenvalue equations on the hyperbolic plane. These results reproduce known features of the low-energy expansion of the four-graviton amplitude in type IIB superstring theory in ten-dimensional flat space and have interesting implications for the structure of the analogous expansion in AdS5× S5.

Localization of the gauged linear sigma model for KK5-branes

Abstract We study quantum aspects of the target space of the non-linear sigma model, which is a low-energy effective theory of the gauged linear sigma model (GLSM). As such, we especially compute the exact sphere partition function of the GLSM for KK$5$-branes whose background geometry is a Taub–NUT space, using the supersymmetric localization technique on the Coulomb branch. From the sphere partition function, we distill the world-sheet instanton effects. In particular, we show that, concerning the single-centered Taub–NUT space, instanton contributions exist only if the asymptotic radius of the $S^1$ fiber in the Taub–NUT space is zero.