scholarly journals Partition functions of superconformal Chern-Simons theories from Fermi gas approach

2014 ◽  
Vol 2014 (11) ◽  
Author(s):  
Sanefumi Moriyama ◽  
Tomoki Nosaka
Keyword(s):  
Fermi Gas ◽  
Partition Functions ◽  
Chern Simons

scholarly journals Fermi gas approach to general rank theories and quantum curves

2020 ◽  
Vol 2020 (10) ◽  
Author(s):  
Naotaka Kubo
Keyword(s):  
Matrix Models ◽  
Critical Role ◽  
Three Dimensional ◽  
Fermi Gas ◽  
Fermi Gases ◽  
Partition Functions ◽  
Density Matrices ◽  
General Rank ◽  
Chern Simons ◽  
The Ideal

Abstract It is known that matrix models computing the partition functions of three-dimensional $$ \mathcal{N} $$ N = 4 superconformal Chern-Simons theories described by circular quiver diagrams can be written as the partition functions of ideal Fermi gases when all the nodes have equal ranks. We extend this approach to rank deformed theories. The resulting matrix models factorize into factors depending only on the relative ranks in addition to the Fermi gas factors. We find that this factorization plays a critical role in showing the equality of the partition functions of dual theories related by the Hanany-Witten transition. Furthermore, we show that the inverses of the density matrices of the ideal Fermi gases can be simplified and regarded as quantum curves as in the case without rank deformations. We also comment on four nodes theories using our results.

scholarly journals Poincaré series, 3d gravity and averages of rational CFT

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
Viraj Meruliya ◽  
Sunil Mukhi ◽  
Palash Singh
Keyword(s):  
Poincaré Series ◽  
Simons Theory ◽  
Partition Functions ◽  
Moody Algebra ◽  
Modular Invariant ◽  
Poincare Series ◽  
Single Genus ◽  
3D Gravity ◽  
Chern Simons ◽  
Chern Simons Theory

Abstract We investigate the Poincaré series approach to computing 3d gravity partition functions dual to Rational CFT. For a single genus-1 boundary, we show that for certain infinite sets of levels, the SU(2)k WZW models provide unitary examples for which the Poincaré series is a positive linear combination of two modular-invariant partition functions. This supports the interpretation that the bulk gravity theory (a topological Chern-Simons theory in this case) is dual to an average of distinct CFT’s sharing the same Kac-Moody algebra. We compute the weights of this average for all seed primaries and all relevant values of k. We then study other WZW models, notably SU(N)1 and SU(3)k, and find that each class presents rather different features. Finally we consider multiple genus-1 boundaries, where we find a class of seed functions for the Poincaré sum that reproduces both disconnected and connected contributions — the latter corresponding to analogues of 3-manifold “wormholes” — such that the expected average is correctly reproduced.

scholarly journals Three-dimensional 𝒩 = 2 supersymmetric gauge theories and partition functions on Seifert manifolds: A review

2019 ◽  
Vol 34 (23) ◽  
pp. 1930011 ◽  
Author(s):  
Cyril Closset ◽  
Heeyeon Kim
Keyword(s):  
Correlation Functions ◽  
Gauge Theories ◽  
Three Dimensional ◽  
Partition Functions ◽  
Exact Results ◽  
Strongly Coupled ◽  
Seifert Manifolds ◽  
Recent Developments ◽  
Supersymmetric Gauge ◽  
Chern Simons

We give a pedagogical introduction to the study of supersymmetric partition functions of 3D [Formula: see text] supersymmetric Chern–Simons-matter theories (with an [Formula: see text]-symmetry) on half-BPS closed three-manifolds — including [Formula: see text], [Formula: see text], and any Seifert three-manifold. Three-dimensional gauge theories can flow to nontrivial fixed points in the infrared. In the presence of 3D [Formula: see text] supersymmetry, many exact results are known about the strongly-coupled infrared, due in good part to powerful localization techniques. We review some of these techniques and emphasize some more recent developments, which provide a simple and comprehensive formalism for the exact computation of half-BPS observables on closed three-manifolds (partition functions and correlation functions of line operators). Along the way, we also review simple examples of 3D infrared dualities. The computation of supersymmetric partition functions provides exceedingly precise tests of these dualities.

scholarly journals Exact results and Schur expansions in quiver Chern-Simons-matter theories

2020 ◽  
Vol 2020 (10) ◽  
Author(s):  
Leonardo Santilli ◽  
Miguel Tierz
Keyword(s):  
Simons Theory ◽  
Partition Functions ◽  
Wilson Loops ◽  
Model Formulation ◽  
Schur Polynomials ◽  
Three Sphere ◽  
The Matrix ◽  
The Masses ◽  
Chern Simons ◽  
Chern Simons Theory

Abstract We study several quiver Chern-Simons-matter theories on the three-sphere, combining the matrix model formulation with a systematic use of Mordell’s integral, computing partition functions and checking dualities. We also consider Wilson loops in ABJ(M) theories, distinguishing between typical (long) and atypical (short) representations and focusing on the former. Using the Berele-Regev factorization of supersymmetric Schur polynomials, we express the expectation value of the Wilson loops in terms of sums of observables of two factorized copies of U(N ) pure Chern-Simons theory on the sphere. Then, we use the Cauchy identity to study the partition functions of a number of quiver Chern-Simons-matter models and the result is interpreted as a perturbative expansion in the parameters tj = −e2πmj , where mj are the masses. Through the paper, we incorporate different generalizations, such as deformations by real masses and/or Fayet-Iliopoulos parameters, the consideration of a Romans mass in the gravity dual, and adjoint matter.

Chern–Simons gauge theory and symplectic quantum mechanics

2015 ◽  
Vol 93 (9) ◽  
pp. 971-973
Author(s):  
Lisa Jeffrey
Keyword(s):  
Quantum Mechanics ◽  
Gauge Theory ◽  
Partition Function ◽  
Partition Functions ◽  
Action Functional ◽  
Symplectic Action ◽  
Chern Simons

We describe the relation between the Chern–Simons gauge theory partition function and the partition function defined using the symplectic action functional as the Lagrangian. We show that the partition functions obtained using these two Lagrangians agree, and we identify the semiclassical formula for the partition function defined using the symplectic action functional. We also compute the semiclassical formulas for the partition functions obtained using the two different Lagrangians: the Chern–Simons functional and the symplectic action functional.

On Chern-Simons and WZW Partition Functions

1999 ◽  
Vol 200 (3) ◽  
pp. 685-698
Author(s):  
Dana Stanley Fine
Keyword(s):  
Partition Functions ◽  
Chern Simons

scholarly journals On partition functions of refined Chern-Simons theories on S3

2021 ◽  
Vol 2021 (10) ◽  
Author(s):  
M.Y. Avetisyan ◽  
R.L. Mkrtchyan
Keyword(s):  
Partition Function ◽  
Gauge Group ◽  
Topological Strings ◽  
Simons Theory ◽  
Partition Functions ◽  
Coupling Constant ◽  
Sine Functions ◽  
Chern Simons ◽  
Arbitrary Gauge ◽  
Chern Simons Theory

Abstract We present a new expression for the partition function of the refined Chern-Simons theory on S3 with an arbitrary gauge group, which is explicitly equal to 1 when the coupling constant is zero. Using this form of the partition function we show that the previously known Krefl-Schwarz representation of the partition function of the refined Chern-Simons theory on S3 can be generalized to all simply laced algebras.For all non-simply laced gauge algebras, we derive similar representations of that partition function, which makes it possible to transform it into a product of multiple sine functions aiming at the further establishment of duality with the refined topological strings.

scholarly journals Topological entanglement and hyperbolic volume

2021 ◽  
Vol 2021 (10) ◽  
Author(s):  
Aditya Dwivedi ◽  
Siddharth Dwivedi ◽  
Bhabani Prasad Mandal ◽  
Pichai Ramadevi ◽  
Vivek Kumar Singh
Keyword(s):  
Density Matrix ◽  
Entanglement Entropy ◽  
Reduced Density Matrix ◽  
Simons Theory ◽  
Partition Functions ◽  
Connected Sum ◽  
Gauge Groups ◽  
Hyperbolic Volume ◽  
Reduced Density ◽  
Chern Simons

AbstractThe entanglement entropy of many quantum systems is difficult to compute in general. They are obtained as a limiting case of the Rényi entropy of index m, which captures the higher moments of the reduced density matrix. In this work, we study pure bipartite states associated with S3 complements of a two-component link which is a connected sum of a knot $$ \mathcal{K} $$ K and the Hopf link. For this class of links, the Chern-Simons theory provides the necessary setting to visualise the m-moment of the reduced density matrix as a three-manifold invariant Z($$ {M}_{{\mathcal{K}}_m} $$ M K m ), which is the partition function of $$ {M}_{{\mathcal{K}}_m} $$ M K m . Here $$ {M}_{{\mathcal{K}}_m} $$ M K m is a closed 3-manifold associated with the knot $$ \mathcal{K} $$ K m, where $$ \mathcal{K} $$ K m is a connected sum of m-copies of $$ \mathcal{K} $$ K (i.e., $$ \mathcal{K} $$ K #$$ \mathcal{K} $$ K . . . #$$ \mathcal{K} $$ K ) which mimics the well-known replica method. We analayse the partition functions Z($$ {M}_{{\mathcal{K}}_m} $$ M K m ) for SU(2) and SO(3) gauge groups, in the limit of the large Chern-Simons coupling k. For SU(2) group, we show that Z($$ {M}_{{\mathcal{K}}_m} $$ M K m ) can grow at most polynomially in k. On the contrary, we conjecture that Z($$ {M}_{{\mathcal{K}}_m} $$ M K m ) for SO(3) group shows an exponential growth in k, where the leading term of ln Z($$ {M}_{{\mathcal{K}}_m} $$ M K m ) is the hyperbolic volume of the knot complement S3\$$ \mathcal{K} $$ K m. We further propose that the Rényi entropies associated with SO(3) group converge to a finite value in the large k limit. We present some examples to validate our conjecture and proposal.

Aharony–Bergman–Jafferis–Maldacena Wilson Loops in the Fermi Gas Approach

2013 ◽  
Vol 68 (1-2) ◽  
pp. 178-209 ◽  
Author(s):  
Albrecht Klemm ◽  
Marcos Mariño ◽  
Masoud Soroush
Keyword(s):  
Vacuum Expectation ◽  
Fermi Gas ◽  
Wilson Loops ◽  
Winding Number ◽  
Airy Functions ◽  
Mechanical Problem ◽  
Expectation Values ◽  
The Matrix ◽  
Statistical Mechanical ◽  
Chern Simons

The matrix model of the Aharony-Bergman-Jafferis-Maldacena theory can be formulated in terms of an ideal Fermi gas with a non-trivial one-particle Hamiltonian. We show that, in this formalism, vacuum expectation values (vevs) of Wilson loops correspond to averages of operators in the statistical-mechanical problem. This makes it possible to calculate these vevs at all orders in 1/N, up to exponentially small corrections, and for arbitrary Chern-Simons coupling, by using the Wentzel- Kramer-Brillouin expansion.We present explicit results for the vevs of 1/6 and the 1/2 Bogomolnyi- Prasad-Sommerfield Wilson loops, at any winding number, in terms of Airy functions. Our expressions are shown to reproduce the low genus results obtained previously in the ’t Hooft expansion.

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