scholarly journals SOFT MATRIX MODELS AND CHERN–SIMONS PARTITION FUNCTIONS

2004 ◽  
Vol 19 (18) ◽  
pp. 1365-1378 ◽  
Author(s):  
M. TIERZ
Keyword(s):  
Partition Function ◽  
Orthogonal Polynomials ◽  
Matrix Models ◽  
Simons Theory ◽  
Partition Functions ◽  
Soft Matrix ◽  
Simple Mapping ◽  
The Moment ◽  
Chern Simons ◽  
Chern Simons Theory

We study the properties of matrix models with soft confining potentials. Their precise mathematical characterization is that their weight function is not determined by its moments. We mainly rely on simple considerations based on orthogonal polynomials and the moment problem. In addition, some of these models are equivalent, by a simple mapping, to matrix models that appear in Chern–Simons theory. The models can be solved with q deformed orthogonal polynomials (Stieltjes–Wigert polynomials), and the deformation parameter turns out to be the usual q parameter in Chern–Simons theory. In this way, we give a matrix model computation of the Chern–Simons partition function on S3 and show that there are infinitely many matrix models with this partition function.

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.

3D MANIFOLDS, GRAPH INVARIANTS AND CHERN-SIMONS THEORY

1992 ◽  
Vol 07 (23) ◽  
pp. 2065-2076 ◽  
Author(s):  
S. KALYANA RAMA ◽  
SIDDHARTHA SEN
Keyword(s):  
Partition Function ◽  
Closed Form ◽  
Three Dimensional ◽  
Simons Theory ◽  
Partition Functions ◽  
Graph Invariants ◽  
Absolute Value ◽  
Dimensional Gravity ◽  
Chern Simons ◽  
Chern Simons Theory

We show how the Turaev-Viro invariant, which is closely related to the partition function of three-dimensional gravity, can be understood within the framework of SU(2) Chern-Simons theory. We also show that, for S3 and RP3, this invariant is equal to the absolute value square of their respective partition functions in SU(2) Chern-Simons theory and give a method of evaluating the latter in a closed form for a class of 3D manifolds, thus in effect obtaining the partition function of three-dimensional gravity for these manifolds. By interpreting the triangulation of a manifold as a graph consisting of crossings and vertices with three lines we also describe a new invariant for certain class of graphs.

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 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.

scholarly journals Discrete and oscillatory matrix models in Chern–Simons theory

2005 ◽  
Vol 731 (3) ◽  
pp. 225-241 ◽  
Author(s):  
Sebastian de Haro ◽  
Miguel Tierz
Keyword(s):  
Matrix Models ◽  
Simons Theory ◽  
Chern Simons ◽  
Chern Simons Theory

scholarly journals Gravitational dual of averaged free CFT’s over the Narain lattice

2020 ◽  
Vol 2020 (11) ◽  
Author(s):  
Alfredo Pérez ◽  
Ricardo Troncoso
Keyword(s):  
Partition Function ◽  
Boundary Conditions ◽  
Three Dimensional ◽  
Simons Theory ◽  
Energy Tensor ◽  
The Family ◽  
Stress Energy ◽  
Higher Spin ◽  
Chern Simons ◽  
Chern Simons Theory

Abstract It has been recently argued that the averaging of free CFT’s over the Narain lattice can be holographically described through a Chern-Simons theory for U (1)D×U (1)D with a precise prescription to sum over three-dimensional handlebodies. We show that a gravitational dual of these averaged CFT’s would be provided by Einstein gravity on AdS3 with U (1)D−1× U (1)D−1 gauge fields, endowed with a precise set of boundary conditions closely related to the “soft hairy” ones. Gravitational excitations then go along diagonal SL (2, ℝ) generators, so that the asymptotic symmetries are spanned by U (1)D× U (1)D currents. The stress-energy tensor can then be geometrically seen as composite of these currents through a twisted Sugawara construction. Our boundary conditions are such that for the reduced phase space, there is a one-to-one map between the configurations in the gravitational and the purely abelian theories. The partition function in the bulk could then also be performed either from a non-abelian Chern-Simons theory for two copies of SL (2, ℝ) × U (1)D−1 generators, or formally through a path integral along the family of allowed configurations for the metric. The new boundary conditions naturally accommodate BTZ black holes, and the microscopic number of states then appears to be manifestly positive and suitably accounted for from the partition function in the bulk. The inclusion of higher spin currents through an extended twisted Sugawara construction in the context of higher spin gravity is also briefly addressed.

scholarly journals Exact Computations in Topological Abelian Chern-Simons and BF Theories

2017 ◽  
Vol 2017 ◽  
pp. 1-6
Author(s):  
Philippe Mathieu
Keyword(s):  
Simons Theory ◽  
Partition Functions ◽  
Topological Invariants ◽  
Deligne Cohomology ◽  
Functional Integrals ◽  
Fibre Bundles ◽  
Bf Theory ◽  
Chern Simons ◽  
Chern Simons Theory

We introduce Deligne cohomology that classifies U1 fibre bundles over 3 manifolds endowed with connections. We show how the structure of Deligne cohomology classes provides a way to perform exact (nonperturbative) computations in U1 Chern-Simons theory (BF theory, resp.) at the level of functional integrals. The partition functions (and observables) of these theories are strongly related to topological invariants well known to the mathematicians.

scholarly journals SOLITONS ON NONCOMMUTATIVE TORUS AS ELLIPTIC CALOGERO–GAUDIN MODELS, BRANES AND LAUGHLIN WAVE FUNCTIONS

2003 ◽  
Vol 18 (14) ◽  
pp. 2477-2500 ◽  
Author(s):  
BO-YU HOU ◽  
DAN-TAO PENG ◽  
KANG-JIE SHI ◽  
RUI-HONG YUE
Keyword(s):  
Quantum Hall ◽  
Moment Map ◽  
Simons Theory ◽  
Noncommutative Torus ◽  
The Moment ◽  
Gaudin Models ◽  
Laughlin Wave Function ◽  
Marked Points ◽  
Chern Simons ◽  
Chern Simons Theory

For the noncommutative torus [Formula: see text], in the case of the noncommutative parameter [Formula: see text], we construct the basis of Hilbert space ℋn in terms of θ functions of the positions zi of n solitons. The wrapping around the torus generates the algebra [Formula: see text], which is the Zn × Zn Heisenberg group on θ functions. We find the generators g of a local elliptic su (n), which transform covariantly by the global gauge transformation of [Formula: see text]. By acting on ℋn we establish the isomorphism of [Formula: see text] and g. We embed this g into the L-matrix of the elliptic Gaudin and Calogero–Moser models to give the dynamics. The moment map of this twisted cotangent [Formula: see text] bundle is matched to the D-equation with the Fayet–Illiopoulos source term, so the dynamics of the noncommutative solitons become that of the brane. The geometric configuration (k, u) of the spectral curve det |L(u) - k| = 0 describes the brane configuration, with the dynamical variables zi of the noncommutative solitons as the moduli T⊗ n/Sn. Furthermore, in the noncommutative Chern–Simons theory for the quantum Hall effect, the constrain equation with quasiparticle source is identified also with the moment map equation of the noncommutative [Formula: see text] cotangent bundle with marked points. The eigenfunction of the Gaudin differential L-operators as the Laughlin wave function is solved by Bethe ansatz.

Sign in / Sign up

Export Citation Format

Share Document