scholarly journals Resurgent analysis of SU(2) Chern-Simons partition function on Brieskorn spheres Σ(2, 3, 6n + 5)

2021 ◽  
Vol 2021 (2) ◽  
Author(s):  
David H. Wu
Keyword(s):  
Partition Function ◽  
Analytic Continuation ◽  
Partition Functions ◽  
Seifert Manifolds ◽  
Resurgent Analysis ◽  
Chern Simons ◽  
Shed Light ◽  
Borel Resummation

Abstract $$ \hat{Z} $$ Z ̂ -invariants, which can reconstruct the analytic continuation of the SU(2) Chern-Simons partition functions via Borel resummation, were discovered by GPV and have been conjectured to be a new homological invariant of 3-manifolds which can shed light onto the superconformal and topologically twisted index of 3d $$ \mathcal{N} $$ N = 2 theories proposed by GPPV. In particular, the resurgent analysis of $$ \hat{Z} $$ Z ̂ has been fruitful in discovering analytic properties of the WRT invariants. The resurgent analysis of these $$ \hat{Z} $$ Z ̂ -invariants has been performed for the cases of Σ(2, 3, 5), Σ(2, 3, 7) by GMP, Σ(2, 5, 7) by Chun, and, more recently, some additional Seifert manifolds by Chung and Kucharski, independently. In this paper, we extend and generalize the resurgent analysis of $$ \hat{Z} $$ Z ̂ on a family of Brieskorn homology spheres Σ(2, 3, 6n + 5) where n ∈ ℤ+ and 6n + 5 is a prime. By deriving $$ \hat{Z} $$ Z ̂ for Σ(2, 3, 6n + 5) according to GPPV and Hikami, we provide a formula where one can quickly compute the non-perturbative contributions to the full analytic continuation of SU(2) Chern-Simons partition function.

scholarly journals Resurgent analysis for some 3-manifold invariants

2021 ◽  
Vol 2021 (5) ◽  
Author(s):  
Hee-Joong Chung
Keyword(s):  
Partition Function ◽  
Jones Polynomial ◽  
Infinite Family ◽  
Simons Theory ◽  
Flat Connections ◽  
Torus Knot ◽  
Seifert Manifolds ◽  
Resurgent Analysis ◽  
Chern Simons ◽  
Chern Simons Theory

Abstract We study resurgence for some 3-manifold invariants when Gℂ = SL(2, ℂ). We discuss the case of an infinite family of Seifert manifolds for general roots of unity and the case of the torus knot complement in S3. Via resurgent analysis, we see that the contribution from the abelian flat connections to the analytically continued Chern-Simons partition function contains the information of all non-abelian flat connections, so it can be regarded as a full partition function of the analytically continued Chern-Simons theory on 3-manifolds M3. In particular, this directly indicates that the homological block for the torus knot complement in S3 is an analytic continuation of the full G = SU(2) partition function, i.e. the colored Jones polynomial.

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.

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.

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 More on topological vertex formalism for 5-brane webs with O5-plane

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
Hirotaka Hayashi ◽  
Rui-Dong Zhu
Keyword(s):  
Gauge Theory ◽  
Partition Function ◽  
Vertex Operator ◽  
Vertex Function ◽  
Gauge Theories ◽  
Partition Functions ◽  
Topological Vertex ◽  
Concrete Form ◽  
Nekrasov Partition Functions ◽  
Chern Simons

Abstract We propose a concrete form of a vertex function, which we call O-vertex, for the intersection between an O5-plane and a 5-brane in the topological vertex formalism, as an extension of the work of [1]. Using the O-vertex it is possible to compute the Nekrasov partition functions of 5d theories realized on any 5-brane web diagrams with O5-planes. We apply our proposal to 5-brane webs with an O5-plane and compute the partition functions of pure SO(N) gauge theories and the pure G2 gauge theory. The obtained results agree with the results known in the literature. We also compute the partition function of the pure SU(3) gauge theory with the Chern-Simons level 9. At the end we rewrite the O-vertex in a form of a vertex operator.

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 Partition functions of Chern-Simons theory on handlebodies by radial quantization

2021 ◽  
Vol 2021 (7) ◽  
Author(s):  
Massimo Porrati ◽  
Cedric Yu
Keyword(s):  
Partition Function ◽  
Riemann Surfaces ◽  
Wilson Loop ◽  
Transition Amplitude ◽  
Simons Theory ◽  
Partition Functions ◽  
Initial State ◽  
Gauge Groups ◽  
Arbitrary Genus ◽  
Chern Simons

Abstract We use radial quantization to compute Chern-Simons partition functions on handlebodies of arbitrary genus. The partition function is given by a particular transition amplitude between two states which are defined on the Riemann surfaces that define the (singular) foliation of the handlebody. The final state is a coherent state while on the initial state the holonomy operator has zero eigenvalue. The latter choice encodes the constraint that the gauge fields must be regular everywhere inside the handlebody. By requiring that the only singularities of the gauge field inside the handlebody must be compatible with Wilson loop insertions, we find that the Wilson loop shifts the holonomy of the initial state. Together with an appropriate choice of normalization, this procedure selects a unique state in the Hilbert space obtained from a Kähler quantization of the theory on the constant-radius Riemann surfaces. Radial quantization allows us to find the partition functions of Abelian Chern-Simons theories for handlebodies of arbitrary genus. For non-Abelian compact gauge groups, we show that our method reproduces the known partition function at genus one.

scholarly journals BPS invariants for 3-manifolds at rational level K

2021 ◽  
Vol 2021 (2) ◽  
Author(s):  
Hee-Joong Chung
Keyword(s):  
Asymptotic Expansion ◽  
Exact Expression ◽  
Roots Of Unity ◽  
Partition Functions ◽  
Knot Invariants ◽  
Volume Conjecture ◽  
Seifert Manifolds ◽  
Coprime Integers ◽  
Bps Invariants ◽  
Chern Simons

Abstract We consider the Witten-Reshetikhin-Turaev invariants or Chern-Simons partition functions at or around roots of unity $$ q={e}^{\frac{2\pi i}{K}} $$ q = e 2 πi K with a rational level K = $$ \frac{r}{s} $$ r s where r and s are coprime integers. From the exact expression for the G = SU(2) Witten-Reshetikhin-Turaev invariants of the Seifert manifolds at a rational level obtained by Lawrence and Rozansky, we provide an expected form of the structure of the Witten-Reshetikhin-Turaev invariants in terms of the homological blocks at a rational level. Also, we discuss the asymptotic expansion of knot invariants around roots of unity where we take a limit different from the limit in the standard volume conjecture.

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 Deconstructing little strings with $\mathcal{N}=1$ gauge theories on ellipsoids

2018 ◽  
Vol 4 (6) ◽  
Author(s):  
Joseph Hayling ◽  
Rodolfo Panerai ◽  
Constantinos Papageorgakis
Keyword(s):  
String Theory ◽  
Partition Function ◽  
Analytic Continuation ◽  
Gauge Theories ◽  
Partition Functions ◽  
Yang Mills ◽  
The Relationship

A formula was recently proposed for the perturbative partition function of certain \mathcal N=1𝒩=1 gauge theories on the round four-sphere, using an analytic-continuation argument in the number of dimensions. These partition functions are not currently accessible via the usual supersymmetric-localisation technique. We provide a natural refinement of this result to the case of the ellipsoid. We then use it to write down the perturbative partition function of an \mathcal N=1𝒩=1 toroidal-quiver theory (a double orbifold of \mathcal N=4𝒩=4 super Yang–Mills) and show that, in the deconstruction limit, it reproduces the zero-winding contributions to the BPS partition function of (1,1) Little String Theory wrapping an emergent torus. We therefore successfully test both the expressions for the \mathcal N=1𝒩=1 partition functions, as well as the relationship between the toroidal-quiver theory and Little String Theory through dimensional deconstruction.

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