scholarly journals Duality and mock modularity

2020 ◽  
Vol 9 (5) ◽  
Author(s):  
Atish Dabholkar ◽  
Pavel Putrov ◽  
Edward Witten
Keyword(s):  
Partition Function ◽  
Elliptic Genus ◽  
Two Dimensions ◽  
Effective Theory ◽  
Coulomb Branch ◽  
Gauge Groups ◽  
Yang Mills ◽  
Holomorphic Anomaly ◽  
Holomorphic Anomaly Equation ◽  
Relevant Field

We derive a holomorphic anomaly equation for the Vafa-Witten partition function for twisted four-dimensional \mathcal{N} =4𝒩=4 super Yang-Mills theory on \mathbb{CP}^{2}ℂℙ2 for the gauge group SO(3)SO(3) from the path integral of the effective theory on the Coulomb branch. The holomorphic kernel of this equation, which receives contributions only from the instantons, is not modular but ‘mock modular’. The partition function has correct modular properties expected from SS-duality only after including the anomalous nonholomorphic boundary contributions from anti-instantons. Using M-theory duality, we relate this phenomenon to the holomorphic anomaly of the elliptic genus of a two-dimensional noncompact sigma model and compute it independently in two dimensions. The anomaly both in four and in two dimensions can be traced to a topological term in the effective action of six-dimensional (2,0)(2,0) theory on the tensor branch. We consider generalizations to other manifolds and other gauge groups to show that mock modularity is generic and essential for exhibiting duality when the relevant field space is noncompact.

scholarly journals Boson-fermion correspondence and holomorphic anomaly equation in 2d Yang-Mills theory on torus

2021 ◽  
Vol 2021 (7) ◽  
Author(s):  
Min-xin Huang
Keyword(s):  
Partition Function ◽  
Yang Mills ◽  
Holomorphic Anomaly ◽  
Holomorphic Anomaly Equation ◽  
Anomaly Equation

Abstract Recently, Okuyama and Sakai proposed a novel holomorphic anomaly equation for the partition function of 2d Yang-Mills theory on a torus, based on an anholomorphic deformation of the propagator in the bosonic formulation. Using the boson-fermion correspondence, we derive the formula for the deformed partition function in fermionic description and give a proof of the holomorphic anomaly equation.

scholarly journals Localization of the gauged linear sigma model for KK5-branes

2021 ◽  
Vol 2021 (3) ◽  
Author(s):  
Yuki Hiraga ◽  
Yuki Sato
Keyword(s):  
Partition Function ◽  
Sigma Model ◽  
Low Energy ◽  
Target Space ◽  
Linear Sigma Model ◽  
Effective Theory ◽  
Coulomb Branch ◽  
World Sheet ◽  
The World ◽  
Gauged Linear Sigma Model

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.

scholarly journals Localization of four-dimensional super-Yang–Mills theories compactified on Riemann surface

2016 ◽  
Vol 31 (36) ◽  
pp. 1650195
Author(s):  
Koichi Nagasaki
Keyword(s):  
Riemann Surface ◽  
Partition Function ◽  
Path Integral ◽  
Elliptic Genus ◽  
Yang Mills ◽  
The One

We consider the partition function of super-Yang–Mills theories defined on [Formula: see text]. This path integral can be computed by the localization. The one-loop determinant is evaluated by the elliptic genus. This elliptic genus gives trivial result in our calculation. As a result, we obtain a theory defined on the Riemann surface.

scholarly journals Double genus expansion for general Ω background

2016 ◽  
Vol 31 (19) ◽  
pp. 1650114
Author(s):  
Andrea Prudenziati
Keyword(s):  
Partition Function ◽  
Physical Interpretation ◽  
Topological String ◽  
Quantitative Description ◽  
Parameter Expansion ◽  
Holomorphic Anomaly ◽  
Holomorphic Anomaly Equation ◽  
Anomaly Equation ◽  
Genus Expansion ◽  
Two Parameter

We will show how the refined holomorphic anomaly equation obeyed by the Nekrasov partition function at generic [Formula: see text], [Formula: see text] values becomes compatible, in a certain two-parameter expansion, with the assumption that both parameters are associated to genus counting. The underlying worldsheet theory will be analyzed and constrained in various ways, and we will provide both physical interpretation and some alternative evidence for this model. Finally, we will use the Gopakumar–Vafa formulation for the refined topological string in order to give a more quantitative description.

scholarly journals A NOTE ON EFFECTIVE STRING THEORY

1993 ◽  
Vol 08 (29) ◽  
pp. 2763-2773
Author(s):  
SHYAMOLI CHAUDHURI ◽  
DJORDJE MINIC
Keyword(s):  
String Theory ◽  
Partition Function ◽  
Target Space ◽  
Effective Theory ◽  
Scaling Exponents ◽  
Random Surfaces ◽  
Yang Mills ◽  
Fixed Area ◽  
Effective String Theory

Motivated by the possibility of an effective string description for the ir limit of pure Yang-Mills theory, we present a toy model for an effective theory of random surfaces propagating in a target space of D > 2. We show that the scaling exponents for the fixed area partition function of the theory are apparently well behaved. We make some observations regarding the usefulness of this toy model.

scholarly journals Curves on K3 surfaces in divisibility 2

2021 ◽  
Vol 9 ◽  
Author(s):  
Younghan Bae ◽  
Tim-Henrik Buelles
Keyword(s):  
Modular Forms ◽  
K3 Surfaces ◽  
Initial Condition ◽  
Smooth Curves ◽  
Holomorphic Anomaly ◽  
Holomorphic Anomaly Equation ◽  
Anomaly Equation ◽  
Degeneration Formula ◽  
The Relationship ◽  
Level 2

Abstract We prove a conjecture of Maulik, Pandharipande and Thomas expressing the Gromov–Witten invariants of K3 surfaces for divisibility 2 curve classes in all genera in terms of weakly holomorphic quasi-modular forms of level 2. Then we establish the holomorphic anomaly equation in divisibility 2 in all genera. Our approach involves a refined boundary induction, relying on the top tautological group of the moduli space of smooth curves, together with a degeneration formula for the reduced virtual fundamental class with imprimitive curve classes. We use double ramification relations with target variety as a new tool to prove the initial condition. The relationship between the holomorphic anomaly equation for higher divisibility and the conjectural multiple cover formula of Oberdieck and Pandharipande is discussed in detail and illustrated with several examples.

scholarly journals Free partition functions and an averaged holographic duality

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Nima Afkhami-Jeddi ◽  
Henry Cohn ◽  
Thomas Hartman ◽  
Amirhossein Tajdini
Keyword(s):  
Partition Function ◽  
Spectral Gap ◽  
Central Charge ◽  
Three Dimensional ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Partition Functions ◽  
General Constraints ◽  
Dimensional Gravity ◽  
Holographic Duality

Abstract We study the torus partition functions of free bosonic CFTs in two dimensions. Integrating over Narain moduli defines an ensemble-averaged free CFT. We calculate the averaged partition function and show that it can be reinterpreted as a sum over topologies in three dimensions. This result leads us to conjecture that an averaged free CFT in two dimensions is holographically dual to an exotic theory of three-dimensional gravity with U(1)c×U(1)c symmetry and a composite boundary graviton. Additionally, for small central charge c, we obtain general constraints on the spectral gap of free CFTs using the spinning modular bootstrap, construct examples of Narain compactifications with a large gap, and find an analytic bootstrap functional corresponding to a single self-dual boson.

scholarly journals Towards the classification of rank-r $$ \mathcal{N} $$ = 2 SCFTs. Part I. Twisted partition function and central charge formulae

2020 ◽  
Vol 2020 (12) ◽  
Author(s):  
Mario Martone
Keyword(s):  
Partition Function ◽  
Central Charge ◽  
Singular Locus ◽  
Companion Paper ◽  
Coulomb Branch ◽  
Energy Limit ◽  
Superconformal Field Theories ◽  
Rank 2 ◽  
Explicit Formulae

Abstract We derive explicit formulae to compute the a and c central charges of four dimensional $$ \mathcal{N} $$ N = 2 superconformal field theories (SCFTs) directly from Coulomb branch related quantities. The formulae apply at arbitrary rank. We also discover general properties of the low-energy limit behavior of the flavor symmetry of $$ \mathcal{N} $$ N = 2 SCFTs which culminate with our $$ \mathcal{N} $$ N = 2 UV-IR simple flavor condition. This is done by determining precisely the relation between the integrand of the partition function of the topologically twisted version of the 4d $$ \mathcal{N} $$ N = 2 SCFTs and the singular locus of their Coulomb branches. The techniques developed here are extensively applied to many rank-2 SCFTs, including new ones, in a companion paper.This manuscript is dedicated to the memory of Rayshard Brooks, George Floyd, Breonna Taylor and the countless black lives taken by US police forces and still awaiting justice. Our hearts are with our colleagues of color who suffer daily the consequences of this racist world.

scholarly journals Gauge theories on compact toric manifolds

2021 ◽  
Vol 111 (3) ◽  
Author(s):  
Giulio Bonelli ◽  
Francesco Fucito ◽  
Jose Francisco Morales ◽  
Massimiliano Ronzani ◽  
Ekaterina Sysoeva ◽  
...  
Keyword(s):  
Partition Function ◽  
Gauge Theories ◽  
Blow Up ◽  
Gauge Coupling ◽  
Piecewise Constant Function ◽  
Partition Functions ◽  
Piecewise Constant ◽  
Toric Manifolds ◽  
Holomorphic Anomaly ◽  
The One

AbstractWe compute the $$\mathcal{N}=2$$ N = 2 supersymmetric partition function of a gauge theory on a four-dimensional compact toric manifold via equivariant localization. The result is given by a piecewise constant function of the Kähler form with jumps along the walls where the gauge symmetry gets enhanced. The partition function on such manifolds is written as a sum over the residues of a product of partition functions on $$\mathbb {C}^2$$ C 2 . The evaluation of these residues is greatly simplified by using an “abstruse duality” that relates the residues at the poles of the one-loop and instanton parts of the $$\mathbb {C}^2$$ C 2 partition function. As particular cases, our formulae compute the SU(2) and SU(3) equivariant Donaldson invariants of $$\mathbb {P}^2$$ P 2 and $$\mathbb {F}_n$$ F n and in the non-equivariant limit reproduce the results obtained via wall-crossing and blow up methods in the SU(2) case. Finally, we show that the U(1) self-dual connections induce an anomalous dependence on the gauge coupling, which turns out to satisfy a $$\mathcal {N}=2$$ N = 2 analog of the $$\mathcal {N}=4$$ N = 4 holomorphic anomaly equations.

scholarly journals Holomorphic anomaly equation for and the Nekrasov-Shatashvili limit of local

2021 ◽  
Vol 9 ◽  
Author(s):  
Pierrick Bousseau ◽  
Honglu Fan ◽  
Shuai Guo ◽  
Longting Wu
Keyword(s):  
Elliptic Curve ◽  
Moduli Spaces ◽  
Topological String ◽  
Holomorphic Anomaly ◽  
Holomorphic Anomaly Equation ◽  
Anomaly Equation ◽  
Witten Theory ◽  
Generating Series ◽  
Maximal Contact ◽  
Higher Genus

Abstract We prove a higher genus version of the genus $0$ local-relative correspondence of van Garrel-Graber-Ruddat: for $(X,D)$ a pair with X a smooth projective variety and D a nef smooth divisor, maximal contact Gromov-Witten theory of $(X,D)$ with $\lambda _g$ -insertion is related to Gromov-Witten theory of the total space of ${\mathcal O}_X(-D)$ and local Gromov-Witten theory of D. Specializing to $(X,D)=(S,E)$ for S a del Pezzo surface or a rational elliptic surface and E a smooth anticanonical divisor, we show that maximal contact Gromov-Witten theory of $(S,E)$ is determined by the Gromov-Witten theory of the Calabi-Yau 3-fold ${\mathcal O}_S(-E)$ and the stationary Gromov-Witten theory of the elliptic curve E. Specializing further to $S={\mathbb P}^2$ , we prove that higher genus generating series of maximal contact Gromov-Witten invariants of $({\mathbb P}^2,E)$ are quasimodular and satisfy a holomorphic anomaly equation. The proof combines the quasimodularity results and the holomorphic anomaly equations previously known for local ${\mathbb P}^2$ and the elliptic curve. Furthermore, using the connection between maximal contact Gromov-Witten invariants of $({\mathbb P}^2,E)$ and Betti numbers of moduli spaces of semistable one-dimensional sheaves on ${\mathbb P}^2$ , we obtain a proof of the quasimodularity and holomorphic anomaly equation predicted in the physics literature for the refined topological string free energy of local ${\mathbb P}^2$ in the Nekrasov-Shatashvili limit.

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