scholarly journals BPS Quivers of Five-Dimensional SCFTs, Topological Strings and q-Painlevé Equations

2021 ◽  
Author(s):  
Giulio Bonelli ◽  
Fabrizio Del Monte ◽  
Alessandro Tanzini
Keyword(s):  
Topological String ◽  
Cluster Algebra ◽  
Painlevé Equations ◽  
Partition Functions ◽  
Quantum Field Theories ◽  
Particle Spectrum ◽  
Yang Mills ◽  
Painleve Equations ◽  
Rank One ◽  
Bilinear Equations

AbstractWe study the discrete flows generated by the symmetry group of the BPS quivers for Calabi–Yau geometries describing five-dimensional superconformal quantum field theories on a circle. These flows naturally describe the BPS particle spectrum of such theories and at the same time generate bilinear equations of q-difference type which, in the rank one case, are q-Painlevé equations. The solutions of these equations are shown to be given by grand canonical topological string partition functions which we identify with $$\tau $$ τ -functions of the cluster algebra associated to the quiver. We exemplify our construction in the case corresponding to five-dimensional SU(2) pure super Yang–Mills and $$N_f=2$$ N f = 2 on a circle.

scholarly journals Domain walls in 4d $$ \mathcal{N} $$ = 1 SYM

2021 ◽  
Vol 2021 (3) ◽  
Author(s):  
Diego Delmastro ◽  
Jaume Gomis
Keyword(s):  
Domain Wall ◽  
Domain Walls ◽  
Simons Theory ◽  
Partition Functions ◽  
Quantum Field Theories ◽  
Yang Mills ◽  
Stable Domain ◽  
Coxeter Number ◽  
Topological Quantum ◽  
Simply Connected

Abstract 4d$$ \mathcal{N} $$ N = 1 super Yang-Mills (SYM) with simply connected gauge group G has h gapped vacua arising from the spontaneously broken discrete R-symmetry, where h is the dual Coxeter number of G. Therefore, the theory admits stable domain walls interpolating between any two vacua, but it is a nonperturbative problem to determine the low energy theory on the domain wall. We put forward an explicit answer to this question for all the domain walls for G = SU(N), Sp(N), Spin(N) and G2, and for the minimal domain wall connecting neighboring vacua for arbitrary G. We propose that the domain wall theories support specific nontrivial topological quantum field theories (TQFTs), which include the Chern-Simons theory proposed long ago by Acharya-Vafa for SU(N). We provide nontrivial evidence for our proposals by exactly matching renormalization group invariant partition functions twisted by global symmetries of SYM computed in the ultraviolet with those computed in our proposed infrared TQFTs. A crucial element in this matching is constructing the Hilbert space of spin TQFTs, that is, theories that depend on the spin structure of spacetime and admit fermionic states — a subject we delve into in some detail.

scholarly journals ON THE STRING ACTIONS FOR THE GENERALIZED TWO-DIMENSIONAL YANG-MILLS THEORIES

1996 ◽  
Vol 11 (20) ◽  
pp. 1675-1685 ◽  
Author(s):  
YUJI SUGAWARA
Keyword(s):  
Topological String ◽  
String Model ◽  
Partition Functions ◽  
Two Dimensional ◽  
Yang Mills ◽  
Large N

We study the structures of partition functions of the large-N generalized two-dimensional Yang-Mills theories (gY M2) by recasting the higher Casimirs. We clarify the appropriate interpretations of them and try to extend the Cordes-Moore-Ramgoolam’s topological string model describing the ordinary4Y M2 to those describing gY M2. We present the expressions of the appropriate operators to reproduce the higher Casimir terms in gY M2. The concept of “deformed gravitational descendants” will be introduced for this purpose.

scholarly journals Trinion conformal blocks from topological strings

2020 ◽  
Vol 2020 (9) ◽  
Author(s):  
Ioana Coman ◽  
Elli Pomoni ◽  
Joerg Teschner
Keyword(s):  
Topological String ◽  
Free Field ◽  
Partition Functions ◽  
Integration Contour ◽  
Topological Vertex ◽  
Field Representation ◽  
Conformal Blocks ◽  
Starting Point ◽  
Topological Recursion ◽  
Rank One

Abstract In this paper we investigate the relation between conformal blocks of Liouville CFT and the topological string partition functions of the rank one trinion theory T2. The partition functions exhibit jumps when passing from one chamber in the parameter space to another. Such jumps can be attributed to a change of the integration contour in the free field representation of Liouville conformal blocks. We compare the partition functions of the T2 theories representing trifundamental half hypermultiplets in N = 2, d = 4 field theories to the partition functions associated to bifundamental hypermultiplets. We find that both are related to the same Liouville conformal blocks up to inessential factors. In order to establish this picture we combine and compare results obtained using topological vertex techniques, matrix models and topological recursion. We furthermore check that the partition functions obtained by gluing two T2 vertices can be represented in terms of a four point Liouville conformal block. Our results indicate that the T2 vertex offers a useful starting point for developing an analog of the instanton calculus for SUSY gauge theories with trifundamental hypermultiplets.

Two variations on ( A 3 × A 1 × A 1 ) (1) type discrete Painlevé equations

2019 ◽  
Vol 475 (2229) ◽  
pp. 20190299
Author(s):  
Yang Shi
Keyword(s):  
Coxeter Groups ◽  
Infinite Order ◽  
Cluster Algebra ◽  
Painlevé Equations ◽  
Group Symmetry ◽  
Parabolic Subgroups ◽  
Reflection Groups ◽  
Kp Hierarchy ◽  
Painleve Equations ◽  
Discrete Painlevé Equations

By considering the normalizers of reflection subgroups of types A (1) 1 and A (1) 3 in W ~ ( D 5 ( 1 ) ) , two subgroups: W ~ ( A 3 × A 1 ) ( 1 ) ⋉ W ( A 1 ( 1 ) ) and W ~ ( A 1 × A 1 ) ( 1 ) ⋉ W ( A 3 ( 1 ) ) can be constructed from a ( A 3  ×  A 1  ×  A 1 ) (1) type subroot system. These two symmetries arose in the studies of discrete Painlevé equations (Kajiwara K, Noumi M, Yamada Y. 2002 q -Painlevé systems arising from q -KP hierarchy. Lett. Math. Phys. 62 , 259–268; Takenawa T. 2003 Weyl group symmetry of type D (1) 5 in the q -Painlevé V equation. Funkcial. Ekvac. 46 , 173–186; Okubo N, Suzuki T. 2018 Generalized q -Painlevé VI systems of type ( A 2 n +1  +  A 1  +  A 1 ) (1) arising from cluster algebra. ( http://arxiv.org/abs/quant-ph/1810.03252 )), where certain non-translational elements of infinite order were shown to give rise to discrete Painlevé equations. We clarify the nature of these elements in terms of Brink-Howlett theory of normalizers of Coxeter groups (Howlett RB. 1980 Normalizers of parabolic subgroups of reflection groups. J. London Math. Soc. (2) 21 , 62–80; Brink B, Howlett RB. 1999 Normalizers of parabolic subgroups in Coxeter groups. Invent. Math. 136 , 323–351). This is the first of a series of studies which investigates the properties of discrete integrable equations via the theory of normalizers.

scholarly journals The properties of solutions for several types of Painlevé equations concerning fixed-points, zeros and poles

2019 ◽  
Vol 17 (1) ◽  
pp. 1014-1024
Author(s):  
Hong Yan Xu ◽  
Xiu Min Zheng
Keyword(s):  
Fixed Points ◽  
Difference Equations ◽  
Painlevé Equations ◽  
Meromorphic Solutions ◽  
Painleve Equations ◽  
Zeros And Poles

Abstract The purpose of this manuscript is to study some properties on meromorphic solutions for several types of q-difference equations. Some exponents of convergence of zeros, poles and fixed points related to meromorphic solutions for some q-difference equations are obtained. Our theorems are some extension and improvements to those results given by Qi, Peng, Chen, and Zhang.

scholarly journals $q$ -Racah Ensemble and Discrete Painlevé Equation

2019 ◽  
Vol 2020 (24) ◽  
pp. 9797-9843 ◽  
Author(s):  
Anton Dzhamay ◽  
Alisa Knizel
Keyword(s):  
Orthogonal Polynomial ◽  
Lax Pair ◽  
Painlevé Equations ◽  
Painlevé Equation ◽  
Painleve Equations ◽  
Symmetry Structure ◽  
The One ◽  
The Relationship ◽  
Gap Probabilities ◽  
Discrete Painlevé Equation

Abstract The goal of this paper is to investigate the missing part of the story about the relationship between the orthogonal polynomial ensembles and Painlevé equations. Namely, we consider the $q$-Racah polynomial ensemble and show that the one-interval gap probabilities in this case can be expressed through a solution of the discrete $q$-P$\left (E_7^{(1)}/A_{1}^{(1)}\right )$ equation. Our approach also gives a new Lax pair for this equation. This Lax pair has an interesting additional involutive symmetry structure.

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