arXiv subjects

Bao Shou

Publications and source records attributed to Bao Shou.

11 recordsLinked to original sources

arXiv2025

Exact $S$-duality Map for Rigid Surface Operators

Surface operators in four-dimensional gauge theories are two-dimensional defects, serving as natural generalizations of Wilson lines and 't Hooft line operators. They act as ideal probes for exploring the non-perturbative structure of the theory. Rigid surface operators are a specific class of surface operators characterized by the absence of continuous deformation parameters. It is expected that a closed $S$-duality map should exist among these rigid operators. While progress has been made on specific examples or subclasses by leveraging invariants and empirical conjectures, a complete picture remains elusive. A significant challenge arises when multiple rigid surface operators share identical invariants, making the determination of $S$-duality relations difficult. More critically, a mismatch exists in the number of rigid surface operators between dual theories when classified by invariants; this is referred to as the \textit{mismatch problem}. This discrepancy suggests the necessity of extending the scope of consideration beyond strictly rigid operators. In this paper, we propose a direct, natural, and precise $S$-duality map for rigid surface operators. Our map is realized by moving the longest row in the pair of partitions defining a surface operator from one factor to the other, with an additional box appended or deleted to balance the total number of boxes. This mapping naturally incorporates non-rigid surface operators, thereby resolving the mismatch problem. The proposed map is applicable to gauge groups of all ranks and clarifies several long-standing puzzles in the field.

hep-th↗

arXiv2019

Symbol Invariant of Partition and the Construction

The symbol is used to describe the Springer correspondence for the classical groups. We propose equivalent definitions of symbols for rigid partitions in the $B_n$, $C_n$, and $D_n$ theories uniformly. Analysing the new definition of symbol in detail, we give rules to construct symbol of a partition, which are easy to remember and to operate on. We introduce formal operations of a partition, which reduce the difficulties in the proof of the construction rules. According these rules, we give a closed formula of symbols for different theories uniformly. As applications, previous results can be illustrated more clearly by the construction rules of symbol.

math.RT↗

arXiv2018

Inner Product in Highest-Weight Representation

In this paper, we study the inner product of states corresponding to weights of finite-dimensional highest-weight representations of classical groups. We prove that the action of the raising operators would reduce a state of hight-weight representation to a linear combination of states of highest-weight representation, with the level decreased by one. Then we propose an iterative algorithm for calculating the inner products of sates efficiently, revealing the intricate structure of the representation. As applications, we discuss the unitarity of the highest-weight representation and propose a conjecture. We determine the norm of a special class of states. And we completely determine the inner products of states of the minuscule representations. The algorithm proposed is applicable to the highest-weight representation of affine Lie algebra without modifications. These findings can be used to study the construction of solutions to Kapustin-Witten equations which are based on the fundamental solutions of Toda systems.

math-ph↗

arXiv2017

Invariants of rigid surface operators

Lusztig used the symbol invariant to describe the Springer correspondence for classical groups. Similarly, the fingerprint invariant can describe the Kazhdan-Lusztig map. Both invariants pertain to rigid semisimple operators labeled by pairs of partitions $(\lambda', \lambda'')$. It is conjectured that the symbol invariant is equivalent to the fingerprint invariant for rigid surface operators. In this study, we provide a proof of this conjecture. We classify the maps that preserve the fingerprint invariant and demonstrate that they also preserve the symbol invariant. Conversely, we classify the maps that preserve the symbol invariant and show that they also preserve the fingerprint invariant. The constructions of the symbol and fingerprint invariants in prior works are crucial to the proof. Additionally, we found that one condition in the definition of the fingerprint invariant is redundant for rigid surface operators. In the appendix, we present an alternative strategy to prove the equivalence of these invariants.

math.RT↗

arXiv2017

Invariants of partitions and representative elements

The symbol invariant is used to describe the Springer correspondence for the classical groups by Lusztig. And the fingerprint invariant can be used to describe the Kazhdan-Lusztig map. They are invariants of rigid semisimple operators described by pairs of partitions $(λ^{'}, λ^{"})$. We construct a nice representative element of the rigid semisimple operators with the same symbol invariant. The fingerprint of the representative element can be obtained immediately. We also discuss the representative element of rigid semisimple operator with the same fingerprint invariant. Our construction can be regarded as the maps between these two invariants.

math.CO↗

arXiv2017

Fingerprint Invariant of Partitions and Construction

The fingerprint invariant of partitions can be used to describe the Kazhdan-Lusztig map for the classical groups. We discuss the basic properties of fingerprint. We construct the fingerprints of rigid partitions in the $B_n$, $C_n$, and $D_n$ theories. To calculate the fingerprint of a rigid semisimple operator $(λ^{'};λ^{"})$, we decompose $λ^{'}+λ^{"}$ into several blocks. We define operators to calculate the fingerprint for each block using the results of fingerprint of the unipotent operators.

math.CO↗

arXiv2017

Rigid Surface Operator and Symbol Invariant of Partitions

The symbol is used to describe the Springer correspondence for the classical groups by Lusztig. We refine the explanation that the $S$-duality maps of the rigid surface operators are symbol preserving maps. And we find that the maps $X_S$ and $Y_S$ used in the construction of $S$-duality maps are essentially the same. We clear up cause of the mismatch problem of the total number of the rigid surface operators between the $B_n$ and $C_n$ theories. And we construct all the $B_n/C_n$ rigid surface operators which can not have a dual. A classification of the problematic surface operators is made.

math-ph↗

arXiv2017

Construction of the Symbol Invariant of Partition

Symbol is used to describe the Springer correspondence for the classical groups. We prove two structure theorems of symbol. We propose a construction of the symbol of the rigid partitions in the $B_n$, $C_n$, and $D_n$ theories. This construction is natural and consists of two basic building blocks. Using this construction, we give closed formulas of symbols for the rigid partitions in the $B_n, C_n$, and $D_n$ theories. One part of the closed formula is universal and other parts are determined by the specific theory. A comparison of between this closed formula and the old one is made. Previous results can be illustrated more clearly by this closed formula.

math.CO↗

arXiv2017

Symbol, Surface operators and $S$-duality

We study rigid surface operators in the $N=4$ supersymmetric Yang-Mills theories with gauge groups $SO(n)$ and $Sp(2n)$. Using maps $X_S$ and $Y_S$ between these two theories, Wyllard made explicit proposals for how the $S$-duality map should act on certain subclasses of surface operators. We study the maps $X_S$ and $Y_S$ further and simplify the construction of symbol invariant of rigid surface operators by a convenient trick. By consistency checks, we recover and extend the $S$-duality maps proposed by Wyllard. We find new subclasses of rigid surface operators related by $S$-duality. We try to explain the exceptions of $S$-duality maps. We also discuss the extension of the techniques used in the $B_n/C_n$ theories to the $D_n$ theories.

hep-th↗

arXiv2017

Solutions of Kapustin-Witten equations for ADE-type groups

Kapustin-Witten (KW) equations are encountered in the localization of the topological N=4 SYM theory. Mikhaylov has constructed model solutions of KW equations for the boundary 't~Hooft operators on a half space. Direct proof of the solutions boils down to check a boundary condition. There are two computational difficulties in explicitly constructing the solutions to Lie algebra of higher rank. The first one is related to the commutation of generators of Lie algebra. We derived an identity which effectively reduces this computational difficulty. The second one involves the number of ways from the highest weights to other weights in the fundamental representation. For ADE-type gauge groups, we found an amazing formula which can be used to rewrite the solutions of KW equations. This new formula of solutions bypass above two computational difficulties.

hep-th↗

arXiv2011

AGT conjecture and AFLT states: a complete construction

A complete construction of the AFLT states is proposed. With this construction and for all the cases we have checked, the AGT conjecture on the equivalence of Nekrasov Instanton Counting (NIC) to the $Vir\oplus u(1)$ conformal block has been verified to be true.

hep-th↗