Quivers, words and fundamentals

Mattioli, Paolo; Ramgoolam, Sanjaye

doi:10.1007/jhep03(2015)105

Cited by 13 publications

(24 citation statements)

References 60 publications

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“…The other way is to tie all the incoming and outgoing legs to a single new node, preserving their orientation. This latter procedure was useful in consideration of the counting of gauge invariant operators [8].…”

Section: Discussionmentioning

confidence: 99%

“…In this paper we study correlation functions of holomorphic and anti-holomorphic gauge invariant operators in quiver gauge theories with flavour symmetries, in the zero coupling limit. This builds on the results in our previous paper focused on enumeration of gauge invariant operators [8] and proceeds to explicit construction of the operators and consideration of free field two and three point functions. These have non-trivial dependences on the structure of the operators and on the ranks of the gauge and flavour symmetries.…”

Section: Introductionmentioning

confidence: 94%

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Gauge invariants and correlators in flavoured quiver gauge theories

Mattioli

Ramgoolam

2016

Nuclear Physics B

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In this paper we study the construction of holomorphic gauge invariant operators for general quiver gauge theories with flavour symmetries. Using a characterisation of the gauge invariants in terms of equivalence classes generated by permutation actions, along with representation theory results in symmetric groups and unitary groups, we give a diagonal basis for the 2-point functions of holomorphic and anti-holomorphic operators. This involves a generalisation of the previously constructed Quiver Restricted Schur operators to the flavoured case. The 3-point functions are derived and shown to be given in terms of networks of symmetric group branching coefficients. The networks are constructed through cutting and gluing operations on the quivers.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 94%

Gauge invariants and correlators in flavoured quiver gauge theories

Mattioli

Ramgoolam

2016

Nuclear Physics B

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“…Hikes can also be defined as the particular heaps of pieces formed from the simple cycles of the graph, see [23]. More recently, their role in gauge theory has been investigated in [16].…”

Section: )mentioning

confidence: 99%

Algebraic Combinatorics on Trace Monoids: Extending Number Theory to Walks on Graphs

Giscard¹,

Rochet²

2017

SIAM J. Discrete Math.

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Partially commutative monoids provide a powerful tool to study graphs, viewing walks as words whose letters, the edges of the graph, obey a specific commutation rule. A particular class of traces emerges from this framework, the hikes, whose alphabet is the set of simple cycles on the graph. We show that hikes characterize undirected graphs uniquely, up to isomorphism, and satisfy remarkable algebraic properties such as the existence and uniqueness of a prime factorization. Because of this, the set of hikes partially ordered by divisibility hosts a plethora of relations in direct correspondence with those found in number theory. Some applications of these results are presented, including a permanantal extension to MacMahon's master theorem and a derivation of the Ihara zeta function.

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“…Recently these results have been generalized to the counting and correlators of quiver gauge theories, where the gauge group is a product of unitary groups and the matter is in bifundamentals or fundamentals [18,25,26]. Links to two dimensional topological field theory have been described and some surprising connections to trace monoids which have applications in computer science have been described.…”

Section: Quiversmentioning

confidence: 99%

Permutations and the combinatorics of gauge invariants for general $N$

Ramgoolam¹

2016

Proceedings of Proceedings of the Corfu Summer Institute 2015 — PoS(CORFU2015)

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Group algebras of permutations have proved highly useful in solving a number of problems in large N gauge theories. I review the use of permutations in classifying gauge invariants in onematrix and multi-matrix models and computing their correlators. These methods are also applicable to tensor models and have revealed a link between tensor models and the counting of branched covers. The key idea is to parametrize U(N) gauge invariants using permutations, subject to equivalences. Correlators are related to group theoretic properties of these equivalence classes. Fourier transformation on symmetric groups by means of representation theory offers nice bases of functions on these equivalence classes. This has applications in AdS/CFT in identifying CFT duals of giant gravitons and their perturbations. It has also lead to general results on quiver gauge theory correlators, uncovering links to two dimensional topological field theory and the combinatorics of trace monoids.

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Quivers, words and fundamentals

Cited by 13 publications

References 60 publications

Gauge invariants and correlators in flavoured quiver gauge theories

Gauge invariants and correlators in flavoured quiver gauge theories

Algebraic Combinatorics on Trace Monoids: Extending Number Theory to Walks on Graphs

Permutations and the combinatorics of gauge invariants for general $N$

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