Second Quantization of the Wilson Loop

Focusing on systems of many identical particles, Chapter 2 introduces appropriate operators to describe their properties in terms of Schwinger’s “measurement symbols.” The latter are then factorized into “creation” and “annihilation” operators, whose fundamental properties and commutation/anticommutation relations are derived in conjunction with the Pauli exclusion principle. This leads to “second quantization” with the Hamiltonian, number, linear and angular momentum operators expressed in terms of the annihilation and creation operators, as well as the occupation number representation. Finally, the concept of coherent states, as eigenstates of the annihilation operator, having minimum uncertainty, is introduced and discussed in detail.

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TOPOLOGY OF THE GRASSMANNIAN σ MODEL ON A LATTICE

Modern Physics Letters A ◽

10.1142/s0217732387000744 ◽

1987 ◽

Vol 02 (08) ◽

pp. 601-608 ◽

Cited By ~ 1

Author(s):

T. FUKAI ◽

M. V. ATRE

Keyword(s):

Partition Function ◽

Strong Coupling ◽

Wilson Loop ◽

Strong Coupling Limit ◽

Topological Term

The Grassmannian σ model with a topological term is studied on a lattice. The θ dependence of the partition function and the Wilson loop are evaluated in the strong coupling limit. The latter is shown to be independent of the area at θ = π, as in the CPN−1 model.

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Exact 1/N expansion of Wilson loop correlators in $$ \mathcal{N} $$ = 4 Super-Yang-Mills theory

Journal of High Energy Physics ◽

10.1007/jhep07(2021)001 ◽

2021 ◽

Vol 2021 (7) ◽

Author(s):

Wolfgang Mück

Keyword(s):

Matrix Model ◽

Wilson Loop ◽

Model Solution ◽

Wilson Loops ◽

Combinatorial Formula ◽

Expectation Value ◽

Yang Mills ◽

Hooft Coupling ◽

The Matrix ◽

Super Yang Mills Theory

Abstract Supersymmetric circular Wilson loops in $$ \mathcal{N} $$ N = 4 Super-Yang-Mills theory are discussed starting from their Gaussian matrix model representations. Previous results on the generating functions of Wilson loops are reviewed and extended to the more general case of two different loop contours, which is needed to discuss coincident loops with opposite orientations. A combinatorial formula representing the connected correlators of multiply wound Wilson loops in terms of the matrix model solution is derived. Two new results are obtained on the expectation value of the circular Wilson loop, the expansion of which into a series in 1/N and to all orders in the ’t Hooft coupling λ was derived by Drukker and Gross about twenty years ago. The connected correlators of two multiply wound Wilson loops with arbitrary winding numbers are calculated as a series in 1/N. The coefficient functions are derived not only as power series in λ, but also to all orders in λ by expressing them in terms of the coefficients of the Drukker and Gross series. This provides an efficient way to calculate the 1/N series, which can probably be generalized to higher-point correlators.

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Entanglement entropy for $$ \mathrm{T}\overline{\mathrm{T}} $$, $$ \mathrm{J}\overline{\mathrm{T}} $$, $$ \mathrm{T}\overline{\mathrm{J}} $$ deformed holographic CFT

Journal of High Energy Physics ◽

10.1007/jhep02(2021)096 ◽

2021 ◽

Vol 2021 (2) ◽

Author(s):

Soumangsu Chakraborty ◽

Akikazu Hashimoto

Keyword(s):

Parameter Space ◽

Wilson Loop ◽

Lorentz Invariance ◽

Entanglement Entropy ◽

Geodesic Equation ◽

Leading Order ◽

Theoretic Analysis ◽

Expectation Value ◽

Spatial Coordinates ◽

Single Trace

Abstract We derive the geodesic equation for determining the Ryu-Takayanagi surface in AdS3 deformed by single trace $$ \mu T\overline{T} $$ μT T ¯ + $$ {\varepsilon}_{+}J\overline{T} $$ ε + J T ¯ + $$ {\varepsilon}_{-}T\overline{J} $$ ε − T J ¯ deformation for generic values of (μ, ε+, ε−) for which the background is free of singularities. For generic values of ε±, Lorentz invariance is broken, and the Ryu-Takayanagi surface embeds non-trivially in time as well as spatial coordinates. We solve the geodesic equation and characterize the UV and IR behavior of the entanglement entropy and the Casini-Huerta c-function. We comment on various features of these observables in the (μ, ε+, ε−) parameter space. We discuss the matching at leading order in small (μ, ε+, ε−) expansion of the entanglement entropy between the single trace deformed holographic system and a class of double trace deformed theories where a strictly field theoretic analysis is possible. We also comment on expectation value of a large rectangular Wilson loop-like observable.

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Wilson loop algebras and quantum K-theory for Grassmannians

Journal of High Energy Physics ◽

10.1007/jhep10(2020)036 ◽

2020 ◽

Vol 2020 (10) ◽

Author(s):

Hans Jockers ◽

Peter Mayr ◽

Urmi Ninad ◽

Alexander Tabler

Keyword(s):

Wilson Loop ◽

Gauge Theories ◽

Wilson Line ◽

Quantum Cohomology ◽

Three Dimensional ◽

Loop Algebra ◽

Loop Algebras ◽

Verlinde Algebra ◽

K Theory ◽

Chern Simons

Abstract We study the algebra of Wilson line operators in three-dimensional $$ \mathcal{N} $$ N = 2 supersymmetric U(M ) gauge theories with a Higgs phase related to a complex Grassmannian Gr(M, N ), and its connection to K-theoretic Gromov-Witten invariants for Gr(M, N ). For different Chern-Simons levels, the Wilson loop algebra realizes either the quantum cohomology of Gr(M, N ), isomorphic to the Verlinde algebra for U(M ), or the quantum K-theoretic ring of Schubert structure sheaves studied by mathematicians, or closely related algebras.

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Notes on cluster algebras and some all-loop Feynman integrals

Journal of High Energy Physics ◽

10.1007/jhep06(2021)119 ◽

2021 ◽

Vol 2021 (6) ◽

Author(s):

Song He ◽

Zhenjie Li ◽

Qinglin Yang

Keyword(s):

Configuration Space ◽

Strong Evidence ◽

Wilson Loop ◽

Cluster Algebra ◽

Cluster Algebras ◽

Gauge Invariant ◽

Feynman Integrals ◽

Cluster Function ◽

Invariant Description ◽

Cluster Configuration

Abstract We study cluster algebras for some all-loop Feynman integrals, including box-ladder, penta-box-ladder, and double-penta-ladder integrals. In addition to the well-known box ladder whose symbol alphabet is $$ {D}_2\simeq {A}_1^2 $$ D 2 ≃ A 1 2 , we show that penta-box ladder has an alphabet of D3 ≃ A3 and provide strong evidence that the alphabet of seven-point double-penta ladders can be identified with a D4 cluster algebra. We relate the symbol letters to the u variables of cluster configuration space, which provide a gauge-invariant description of the cluster algebra, and we find various sub-algebras associated with limits of the integrals. We comment on constraints similar to extended-Steinmann relations or cluster adjacency conditions on cluster function spaces. Our study of the symbol and alphabet is based on the recently proposed Wilson-loop d log representation, which allows us to predict higher-loop alphabet recursively; by applying it to certain eight-point and nine-point double-penta ladders, we also find D5 and D6 cluster functions respectively.

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