QUANTUM YANG-MILLS THEORY ON ARBITRARY SURFACES

1992 ◽  
Vol 07 (16) ◽  
pp. 3781-3806 ◽  
Author(s):  
MATTHIAS BLAU ◽  
GEORGE THOMPSON
Keyword(s):  
Partition Function ◽  
Path Integral ◽  
Two Dimensional ◽  
Expectation Values ◽  
Gauge Transformations ◽  
Yang Mills ◽  
Integral Methods ◽  
Closed Surfaces ◽  
Abelian Case

We study quantum Maxwell and Yang-Mills theory on orientable two-dimensional surfaces with an arbitrary number of handles and boundaries. Using path integral methods we derive general and explicit expressions for the partition function and expectation values of contractible and noncontractible Wilson loops on closed surfaces of any genus, as well as for the kernels on manifolds with handles and boundaries. In the Abelian case we also compute correlation functions of intersecting and self-intersecting loops on closed surfaces, and discuss the role of large gauge transformations and topologically nontrivial bundles.

scholarly journals From $$ \mathcal{N} $$ = 4 Super-Yang-Mills on ℝℙ4 to bosonic Yang-Mills on ℝℙ2

2021 ◽  
Vol 2021 (3) ◽  
Author(s):  
Yifan Wang
Keyword(s):  
Partition Function ◽  
Dirac Operator ◽  
Large Class ◽  
Integral Form ◽  
Two Dimensional ◽  
Expectation Values ◽  
Yang Mills ◽  
Spacetime Manifold ◽  
Agt Correspondence ◽  
Potential Applications

Abstract We study the four-dimensional $$ \mathcal{N} $$ N = 4 super-Yang-Mills (SYM) theory on the unorientable spacetime manifold ℝℙ4. Using supersymmetric localization, we find that for a large class of local and extended SYM observables preserving a common supercharge $$ \mathcal{Q} $$ Q , their expectation values are captured by an effective two-dimensional bosonic Yang-Mills (YM) theory on an ℝℙ2 submanifold. This paves the way for understanding $$ \mathcal{N} $$ N = 4 SYM on ℝℙ4 using known results of YM on ℝℙ2. As an illustration, we derive a matrix integral form of the SYM partition function on ℝℙ4 which, when decomposed into discrete holonomy sectors, contains subtle phase factors due to the nontrivial η-invariant of the Dirac operator on ℝℙ4. We also comment on potential applications of our setup for AGT correspondence, integrability and bulk-reconstruction in AdS/CFT that involve cross-cap states on the boundary.

scholarly journals Schwarzian quantum mechanics as a Drinfeld-Sokolov reduction of BF theory

2020 ◽  
Vol 2020 (12) ◽  
Author(s):  
Fridrich Valach ◽  
Donald R. Youmans
Keyword(s):  
Quantum Mechanics ◽  
Partition Function ◽  
Gauge Group ◽  
Path Integral ◽  
Coadjoint Orbits ◽  
Two Dimensional ◽  
Edge States ◽  
Class Constraint ◽  
Action Functional ◽  
Bf Theory

Abstract We give an interpretation of the holographic correspondence between two-dimensional BF theory on the punctured disk with gauge group PSL(2, ℝ) and Schwarzian quantum mechanics in terms of a Drinfeld-Sokolov reduction. The latter, in turn, is equivalent to the presence of certain edge states imposing a first class constraint on the model. The constrained path integral localizes over exceptional Virasoro coadjoint orbits. The reduced theory is governed by the Schwarzian action functional generating a Hamiltonian S1-action on the orbits. The partition function is given by a sum over topological sectors (corresponding to the exceptional orbits), each of which is computed by a formal Duistermaat-Heckman integral.

scholarly journals TWO-DIMENSIONAL YANG-MILLS THEORIES ARE STRING THEORIES

1993 ◽  
Vol 08 (23) ◽  
pp. 2223-2235 ◽  
Author(s):  
STEPHEN G. NACULICH ◽  
HAROLD A. RIGGS ◽  
HOWARD J. SCHNITZER
Keyword(s):  
String Theory ◽  
Partition Function ◽  
Arbitrary Number ◽  
Closed String ◽  
Two Dimensional ◽  
Branched Covers ◽  
Yang Mills ◽  
String Worldsheet ◽  
String Theories

We show that two-dimensional SO (N) and Sp (N) Yang-Mills theories without fermions can be interpreted as closed string theories. The terms in the 1/N expansion of the partition function on an orientable or nonorientable manifold ℳ can be associated with maps from a string worldsheet onto ℳ. These maps are unbranched and branched covers of ℳ with an arbitrary number of infinitesimal worldsheet cross-caps mapped to points in ℳ. These string theories differ from SU (N) Yang-Mills string theory in that they involve odd powers of 1/N and require both orientable and nonorientable worldsheets.

scholarly journals Exact expressions for n-point maximal U(1)Y-violating integrated correlators in SU(N) $$ \mathcal{N} $$ = 4 SYM

2021 ◽  
Vol 2021 (11) ◽  
Author(s):  
Daniele Dorigoni ◽  
Michael B. Green ◽  
Congkao Wen
Keyword(s):  
Modular Forms ◽  
Free Field ◽  
Perturbation Expansion ◽  
Flat Space ◽  
Low Energy ◽  
Two Dimensional ◽  
Dimensional Lattice ◽  
Yang Mills ◽  
Infinite Sums

Abstract The exact expressions for integrated maximal U(1)Y violating (MUV) n-point correlators in SU(N) $$ \mathcal{N} $$ N = 4 supersymmetric Yang-Mills theory are determined. The analysis generalises previous results on the integrated correlator of four superconformal primaries and is based on supersymmetric localisation. The integrated correlators are functions of N and τ = θ/(2π) + 4πi/$$ {g}_{YM}^2 $$ g YM 2 , and are expressed as two-dimensional lattice sums that are modular forms with holomorphic and anti-holomorphic weights (w, −w) where w = n − 4. The correlators satisfy Laplace-difference equations that relate the SU(N+1), SU(N) and SU(N−1) expressions and generalise the equations previously found in the w = 0 case. The correlators can be expressed as infinite sums of Eisenstein modular forms of weight (w, −w). For any fixed value of N the perturbation expansion of this correlator is found to start at order ($$ {g}_{YM}^2 $$ g YM 2 N)w. The contributions of Yang-Mills instantons of charge k > 0 are of the form qkf(gYM), where q = e2πiτ and f(gYM) = O($$ {g}_{YM}^{-2w} $$ g YM − 2 w ) when $$ {g}_{YM}^2 $$ g YM 2 ≪ 1. Anti-instanton contributions have charge k < 0 and are of the form $$ {\overline{q}}^{\left|k\right|}\hat{f}\left({g}_{YM}\right) $$ q ¯ k f ̂ g YM , where $$ \hat{f}\left({g}_{YM}\right)=O\left({g}_{YM}^{2w}\right) $$ f ̂ g YM = O g YM 2 w when $$ {g}_{YM}^2 $$ g YM 2 ≪ 1. Properties of the large-N expansion are in agreement with expectations based on the low energy expansion of flat-space type IIB superstring amplitudes. We also comment on the identification of n-point free-field MUV correlators with the integrands of (n − 4)-loop perturbative contributions to the four-point correlator. In particular, we emphasise the important rôle of SL(2, ℤ)-covariance in the construction.

scholarly journals SYMMETRIES OF p-BRANES

1993 ◽  
Vol 08 (30) ◽  
pp. 5367-5381 ◽  
Author(s):  
R. PERCACCI ◽  
E. SEZGIN
Keyword(s):  
Global Symmetries ◽  
Sigma Model ◽  
Space Time ◽  
Gauge Transformations ◽  
Yang Mills ◽  
Infinite Dimensional ◽  
Invariance Properties ◽  
Group Manifold ◽  
Mills Field

Using canonical methods, we study the invariance properties of a bosonic p-brane propagating in a curved background locally diffeomorphic to M×G, where M is space-time and G a group manifold. The action is that of a gauged sigma model in p+1 dimensions coupled to a Yang-Mills field and a (p+1) form in M. We construct the generators of Yang-Mills and tensor gauge transformations and exhibit the role of the (p+1) form in canceling the potential Schwinger terms. We also discuss the Noether currents associated with the global symmetries of the action and the question of the existence of infinite-dimensional symmetry algebras, analogous to the Kac-Moody symmetry of the string.

scholarly journals TWO-DIMENSIONAL LATTICE GRAVITY AS A SPIN SYSTEM

1994 ◽  
Vol 05 (02) ◽  
pp. 359-361 ◽  
Author(s):  
W. BEIRL ◽  
H. MARKUM ◽  
J. RIEDLER
Keyword(s):  
Partition Function ◽  
External Field ◽  
Path Integral ◽  
Spin System ◽  
Higher Dimensions ◽  
Two Dimensional ◽  
Path Integral Formulation ◽  
Regge Calculus ◽  
Dimensional Lattice ◽  
Matter Fields

Quantum gravity is studied in the path integral formulation applying the Regge calculus. Restricting the quadratic link lengths of the originally triangular lattice the path integral can be transformed to the partition function of a spin system with higher couplings on a Kagomé lattice. Various measures acting as external field are considered. Extensions to matter fields and higher dimensions are discussed.

scholarly journals EMERGENCE OF THE HAAR MEASURE IN THE STANDARD FUNCTIONAL INTEGRAL REPRESENTATION OF THE YANG-MILLS PARTITION FUNCTION

1996 ◽  
Vol 11 (30) ◽  
pp. 2451-2461 ◽  
Author(s):  
H. REINHARDT
Keyword(s):  
Partition Function ◽  
Path Integral ◽  
Haar Measure ◽  
Transition Amplitude ◽  
Functional Integral ◽  
Abelian Gauge ◽  
Gauge Invariant ◽  
Yang Mills ◽  
Gauge Fixing ◽  
Popov Method

The conventional path integral expression for the Yang–Mills transition amplitude with flat measure and gauge-fixing built in via the Faddeev-Popov method has been claimed to fall short of guaranteeing gauge invariance in the nonperturbative regime. We show, however, that it yields the gauge-invariant partition function where the projection onto gauge-invariant wave functions is explicitly performed by integrating over the compact gauge group. In a variant of maximal Abelian gauge the Haar measure arises in the conventional Yang-Mills path integral from the Faddeev-Popov determinant.

scholarly journals PERTURBATIVE VACUA FROM IIB MATRIX MODEL

2008 ◽  
Vol 23 (14n15) ◽  
pp. 2279-2280
Author(s):  
HIKARU KAWAI ◽  
MATSUO SATO
Keyword(s):  
String Theory ◽  
Path Integral ◽  
Matrix Model ◽  
Moduli Space ◽  
Two Dimensional ◽  
Reduced Models ◽  
Yang Mills ◽  
Large N ◽  
Super Yang Mills Theory

It has not been clarified whether a matrix model can describe various vacua of string theory. In this talk, we show that the IIB matrix model includes type IIA string theory1. In the naive large N limit of the IIB matrix model, configurations consisting of simultaneously diagonalizable matrices form a moduli space, although the unique vacuum would be determined by complicated dynamics. This moduli space should correspond to a part of perturbatively stable vacua of string theory. Actually, one point on the moduli space represents type IIA string theory. Instead of integrating over the moduli space in the path-integral, we can consider each of the simultaneously diagonalizable configurations as a background and set the fluctuations of the diagonal elements to zero. Such procedure is known as quenching in the context of the large N reduced models. By quenching the diagonal elements of the matrices to an appropriate configuration, we show that the quenched IIB matrix model is equivalent to the two-dimensional large N [Formula: see text] super Yang-Mills theory on a cylinder. This theory is nothing but matrix string theory and is known to be equivalent to type IIA string theory. As a result, we find the manner to take the large N limit in the IIB matrix model.

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.

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