The moduli space of flat connections for maximally supersymmetric Yang-Mills theories, in a space-time of the form T^3xR, contains isolated points, corresponding to normalizable zero energy states, for certain simple gauge groups G. We consider the low energy effective field theories in the weak coupling limit supported on such isolated points and find that when quantized they consist of an infinite set of harmonic oscillators whose angular frequencies are completely determined by the Lie algebra of G. We then proceed to find the isolated flat connections for all simple G and subsequently specify the corresponding effective field theories.
An explicit proof of the existence of nontrivial vacua in the pure supersymmetric Yang-Mills theories with higher orthogonal SO(N), N>=7 or the G_2 gauge group defined on a 3-torus with periodic boundary conditions is given. Extra vacuum states are separated by an energy barrier from the perturbative vacuum A_i=0 and its gauge copies.
Classical vacuum - pure gauge - solutions of Euclidean two-dimensional SU(2) Yang-Mills theories are studied. Topologically non-trivial vacua are found in a class of gauge group elements isomorphic to $S_2$. These solutions are unexpectedly related to the solution of the non-linear O(3) model and to the motion of a particle in a periodic potential.
Reducing Supersymmetric Yang-Mills Field Theory to a single point in the three dimensional space results in the Supersymmetric Yang-Mills Quantum Mechanics (SYMQM) which basically is the effective quantum mechanics of zero momentum modes of the original theory. Such a system is still quite non-trivial and usually inherits many properties of the original field theory. In this talk some beautiful features of the three-dimensional model will be reviewed and illustrated with the aid of the recent, quantitative solution. In particular the structure of the supersymmetric vacua and condensates will b…
We compute the number of massive vacua of N=4 supersymmetric Yang-Mills theory mass-deformed to preserve N=1 supersymmetry, for any gauge group G. We use semi-classical techniques and efficiently reproduce the known counting for A,B and C-type gauge groups, present the generating function for both O(2n) and SO(2n), and compute the supersymmetric index for gauge groups of exceptional type. A crucial role is played by the classification of nilpotent orbits, as well as global properties of their centralizers. We give illustrative examples of new features of our analysis for the D-type algebras.
We provide a simple derivation of the extremal values of the superpotential in massive vacua of N=1* SYM, making use of the required modular weight for the central charge of BPS walls interpolating between these vacua. This modular weight descends from the action of S-duality on the N=4 superalgebra which in turn is inherited from its classical action on the dyon spectrum. We show that this kinematic information, combined with minimal knowledge of the weak coupling asymptotics, is sufficient to determine the exact vacuum superpotentials in terms of Eisenstein series.
We investigate the vacuum structure of pure SU(N) N=1 super Yang-Mills. The theory is expected to possess N vacua with associated domain walls. We show that the newly extended version of the low energy effective Lagrangian for super Yang-Mills supports the BPS domain wall solutions associated with any two vacua aligned with the origin of the moduli space. For the two color theory the domain wall analysis is complete. We also find new non BPS domain wall solutions connecting any two vacua of the underling SU(N) super Yang-Mills theory not necessarely aligned. When two vacua are aligned with the…
We discuss the existence of $θ$-vacua in pure Yang-Mills theory in two space-time dimensions. More precisely, a procedure is given which allows one to classify the distinct quantum theories possessing the same classical limit for an arbitrary connected gauge group G and compact space-time manifold M (possibly with boundary) possessing a special basepoint. For any such G and M it is shown that the above quantizations are in one-to-one correspondence with the irreducible unitary representations (IUR's) of $π_1(G)$ if M is orientable, and with the IUR's of $π_1(G)/2π_1(G)$ if M is nonorientable.
We consider generic properties of the moduli space of vacua in $N=2$ supersymmetric Yang--Mills theory recently studied by Seiberg and Witten. We find, on general grounds, Picard--Fuchs type of differential equations expressing the existence of a flat holomorphic connection, which for one parameter (i.e. for gauge group $G=SU(2)$), are second order equations. In the case of coupling to gravity (as in string theory), where also ``gravitational'' electric and magnetic monopoles are present, the electric--magnetic S duality, due to quantum corrections, does not seem any longer to be related to $S…
Two-dimensional SU(N) Yang-Mills theory is endowed with a non-trivial vacuum structure (k-sectors). The presence of k-sectors modifies the energy spectrum of the theory and its instanton content, the (Euclidean) space-time being compactified on a sphere. For the exact solution, in the limit in which the sphere is decompactified, a k-sector can be mimicked by the presence of k-fundamental charges at infinity, according to a Witten's suggestion. However, this property neither holds before decompactification nor for the genuine perturbative solution which corresponds to the zero-instanton contrib…
In this thesis we discuss recent new insights in the structure of the moduli space of flat connections of Yang-Mills theory on a 3-torus. Chapter 2 discusses the computation of Witten's index for 4-dimensional gauge theories, and the paradox that arises in comparing various computations. This was resolved by the discovery that for orthogonal and exceptional gauge groups, periodic flat connections exist that are contained in seperate, disconnected components of the moduli space. Chapter 3 and 4 discuss some aspects of the construction of holonomies parametrising vacua on such disconnected compo…
It has recently been argued that the confining vacua of Yang-Mills theory in the far infrared can have topological degrees of freedom given by magnetic $\mathbb{Z}_q$ gauge field, both in the non-supersymmetric case and in the N=1 supersymmetric case. In this short note we give another piece of evidence by computing and matching the supersymmetric index of the pure super Yang-Mills theory both in the ultraviolet and in the infrared.
We study the N=1* supersymmetric gauge theories with gauge groups Sp(2n) and SO(2n+1). These theories are obtained from the corresponding N=4 supersymmetric Yang-Mills theories via a mass deformation. We show that the number of quantum vacua in the Sp(2n) theory is equal to the number of quantum vacua in the SO(2n+1) theory. This constitutes non-trivial support for S-duality between these theories. The verification of the equality of the number of quantum vacua involves a rather esoteric identity due to Ramanujan.
In the (non-supersymmetric) Yang-Mills theory in the large N limit there exists an infinite set of non-degenerate vacua. The distinct vacua are separated by domain walls whose tension determines the decay rate of the false vacua. I discuss the phenomenon from a field-theoretic point of view, starting from supersymmetric gluodynamics and then breaking supersymmetry, by introducing a gluino mass. By combining previously known results, the decay rate of the excited vacua is estimated, Γ\sim \exp (-const \times N^4). The fourth power of N in the exponent is a consequence of the fact that the wall …
$QCD_2$ with fermions in the adjoint representation is invariant under $SU(N)/Z_N$ and thereby is endowed with a non-trivial vacuum structure (k-sectors). The static potential between adjoint charges, in the limit of infinite mass, can be therefore obtained by computing Wilson loops in the pure Yang-Mills theory with the same non-trivial structure. When the (Euclidean) space-time is compactified on a sphere $S^2$, Wilson loops can be exactly expressed in terms of an infinite series of topological excitations (instantons). The presence of k-sectors modifies the energy spectrum of the theory and…
Existence of theta-vacuum states in Yang--Mills theories defined over asymptotically flat space-times examined taking into account not only the topology but the complicated causal structure of these space-times, too. By a result of Galloway apparently causality makes all vacuum states, seen by a distant observer, homotopically equivalent making the introduction of theta-terms unnecessary. But a more careful analysis shows that certain twisted classical vacuum states survive even in this case eventually leading to the conclusion that the concept of ``theta-vacua'' is meaningful in the case of g…
Flat connections for unitary gauge groups on a 3--torus with twisted boundary conditions as well as recently discovered periodic nontrivial flat connections with ``nondiagonalizable'' triples of holonomies for higher orthogonal and exceptional groups are constructed explicitly in terms of Jacobi theta functions with rational characteristics. The (fractional) Chern-Simons numbers of these vacuum gauge field configurations are verified by direct computation.
Instantons and their quantisation in pure Yang-Mills theory formulated in the background of de Sitter spacetime represented by spatially-closed ($k = 1$) Friedmann-Robertson-Walker metric are discussed. As for the classical treatment of the instanton physics, first, explicit instanton solutions are found and next, quantities like Pontryagin index and the semiclassical approximation to the inter-vacua tunnelling amplitude are evaluated. The Atiyah-Patodi-Singer index theorem is checked as well by constructing explicitly the normalizable fermion zero modes in this de Sitter spacetime instanton b…
I consider N=1 U(N) gauge theory with matter in the adjoint, fundamental and anti-fundamental representations. Focusing on the equations defining the Riemann surface that describes the quantum theory, the gaugino condensates (and related superpotentials) are calculated in the limit of SU(N) gauge group, both in the pure theory and in the presence of matter. In the case without fundamental matter it is investigated the structure of the space of vacua. In particular it is discussed how different vacua can be related, in a way which finally helps to count them.
We give a detailed study of the critical points of the potentials of the simplest non-trivial N=2 gauged Yang-Mills/Einstein supergravity theories with tensor multiplets. The scalar field target space of these examples is SO(1,1)XSO(2,1)/SO(2). The possible gauge groups are SO(2)XU(1)_R and SO(1,1)XU(1)_R, where U(1)_R is a subgroup of the R-symmetry group SU(2)_R, and SO(2) and SO(1,1) are subgroups of the isometry group of the scalar manifold. The scalar potentials of these theories consist of a contribution from the U(1)_R gauging and a contribution that is due to the presence of the tensor…
Recent analytical and numerical solutions of the above systems are reviewed. Discussed results include: a) exact construction of the supersymmetric vacua in two space-time dimensions, and b) precise numerical calculations of the coexisting continuous and discrete spectra in the four-dimensional system, together with the identification of dynamical supermultiplets and SUSY vacua. New construction of the gluinoless SO(9) singlet state, which is vastly different from the empty state, in the ten-dimensional model is also briefly summarized.
We find an argument related to the existence of a Z_2-symmetry for the renormalization group flow derived from the bare Yang-Mills Lagrangian, and show that the cancellation of the vacuum energy may arise motivated both from the renormalization group flow solutions and the effective Yang-Mills action. In the framework of the effective Savvidy's action, two Mirror minima are allowed, with exactly equal and hold opposite sign energy densities. At the cosmological level, we explore the stability of the electric and magnetic attractor solutions, both within and beyond the perturbation theory, and …
At a generic point in the moduli space of vacua of an N=4 supersymmetric gauge theory with arbitrary gauge group the Higgs force does not cancel the magneto-static force between magnetic monopoles of distinct charge. As a consequence the moduli space of magnetically charged solutions is related in a simple way to those of the SU(2) theory. This leads to a rather simple test of S-duality. On certain subspaces of the moduli space of vacua the forces between distinct monopoles cancel and the test of S-duality becomes more complicated.
We analyze a particular SU(2) invariant sector of the scalar manifold of gauged N=8 supergravity in five dimensions, and find all the critical points of the potential within this sector. The critical points give rise to Anti-de Sitter vacua, and preserve at least an SU(2) gauge symmetry. Consistent truncation implies that these solutions correspond to Anti-de Sitter compactifications of IIB supergravity, and hence to possible near-horizon geometries of 3-branes. Thus we find new conformal phases of softly broken N=4 Yang--Mills theory. One of the critical points preserves N=2 supersymmetry in …
$4d$ ${\mathcal N}=1$ 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 $G_2$, and for the minimal domain wall connecting neighboring vacua for arbitrary $G$. We propose that t…
We consider a pair of noncommutative lumps in the noncommutative Yang--Mills/M(atrix) model. In the case when the lumps are separated by a finite distance their ``polarisations'' do not belong to orthogonal subspaces of the Hilbert space. In this case the interaction between lumps is nontrivial. We analyse the dynamics arisen due to this interaction in both naive approach of rigid lumps and exactly as described by the underlying gauge model. It appears that the exact description is given in terms of finite matrix models/multidimensional mechanics whose dimensionality depends on the initial con…
We present the configurations of intersecting branes in type IIA string theory corresponding to the meta-stable supersymmetry breaking vacua(hep-th/0608063) in the four-dimensional N=1 supersymmetric Yang-Mills theory coupled massive flavors with adjoint matter where the superpotential has three deformed terms.
We consider classical, pure Yang-Mills theory in a box. We show how a set of static electric fields that solve the theory in an adiabatic limit correspond to geodesic motion on the space of vacua, equipped with a particular Riemannian metric that we identify. The vacua are generated by spontaneously broken global gauge symmetries, leading to an infinite number of conserved momenta of the geodesic motion. We show that these correspond to the soft multipole charges of Yang-Mills theory.
We provide explicit formulas for the number of vacua of four-dimensional pure N=1 super Yang-Mills theories on a circle, with any simple gauge algebra and any choice of center and spectrum of line operators. These form a key ingredient in the semi-classical calculation of the number of massive vacua of N=1* gauge theories with gauge algebra su(n) compactified on a circle. Using arithmetic, we express that number in an SL(2,Z) duality invariant manner. We confirm our tally of massive vacua of the N=1* theories by a count of inequivalent extrema of the exact superpotential. Furthermore, we compu…
One can define generalized models of gaugino condensation as theories which dynamically break a discrete R-symmetry, but do not break supersymmetry. We consider general examples consisting of gauge and matter fields, and the minimal number of gauge singlet fields to avoid flat directions in the potential. We explore which R-symmetries can arise, and their spontaneous breaking. In general, we find that the discrete symmetry is $\mathbb{Z}_{2b_0R}$ and the number of supersymmetric vacua is $b_0$, where $b_0$ is the coefficient of the one-loop beta function. Results are presented for various grou…
We study vacuum fluctuation properties of an ensemble of $SU(N)$ gauge theory configurations, in the limit of large number of colors, \textit{viz.} $N_c \rightarrow \infty$, and explore statistical nature of the topological susceptibility by analyzing its critical behavior at a nonzero vacuum parameter $θ$ and temperature $T$. We find that the system undergoes a vacuum phase transition at the chiral symmetry restoration temperature as well as at an absolute value of $θ$. On the other hand, the long range correlation length solely depends on $θ$ for the theories having critical exponent $e=2$ o…
We present families of algebraic curves describing the moduli-space of $N\!=\!2$ supersymmetric Yang-Mills theory with gauge group $SO(2n)$. We test our curves by computing the weak coupling monodromies and the number of $N\!=\!1$ vacua.
We consider instanton contributions to chiral correlators, such as <0| Tr λ^2 (x) Tr λ^2(x') |0>, in N=1 supersymmetric Yang-Mills theory with either light adjoint or fundamental matter. Within the former model, extraction of the gluino condensate from a connected 1-instanton diagram, evaluated at strong coupling, can be contrasted with expectations from the Seiberg-Witten solution perturbed to an N=1 vacuum. We observe a numerical discrepancy, coinciding with that observed previously in N=1 SQCD. Moreover, since knowledge of the vacuum structure is complete for softly broken N=2 Yang-Mills, t…
We analyze the classical and quantum vacua of 2d $\mathcal{N}=(8,8)$ supersymmetric Yang-Mills theory with $SU(N)$ and $U(N)$ gauge group, describing the worldvolume interactions of $N$ parallel D1-branes with flat transverse directions $\mathbb{R}^8$. We claim that the IR limit of the $SU(N)$ theory in the superselection sector labeled $M \pmod{N}$ --- identified with the internal dynamics of $(M,N)$-string bound states of Type IIB string theory --- is described by the symmetric orbifold $\mathcal{N}=(8,8)$ sigma model into $(\mathbb{R}^8)^{D-1}/\mathbb{S}_D$ when $D=\gcd(M,N)>1$, and by a si…
Multicenter supergravity solutions corresponding to Higgs phases of supersymmetric Yang-Mills theories are considered. For NS5 branes we identify three cases where there is a description in terms of exact conformal field theories. Other supergravity solutions, such as D3-branes with angular momentum, are understood as special limits of multicenter ones. Within our context we also consider 4-dim gravitational multi-instantons.
We present a construction of domain walls in string theory. The domain walls can bridge both Minkowski and AdS string vacua. A key ingredient in the construction are novel classical Yang-Mills configurations, including instantons, which interpolate between toroidal Yang-Mills vacua. Our construction provides a concrete framework for the study of inflating metrics in string theory. In some cases, the accelerating space-time comes with a holographic description. The general form of the holographic dual is a field theory with parameters that vary over space-time.
We study the algebraic and geometric structures that underly the space of vacua of N=1 super Yang-Mills theories at the non-perturbative level. Chiral operators are shown to satisfy polynomial equations over appropriate rings, and the phase structure of the theory can be elegantly described by the factorization of these polynomials into irreducible pieces. In particular, this idea yields a powerful method to analyse the possible smooth interpolations between different classical limits in the gauge theory. As an application in U(Nc) theories, we provide a simple and completely general proof of …
We construct the 4-dimensional ${\cal N}=\frac12$ and ${\cal N}=1$ inhomogeneously mass-deformed super Yang-Mills theories from the ${\cal N} =1^*$ and ${\cal N} =2^*$ theories, respectively, and analyse their supersymmetric vacua. The inhomogeneity is attributed to the dependence of background fluxes in the type IIB supergravity on a single spatial coordinate. This gives rise to inhomogeneous mass functions in the ${\cal N} =4$ super Yang-Mills theory which describes the dynamics of D3-branes. The Killing spinor equations for those inhomogeneous theories lead to the supersymmetric vacuum equa…
It is believed that in $SU(N)$ Yang-Mills theory observables are $N$-branched functions of the topological $θ$ angle. This is supposed to be due to the existence of a set of locally-stable candidate vacua, which compete for global stability as a function of $θ$. We study the number of $θ$ vacua, their interpretation, and their stability properties using systematic semiclassical analysis in the context of adiabatic circle compactification on $\mathbb{R}^3 \times S^1$. We find that while observables are indeed N-branched functions of $θ$, there are only $\approx N/2$ locally-stable candidate vac…
We propose a deformation principle of gauge theories in three dimensions that can describe topologically stable self-dual gauge fields, i.e., vacua configurations that in spite of their masses do not deform the background geometry and are locally undetected by charged particles. We interpret these systems as describing boundary degrees of freedom of a self-dual Yang-Mills field in $2+2$ dimensions with mixed boundary conditions. Some of these fields correspond to Abrikosov-like vortices with an exponential damping in the direction penetrating into the bulk. We also propose generalizations of t…
Cosmological solutions with a homogeneous Yang-Mills field which oscillates and passes between topologically distinct vacua are discussed. These solutions are used to model the collapsing Bartnik-McKinnon gravitational sphaleron and the associated anomalous production of fermions. The Dirac equation is analyzed in these backgrounds. It is shown explicitly that a fermion energy level crosses from the negative to positive energy spectrum as the gauge field evolves between the topologically distinct vacua. The cosmological solutions are also generalized to include an axion field.
An announcement of some results of a longer paper where the supersymmetric vacua of two dimensional N=2 susy gauge theories with matter are shown to be in one-to-one correspondence with the eigenstates of integrable spin chain Hamiltonians. The Heisenberg spin chain is mapped to the two dimensional U(N) theory with fundamental hypermultiplets, the XXZ spin chain is mapped to the analogous three dimensional super-Yang-Mills theory compactified on a circle, the XYZ spin chain and eight-vertex model are related to the four dimensional theory compactified on a torus. The correspondence extends to …
A generalization of the two-dimensional Yang-Mills and generalized Yang-Mills theory is introduced in which the building B-F theory is nonlocal in the auxiliary field. The classical and quantum properties of this nonlocal generalization are investigated and it is shown that for large gauge groups, there exist a simple correspondence between the properties a nonlocal theory and its corresponding local theory.
We consider the N=1 super Yang-Mills theory with gauge group U(Nc) or SU(Nc) and one adjoint Higgs field with an arbitrary polynomial superpotential. We provide a purely field theoretic derivation of the exact effective superpotential W(S) for the glueball superfield S in the confining vacua. We show that the result matches with the Dijkgraaf-Vafa matrix model proposal. The proof brings to light a deep relationship between non-renormalization theorems first discussed by Intriligator, Leigh and Seiberg, and the fact that W(S) is given by a sum over planar diagrams.
Using an infinite number of fields, we construct actions for D=4 self-dual Yang-Mills with manifest Lorentz invariance and for D=10 super-Yang-Mills with manifest super-Poincaré invariance. These actions are generalizations of the covariant action for the D=2 chiral boson which was first studied by McClain, Wu, Yu and Wotzasek.
We will discuss an integrable structure for weakly coupled superconformal Yang-Mills theories, describe certain equivalences for the Yangian algebra, and fill a technical gap in our previous study of this subject.
We examine the vacua of the scalar potential of the effective 4-d action, obtained after the dimensional reduction of the 10-d $\mathcal{N}=1$ heterotic supergravity coupled to an $\mathcal{N}=1$ Yang-Mills sector. The (Coset Space) dimensional reduction takes place over the three 6-d nearly-Kähler manifolds, namely the homogeneous 6-d non-symmetric coset spaces, $G_2/SU(3)$, $Sp_4/SU(2)\times U(1)$ and $SU(3)/U(1)\times U(1)$. The current work consists a complete catalogue of the kinds of vacua of theories obtained after the reduction of the heterotic string over the 6-d non-symmetric coset s…
The planar Yang-Mills theory in three spatial dimensions is examined in a particular representation which explicitly embodies factorization. The effective planar Yang-Mills theory Hamiltonian is constructed in this representation.
The five dimensional stable de Sitter ground states in N=2 supergravity obtained by gauging SO(1,1) symmetry of the real symmetric scalar manifold (in particular a generic Jordan family manifold of the vector multiplets) simultaneously with a subgroup R_s of the R-symmetry group descend to four dimensional de Sitter ground states under certain conditions. First, the holomorphic section in four dimensions has to be chosen carefully by using the symplectic freedom in four dimensions; and second, a group contraction is necessary to bring the potential into a desired form. Under these conditions, …
Recent developments in the determination of the planar S-matrix of N=4 Super Yang-Mills are closely related to its Yangian symmetry. Here we provide evidence for a yet unobserved additional symmetry: the Yangian level-one helicity operator.
The BCFW recursion relation in $\mathcal{N}=4$ super-Yang-Mills theory is solved using Yang-Baxter $R$-operators in the NMHV sector. Explicit expressions for $\mathcal{R}$-invariants are obtained in terms of the chains of $R$-operators acting on an appropriate basic state.
The large-N behavior of Yang-Mills and generalized Yang-Mills theories in the double-scaling limit is investigated. By the double-scaling limit, it is meant that the area of the manifold on which the theory is defined, is itself a function of N. It is shown that phase transitions of different orders occur, depending on the functional dependence of the area on N. The finite-size scalings of the system are also investigated. Specifically, the dependence of the dominant representation on A, for large but finite N is determined.
We present new relations for scattering amplitudes of color ordered gluons and gravitons in Einstein-Yang-Mills theory. Tree-level amplitudes of arbitrary multiplicities and polarizations involving up to three gravitons and up to two color traces are reduced to partial amplitudes of pure Yang-Mills theory. In fact, the double-trace identities apply to Einstein-Yang-Mills extended by a dilaton and a B-field. Our results generalize recent work of Stieberger and Taylor for the single graviton case with a single color trace. As the derivation is made in the dimension-agnostic Cachazo-He-Yuan forma…
We analyze in an effective Lagrangian framework the connection between Super QCD (Super Yang Mills) and QCD (Yang Mills) by highlighting the crucial role that the zero modes play in the process of decoupling gluinos and squarks.
We show that the solutions of SU(2) Yang-Mills-dilation and Einstein-Yang-Mills-dilaton theories described in a sequence of papers by Kleihaus and Kunz are not regular in the gauge field part.
The existence of a "bonus" U(1) level-one Yangian symmetry of N=4 super-Yang-Mills has recently been proposed. We provide evidence for this proposal by constructing the BRST-invariant nonlocal charge in the pure spinor sigma model corresponding to this bonus level-one symmetry. We also construct analogous charges for bonus U(1) symmetries at all odd levels of the Yangian.
We develop a systematic method to describe the moduli space of vacua of four dimensional $\mathcal{N}=2$ class ${\cal S}$ theories including Coulomb branch, Higgs branch and mixed branches. In particular, we determine the Higgs and mixed branch roots, and the dimensions of the Coulomb and Higgs components of mixed branches. They are derived by using generalized Hitchin's equations obtained from twisted compactification of 5d maximal Super-Yang-Mills, with local degrees of freedom at punctures given by (nilpotent) orbits. The crucial thing is the holomorphic factorization of the Seiberg-Witten …
