Supersymmetric Gauge Theory

Non-perturbative effects of constant magnetic fields in a Higgs-Yukawa gauge model are studied using the extremum equations of the effective action for composite operators. It is found that the magnetic field induces a Higgs condensate, a fermion-antifermion condensate, and a fermion dynamical mass, hence breaking the discrete chiral symmetry of the theory. The results imply that for a non-simple group extension of the present model, the external magnetic field would either induce or reinforce gauge symmetry breaking. Possible cosmological applications of these results in the electroweak phase transition are suggested. Symmetry behavior in quantum field theories under the influence of external fields has long been a topic of intensive study in theoretical physics . In the present paper we are interested in particular in non-perturbative effects produced by external magnetic fields in gauge theories with scalars. Our main claim is that for theories with non-simple gauge group, scalar-scalar and scalar-fermion interactions, the magnetic field reinforces gauge symmetry breaking. The observation of large-scale galactic magnetic fields in a number of galaxies, in galactic halos, and in clusters of galaxies has recently stimulated a large number of works trying to explain the physical mechanism responsible for the origin of these fields. Many of the proposed generating mechanisms have compelling arguments in favor of the existence of strong primordial magnetic fields (for a review of cosmological generating mechanisms see [3] and references therein). Since primordial magnetic fields could play a significant role in particle cosmology, the investigations on the theme have recently boomed. In this context, the implications of a magnetic-field-driven gauge symmetry breaking mechanism may be important. Several years before this renewed interest in cosmological magnetic fields, Ambøjrn and Olesen [4] considered the electroweak model in the presence of a constant magnetic field. Assuming certain special values of the couplings, they obtained a W-and Z-condensate solution forming a lattice of abelian vortex lines for a range of magnetic fields lying between m 2 W e and m 2 W cos 2 θ e

The Source Energy Field Theory (SEFT) extends the freedom to define spacetime itself as a manifestation of the fundamental energy field. This enhanced degree of freedom allows the Standard Model gauge symmetries and the Higgs mechanism to emerge as natural geometric consequences, rather than imposed assumptions. While its application to cosmology has successfully explained supernova redshiftdistance relations, galactic rotation curves, and the cosmic microwave background, its implications for particle physics have remained unexplored. In this paper, we extend SEFT to the domain of particle physics and demonstrate that and preserving the custodial relation ρ = 1. These results establish SEFT as a framework in which the core architecture of the Standard Model is embedded in the dynamics of a single fundamental field, thus providing the particle-physics counterpart to the cosmological results previously derived in SEFT.

We show that noncommutative gauge theories with arbitrary compact gauge group defined by means of the Seiberg-Witten map have the same one-loop anomalies as their commutative counterparts. This is done in two steps. By explicitly calculating the µ 1 µ 2 µ 3 µ 4 part of the renormalized effective action, we first find the would-be one-loop anomaly of the theory to all orders in the noncommutativity parameter θ µν . And secondly we isolate in the would-be anomaly radiative corrections which are not BRS trivial. This gives as the only true anomaly occurring in the theory the standard Bardeen anomaly of commutative spacetime, which is set to zero by the usual anomaly cancellation condition.

We continue the effort of defining and evaluating the quantum entropy function for supersymmetric black holes in 4d $$ \mathcal{N} $$ N = 2 gauged supergravity, initiated in [1]. The emphasis here is on the missing steps in the previous localization analysis, mainly dealing with one-loop determinants for abelian vector multiplets and hypermultiplets on the non-compact space ℍ2× Σg with particular boundary conditions. We use several different techniques to arrive at consistent results, which have a most direct bearing on the logarithmic correction terms to the Bekenstein-Hawking entropy of said black holes.

We classify all N = 2 rigid supersymmetric backgrounds in four dimensions with both Lorentzian and Euclidean signature that preserve eight real supercharges, up to discrete identifications. Among the backgrounds we find specific warpings of S 3 × R and AdS 3 × R, AdS 2 × S 2 and H 2 × S 2 with generic radii, and some more exotic geometries. We provide the generic two-derivative rigid vector and hypermultiplet actions and analyze the conditions imposed on the special Kähler and hyperkähler target spaces.

The conditions for fully supersymmetric backgrounds of general N = 2 locally supersymmetric theories are derived based on the off-shell superconformal multiplet calculus. This enables the derivation of a non-renormalization theorem for a large class of supersymmetric invariants with higher-derivative couplings. The theorem implies that the invariant and its first order variation must vanish in a fully supersymmetric background. The conjectured relation of one particular higher-derivative invariant with a specific fivedimensional invariant containing the mixed gauge-gravitational Chern-Simons term is confirmed.

A new class of N = 2 locally supersymmetric higher-derivative invariants is constructed based on logarithms of conformal primary chiral superfields. They characteristically involve a coupling to R µν 2 -1 3 R 2 , which equals the non-conformal part of the Gauss-Bonnet term. Upon combining one such invariant with the known supersymmetric version of the square of the Weyl tensor one obtains the supersymmetric extension of the Gauss-Bonnet term. The construction is carried out in the context of both conformal superspace and the superconformal multiplet calculus. The new class of supersymmetric invariants resolves two open questions. The first concerns the proper identification of the 4D supersymmetric invariants that arise from dimensional reduction of the 5D mixed gauge-gravitational Chern-Simons term. The second is why the pure Gauss-Bonnet term without supersymmetric completion has reproduced the correct result in calculations of the BPS black hole entropy in certain models.

This paper develops the concept of autonegation as the destruction of supersymmetry within the framework of Meta-Monism ontology. Unlike Hegelian "negation of negation," autonegation is not a logical move, but a primordial ontological act in which chaos denies its own identity. Chaos is interpreted as absolute negativity: not a substance with attributes, but negativity itself, eternally negating itself. This act generates duality, relations, time, and dissipation. The argument unfolds in six steps: (1) chaos as supersymmetry (absolute selfidentity), (2) autonegation as its destruction, (3) duality (Being/Non-Being), (4) emergence of relations (AND/OR), (5) time as the scene of differentiation, and (6) dissipation as the tendency toward equilibrium. The paper situates this ontological model in dialogue with physics (symmetry breaking in supersymmetry, entropy, Seiberg-Witten theory), philosophy (Nagarjuna, Hegel, Schelling, Kant), and phenomenology of consciousness. Its unique contribution is the consistent assertion of negativity as the fundamental principle of reality. It offers not only an ontological explanation of the emergence of order from chaos, but also an ethical orientation: creativity as conscious selfnegation, the destruction of obsolete forms to allow the birth of new ones.

This paper proposes a first-principles explanation, within the framework of the Information-Causal Compression Field (ICCF) unified field theory, for why the Standard Model of particle physics adopts the specific gauge symmetry group SU(3)×SU(2)×U(1). Rather than treating this structure as an arbitrary postulate, we argue that it emerges as the universe’s unique and inevitable “Cosmic Resonance” mode, arising from the interplay of three fundamental principles: energy conservation, entropy increase, and the principle of least action. We identify SU(3), SU(2), and U(1) respectively with the three most basic resonance topologies in ICCF: causal saturation, causal reconstruction, and causal propagation. We further demonstrate that any deviation from this “triadic chord” of symmetries leads to instability, higher energy dissipation, and eventual elimination by the principle of least action. Thus, the Standard Model’s gauge structure is not an accident of nature, but the only stable resonance solution capable of sustaining a universe with stability (SU(3)), evolution (SU(2)), and communication (U(1)). Finally, the ICCF framework yields testable predictions, including harmonic mass spectra, coupling constant modulations, and observable “dissonances” in extreme cosmological events, positioning this theory as a physically grounded alternative to traditional symmetry-origin frameworks such as GUTs and string theory.

We demonstrate Yokoyama gaugeon formalism for the Abelian one-form gauge (Maxwell) as well as for Abelian two-form gauge theory in the very special relativity (VSR) framework. In VSR scenario, the extended action due to introduction of gaugeon fields also possesses form invariance under quantum gauge transformations. It is observed that the gaugeon field together with gauge field naturally acquire mass, which is different from the conventional Higgs mechanism. The quantum gauge transformation implements a shift in gauge parameter. Further, we analyse the BRST symmetric gaugeon formalism in VSR which embeds only one subsidiary condition rather than two.

In a recent letter it has been shown that gauge theory with a dilaton provides linearly increasing gauge potentials from static or uniformly moving pointlike colour sources. This ensures confinement in the framework of no-pair equations. Here I would like to point out that a dilaton coupling both to the gauge curvature term and to fermion masses yields a linear potential with a scalar component and a dominant vector contribution.

The Fermi coupling constant G F is a fundamental parameter of the weak interaction. Traditionally determined experimentally, it defines the strength of processes such as beta decay and neutrino scattering. In this work, we present an analytical derivation of G F within the Mo framework, where it emerges naturally from the Boltzmann constant k B and the unifying Mo mass scale M o. The derived expression reproduces the experimentally measured value with high precision. This result highlights the potential of Mo Theory to connect disparate physical domains through a single fundamental constant.

This work concludes the comparative analysis of the Tensor Model of Discrete Dynamics (TMDD) and the Standard Model (SM) of particle physics, initiated in Part One. A top-down approach is implemented: the SM Lagrangian is discretized on a space-time lattice, followed by a direct structural comparison of the resulting functional S discr SM with the original action functional of TMDD, S TMDD , dened on a network of θ-cells. This approach enables a comparison at a unied ontological level of fundamental discreteness, avoiding the approximations inherent in taking the continuum limit. The ontological "tail" of TMDD is clearly identied and comprehensively investigated. This tail constitutes the set of terms in the action functional that lack direct analogues in the SM. It is shown that the tail forms an internally consistent and stable dynamic structure responsible for phenomena such as dark matter, dark energy, and the mechanism of spontaneous symmetry breaking. An exact correspondence is established between the parametric and spectral representations of the TMDD's ontological core, demonstrating its integrity. The consensus tensor is investigated as the central operator of the network's dynamics. Based on this, spontaneous symmetry breaking is interpreted as a process of dynamic basis restructuring at a saddle point of the action landscape. The results of this work serve as independent conrmation of the conclusions from Part One and lay the foundation for subsequent model calibration and quantitative verication.

In the usual and current understanding of planar gauge choices for Abelian and non-Abelian gauge fields, the external defining vector nμ can either be space-like (n2 < 0) or time-like (n2>0) but not light-like (n2=0). In this work we propose a light-like planar gauge that consists of defining a modified gauge-fixing term, ℒ GF , whose main characteristic is a two-degree violation of Lorentz covariance arising from the fact that four-dimensional space–time spanned entirely by null vectors as basis necessitates two light-like vectors, namely nμ and its dual mμ, with n2=m2=0, n·m≠0, say, e.g. normalized to n·m=2.

Starting from the generalized Konishi anomaly equations at the non-perturbative level, we demonstrate that the algebraic consistency of the quantum chiral ring of the N = 1 super Yang-Mills theory with gauge group U(N), one adjoint chiral superfield X and N f ≤ 2N flavours of quarks implies that the periods of the meromorphic one-form Tr dz z-X must be quantized. This shows in particular that identities in the open string description of the theory, that follow from the fact that gauge invariant observables are expressed in terms of gauge variant building blocks, are mapped onto non-trivial dynamical equations in the closed string description.

Using generalized Konishi anomaly equations, it is known that one can express, in a large class of supersymmetric gauge theories, all the chiral operators expectation values in terms of a finite number of a priori arbitrary constants. We show that these constants are fully determined by the requirement of gauge invariance and an additional anomaly equation. The constraints so obtained turn out to be equivalent to the extremization of the Dijkgraaf-Vafa quantum glueball superpotential, with all terms (including the Veneziano-Yankielowicz part) unambiguously fixed. As an application, we fill non-trivial gaps in existing derivations of the mass gap and confinement properties in super Yang-Mills theories.

We solve a generalization of ordinary N = 1 super Yang-Mills theory with gauge group U(N) and an adjoint chiral multiplet X for which we turn on both an arbitrary tree-level superpotential term d 2 θ Tr W (X) and an arbitrary field-dependent gauge kinetic term d 2 θ Tr V (X)W α W α . When W = 0, the model reduces to the extended Seiberg-Witten theory recently studied by Marshakov and Nekrasov. We use two different points of view: a "macroscopic" approach, using generalized anomaly equations, the Dijkgraaf-Vafa matrix model and the glueball superpotential; and the recently proposed "microscopic" approach, using Nekrasov's sum over colored partitions and the quantum microscopic superpotential. The two formalisms are based on completely different sets of variables and statistical ensembles. Yet it is shown that they yield precisely the same gauge theory correlators. This beautiful mathematical equivalence is a facet of the open/closed string duality. A full microscopic derivation of the non-perturbative N = 1 gauge dynamics follows.

We explain how to generalize Nekrasov's microscopic approach to N = 2 gauge theories to the N = 1 case, focusing on the typical example of the U(N) theory with one adjoint chiral multiplet X and an arbitrary polynomial tree-level superpotential Tr W (X). We provide a detailed analysis of the generalized glueball operators and a non-perturbative discussion of the Dijkgraaf-Vafa matrix model and of the generalized Konishi anomaly equations. We compute in particular the non-trivial quantum corrections to the Virasoro operators and algebra that generate these equations. We have performed explicit calculations up to two instantons, that involve the next-to-leading order corrections in Nekrasov's Ω-background.

We consider the N = 1 super Yang-Mills theory with gauge group U(N), adjoint chiral multiplet X and tree-level superpotential Tr W (X). We compute the quantum effective superpotential W mic as a function of arbitrary off-shell boundary conditions at infinity for the scalar field X. This effective superpotential has a remarkable property: its critical points are in one-to-one correspondence with the full set of quantum vacua of the theory, providing in particular a unified picture of solutions with different ranks for the low energy gauge group. In this sense, W mic is a good microscopic effective quantum superpotential for the theory. This property is not shared by other quantum effective superpotentials commonly used in the literature, like in the strong coupling approach or the glueball superpotentials. The result of this paper is a first step in extending Nekrasov's microscopic derivation of the Seiberg-Witten solution of N = 2 super Yang-Mills theories to the realm of N = 1 gauge theories.

In this note, we solve an extended version of the N = 1 super Yang-Mills theory with gauge group U(N ), an adjoint chiral multiplet and N f flavors of quarks, by using the N = 1 microscopic formalism based on Nekrasov's sums over colored partitions. Our main new result is the computation of the general mesonic operators. We prove that the generalized Konishi anomaly equations with flavors are satisfied at the nonperturbative level. This yields in particular a microscopic, first principle derivation of the matrix model disk diagram contributions that must be included in the Dijkgraaf-Vafa approach.

Classifying the phases of gauge theories is hindered by the lack of local order parameters. In particular, the standard Wilson's and 't Hooft's non-local order parameters are known to be insufficient to explain the existence of the plethora of phases that are found in supersymmetric gauge theories. Motivated by these observations, we reanalyze the concept of gauge symmetry breaking using Galois theory. Unlike the ordinary classical notion of unbroken gauge group, the Galois symmetry makes sense in the full quantum theory and must be a phase invariant. The algebraic structure underlying the space of vacua of supersymmetric gauge theories, that we have developed recently, is precisely designed to allow a rigorous mathematical implementation of these ideas.

We study the eigenvalue problem for some special class of antitriangular matrices. Though the eigenvalue problem is quite classical, as far as we know, almost nothing is known about properties of eigenvalues for anti-triangular matrices. In this paper, we show that there is a nice class of anti-triangular matrices whose eigenvalues are given explicitly by their elements. Moreover, this class contains several interesting subclasses which we characterize in terms of probability measures. We also discuss the application of our main theorem to the study of interacting particle systems, which are stochastic processes studied in extensive literature.

This paper develops a rigorous mathematical framework for non-perturbative dualities in type IIB superstring theory, deriving generalized S-duality transformations for D3-branes coupled to topological defects. We formulate supergravity solutions, compute modular invariants, and derive new constraints on holographic gauge theories via conformal anomalies and Wilson loop operators. Novel results include a non-linear axion-dilaton coupling and defect-induced anomalies, advancing our understanding of quantum gravity and topological quantum computing.

We study the integrability properties of planar N = 2 superconformal field theories in four dimensions. We show that the spin chain associated to the planar dilation operator of N = 2 superconformal QCD fails to be integrable at two loops. In our analysis we focus on a closed SU (2|1) sector, whose two-loop spin chain we fix by symmetry arguments (up to a few undetermined coefficients). It turns out that the Yang-Baxter equation for magnon scattering is not satisfied in this sector. On the other hand, we suggest that the closed SU (2, 1|2) sector, which exists in any N = 2 superconformal gauge theory, may be integrable to all loops. We summarize the known results in the literature that are consistent with this conjecture.

We report a novel BPS jumping phenomenon of 5d N = 1 supersymmetric gauge theories whose brane configuration is equipped with an O7-plane. The study of the relation between O7 +-plane and O7-plane reveals that such BPS jumps take place when the Higgsing is triggered near the O7-plane upon a particular parameter tuning of the theories. We propose two types of gauge theories whose BPS spectra jump. One is the SU(2N + 8) gauge theory with a symmetric hypermultiplet converted to the SU(2N) gauge theory with an antisymmetric hypermultiplet. The other is pure SO(2N + 8) gauge theory jumping to pure Sp(N) gauge theory. We explicitly confirm our proposal through the (un)refined instanton partition functions. Furthermore, we discuss feasible generalizations involving an Op-plane for supersymmetric gauge theories of eight supercharges in four and three dimensions (p = 6, 5 respectively).

We revisit the instanton partition function for 5d N = 1 SO(N) gauge theories compactified on S 1 , computed from the topological vertex formalism with the O-vertex based on a 5-brane web diagram with an O5-plane. We introduce an identity that enables us to rewrite the unrefined partition function into a new expression in terms of the Nekrasov factors summed over Young diagrams, which can be interpreted as the freezing of an O7-plane. Based on this, we propose topological vertex formalism with an O7 +-plane.

Pierce identified 3 invariants of a compact metrisable Boolean space, derived from its Cantor-Bendixson sequence, that determine the space up to homeomorphism. For locally compact spaces we define an additional invariant, the compact rank, and show that these 4 invariants determine a locally compact metrisable Boolean space up to homeomorphism. We also identify which combinations of the 4 invariants can arise in practice. A Boolean ring and its associated Boolean space are primitive if the ring is disjointly generated by its pseudo-indecomposable (PI) elements. Spaces in this important subclass of Boolean spaces can be well described (uniquely in the case of compact spaces) by an extended PO system (poset with a distinguished subset). We define the Cantor-Bendixson sequence and associated invariants for a PO system, and show that almost all of the invariant information for a primitive space can be recovered from that of an associated extended PO system. We also show how the primitivity of a Boolean space corresponds to a notion of primitivity of the additive measure associated with the rank function of a space, which in turn depends on the additive measure being sufficiently "self-similar". We use these ideas to develop a method for constructing non-primitive spaces.

Within the AdS/CFT correspondence, we review the studies of field theories with a large number of adjoint and fundamental fields, in the Veneziano limit. We concentrate in set-ups where the fundamentals are introduced by a smeared set of D-branes. We make emphasis on the general ideas and then in subsequent chapters that can be read independently and describe particular considerations in various different models. Some new material is presented along the various sections.

In the holographic model of QCD, θ dependence sharply changes at the point of confinementdeconfinement phase transition. In large N QCD such a change in θ behavior can be related to the breakdown of the instanton expansion at some critical temperature Tc. Associating this temperature with confinement-deconfinement phase transition leads to the description of the latter in terms of dissociation of instantons into the fractionally charged instanton quarks. To elucidate this picture, we introduce the nonvanishing chiral condensate in the deconfining phase and assume a specific lagrangian for the η ′ field in the confining phase. In the resulting picture the high temperature phase of the theory consists of the dilute gas of instantons, while the low temperature phase is described in terms of freely moving fractional instanton quarks.

In a number of field theoretical models the vacuum angle θ enters physics in the combination θ/N, where N stands generically for the number of colors or flavors, in an apparent contradiction with the expected 2π periodicity in θ. We argue that a resolution of this puzzle is related to the existence of a number of different θ dependent sectors in a finite volume formulation, which can not be seen in the naive thermodynamic limit V → ∞. It is shown that, when the limit V → ∞ is properly defined, physics is always 2π periodic in θ for any integer, and even rational, values of N, with vacuum doubling at certain values of θ. We demonstrate this phenomenon in both the multi-flavor Schwinger model with the bosonization technique, and four-dimensional gluodynamics with the effective Lagrangian method. The proposed mechanism works for an arbitrary gauge group.

We suggest that the topological susceptibility in gluodynamics can be found in terms of the gluon condensate using renormalizability and heavy fermion representation of the anomaly. Analogous relations can also be obtained for other zero-momentum correlation functions involving the topological density operator. Using these relations, we find the θ dependence of the condensates , [Formula: see text] and of the partition function for small θ and an arbitrary number of colors.

The idea of grand unification in a minimal supersymmetric SU(5) × SU(5) framework is revisited. It is shown that the unification of gauge couplings into a unique coupling constant can be achieved at a high-energy scale compatible with proton decay constraints. This requires the addition of a minimal particle content at intermediate energy scales. In particular, the introduction of the SU(2) L triplets belonging to the (15, 1) + (15, 1) representations, as well as of the scalar triplet Σ 3 and octet Σ 8 in the (24, 1) representation, turns out to be crucial for unification. The masses of these intermediate particles can vary over a wide range, and even lie in the TeV region. In contrast, the exotic vector-like fermions must be heavy enough and have masses above 10 10 GeV. We also show that, if the SU(5) × SU(5) theory is embedded into a heterotic string scenario, it is not possible to achieve gauge coupling unification with gravity at the perturbative string scale.

The idea of grand unification in a minimal supersymmetric SU(5) x SU(5) framework is revisited. It is shown that the unification of gauge couplings into a unique coupling constant can be achieved at a high-energy scale compatible with proton decay constraints. This requires the addition of a minimal particle content at intermediate energy scales. In particular, the introduction of the SU(2)(L) triplets belonging to the (15, 1)+((15) over bar, 1) representations, as well as of the scalar triplet Sigma(3) and octet Sigma(8) in the (24, 1) representation, turns out to be crucial for unification. The masses of these intermediate particles can vary over a wide range, and even lie in the TeV region. In contrast, the exotic vector-like fermions must be heavy enough and have masses above 10(10) GeV. We also show that, if the SU(5) x SU(5) theory is embedded into a heterotic string scenario, it is not possible to achieve gauge coupling unification with gravity at the perturbative string scale.

It is a familiar fact to develop inequalities using the popular method by adopting fractional operators, and such study of methods is the main core of modern research in recent year. Fuzzy interval valued (FIV) mappings not only used to generalize of different convex mappings but also developed fractional operators. In this paper, we investigate fuzzy fractional inequalities for different fuzzy convexities by successfully implementing generalized fuzzy fractional operators (G-FFO). We discuss the extension of Hermite-Hadamard, trapezoid-type inequalities on the basis of fuzzy convexities and fuzzy fractional operators. Moreover, we establish the Fejér and midpoint type fuzzy inequalities for (η 1 , η 2)-convex fuzzy function.

We consider a Bianchi type III axisymmetric geometry in the presence of an electromagnetic field. A first result at the classical level is that the symmetry of the geometry need not be applied on the electromagnetic tensor F µν ; the algebraic restrictions, implied by the Einstein field equations to the stress energy tensor T µν , suffice to reduce the general F µν to the appropriate form. The classical solution thus found contains a time dependent electric and a constant magnetic charge. The solution is also reachable from the corresponding mini-superspace action, which is strikingly similar to the Reissner-Nordström one. This points to a connection between the black hole geometry and the cosmological solution here found, which is the analog of the known correlation between the Schwarzschild and the Kantowski-Sachs metrics. The configuration space is drastically modified by the presence of the magnetic charge from a 3D flat to a 3D pp wave geometry. We map the emerging linear and quadratic classical integrals of motion, to quantum observables. Along with the Wheeler-DeWitt equation these observables provide unique, up to constants, wave functions. The employment of a Bohmian interpretation of these quantum states results in deterministic (semi-classical) geometries most of which are singularity free.

The use of automorphisms of the various Bianchi-type Lie algebras as Lie-point symmetries of the corresponding Einstein field equations entails a reduction of their order and ultimately leads to the entire solution space. When a valid reduced action principle exists, the symmetries of the configuration mini-supermetric space can also be used, in conjunction with the constraints, to provide local or non-local constants of motion. At the classical level, depending on their number, these integrals can even secure the acquisition of the entire solution space without any further solving of the dynamical equations. At the quantum level, their operator analogues can be used, along with the Wheeler-DeWitt equation, to define unique wave functions that exhibit singularity-free behavior at a semi-classical level.

We study N = 1 supersymmetric SU (N ) gauge theories with an antisymmetric tensor and F flavors using the recent proposal of a-maximization by Intriligator and Wecht. This theory had previously been studied using the method of "deconfinement," but such an analysis was not conclusive since anomalous dimensions in the non-perturbative regime could not be calculated. Using a-maximization we show that for a large range of F the theory is at an interacting superconformal fixed point. However, we also find evidence that for a range of F the theory in the IR splits into a free "magnetic" gauge sector and an interacting superconformal sector.

We study N = 1 supersymmetric SU (N ) gauge theories with an antisymmetric tensor and F flavors using the recent proposal of a-maximization by Intriligator and Wecht. This theory had previously been studied using the method of "deconfinement," but such an analysis was not conclusive since anomalous dimensions in the non-perturbative regime could not be calculated. Using a-maximization we show that for a large range of F the theory is at an interacting superconformal fixed point. However, we also find evidence that for a range of F the theory in the IR splits into a free "magnetic" gauge sector and an interacting superconformal sector.

We show that noncommutative gauge theories with arbitrary compact gauge group defined by means of the Seiberg-Witten map have the same one-loop anomalies as their commutative counterparts. This is done in two steps. By explicitly calculating the µ 1 µ 2 µ 3 µ 4 part of the renormalized effective action, we first find the would-be one-loop anomaly of the theory to all orders in the noncommutativity parameter θ µν . And secondly we isolate in the would-be anomaly radiative corrections which are not BRS trivial. This gives as the only true anomaly occurring in the theory the standard Bardeen anomaly of commutative spacetime, which is set to zero by the usual anomaly cancellation condition.

In this paper we first investigate the Ansatz of one of the present authors for K(Ψ, Ψ), the adimensional modular invariant non-holomorphic correction to the Wilsonian effective Lagrangian of an N = 2 globally supersymmetric gauge theory. The renormalisation group β-function of the theory crucially allows us to express K(Ψ, Ψ) in a form that easily generalises to the case in which the theory is coupled to N F hypermultiplets in the fundamental representation of the gauge group. This function satisfies an equation which should be viewed as a fully non-perturbative "non-chiral superconformal Ward identity". We also determine its renormalisation group equation. Furthermore, as a first step towards checking the validity of this Ansatz, we compute the contribution to K(Ψ, Ψ) from instantons of winding number k = 1 and k = 2. As a by-product of our analysis we check a non-renormalisation theorem for N F = 4.

Higher gauge theory arises naturally in superstring theory, but many of its features remain obscure. In this thesis, after an exposition of the bacis tools in local higher gauge theory, a higher gauge Chern-Simons model is defined. We discuss the classical equations of motion as well as the behaviour of the gauge anomaly. We perform canonical quantization and we introduce two possible quantization schemes for the model. We also expound higher parallel transport in higher gauge theory, and we speculate that it can provide Wilson surfaces as topological observables for the higher gauge Chern-Simons theory.

A quantum mechanical model that realizes the Z 2 × Z 2 -graded generalization of the one-dimensional supertranslation algebra is proposed. This model shares some features with the well-known Witten model and is related to parasupersymmetric quantum mechanics, though the model is not directly equivalent to either of these. The purpose of this paper is to show that novel "higher gradings" are possible in the context of non-relativistic quantum mechanics.

A standard procedure is proposed for inducing interactions between pure gauge theories. To exhibit the method the Kalb-Ramond theory and the Green-Schwarz superstring theory are considered. When "big" gauge transformations are considered a quantization condition arises in disagreement with the values of the constants of the superstring theory. Pure gauge theories are usually formulated in terms of one-form potentials. Generalizations of these theories with a p-form as the fundamental object have been proposed by Nambu . Interactions between these gauge theories may be established by introducing suitable generalizations of the gauge transformations. The simplest example of such an interacting gauge theory was proposed by Kalb and Ramond [2] within the context of the dual string model. This theory involves an abelian vector potential A ° and a second-rank antisymmetric tensor a~v(x ). Recently [3] other coupling between oneand two-form~ has beenpropov, edand later, generalized to the non-abelian case by Chapline and Manton [4]. This interaction also appears, in the context of superstring theories in ten dimensions, as shown by Green and Schwarz [5]. The Chern-Simons densities play a fundamental role in these theories. In particular, they lead to the cancellation of the anomalies in the

We derive the partition function for the Vafa-Witten twist of the N = 4 supersymmetric gauge theory with gauge group SU(N) (for prime N) and arbitrary values of the 't Hooft fluxes v ∈ H 2 (X, Z Z N ) on Kähler four-manifolds with b + 2 > 1.

The state of art in spin glass field theory is reviewed. We start from an Edwards-Anderson-type model in finite dimensions, with finite but long range forces, construct the effective field theory that allows one to extract the long wavelength behaviour of the model, and set up an expansion scheme (the loop expansion) in the inverse range of the interaction. At the zeroth order we recover mean field theory. We evaluate systematic corrections to this around Parisi's replica symmetry broken solution. At the level of quadratic fluctuations we derive a set of coupled integral equations for the free propagators of the theory and show how they can be solved for short, intermediate and extreme long distances. To reveal the physical meaning of these results, we relate the various propagator components to overlaps of spin-spin correlation functions inside a single phase space valley resp. between different valleys. Next we calculate the first loop corrections to the theory above 8 dimensions, where we find that it maps back onto mean field theory, with basically temperature independent renormalization of the coupling constants, thereby demonstrating that Parisi's mean field theory is, at least perturbatively, stable against finite range corrections. In the range between six and eight dimensions various physical quantities pick up nontrivial temperature dependences which can, however, still be determined exactly. Upon approaching the upper critical dimension (d = 6) of the model, scaling which is badly violated in Parisi's mean field theory is gradually restored. Below 6 dimensions one should apply renormalization group methods. Unfortunately, the structure of RG is not completely understood in spin glass theory. Nevertheless, the first corrections in 6 -d to e.g. the exponent of the order parameter can still be calculated, moreover exponentiation to this power can be checked at the next order. The theory is, however, plagued by infrared divergences due to the presence of zero modes and soft modes. Systematic methods (like those developed in the O(n) model) to handle these infrared singularities are not yet available in spin glass theory.

A modern review of Kaluza-Klein theories is presented. We adopt the version where the whole space is a principal fiber bundle with the four-dimensional spacetime as base, and as typical fiber a G Lie group. It is a natural generalization of gauge theories which metric is just the Kaluza-Klein metric. For the five dimensional theory we give an invariant formulation of the axisymmetric-stationary case. Some techniques for obtaining exact solutions and cosmology in specific dimensions are studied. Finally the method of spontaneus compactification is outlined.