We construct supersymmetric Lifshitz field theories with four real supercharges in a general number of space dimensions. The theories consist of complex bosons and fermions and exhibit a holomorphic structure and non-renormalization properties of the superpotential. We study the theories in a diverse number of space dimensions and for various choices of marginal interactions. We show that there are lines of quantum critical points with an exact Lifshitz scale invariance and a dynamical critical exponent that depends on the coupling constants.
We consider field theories that exhibit a supersymmetric Lifshitz scaling with two real supercharges. The theories can be formulated in the language of stochastic quantization. We construct the free field supersymmetry algebra with rotation singlet fermions for an even dynamical exponent $z=2k$ in an arbitrary dimension. We analyze the classical and quantum $z=2$ supersymmetric interactions in $2+1$ and $3+1$ spacetime dimensions and reveal a supersymmetry preserving quantum diagrammatic cancellation. Stochastic quantization indicates that Lifshitz scale invariance is broken in the $(3+1)$-dim…
We show that there exist supersymmetric Lifshitz vacua in off-shell Einstein-Weyl supergravity, in addition to the BPS AdS_4 vacuum. The Lifshitz exponents are determined by the product of the cosmological constant and the coupling of the Weyl-squared term. We then obtain the equations of the supersymmetric solutions that are asymptotic to the AdS or Lifshitz vacua. We obtain many examples of exact solutions as well as numerical ones. We find examples of extremal AdS black branes whose near-horizon geometry is AdS_2\times T^2. We also find an extremal Lifshitz black hole with z=-2, whose horiz…
We discuss a class of supersymmetric type II non-relativistic solutions with exact or asymptotic scale invariance. As already emerged from previous investigations, we find a clear correspondence between anisotropic d-dimensional vacua and relativistic solutions in (d + 1)-dimensions. We will show that supersymmetric four-dimensional Poincare' invariant backgrounds in type IIB can descend to analogous solutions with anisotropic scaling in t and (x, y). This result can be applied to scale invariant theories, domain walls interpolating between four-dimensional Lifshitz vacua and more general solu…
Within the superfield approach, we formulate two different extensions of the Wess-Zumino model and super-QED with Horava-Lifshitz-like additive terms, discuss their quantum properties and calculate lower contributions to the effective action. In the case of the gauge theory, the one-loop effective potential turns out to be gauge independent.
We construct two interpolating solutions in type II string theory which interpolate between an AdS$_5$ in the UV and a hyperscaling violating three (spatial) dimensional Lifshitz space-time in the IR. The first solution is non-supersymmetric and is obtained from a known intersecting non-supersymmetric D3-brane with chargeless D0-brane solution of type IIB string theory, by restricting some parameters characterizing the solution and going to a new coordinate. In the IR the dilaton is non-constant in general and the metric is three (spatial) dimensional hyperscaling violating Lifshitz with dynam…
The quantum Lifshitz model provides an effective description of a quantum critical point. It has been shown that even though non--Lorentz invariant, the action admits a natural supersymmetrization. In this note we introduce a perturbative framework and show how the supersymmetric structure can be used to greatly simplify the Feynman rules and thus the study of the model.
We study the renormalization of an N = 1 supersymmetric Lifshitz sigma model in three dimensions. The sigma model exhibits worldvolume anisotropy in space and time around the high-energy z = 2 Lifshitz point, such that the worldvolume is endowed with a foliation structure along a preferred time direction. In curved backgrounds, the target-space geometry is equipped with two distinct metrics, and the interacting sigma model is power-counting renormalizable. At low energies, the theory naturally flows toward the relativistic sigma model where Lorentz symmetry emerges. In the superspace formalism…
We consider supersymmetric configurations in Type IIB supergravity obtained by the beackreaction of fundamental strings ending on a stack of D3-branes and smeared uniformly in the three spatial directions along the D3-branes. These automatically include a distribution of D5-brane baryon vertices necessary to soak up string charge. The backgrounds are static, preserving eight supersymmetries, an SO(5) global symmetry and symmetry under spatial translations and rotations. We obtain the most general BPS configurations consistent with the symmetries. We show that the solutions to the Type IIB fiel…
We discuss two classes of non-supersymmetric interpolating solutions in N=2, D=4 gauged supergravity, that flow from either a z=2 Lifshitz geometry or a conformal AdS background to the near-horizon geometry of a Nernst brane. We obtain these solutions by constructing a z=2 supersymmetric Lifshitz solution in the STU model from a first-order rewriting of the action, then lifting it up to a five-dimensional background and subsequently modifying this five-dimensional solution by a two-parameter family of deformations. Under reduction, these give four-dimensional non-supersymmetric Nernst brane so…
The Wilson (exact) renormalization group equations are used to determine the evolution of a general low energy N=1 supersymmetric action containing a U(1) gauge vector multiplet and a neutral chiral multiplet. The effective theory evolves towards satisfying a fixed relation where the Kähler potential and effective gauge coupling are obtained from a N=2 supersymmetric holomorphic prepotential.
We examine a class of gravity backgrounds obtained by considering the backreaction of a spatially uniform density of mutually BPS Wilson lines or heavy quarks in N=4 SUSY Yang-Mills theory. The configurations preserve eight supercharges and an SO(5) subgroup of the SO(6) R-symmetry. They are obtained by considering the 1/4-BPS geometries associated to smeared string/D3-brane (F1-D3) intersections. We argue that for the (partially) localized intersection, the geometry exhibits a flow from AdS_5 x S^5 in the UV to a novel IR scaling solution displaying anisotropic Lifshitz-like scaling with dyna…
We present a Lorentz-breaking supersymmetric algebra characterized by a critical exponent $z$. Such construction requires a non trivial modification of the supercharges and superderivatives. The improvement of renormalizability for supersymmetric scalar QED is shown and the Kählerian effective potentials are calculated in different cases. We also show how the theory flows naturally to the Lorentz symmetric case at low energies.
Under general assumptions, we show that a gravitational theory in d+1 dimensions admitting an AdS solution can be reduced to a d-dimensional theory containing a Lifshitz solution with dynamical exponent z=2. Working in a d=4, N=2 supergravity setup, we prove that if the AdS background is N=2 supersymmetric, then the Lifshitz geometry preserves 1/4 of the supercharges, and we construct the corresponding Killing spinors. We illustrate these results in examples from supersymmetric consistent truncations of type IIB supergravity, enhancing the class of known 4-dimensional Lifshitz solutions of str…
We propose a novel mechanism of SUSY breaking by coupling a Lorentz-invariant supersymmetric matter sector to non-supersymmetric gravitational interactions with Lifshitz scaling. The improved UV properties of Lifshitz propagators moderate the otherwise uncontrollable ultraviolet divergences induced by gravitational loops. This ensures that both the amount of induced Lorentz violation and SUSY breaking in the matter sector are controlled by Lambda_HL^2/M_P^2, the ratio of the Horava-Lifshitz cross-over scale Lambda_HL to the Planck scale M_P. This ratio can be kept very small, providing a novel…
This work is dedicated to the study of both large-$N$ and perturbative quantum behaviors of Lifshitz nonlinear sigma models with dynamical critical exponent $z=2$ in 2+1 dimensions. We discuss renormalization and renormalization group aspects with emphasis on the possibility of emergence of Lorentz invariance at low energies. Contrarily to the perturbative expansion, where in general the Lorentz symmetry restoration is delicate and may depend on stringent fine-tuning, our results provide a more favorable scenario in the large-$N$ framework. We also consider supersymmetric extension in this non…
Recently, it has been shown that the holomorphic anomaly of unitarity cuts can be used as a tool in determining the one-loop amplitudes in N=4 super Yang-Mills theory. It is interesting to examine whether this method can be applied to more general cases. We present results for a non-MHV N=1 supersymmetric one-loop amplitude. We show that the holomorphic anomaly of each unitarity cut correctly reproduces the action on the amplitude's imaginary part of the differential operators corresponding to collinearity in twistor space. We find that the use of the holomorphic anomaly to evaluate the amplit…
We introduce a natural generalisation of holomorphic curves to morphisms of supermanifolds, referred to as holomorphic supercurves. More precisely, supercurves are morphisms from a Riemann surface, endowed with the structure of a supermanifold which is induced by a holomorphic line bundle, to an ordinary almost complex manifold. They are called holomorphic if a generalised Cauchy-Riemann condition is satisfied. We show, by means of an action identity, that holomorphic supercurves are special extrema of a supersymmetric action functional.
We explore the constant curvature holomorphic solutions of the supersymmetric grassmannian sigma model $G(M,N)$ using in particular the gauge invariance of the model. Supersymmetric invariant solutions are constructed via generalizing a known result for ${C}P^{N-1}$. We show that some other such solutions also exist. Indeed, considering the simplest case of $G(2,N)$ model, we give necessary and sufficient conditions for getting the constant curvature holomorphic solutions. Since, all the constant curvature holomorphic solutions of the bosonic $G(2,4)$ $σ$-model are known, we treat this example…
Twisted supersymmetric theories on a product of two Riemann surfaces possess non-local holomorphic currents in a BRST cohomology. The holomorphic currents act as vector fields on the chiral ring. The OPE's of these currents are invariant under the renormalization group flow up to BRST-exact terms. In the context of electric-magnetic duality, the algebra generated by the holomorphic currents in the electric theory is isomorphic to the one on the magnetic side. For the currents corresponding to global symmetries this isomorphism follows from 't Hooft anomaly matching conditions. The isomorphism …
We construct fermionic Lagrangians with anisotropic scaling z=2, the natural counterpart of the usual z=2 Lifshitz field theories for scalar fields. We analyze the issue of chiral symmetry, construct the Noether axial currents and discuss the chiral anomaly giving explicit results for two-dimensional case. We also exploit the connection between detailed balance and the dynamics of Lifshitz theories to find different z=2 fermionic Lagrangians and construct their supersymmetric extensions.
We construct infinite families of Lifshitz solutions of D=10 and D=11 supergravity with dynamical exponent z=2. The new solutions are based on five- and seven-dimensional Einstein manifolds and are dual to theories in 1+2 and 1+1 spacetime dimensions, respectively. When the Einstein spaces are Sasaki-Einstein, the solutions are supersymmetric.
The most recent studies on the supersymmetric localization reveal many non-trivial features of supersymmetric field theories in diverse dimensions, and 3d gauge theory provides a typical example. It was conjectured that the index and the partition function of a 3d N=2 theory are constructed from a single component: the holomorphic block. We prove this conjecture for non-abelian gauge theories by computing exactly the 3d partition functions and holomorphic blocks.
We consider strongly coupled supersymmetric gauge theories softly broken by the addition of gaugino masses $m_λ$ and (non-holomorphic) scalar masses $m^2$, taken to be small relative to the dynamical scale $Λ$. For theories with a weakly coupled dual description in the infrared, we compute exactly the leading soft masses for the "magnetic" degrees of freedom, with uncalculable corrections suppressed by powers of $(m_λ/Λ), (m/Λ)$. The exact relations hold between the infrared fixed point "magnetic" soft masses and the ultraviolet fixed point "electric" soft masses, and correspond to a duality m…
We construct a class of supersymmetric boundary interactions in N=2 field theories on the half-space, which depend on parameters that are not at all renormalized or not renormalized in perturbation theory beyond one-loop. This can be used to study D-branes wrapped on a certain class of Lagrangian submanifolds as well as holomorphic cycles. The construction of holomorphic D-branes is in close relationship with the background independent open string field theory approach to brane/anti-brane systems. As an application, mirror pairs of Lagrangian and holomorphic D-branes are identified. The mirror…
In four-dimensional N=2 compactifications of string theory or M-theory, modifications of the Bekenstein-Hawking area law for black hole entropy in the presence of higher-derivative interactions are crucial for finding agreement between the macroscopic entropy obtained from supergravity and subleading corrections to the microscopic entropy obtained via state counting. Here we compute the modifications to the area law for various classes of black holes, such as heterotic black holes, stemming from certain higher-derivative gravitational Wilsonian coupling functions. We consider the extension to …
Based on our idea of an alternative supersymmetrization of the Nambu--Jona-Lasinio model for dynamical symmetry breaking, we analyze the resulted new model with a holomorphic dimension-five operator in the superpotential. The approach provides a new direction for modeling dynamical symmetry breaking in a supersymmetric setting. In particular, we adopt the idea to formulate a model that gives rise to the Minimal Supersymmetric Standard Model as the low energy effective theory with both Higgs superfields as composites. A renormalization group analysis is performed to establish the phenomenologic…
We construct an N=1 supersymmetric gauge theory from z=3 Lifshitz field theory. By modifying the supersymmetry (susy) algebra based on the spacetime symmetry SO(3) $\times$ scaling symmetry, we get a supersymmetric Lagrangian with scalar, fermion and gauge fields, all of whom have the same limiting speed. This solves some naturalness problems of the original Lifshitz theory which is characterized by Lorentz symmetry violation. In order that the susy breaking does not introduce any disastrous terms into the theory, the susy breaking scale is required to be smaller than the scale of Lorentz symm…
We consider supersymmetric field theories on compact manifolds M and obtain constraints on the parameter dependence of their partition functions Z_M. Our primary focus is the dependence of Z_M on the geometry of M, as well as background gauge fields that couple to continuous flavor symmetries. For N=1 theories with a U(1)_R symmetry in four dimensions, M must be a complex manifold with a Hermitian metric. We find that Z_M is independent of the metric and depends holomorphically on the complex structure moduli. Background gauge fields define holomorphic vector bundles over M and Z_M is a holomo…
We show that various holomorphic quantities in supersymmetric gauge theories can be conveniently computed by configurations of D4-branes and D6-branes. These D-branes intersect along a Riemann surface that is described by a holomorphic curve in a complex surface. The resulting I-brane carries two-dimensional chiral fermions on its world-volume. This system can be mapped directly to the topological string on a large class of non-compact Calabi-Yau manifolds. Inclusion of the string coupling constant corresponds to turning on a constant B-field on the complex surface, which makes this space non-…
In a previous paper we had proposed a specific route to relating the entropy of two charge black holes to the degeneracy of elementary string states in N=4 supersymmetric heterotic string theory in four dimensions. For toroidal compactification this proposal works correctly to all orders in a power series expansion in inverse charges provided we take into account the corrections to the black hole entropy formula due to holomorphic anomaly. In this paper we demonstrate that similar agreement holds also for other N=4 supersymmetric heterotic string compactifications.
A class of 3d $\mathcal{N}=2$ supersymmetric gauge theories are constructed and shown to encode the simplicial geometries in 4-dimensions. The gauge theories are defined by applying the Dimofte-Gaiotto-Gukov construction in 3d/3d correspondence to certain graph complement 3-manifolds. Given a gauge theory in this class, the massive supersymmetric vacua of the theory contain the classical geometries on a 4d simplicial complex. The corresponding 4d simplicial geometries are locally constant curvature (either dS or AdS), in the sense that they are made by gluing geometrical 4-simplices of the sam…
We study the dependence of supersymmetric partition functions on continuous parameters for the flavor symmetry group, $G_F$, for 2d $\mathcal{N} = (0,2)$ and 4d $\mathcal{N}=1$ supersymmetric quantum field theories. In any diffeomorphism-invariant scheme and in the presence of $G_F$ 't Hooft anomalies, the supersymmetric Ward identities imply that the partition function has a non-holomorphic dependence on the flavor parameters. We show this explicitly for the 2d torus partition function, $Z_{T^2}$, and for a large class of 4d partition functions on half-BPS four-manifolds, $Z_{\mathcal{M}_4}$-…
We study the theoretical features in relation to dynamical mass generation and symmetry breaking for the recently proposed holomorphic supersymmetric Nambu--Jona-Lasinio model. The basic model has two different chiral superfields (multiplets) with a strongly coupled dimension five four-superfield interaction. In addition to the possibility of generation of Dirac mass between the pair established earlier, we show here the new option of generation of Majorana masses for each chiral superfield. We also give a first look at what condition may prefer Dirac over Majorana mass, illustrating that a sp…
In this paper, we will analyse a supersymmetric field theory deformed by generalized uncertainty principle and Lifshitz scaling. It will be observed that this deformed supersymmetric field theory contains non-local fractional derivative terms. In order to construct such deformed N=1 supersymmetric theory, a harmonic extension of functions will be used. However, the supersymmetry will be only preserved for a free theory and will be broken by the inclusion of interaction terms.
The paper is a brief review of the works devoted to problem of effective action in N=2 super Yang-Mills theories in harmonic superspace approach. The formulation of N=2 superfield models in harmonic superspace is discussed, background field method for N=2 super Yang-Mills theory is constructed and general structure of effective action in harmonic superspace is investigated. It is shown how the holomorphic and non-holomorphic contributions to effective action can be calculated within the harmonic superspace approach.
We consider finite deformations of the Hull--Strominger system. Starting from the heterotic superpotential, we identify complex coordinates on the off-shell parameter space. Expanding the superpotential around a supersymmetric vacuum leads to a third-order Maurer--Cartan equation that controls the moduli. The resulting complex effective action generalises that of both Kodaira--Spencer and holomorphic Chern--Simons theory. The supersymmetric locus of this action is described by an $L_3$ algebra.
We describe the first order moduli space of heterotic string theory compactifications which preserve $N=1$ supersymmetry in four dimensions, that is, the infinitesimal parameter space of the Strominger system. We establish that if we promote a connection on $TX$ to a field, the moduli space corresponds to deformations of a holomorphic structure $\bar D$ on a bundle $\cal Q$. The bundle $\cal Q$ is constructed as an extension by the cotangent bundle $T^*X$ of the bundle $E= {\rm End}(V) \oplus {\rm End}(TX) \oplus TX$ with an extension class $\cal H$ which precisely enforces the anomaly cancela…
In the recent paper hep-th/0502076, it was argued that the open topological B-model whose target space is a complex (2|4)-dimensional mini-supertwistor space with D3- and D1-branes added corresponds to a super Yang-Mills theory in three dimensions. Without the D1-branes, this topological B-model is equivalent to a dimensionally reduced holomorphic Chern-Simons theory. Identifying the latter with a holomorphic BF-type theory, we describe a twistor correspondence between this theory and a supersymmetric Bogomolny model on R^3. The connecting link in this correspondence is a partially holomorphic…
Supersymmetric terms in the effective action of N=2 supergravity in four dimensions are generically classified into chiral-superspace integrals and full-superspace integrals. For a theory of N=2 vector multiplets coupled to supergravity, a special class of couplings is given by chiral-superspace integrals that are governed by a holomorphic prepotential function. The quantum entropy of BPS black holes in such theories depends on the prepotential according to a known integral formula. We show, using techniques of localization, that a large class of full-superspace integrals in the effective acti…
We give a recipe relating holomorphic quantities in supersymmetric field theory to their descendants in non-supersymmetric Z_2 orbifold field theories. This recipe, consistent with a recent proposal of Strassler, gives exact results for bifermion condensates, domain wall tensions and gauge coupling constants in the planar limit of the orbifold theories.
Supersymmetric vacua (`universes') of string/M theory may be identified with certain critical points of a holomorphic section (the `superpotential') of a Hermitian holomorphic line bundle over a complex manifold. An important physical problem is to determine how many vacua there are and how they are distributed. The present paper initiates the study of the statistics of critical points $\nabla s = 0$ of Gaussian random holomorphic sections with respect to a connection $\nabla$. Even the expected number of critical points depends on the curvature of $\nabla$. The principal results give formulas…
Supersymmetric pure Yang-Mills theory is formulated with a local, i.e. space-time dependent, complex coupling in superspace. Super-Yang-Mills theories with local coupling have an anomaly, which has been first investigated in the Wess-Zumino gauge and there identified as an anomaly of supersymmetry. In a manifest supersymmetric formulation the anomaly appears in two other identities: The first one describes the non-renormalization of the topological term, the second relates the renormalization of the gauge coupling to the renormalization of the complex supercoupling. Only one of the two identit…
We study renormalization group flows in the Lifshitz-like $N$-flavour four fermi model discussed in 0905.2928. In the large-$N$ limit, a nontrivial flow occurs in only one of all possible marginal couplings and one relevant coupling, which provides the scale for Lorentz invariance violations. We discuss in detail the phase diagram and RG flows in the space of couplings, which includes the Lifshitz fixed point, the free field fixed point and a new fixed point characterized by $z=1$ scaling and a violation of Lorentz invariance, which cannot be tuned away by adjusting a parameter. In the broken …
We introduce a class of supersymmetric cycles in spacetimes of the form AdS times a sphere or $T^{1,1}$ which can be considered as generalizations of the giant gravitons. Branes wrapped on these cycles preserve $1\over 2$, $1\over 4$ or $1\over 8$ of the supersymmetry. On the CFT side these configurations correspond to superpositions of the large number of BPS states.
In this paper we discuss the strong coupling limit of chiral N=1 supersymmetric gauge theory via their embedding into M-theory. In particular we focus on the brane box models of Hanany and Zaffaroni and show that after a T-duality transformation their M-theory embedding is described by supersymmetric 3-cycles; its geometry will encode the holomorphic non-perturbative information about the gauge theory.
Holomorphic gauge fields in N=1 supersymmetric heterotic compactifications can constrain the complex structure moduli of a Calabi-Yau manifold. In this paper, the tools necessary to use holomorphic bundles as a mechanism for moduli stabilization are systematically developed. We review the requisite deformation theory -- including the Atiyah class, which determines the deformations of the complex structure for which the gauge bundle becomes non-holomorphic and, hence, non-supersymmetric. In addition, two equivalent approaches to this mechanism of moduli stabilization are presented. The first is…
We construct D=11 supergravity solutions dual to the twisted field theories arising when M-theory fivebranes wrap supersymmetric cycles. The cases considered are M-fivebranes wrapped on (i) a complex Lagrangian four-cycle in a D=8 hyper-Kaehler manifold corresponding to a D=2 field theory with (2,1) supersymmetry (ii) a product of two holomorphic two-cycles in a product of two Calabi-Yau two-folds corresponding to a D=2 field theory with (2,2) supersymmetry and (iii) a product of a holomorphic two-cycle and a SLAG three-cycle in a product of a Calabi-Yau two-fold and a Calabi-Yau three-fold co…
We consider four-dimensional Omega-deformed N=2 supersymmetric SU(2) gauge theory on A1 space and its lift to five dimensions. We find that the partition functions can be reproduced via special geometry and the holomorphic anomaly equation. Schwinger type integral expressions for the boundary conditions at the monopole/dyon point in moduli space are inferred. The interpretation of the five-dimensional partition function as the partition function of a refined topological string on A1x(local P1xP1) is suggested.
We explicitly construct the metric and torsion couplings of two-dimensional (4,0)-super\-sym\-metric sigma models with target space a four-manifold that are invariant under a $U(1)$ symmetry generated by a tri-holomorphic Killing vector field that leaves in addition the torsion invariant. We show that the metric couplings arise from magnetic monopoles on the three-sphere which is the space of orbits of the group action generated by the tri-holomorphic Killing vector field on the sigma model target manifold. We also examine the global structure of a subclass of these metrics that are in additio…
Lorentz-invariant scalar field theories in d+1 dimensions with second-order derivative terms are unable to support static soliton solutions that are both finite in energy and stable for d>2, a result known as Derrick's theorem. Lifshitz theories, which introduce higher-order spatial derivatives, need not obey Derrick's theorem. We construct stable, finite-energy, static soliton solutions in Lifshitz scalar field theories in 3+1 dimensions with dynamical critical exponent z=2. We exhibit three generic types: non-topological point defects, topological point defects, and topological strings. We f…
A large class of supersymmetric quantum field theories, including all theories with $\mathcal{N} = 2$ supersymmetry in three dimensions and theories with $\mathcal{N} = 2$ supersymmetry in four dimensions, possess topological-holomorphic sectors. We formulate Poisson vertex algebras in such topological-holomorphic sectors and discuss some examples. For a four-dimensional $\mathcal{N} = 2$ superconformal field theory, the associated Poisson vertex algebra is the classical limit of a vertex algebra generated by a subset of local operators of the theory.
We review modifications of the Bekenstein-Hawking area law for black hole entropy in the presence of higher-derivative interactions. In four-dimensional N=2 compactifications of string theory or M-theory these modifications are crucial for finding agreement between the macroscopic entropy obtained from supergravity and the microscopic entropy obtained by counting states in string or M-theory. Our discussion is based on the effective Wilsonian action, which in the context of N=2 supersymmetric theories is defined in terms of holomorphic quantities. At the end we briefly indicate how to incorpor…
Generalized matter couplings to four-dimensional supersymmetric sigma models on general Kaehler manifolds are presented, preserving all holomorphic symmetries. Our generalization allows assignment of arbitrary U(1) charges to additional matter fermions, in all representations of (the holomorphic part of) the isometry group. This can be used to eliminate unwanted gamma_5 anomalies, in particular for the U(1) symmetry arising from the complex structure of the target space. A consistent gauging of this isometry group, or any of its subgroups, then becomes possible. When gauged in the presence of …
We show that the holomorphic Wilsonian beta-function of a renormalizable asymptotically free supersymmetric gauge theory with an arbitrary semi-simple gauge group, matter content, and renormalizable superpotential is exhausted at 1-loop with no higher loops and no non-perturbative contributions. This is a non-perturbative extension of the well known result of Shifman and Vainshtein.
We revise the sequences of SUSY for a cyclic adiabatic evolution governed by the supersymmetric quantum mechanical Hamiltonian. The condition (supersymmetric adiabatic evolution) under which the supersymmetric reductions of Berry (nondegenerated case) or Wilczek-Zee (degenerated case) phases of superpartners are taking place is pointed out. The analogue of Witten index (supersymmetric Berry index) is determined. As the examples of suggested concept of supersymmetric adiabatic evolution the Holomorphic quantum mechanics on complex plane and Meromorphic quantum mechanics on Riemann surface are c…
We study the gauge invariance of the supersymmetric grassmannian sigma model $G(M,N)$. It is richer then its purely bosonic submodel and we show how to use it in order to reduce some constant curvature holomorphic solutions of the model into simpler expressions.
Quantum dynamical semigroups are applied to the study of the time evolution of harmonic oscillators, both bosonic and fermionic. Explicit expressions for the density matrices describing the states of these systems are derived using the holomorphic representation. Bosonic and fermionic degrees of freedom are then put together to form a supersymmetric oscillator; the conditions that assure supersymmetry invariance of the corresponding dynamical equations are explicitly derived.
