We have performed a direct calculation of Witten index in N = 1,2,3 supersymmetric Yang-Mills Chern-Simons 3d theories. We do it in the framework of Born-Oppenheimer (BO) approach by putting the system into a small spatial box and studying the effective Hamiltonian depending on the zero field harmonics. At the tree level, our results coincide with the results of Witten, but there is a difference in the way the loop effects are implemented. In Witten's approach, one has only take into account the fermion loops, which bring about a negative shift of the (chosen positive at the tree level) Chern-Simons coupling k. As a result, Witten index vanishes and supersymmetry is broken at small k. In the effective BO Hamiltonian framework, fermion, gluon and ghost loops contribute on an equal footing. Fermion loop contribution to the effective Hamiltonian can be evaluated exactly, and their effect amounts to the negative shift k -> k - h/2 for N =1 and k -> k - h for N = 2,3 in the tree-level formulae for the index. In our approach, with rather natural assumptions on the structure of bosonic corrections, the shift k -> k + h brought about by the gluon loops also affects the index. Since the total shift of k is positive or zero, Witten index appears to be nonzero at nonzero k, and supersymmetry is not broken. We discuss possible reasons for such disagreement.
We provide a new proof of Cachazo-Svrcek-Witten rules for tree-level gluonic amplitudes. As a key step, we explicitly show the cancellation of spurious poles originating from the maximally helicity violating vertices in these rules. To achieve this, we introduce specially-defined two-off-shell-line sub-amplitudes and study their residues at spurious poles.
We attempt to deal with the orbifold singularities in the moduli space of flat connections for supersymmetric gauge theories on the torus. The fields are restricted to the fundamental domain, containing no gauge copies, but requiring a boundary condition in field space.
We develop a new method for decomposing Witten diagrams into conformal blocks. The steps involved are elementary, requiring no explicit integration, and operate directly in position space. Central to this construction is an appealingly simple answer to the question: what object in AdS computes a conformal block? The answer is a "geodesic Witten diagram," which is essentially an ordinary exchange Witten diagram, except that the cubic vertices are not integrated over all of AdS, but only over bulk geodesics connecting the boundary operators. In particular, we consider the case of four-point func…
We provide a uniform solution to 4d N=2 gauge theory with a single gauge group G=A,D,E when the one-loop contribution to the beta function from any irreducible component R of the hypermultiplets is less than or equal to half of that of the adjoint representation. The solution is given by a non-compact Calabi-Yau geometry, whose defining equation is built from explicitly known polynomials W_G and X_R, associated respectively to the gauge group G and each irreducible component R. We provide many pieces of supporting evidence, for example by analyzing the system from the point of view of the 6d N…
We propose a method for the evaluation of Witten index in D=2 supersymmetric Yang-Mills quantum mechanics. We rederive a known result for the SU(2) gauge group and generalize it to any SU(N) gauge group.
A reliable method to construct Supersymmetric Noether currents is presented. As the most important application the central charge of the N=2 Supersymmetric Yang-Mills effective theory, known as Seiberg-Witten (SW) theory, is computed. The analisys is carried out in the SW low energy U(1) effective Sector, as well as in the SW high energy SU(2) effective Sector.
The purpose of this paper is twofold. The first purpose is to review a systematic construction of Noether currents for supersymmetric theories, especially effective supersymmetric theories. The second purpose is to use these currents to derive the mass-formula for the quantized Seiberg-Witten model from the supersymmetric algebra. We check that the mass-formula of the low-energy theory agrees with that of the full theory (in the broken phase).
In these notes we attempt to give a pedagogical introduction to the work of Seiberg and Witten on S-duality and the exact results of N=2 supersymmetric gauge theories with and without matter. The first half is devoted to a review of monopoles in gauge theories and the construction of supersymmetric gauge theories. In the second half, we describe the work of Seiberg and Witten.
We examine supersymmetric quantum mechanics on $SO(4)$ to realize Witten's idea. We find instanton solutions connecting approximate vacuums. We calculate Hessian matrices for these solutions to determine true vacuums. Our result is in agreement with de Rham cohomology of $SO(4)$. We also give a criterion for cancellation of instanton effects for a pair of instanton paths.
The explicit form of non-Abelian noncommutative supersymmetric (SUSY) chiral anomaly is calculated, the Wess-Zumino consistency condition is verified and the correspondence of the Yang-Mills sector to the previously obtained results is shown. We generalize the Seiberg-Witten map to the case of N=1 SUSY Yang-Mills theory and calculations up to the second order in the noncommutativity parameter are done.
In this work we extend the results of previous derivations of Seiberg-like dualities (level-rank duality) between gauged Wess-Zumino-Witten theories. The arguments in use to identify a potential dual for the supersymmetric WZW theory based on the coset $\frac{U(N+M)_k}{U(N)_k}$ can be extended to be applied to a wider variety of gauge groups, notably $\frac{USp(2N+2M)_{2k}}{USp(2N)_{2k}}$ and $\frac{SO(2N+2M)_{2k}}{SO(2N)_{2k}}$, which will be dealt with briefly. Most interestingly, non-supersymmetric versions of the latter theories can also be shown to have duals in a similar fashion. These r…
A new, recursive method of calculating matrix elements of polynomial hamiltonians is proposed. It is particularly suitable for the recent algebraic studies of the supersymmetric Yang-Mills quantum mechanics in any dimensions. For the D=2 system with the SU(2) gauge group, considered here, the technique gives exact, closed expressions for arbitrary matrix elements of the hamiltonian and of the supersymmetric charge, in the occupation number representation. Subsequent numerical diagonalization provides the spectrum and restricted Witten index of the system with very high precision (taking into a…
We argue that, contrary to previous claims, the supersymmetric sine-Gordon model with boundary has a two-parameter family of boundary interactions which preserves both integrability and supersymmetry. We also propose the corresponding boundary S matrix for the first supermultiplet of breathers.
We study the Donaldson-Witten function in four-dimensional topological gauge theory which is constructed from N=2 supersymmetric SU(2) gauge theory with $N_f < 4$ massless fundamental hypermultiplets. When $N_f = 2,3$, the strong-coupling singularities with multiple massless monopoles appear in the moduli space (the u-plane) of the Coulomb branch. We show that the invariants made out of such singularities exhibit a property which is similar to the one expected for four-manifolds of generalized simple type.
We construct a three dimensional deconfinement method which enables us to find new three-dimensional dualities and we apply various techniques developed in four dimensional supersymmetric gauge theories, such as the product gauge groups and Seiberg-Witten curves to the three dimensional $\mathcal{N}=2$ supersymmetric gauge theories. Dual descriptions of three dimensional $\mathcal{N}=2$ supersymmetric gauge theories which involve two-index matters, for example, adjoint, symmetric, and anti-symmetric matters without superpotentials can be obtained. These matters are described in terms of s-conf…
We discuss two distinct aspects in supersymmetric quantum mechanics. First, we introduce a new class of operators A and $\bar{A}$ in terms of anticommutators between the momentum operator and N+1 arbitrary superpotentials. We show that these operators reduce to the conventional ones which are the starting point in standard supersymmetric quantum mechanics. In this context, we argue furthermore that supersymmetry does not only connect Schrödinger-like operators, but also a more general class of differential operators. Second, we revisit the supersymmetric $\varepsilon$-system recently introduce…
Using the Mathai-Quillen formalism we reexamine the twisted N=4 supersymmetric model of Vafa-Witten theory. Smooth out the relation between the supersymmetric action and the path integral representation of the Thom class.
{}~~~We show that the recently constructed $~N=4$~ supersymmetric self-dual Yang-Mills theory as the consistent background of \hbox{$~N=2$} open superstring will generate Witten's topological field theory in two-dimensions as a descendant theory after appropriate twisted dimensional reduction/truncations. We also show that this topological field theory further generates supersymmetric Korteweg de Vries equations, $~SL(n)\-$Toda theory and $~W_\infty\-$gravity in the $~n\rightarrow\infty$~ limit. Considering also that this topological field theory is to generate other integrable and topological…
We determine the non-abelian Born-Infeld action, including fermions, as it results from the four-point tree-level open superstring scattering amplitudes at order alpha'^2. We find that, after an appropriate field redefinition all terms at this order can be written as a symmetrised trace. We confront this action with the results that follow from kappa-symmetry and conclude that the recently proposed non-abelian kappa-symmetry cannot be extended to cubic orders in the Born-Infeld curvature.
We simplify the classification of supersymmetric solutions with compact holonomy of the Killing spinor equations of heterotic supergravity using the field equations and the additional assumption that the 3-form flux is closed. We determine all the fractions of supersymmetry that the solutions preserve and find that there is a restriction on the number of supersymmetries which depends on the isometry group of the background. We examine the geometry of spacetime in all cases. We find that the supersymmetric solutions of heterotic supergravity are associated with a large number of geometric struc…
We recently obtained the conditions on the couplings of the general two-dimensional massive sigma-model required by (p,q)-supersymmetry. Here we compute the Poisson bracket algebra of the supersymmetry and central Noether charges, and show that the action is invariant under the automorphism group of this algebra. Surprisingly, for the (4,4) case the automorphism group is always a subgroup of SO(3), rather than SO(4). We also reanalyse the conditions for the (2,2) and (4,4) supersymmetry of the zero torsion models without assumptions about the central charge matrix.
In [arXiv:0808.1677] 3 dimensional field theories are proposed to on-shell represent N=8 SUSY algebra and SO(7) or SO(4)xSO(4) subgroups of full SO(8) automorphism. They are theories of 8 scalar and 8 spinor matrix fields with Yukawa, quartic and sextic self-interactions. This note is devoted to superfield and off-shell formulation of the theory. Scalar superfields are proposed to represent scalar supermultiplets. A renormalizable action with quadratic and quartic self-interactions is proposed to govern their dynamics.
We discuss various properties of the Seiberg-Witten curve for the E-string theory which we have obtained recently in hep-th/0203025. Seiberg-Witten curve for the E-string describes the low-energy dynamics of a six-dimensional (1,0) SUSY theory when compactified on R^4 x T^2. It has a manifest affine E_8 global symmetry with modulus τand E_8 Wilson line parameters {m_i},i=1,2,...,8 which are associated with the geometry of the rational elliptic surface. When the radii R_5,R_6 of the torus T^2 degenerate R_5,R_6 --> 0, E-string curve is reduced to the known Seiberg-Witten curves of four- and fiv…
We study two-dimensional supersymmetric non-linear sigma-models with boundaries. We derive the most general family of boundary conditions in the non-supersymmetric case. Next we show that no further conditions arise when passing to the N=1 model. We present a manifest N=1 off-shell formulation. The analysis is greatly simplified compared to previous studies and there is no need to introduce non-local superspaces nor to go (partially) on-shell. Whether or not torsion is present does not modify the discussion. Subsequently, we determine under which conditions a second supersymmetry exists. As fo…
Inspired by ideas regarding Hermitian NxN matrix fields obeying a non-associative algebra, 3-dimensional N=8 SUSic field theories are proposed to on-shell represent subalgebras of OSp(8|2) and OSp(8|4) groups of SUSY transformations. They are theories of 8 scalar and 8 spinor fields with Yukawa, quartic and sextic self-interactions. The actions as their R-symmetry exhibit only SO(7) or SO(4)xSO(4) subgroups of full SO(8) automorphisms. It is argued that the number of degrees of freedom scale like N^{3/2}. There also exists an extra S_N permutation symmetry group.
Let $n$ be any natural number. Let $K$ be any $n$-dimensional knot in $S^{n+2}$. We define a supersymmetric quantum system for $K$ with the following properties. We firstly construct a set of functional spaces (spaces of fermionic \{resp. bosonic\} states) and a set of operators (supersymmetric infinitesimal transformations) in an explicit way. Thus we obtain a set of the Witten indexes for $K$. Our Witten indexes are topological invariants for $n$-dimensional knots. Our Witten indexes are not zero in general. If $K$ is equivalent to the trivial knot, all of our Witten indexes are zero. Our Wi…
We consider the theories obtained by dimensional reduction to D=1,2,3 of 4D supersymmetric Yang--Mills theories and calculate there the effective low-energy lagrangia describing moduli space dynamics -- the low-dimensional analogs of the Seiberg--Witten effective lagrangian. The effective theories thus obtained are rather beautiful and interesting from mathematical viewpoint. In addition, their study allows one to understand better some essential features of 4D supersymmetric theories, in particular -- the nonrenormalisation theorems.
This note addresses the question of the number of normalizable vacuum states in supersymmetric quantum mechanics with sixteen supercharges and arbitrary semi-simple compact gauge group, up to rank three. After evaluating certain contour integrals obtained by appropriately adapting BRST deformation techniques we propose novel rational values for the bulk indices. Our results demonstrate that an asymptotic method for obtaining the boundary contribution to the index, originally due to Green and Gutperle, fails for groups other than SU(N). We then obtain likely values for the number of ground stat…
In this sequel to [L-Y1], [L-L-S-Y], and [L-Y2] (respectively arXiv:0709.1515 [math.AG], arXiv:0809.2121 [math.AG], and arXiv:0901.0342 [math.AG]), we study a D-brane probe on a conifold from the viewpoint of the Azumaya structure on D-branes and toric geometry. The details of how deformations and resolutions of the standard toric conifold $Y$ can be obtained via morphisms from Azumaya points are given. This should be compared with the quantum-field-theoretic/D-brany picture of deformations and resolutions of a conifold via a D-brane probe sitting at the conifold singularity in the work of Kle…
The one-loop contribution to the superpotential, in particular the Veneziano-Yankielowicz potential in N=1 supersymmetric Yang-Mills model is discussed from an elementary field theory method and the matrix model point of view. Both approaches are based on the Renormalization Group variation of the superconformal N=4 supersymmetric Yang-Mills model.
We introduce in supersymmetric QCD with $N_f<N$ additional meson singlet degrees of freedom made of $2N_f-2$ supermultiplets. We construct an alternative superpotential with $N_f^2$ constraints and show its consistency. Based on this superpotential we prove that for $\frac{N}{2}<N_f<N$ the supersymmetry is unbroken whereas for $N_f\leq\frac{N}{2}$ is dynamically broken.
We showed that there is no SU(2) Witten anomaly in a large class of 4d N = 2 supersymmetric Heterotic string compactifications. The consistency conditions we consider are the modularity of the new supersymmetric index, the integrality of BPS indices, and the discrete Peccei-Quinn shift symmetries. We also found an example where these conditions are not sufficient to show that the theory is anomaly free. This suggests that there are more conditions in the worldsheet SCFT essential for consistent string compactifications.
An essential assumption in the Vafa and Witten's theorem on P and CT realization in vector-like theories concerns the existence of a free energy density in Euclidean space in the presence of any external hermitian symmetry breaking source. We show how this requires the previous assumption that the symmetry is realized in the vacuum. Even if Vafa and Witten's conjecture is plausible, actually a theorem is still lacking.
In my lecture I consider integrals over moduli spaces of supersymmetric gauge field configurations (instantons, Higgs bundles, torsion free sheaves). The applications are twofold: physical and mathematical; they involve supersymmetric quantum mechanics of D-particles in various dimensions, direct computation of the celebrated Seiberg-Witten prepotential, sum rules for the solutions of the Bethe ansatz equations and their relation to the Laumon's nilpotent cone. As a by-product we derive some combinatoric identities involving the sums over Young tableaux.
A new method to obtain the Picard-Fuchs equations of effective, N=2 supersymmetric gauge theories with massive matter hypermultiplets in the fundamental representation is presented. It generalises a previously described method to derive the Picard-Fuchs equations of both pure super Yang-Mills and supersymmetric gauge theories with massless matter hypermultiplets. The techniques developed are well suited to symbolic computer calculations.
We present a systematic study of N=1 supersymmetric gauge theories which are in the Coulomb phase. We show how to find all such theories based on a simple gauge group and no tree-level superpotential. We find the low-energy solution for the new theories in terms of a hyperelliptic Seiberg-Witten curve. This work completes the study of all N=1 supersymmetric gauge theories where the Dynkin index of the matter fields equals the index of the adjoint (mu=G), and consequently all theories for which mu<G.
The Yukawa interaction sector of superstring inspired models that give superconducting strings, can be described in terms of a supersymmetric quantum mechanics algebra. We relate the Witten index of susy quantum mechanics with an index characteristic to superconducting string models.
Superspace power-counting rules give estimates for the loop order at which divergences can first appear in non-renormalisable supersymmetric field theories. In some cases these estimates can be improved if harmonic superspace, rather than ordinary superspace, is used. The new estimates are in agreement with recent results derived from unitarity calculations for maximally supersymmetric Yang-Mills theories in five and six dimensions. For N=8 supergravity in four dimensions, we speculate that the onset of divergences may correspondingly occur at the six loop order.
We show that parity is conserved in vector-like supersymmetric theories, such as supersymmetric QCD with massive quarks with no cubic couplings among chiral multiplets, based on fermionic path-integrals, originally developed by Vafa and Witten. We also look into the effect of supersymmetric breaking through gluino masses, and see that the parity-conservation is intact also in this case. Our conclusion is valid, when only bosonic parity-breaking observable terms are considered in path-integrals like the original Vafa-Witten formulation.
I discuss the symmetry structure of the N=2 supersymmetric extension of the Born-Infeld action in four dimensions, and confirm its interpretation as the Goldstone-Maxwell action associated with partial breaking of N=4 extended supersymmetry down to N=2, by revealing a hidden invariance of the action with respect to two non-linearly realized supersymmetries and two spontaneously broken translations. I also argue about the uniqueness of supersymmetric extension of the Born-Infeld action, and its possible relation to noncommutative geometry.
In this work a thorough study of a number of specific supersymmetric sigma-models with extended supersymmetry is performed within the context of generalised complex geometry. More specifically the supersymmetric Wess-Zumino-Witten model on a variety of group manifolds was considered. By explicitly calculating the admissible complex structures and the associated pure spinors on the target manifold a full characterisation of the different possible geometries is provided. By using this approach the various aspects of generalised Kähler geometry can be studied in detail. Also considered are the va…
Electric-magnetic duality allows to calculate the central charges of N=2 supersymmetric theories with massless hypermultiplets as derivatives of simple modular forms. The procedure reproduces the Seiberg-Witten results for N_f=0,2,3 in a uniform way, but indicates open problems for N_f=1.
We show that there exist supersymmetric Minkowski vacua on Type IIB toroidal orientifold with general flux compactifications where the RR tadpole cancellation conditions can be relaxed and the Freed-Witten anomaly can be cancelled elegantly. We present a realistic Pati-Salam like flux model without the Freed-Witten anomaly. At the string scale, we can break the gauge symmetry down to the Standard Model (SM) gauge symmetry, achieve the gauge coupling unification naturally, and decouple all the extra chiral exotic particles. Thus, we have the supersymmetric SMs with/without SM singlet(s) below t…
The instanton-induced effective vertex is derived for N=2 supersymmetric QCD (SQCD) with arbitrary mass matter hypermultiplets for the case of SU(2). The leading term of the low energy effective lagrangian obtained from this vertex agrees with one-instanton effective term of the Seiberg-Witten result.
In general, Whitham dynamics involves infinitely many parameters called Whitham times, but in the context of N=2 supersymmetric Yang-Mills theory it can be regarded as a finite system by restricting the number of Whitham times appropriately. For example, in the case of SU(r+1) gauge theory without hypermultiplets, there are r Whitham times and they play an essential role in the theory. In this situation, the generating meromorphic 1-form of the Whitham hierarchy on Seiberg-Witten curve is represented by a finite linear combination of meromorphic 1-forms associated with these Whitham times, but…
In this article we expound a discovery of the quantum equivalence/duality of U(N) noncommutative quantum field theories (NC QFT) related by the theta-exact Seiberg-Witten (SW) maps and at all orders in the perturbation theory with respect to the coupling constant. We show that this proof holds for Super Yang-Mills (SYM) theories with N=0,1,2,4$ supersymmetry. In short, Seiberg-Witten map does commute with the quantization of the U(N) NCQFT independently, with or without supersymmetry.
Description of two three-dimensional topological quantum field theories of Witten type as twisted supersymmetric theories is presented. Low-energy effective action and a corresponding topological invariant of three-dimensional manifolds are considered.
We estimate the Witten index for a supersymmetric gauge theory with or without massless matter. For pure supersymmetric Yang Mills our result reproduces the results in the literature. For the known theories with dynamical symmetry breaking the Witten index is determined to be exactly zero. Our calculation sheds light on the possibility of dynamical supersymmetry breaking for those theories considered up to now unsolvable.
We compute the three-graviton tree amplitude in Type IIB superstring theory compactified to six dimensions using the manifestly (6d) supersymmetric Berkovits-Vafa-Witten worldsheet variables. We consider two cases of background geometry: the flat space example R6xK3, and the curved example AdS3xS3xK3 with Ramond flux, and compute the correlation functions in the bulk.
The problem of counting the vacuum states in the supersymmetric 3d Yang-Mills-Chern-Simons theory is reconsidered. We resolve the controversy between its original calculation by Witten at large volumes and the calculation based on the evaluation of the effective Lagrangian in the small volume limit. We show that the latter calculation suffers from uncertainties associated with the singularities in the moduli space of classical vacua where the Born-Oppenheimer approximation breaks down. We also show that these singularities can be accurately treated in the Hamiltonian Born-Oppenheimer method, w…
Higher-derivative generalization of the supersymmetric quantum mechanics is proposed. It is formally based on the standard superalgebra but supercharges involve differential operators of the order $n$. As a result, their anticommutator entails polynomial of a Hamiltonian. The Witten index does not characterize spontaneous supersymmetry breaking in such models.
In this letter we establish that the supersymmetric R^2 effective action for the heterotic string, obtained from the supersymmetrisation of the Lorentz Chern-Simons term, is to order $α'$ equivalent modulo field redefinitions to heterotic string effective actions computed by different methods.
We give an elementary introduction to the recent solution of $N=2$ supersymmetric Yang-Mills theory. In addition, we review how it can be re-derived from string duality.
