We propose an embedding of $U(N_c)$ QCD$_3$ with a Chern-Simons term in string theory. The UV gauge theory lives on the worldvolume of a Hanany-Witten brane configuration in type 0B string theory in the presence of Sagnotti's O$'3$ orientifold. We use the brane configuration to propose a magnetic Seiberg dual. We identify various phases of the magnetic theory with conjectured phases of QCD$_3$. In particular the symmetry breaking and bosonization phases are both associated with condensation of the dual squark field. We also discuss the abelian theory without Chern-Simons term and argue that flavour symmetry is not broken. Finally, we also predict novel type 0B string dynamics from QCD dynamics.
We argue that N=2 supersymmetric Chern-Simons theories exhibit a strong-weak coupling Seiberg-type duality. We also discuss supersymmetry breaking in these theories.
We show that not all $(2+1)$ dimensional toric phases are Seiberg-like duals. Particularly, we work out superconformal indices for the toric phases of Fanos ${\cal{C}}_3$, ${\cal{C}}_5$ and ${\cal{B}}_2$. We find that the indices for the two toric phases of Fano ${\cal{B}}_2$ do not match, which implies that they are not Seiberg-like duals. We also take the route of acting Seiberg-like duality transformation on toric quiver Chern-Simons theories to obtain dual quivers. We study two examples and show that Seiberg-like dual quivers are not always toric quivers.
Toric Duality arises as an ambiguity in computing the quiver gauge theory living on a D3-brane which probes a toric singularity. It is reviewed how, in simple cases Toric Duality is Seiberg Duality. The set of all Seiberg Dualities on a single node in the quiver forms a group which is contained in a larger group given by a set of Picard-Lefschetz transformations. This leads to elements in the group (sometimes called fractional Seiberg Duals) which are not Seiberg Duality on a single node, thus providing a new set of gauge theories which flow to the same universality class in the Infra Red.
We show how SU(2) $N_f=4$ Seiberg-Witten prepotentials are derived form $\hat{\mathfrak{sl}}_{2,k}$ ($k\to2$) four-point conformal blocks via considering Langlands duality.
We study the equivalence/duality between various non-commutative gauge models at the classical and quantum level. The duality is realised by a linear Seiberg-Witten-like map. The infinitesimal form of this map is analysed in more details.
We propose a Seiberg duality for a 3d $\mathcal{N}=2$ $Spin(7)$ gauge theory with $F$ spinor matters. For $F \ge 6$, the theory allows a magnetic dual description with an $SU(F-4)$ gauge group. The matter content on the magnetic side is ``chiral'' and the duality connects ``chiral'' and ``non-chiral'' 3d gauge theories. As a corollary, we can construct a Seiberg duality for a 3d $\mathcal{N}=2$ $G_2$ gauge theory with fundamental matters.
We analyze three dimensional gauge theories with $Sp$ gauge group. We find that in some regime the theory should be described in terms of a dual theory, very much in the spirit of Seiberg duality in four dimensions. This duality does not coincide with mirror symmetry.
Duality between N=1 supersymmetric gauge theories(Seiberg's duality) is geometrized, in the framework of AdS/CFT correspodences. It is shown that Seiberg's duality corresponds to monodromy of wrapped D5 branes on the homology cycles of a generalized conifold where D3 branes are located. The celebrated \tilde{N}_c=N_f-N_c, \tilde{N}_f=N_f rule is reproduced and a braid group structure in a sequence of dualities, is found.
Starting from the Seiberg-Witten solution of N=2 SQCD with the U(N) gauge group and N_f quark flavors we construct the so-called μ-dual N=1 theory in the r vacua in the regime analogous to that existing to the left of the left edge of the Seiberg conformal window (here r is the number of condensed quarks). The strong-weak coupling duality is shown to exist in the so-called zero vacua which can be found at r< N_f-N. We show that the μ-dual theory matches the Seiberg dual in the zero vacua.
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 propose a duality for N=2 d=3 Chern-Simons gauge theories with orthogonal gauge groups and matter in the vector representation. This duality generalizes level-rank duality for pure Chern-Simons gauge theories with orthogonal gauge groups and is reminiscent of Seiberg duality in four dimensions. We perform extensive checks by comparing partition functions of theories related by dualities. We also determine the conformal dimensions of fields using Z-extremization.
We use field theory and brane diamond techniques to demonstrate that Toric Duality is Seiberg duality for N=1 theories with toric moduli spaces. This resolves the puzzle concerning the physical meaning of Toric Duality as proposed in our earlier work. Furthermore, using this strong connection we arrive at three new phases which can not be thus far obtained by the so-called ``Inverse Algorithm'' applied to partial resolution of C^3/Z_3 x Z_3. The standing proposals of Seiberg duality as diamond duality in the work by Aganagic-Karch-Lüst-Miemiec are strongly supported and new diamond configurati…
In this paper, we use the matrix model of pure fundamental flavors (without the adjoint field) to check the Seiberg duality in the case of complete mass deformation. We show that, by explicit integration at both sides of electric and magnetic matrix models, the results agree with the prediction in the field theory.
We propose a general method to study open/closed string dualities from transitions in M theory which is valid for a large class of geometrical configurations. By T-duality we can transform geometrically engineered configurations into N = 1 brane configurations and study the transitions of the corresponding branes by lifting the configurations to M-theory. We describe the transformed degenerated M5 branes and extract the field theory information on gluino condensation by factorization of the Seiberg-Witten curve. We also include massive flavors and orientifolds and discuss Seiberg duality which…
Branes at a $\mathbb{F}_0$ singularity give rise to two different toric quiver gauge theories, which are related by Seiberg duality. We study where in the Kähler moduli space each of them is physically realized.
In this note, we give a method to derive the Seiberg duality by the matrix model. The key fact we used is that the effective actions given by matrix model method should be identical for both electric and magnetic theories. We demonstrate our method for SQCD with U(N), SO(N) and Sp(N) gauge groups.
The low energy gauge theory living on D-branes probing a del Pezzo singularity of a non-compact Calabi-Yau manifold is not unique. In fact there is a large equivalence class of such gauge theories related by Seiberg duality. As a step toward characterizing this class, we show that Seiberg duality can be defined consistently as an admissible mutation of a strongly exceptional collection of coherent sheaves.
These lectures are devoted to the low energy limit of \N2 SUSY gauge theories, which is described in terms of integrable systems. A special emphasis is on a duality that naturally acts on these integrable systems. The duality turns out to be an effective tool in constructing the double elliptic integrable system which describes the six-dimensional Seiberg-Witten theory. At the same time, it implies a series of relations between other Seiberg-Witten systems.
We investigate features of duality in three dimensional N=2 Chern-Simons matter theories conjectured to describe M2 branes at toric Calabi Yau four-fold singularities. For 3D theories with non-chiral 4D parents we propone a Seiberg-like duality which turns out to be a toric duality. For theories with chiral 4D parents we discuss the conditions under which that Seiberg-like duality leads to toric duality. We comment on such duality in 3D theories without 4D parents.
We use localization techniques to study several duality proposals for supersymmetric gauge theories in three dimensions reminiscent of Seiberg duality. We compare the partition functions of dual theories deformed by real mass terms and FI parameters. We find that Seiberg-like duality for N=3 Chern-Simons gauge theories proposed by Giveon and Kutasov holds on the level of partition functions and is closely related to level-rank duality in pure Chern-Simons theory. We also clarify the relationship between the Giveon-Kutasov duality and a duality in theories of fractional M2 branes and propose a …
We work out the superconformal index for N=2 supersymmetric Chern-Simons matter theories exhibiting Seiberg-like dualities proposed by Giveon and Kutasov. We consider $U(N)/Sp(2N)/O(N)$ gauge theories of QCD type and find the perfect agreements for proposed dual pairs.
It is known that electric-magnetic duality transformations are symmetries of the generalized Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations. In Seiberg-Witten theory the solutions to these equations come in certain sets according to the gauge group. We show that the duality transformations transform solutions within a set to another solution within the same set, by using a subset of the Picard-Fuchs equations on the Seiberg-Witten family of Riemann surfaces. The electric-magnetic duality transformations can be thought of as changes of a canonical homology basis on the surfaces which in ou…
We consider a general class of symmetries of hyper-Kahler quotients which can be interpreted as classical analogs of Seiberg duality for N=2 supersymmetric quiver gauge theories in the baryonic Higgs branch. Along the way we find that a limit application of this duality yields new exotic realizations of ALE spaces as hyper-Kahler quotients. We also discuss how these results admit in some particular cases a natural interpretation in string theory. We finally comment on a possible relationship with Fourier-Mukai tranforms.
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 explore the dynamics of three-dimensional Chern-Simons gauge theories with N=2 supersymmetry and matter in the fundamental and adjoint representations of the gauge group. Realizing the gauge theories of interest in a setup of threebranes and fivebranes in type IIB string theory we argue for a Seiberg duality that relates Chern-Simons theories with non-trivial superpotentials.
I review the proposal of Berenstein-Douglas for a completely general definition of Seiberg duality. To give evidence for their conjecture I present the first example of a physical dual pair and explicitly check that it satisfies the requirements. Then I explicitly show that a pair of toric dual quivers is also dual according to their proposal. All these computations go beyond tilting modules, and really work in the derived category. I introduce all necessary mathematics where needed.
It is known that a 4d N = 1 SCFT lives on D3-branes probing a local del Pezzo Calabi-Yau singularity. The Seiberg (or toric) duality of this SCFT arises from the Picard-Lefshetz transformation of the affine E_N 7-brane background that is associated with the Calabi-Yau threefold. In this paper we study the duality of the affine E_N background itself and a 5-brane probing it. We then find that many different Type IIB 5-brane webs describe the same SCFT in 5d. We check this duality by comparing the Nekrasov partition functions of these 5-brane web configurations.
We analyze the action of Toric (Seiberg) duality on the combined mesonic and baryonic moduli space of quiver gauge theories obtained from D3 branes at Calabi-Yau singularities. We analyze in particular the structure of the master space, the complete moduli space for one brane, for different toric phases of a given singularity. We show that the Hilbert Series for the largest component of the master space of different phases is the same, when refined with all the non anomalous charges. This reflects the fact that the quiver gauge theories associated with different phases are related by Seiberg d…
In this paper we continue the investigation, within the context of the Dijkgraaf-Vafa Programme, of Seiberg duality in matrix models as initiated in hep-th/0211202, by allowing degenerate mass deformations. In this case, there are some massless fields which remain and the theory has a moduli space. With this illustrative example, we propose a general methodology for performing the relevant matrix model integrations and addressing the corresponding field theories which have non-trivial IR behaviour, and which may or may not have tree-level superpotentials.
We study Seiberg duality of quiver gauge theories associated to the complex cone over the second del Pezzo surface. Homomorphisms in the path algebra of the quivers in each of these cases satisfy relations which follow from a superpotential of the corresponding gauge theory as F-flatness conditions. We verify that Seiberg duality between each pair of these theories can be understood as a derived equivalence between the categories of modules of representation of the path algebras of the quivers. Starting from the projective modules of one quiver we construct tilting complexes whose endomorphism…
We study higher symmetries and anomalies of 4d $\mathfrak{so}(2n_c)$ gauge theory with $2n_f$ flavors. We find that they depend on the parity of $n_c$ and $n_f$, the global form of the gauge group, and the discrete theta angle. The contribution from the fermions plays a central role in our analysis. Furthermore, our conclusion applies to $\mathcal{N}=1$ supersymmetric cases as well, and we see that higher symmetries and anomalies match across the Intriligator-Seiberg duality between $\mathfrak{so}(2n_c)\leftrightarrow\mathfrak{so}(2n_f-2n_c+4)$.
We study the duality of $\mathcal{N}=1$ gauge theories in the presence of a massless adjoint field and massive quarks by calculating the superpotential using the Dighkgraaf-Vafa matrix model and by comparing with the previous result coming from Kutasov duality. The Kutasov duality method gives a result in which one instanton term is absent. The matrix model method confirms it and also show that the absence of the one instanton term is related to the masslessness of the adjoint field.
In this paper we discuss some aspects of the behavior of superconformal N=1 models under Seiberg's duality. Our claim is that if an electric gauge theory is superconformal on some marginal subspace of all coupling constants then its magnetic dual must be also superconformal on a corresponding moduli space of dual couplings. However this does not imply that the magnetic dual of a completely finite N=1 gauge theory is again finite. Moreover we generalize this statement conjecturing that also for non-superconformal N=1 models the determinant of the beta-function equations is invariant under Seibe…
An elementary introduction into the Seiberg-Witten theory is given. Many efforts are made to get it as pedagogical as possible, within a reasonable size. The selection of the relevant material is heavily oriented towards graduate students. The basic ideas about solitons, monopoles, supersymmetry and duality are reviewed from first principles, and they are illustrated on the simplest examples. The exact Seiberg-Witten solution to the low-energy effective action of the four-dimensional N=2 supersymmetric pure Yang-Mills theory with the gauge group SU(2) is the main subject of the review. Other g…
Using the generalized Konishi anomaly (GKA) equations, we derive the effective superpotential of four-dimensional N=1 supersymmetric SU(n) gauge theory with n+2 fundamental flavors. We find, however, that the GKA equations are only integrable in the Seiberg dual description of the theory, but not in the direct description of the theory. The failure of integrability in the direct, strongly coupled, description suggests the existence of non-perturbative corrections to the GKA equations.
We define and study a new class of 4d N=1 superconformal quiver gauge theories associated with a planar bipartite network. While UV description is not unique due to Seiberg duality, we can classify the IR fixed points of the theory by a permutation, or equivalently a cell of the totally non-negative Grassmannian. The story is similar to a bipartite network on the torus classified by a Newton polygon. We then generalize the network to a general bordered Riemann surface and define IR SCFT from the geometric data of a Riemann surface. We also comment on IR R-charges and superconformal indices of …
We develop techniques to calculate an index for four dimensional superconformal field theories. This superconformal index is counting BPS operators which preserve only one supercharge. To calculate the superconformal index we quantize the field theory on S^3 X R and show that the twisted theory has an appropriate mass gap. This allows for the interactions to be switched off continuously without the superconformal index being changed. We test those techniques for theories which go through a non-trivial RG flow and for Seiberg dual theories. This leads to the conjecture of some group/number theo…
We study half-BPS surface operators in N=2supersymmetric asymptotically conformal gauge theories in four dimensions with SU(N) gauge group and 2N fundamental flavours using localization methods and coupled 2d/4d quiver gauge theories. We show that contours specified by a particular Jeffrey-Kirwan residue prescription in the localization analysis map to particular realizations of the surface operator as flavour defects. Seiberg duality of the 2d/4d quivers is mapped to contour deformations of the localization integral which in this case involves a residue at infinity. This is reflected as a mod…
We study four N=1 SU(N)^6 gauge theories, with bi-fundamental chiral matter and a superpotential. In the infrared, these gauge theories all realize the low-energy world-volume description of N coincident D3-branes transverse to the complex cone over a del Pezzo surface dP_3 which is the blowup of P^2 at three generic points. Therefore, the four gauge theories are expected to fall into the same universality class--an example of a phenomenon that has been termed "toric duality." However, little independent evidence has been given that such theories are infrared-equivalent. In fact, we show that …
These are notes from introductory lectures given at the Ecole Normale in Paris and at the Strasbourg meeting dedicated to the memory of Claude Itzykson. I review in considerable detail and in a hopefully pedagogical way the work of Seiberg and Witten on $N=2$ supersymmetric $SU(2)$ gauge theory without extra matter. This presentation basically follows their original work, except in the last section where the low-energy effective action is obtained emphasizing more the relation between monodromies and differential equations rather than using elliptic curves.
We review the reduction of four-dimensional N=1 Seiberg duality to three dimensions focusing on the D brane engineering approach. We start with an overview of four-dimensional Seiberg duality for theories with various types of gauge groups and matter content both from a field-theoretic and a brane engineering point of view. Then we describe two families of N=2 three-dimensional dualities, namely Giveon-Kutasov-like and Aharony-like dualities. The last part of our discussion is devoted to the 4D/3D reduction of the dualities studied above. We discuss both the analysis at finite radius, crucial …
Chern-Simons quivers for M2-branes at Calabi-Yau singularities are best understood as the low energy theory of D2-branes on a dual type IIA background. We show how the D2-brane point of view naturally leads to three dimensional Seiberg dualities for Chern-Simons quivers with chiral matter content: They arise from a change of brane basis (or mutation), in complete analogy with the better known Seiberg dualities for D3-brane quivers. This perspective reproduces the known rules for Seiberg dualities in Chern-Simons-Yang-Mills theories with unitary gauge groups. We provide explicit examples of dua…
A popular way to study N=1 supersymmetric gauge theories is to realize them geometrically in string theory, as suspended brane constructions, D-branes wrapping cycles in Calabi-Yau manifolds, orbifolds, and otherwise. Among the applications of this idea are simple derivations and generalizations of Seiberg duality for the theories which can be so realized. We abstract from these arguments the idea that Seiberg duality arises because a configuration of gauge theory can be realized as a bound state of a collection of branes in more than one way, and we show that different brane world-volume theo…
In this talk I review and generalize an idea of Seiberg that an N=1 supersymmetric gauge theory shows confinement without breaking of chiral symmetry when the gauge symmetry of its magnetic dual is completely broken by the Higgs effect. It is shown how the confining spectrum of a supersymmetric gauge theory can easily be derived when a magnetic dual is known and this method is applied to many models containing fields in second rank tensor representations and an appropriate tree-level superpotential.
For a quiver with potential, Derksen, Weyman and Zelevinsky defined a combinatorial transformation - mutations. Mukhopadhyay and Ray, on the other hand, tell us how to compute Seiberg dual quivers for some quivers with potentials through a tilting procedure, thus obtaining derived equivalent algebras. In this text, we compare mutations with the concept of Seiberg duality, concluding that for a certain class of potentials (the good ones) mutations coincide with Seiberg duality, therefore giving derived equivalences.
We propose an electric-magnetic duality and conjecture an exact conformal window for a class of non-supersymmetric U(N_c) gauge theories with fermions in the (anti)symmetric representation of the gauge group and N_f additional scalar and fermion flavors. The duality exchanges N_c with N_f -N_c \mp 4 leaving N_f invariant, and has common features with Seiberg duality in N=1 SQCD with SU or SO/Sp gauge group. At large N the duality holds due to planar equivalence with N=1 SQCD. At finite N we embed these gauge theories in a setup with D-branes and orientifolds in a non-supersymmetric, but tachyo…
We propose a new chiral model of metastable supersymmetry breaking in the conformal window via chiral/nonchiral Seiberg duality, which is focused on Spin(7)/SU(N_f-4) duality in this letter. Following the approach of Intriligator-Seiberg-Shih model, a small mass perturbation in the nonchiral theory leads to supersymmetry breaking by rank condition in the chiral dual description.
Systematizing our results on r duality obtained previously we focus on comparing r duality with the generalized Seiberg duality in the r vacua of N=2 and N=1 super-Yang-Mills theories with the U(N) gauge group and N_f matter flavors (N_f>N). The number of condensed (s)quarks r is assumed to be in the interval 2N_f/3 < r \le N. To pass to N=1 we introduce an N=2-breaking deformation, a mass term μfor the adjoint matter, eventually decoupling the adjoint matter in the limit of large μ. If one starts from a large value of the parameter ξ\simμm, where the original theory is at weak coupling, and d…
Seiberg-like duality of three dimensional N=2 Chern-Simons-Matter quiver gauge theory is shown to have a chiral double tropical cluster algebra structure. We use cluster algebra results to study combinatorial aspects of these theories such as classification, supersymmetry breaking, etc.
We consider an N=1 SU(N_c) SUSY gauge theory with N_f \geq N_c matter multiplets transforming in the fundamental and antifundamental representations of the gauge group. Using the Konishi anomaly and a non-anomalous conservation law, we derive a system of partial differential equations that determine the low energy effective superpotential as a function of the mesonic and baryonic vacuum expectation values. We apply the formalism to the cases of N_f = N_c and N_f = N_c +1 where the equations are easily integrated and recover the known results. We further apply the formalism to derive a system o…
We show that in the perturbative regime defined by the coupling constant, the theta-exact Seiberg-Witten map applied to noncommutative U(N) Yang-Mills --with or without Supersymmetry-- gives an ordinary gauge theory which is, at the quantum level, dual to the former. We do so by using the on-shell DeWitt effective action and dimensional regularization. We explicitly compute the one-loop two-point function contribution to the on-shell DeWitt effective action of the ordinary U(1) theory furnished by the theta-exact Seiberg-Witten map. We find that the non-local UV divergences found in the propag…
We study a three-dimensional U(k) Yang-Mills Chern-Simons theory with adjoint matter preserving two supersymmetries. According to Acharya and Vafa, this theory describes the low-energy worldvolume dynamics of BPS domain walls in four-dimensional N=1 SYM theory. We demonstrate how to obtain the same theory in a brane configuration of type IIB string theory that contains threebranes and fivebranes. A combination of string and field theory techniques allows us to re-formulate some of the well-known properties of N=1 SYM domain walls in a geometric language and to postulate a Seiberg-like duality …
We study low-energy aspects of 3d $\mathcal{N}=2$ $Spin(N)$ gauge theories with matters in vector and (conjugate) spinor representations. Extending the construction of the 4d $\mathcal{N}=1$ $Spin(N)$ Seiberg duality, we find 3d magnetic dual descriptions with tree-level superpotentials slightly different from the 4d ones. We test various consistency checks including RG flows to known 3d dualities and supersymmetry enhancement deformation which leads to a 3d $\mathcal{N}=4$ duality between $SU(2)$ with three hypermultiplets and $U(1)$ with four hypermultiplets.
We consider a non-supersymmetric USp Yang-Mills Chern-Simons gauge theory coupled to fundamental flavours. The theory is realised in type IIB string theory via an embedding in a Hanany-Witten brane configuration which includes an orientifold and anti-branes. We argue that the theory admits a magnetic Seiberg dual. Using the magnetic dual we identify dynamics in field theory and brane physics that correspond to various phases, obtaining a better understanding of 3d bosonization and dynamical breaking of flavour symmetry in USp QCD$_3$ theory. In field theory both phases correspond to magnetic '…
We obtain a Seiberg-Witten map for the gauge sector of multiple D$p$-branes in a large R-R $(p-1)$-form field background up to the first-order in the inverse R-R field background. By applying the Seiberg-Witten map and then electromagnetic duality on the non-commutative D3-brane theory in the large R-R 2-form background, we find the expected commutative diagram of the Seiberg-Witten map and electromagnetic duality. Extending the U(1) gauge group to the U($N$) gauge group, we obtain a commutative description of the D-branes in the large R-R field background. This construction is different from …
Duality in supersymmetric SU(N) gauge theory with a symmetric tensor is studied using the technique of deconfining and Seiberg's duality. By construction the gauge group of the dual theory necessarily becomes a product group. In order to check the duality, several nontrivial consistency conditions are examined. In particular we find that by deforming along a flat direction, the duality flows to the Seiberg's duality of SO(N) gauge theory.
We study supersymmetric Wilson loops in the ${\cal N} = 6$ supersymmetric $U(N_1)_k\times U(N_2)_{-k}$ Chern-Simons-matter (CSM) theory, the ABJ theory, at finite $N_1$, $N_2$ and $k$. This generalizes our previous study on the ABJ partition function. First computing the Wilson loops in the $U(N_1) \times U(N_2)$ lens space matrix model exactly, we perform an analytic continuation, $N_2$ to $-N_2$, to obtain the Wilson loops in the ABJ theory that is given in terms of a formal series and only valid in perturbation theory. Via a Sommerfeld-Watson type transform, we provide a nonperturbative com…
