Hyperkahler Sigma Model and Field Theory on Gibbons-Hawking Spaces

We describe a novel deformation of the 3-dimensional sigma model with hyperk\"ahler target, which arises naturally from the compactification of a 4-dimensional $\mathcal{N}=2$ theory on a hyperk\"ahler circle bundle (Gibbons-Hawking space). We derive the condition for which the deformed sigma model preserves 4 out of the 8 supercharges. We also study the contribution from a NUT center to the sigma model path integral, and find that supersymmetry implies it is a holomorphic section of a certain holomorphic line bundle over the hyperk\"ahler target. We study explicitly the case where the original 4-dimensional theory is pure $U(1)$ super Yang-Mills, and show that the contribution from a NUT center in this case is simply the Jacobi theta function.Comment: 48 page

Good IR Duals of Bad Quiver Theories

The infrared dynamics of generic 3d N=4 bad theories (as per the good-bad-ugly classification of Gaiotto and Witten) are poorly understood. Examples of such theories with a single unitary gauge group and fundamental flavors have been studied recently, and the low energy effective theory around some special point in the Coulomb branch was shown to have a description in terms of a good theory and a certain number of free hypermultiplets. A classification of possible infrared fixed points for bad theories by Bashkirov, based on unitarity constraints and superconformal symmetry, suggest a much richer set of possibilities for the IR behavior, although explicit examples were not known. In this note, we present a specific example of a bad quiver gauge theory which admits a good IR description on a sublocus of its Coulomb branch. The good description, in question, consists of two decoupled quiver gauge theories with no free hypermultiplets.Comment: 20 pages, 3 figures. Discussion of the Coulomb branch singularities is further extende

Three Dimensional Mirror Symmetry and Partition Function on S3S^3

We provide non-trivial checks of $\mathcal{N}=4, D=3$ mirror symmetry in a large class of quiver gauge theories whose Type IIB (Hanany-Witten) descriptions involve D3 branes ending on orbifold/orientifold 5-planes at the boundary. From the M-theory perspective, such theories can be understood in terms of coincident M2 branes sitting at the origin of a product of an A-type and a D-type ALE (Asymtotically Locally Euclidean) space with G-fluxes. Families of mirror dual pairs, which arise in this fashion, can be labeled as $(A_{m-1},D_n)$, where $m$ and $n$ are integers. For a large subset of such infinite families of dual theories, corresponding to generic values of $n\geq 4$, arbitrary ranks of the gauge groups and varying $m$, we test the conjectured duality by proving the precise equality of the $S^3$ partition functions for dual gauge theories in the IR as functions of masses and FI parameters. The mirror map for a given pair of mirror dual theories can be read off at the end of this computation and we explicitly present these for the aforementioned examples. The computation uses non-trivial identities of hyperbolic functions including certain generalizations of Cauchy determinant identity and Schur's Pfaffian identity, which are discussed in the paper.Comment: 45 pages, 9 figure

Exploring Seiberg-like Dualities with Eight Supercharges

We propose a family of IR dualities for 3d $\mathcal{N}=4$ $U(N)$ SQCD with $N_f$ fundamental flavors and $P$ Abelian hypermultiplets i.e. $P$ hypermultiplets in the determinant representation of the gauge group. These theories are good in the Gaiotto-Witten sense if the number of fundamental flavors obeys the constraint $N_f \geq 2N-1$ with generic $P \geq 1$, and in contrast to the standard $U(N)$ SQCD, they do not admit an ugly regime. The IR dualities in question arise in the window $N_f=2N+1,2N,2N-1,$ with $P=1$ in the first case and generic $P \geq 1$ for the others. The dualities involving $N_f=2N \pm 1$ are characterized by an IR enhancement of the Coulomb branch global symmetry on one side of the duality, such that the rank of the emergent global symmetry group is greater than the rank of the UV global symmetry. The dual description makes the rank of this emergent global symmetry manifest in the UV. In addition, one can read off the emergent global symmetry itself from the dual quiver. We show that these dualities are related by certain field theory operations and assemble themselves into a duality web. Finally, we show that the $U(N)$ SQCDs with $N_f \geq 2N-1$ and $P$ Abelian hypers have Lagrangian 3d mirrors, and this allows one to explicitly write down the 3d mirror associated with a given IR dual pair. This paper is the first in a series of four papers on 3d $\mathcal{N}=4$ Seiberg-like dualities.Comment: v1:41 pages + references. The quiver diagrams in the paper are color-coded, v2: References updated, v3: Minor typos fixed, v4: journal versio

Argyres-Douglas Theories, IR N-ality and Complete Graphs

We show that for a large subclass of Argyres-Douglas-type theories, the Higgs branch admits multiple hyperkahler quotient realizations as Higgs branches of three dimensional $\mathcal{N}=4$ quiver gauge theories, which are related by a sequence of Seiberg-like IR dualities. We refer to this phenomenon as the Hyperkahler Quotient N-ality of the four dimensional Higgs branch. The associated set of 3d theories contains a special subset of maximal unitary quivers: quiver gauge theories for which the resolution/deformation parameters of the Higgs branch are manifest in the Lagrangian as Fayet-Iliopoulas parameters. Starting from the Type IIB description for a given SCFT, we present an explicit construction to determine the aforementioned set of 3d quivers, including the subset of maximal unitary quivers. As a byproduct, we find a simple method for constructing the three dimensional mirror associated with the SCFT. We demonstrate the construction for the $(A_k, A_k)$ theories of Cecotti, Neitzke and Vafa, focusing on the cases $k=3$ and $k=4$. The associated maximal unitary quiver is unique up to field redefinitions and turns out to be an Abelian quiver gauge theory. The three dimensional mirror obtained in this fashion reproduces the well-known complete graph. In the appendices to the main paper, we study the quotient N-ality in the closely related family of $D^b_p (SU(N))$ SCFTs, for which both the maximal unitary quiver as well as the 3d mirror turn out to be non-Abelian gauge theories genericallyComment: 10 pages + References + 5 pages. Quiver diagrams are color-code

Three dimensional Mirrors from IR N-ality

We discuss an explicit field theory construction of three dimensional mirrors for a large sub-class of quiver gauge theories involving unitary and special unitary gauge nodes with matter in fundamental and bifundamental representations. For this sub-class of theories, one can deploy a sequence of IR dualities and certain Abelian field theory operations to reduce the aforementioned problem to a problem of finding the 3d mirror of a simpler unitary quiver gauge theory. We illustrate the construction with a linear quiver consisting of unitary and special unitary gauge nodes.Comment: 6 pages, 6 figures, figures are color-code

Exploring Seiberg-like N-alities with Eight Supercharges

We show that a large subclass of 3d $\mathcal{N}=4$ quiver gauge theories consisting of unitary and special unitary gauge nodes with only fundamental/bifundamental matter have multiple Seiberg-like IR duals. A generic quiver $\mathcal{T}$ in this subclass has a non-zero number of balanced special unitary gauge nodes and it is a good theory in the Gaiotto-Witten sense. We refer to this phenomenon as "IR N-ality" and the set of mutually IR dual theories as the "N-al set" associated with the quiver $\mathcal{T}$. Starting from $\mathcal{T}$, we construct a sequence of dualities by step-wise implementing a set of quiver mutations which act locally on the gauge nodes. The associated N-al theories can then be read off from this duality sequence. The quiver $\mathcal{T}$ generically has an emergent Coulomb branch global symmetry in the IR, such that the rank of the IR symmetry is always greater than the rank visible in the UV. We show that there exists at least one theory in the N-al set for which the rank of the IR symmetry precisely matches the rank of the UV symmetry. In certain special cases that we discuss in this work, the correct emergent symmetry algebra itself may be read off from this preferred theory (or theories) in addition to the correct rank. Finally, we give a recipe for constructing the 3d mirror associated with a given N-al set and show how the emergent Coulomb branch symmetry of $\mathcal{T}$ is realized as a UV-manifest Higgs branch symmetry of the 3d mirror. This paper is the second in a series of four papers on 3d $\mathcal{N}=4$ Seiberg-like dualities, preceded by the work [arXiv:2210.04921].Comment: v1:48 pages + references. Quiver diagrams are color-coded, v2: References updated, v3: An error corrected in figure 2 and figure 10, section 2.3 on quiver mutations expanded for better presentation, minor typos fixed, v4: An important typo fixe

Readdressing the hierarchy problem in a Randall-Sundrum scenario with bulk Kalb-Ramond background

We re-examine the fine tuning problem of the Higgs mass, when an antisymmetric two form Kalb-Ramond (KR) field is present in the bulk of a Randall-Sundrum (RS) braneworld. Taking into account the back-reaction of the KR field, we obtain the exact correction to the RS metric. The modified metric also warps the Higgs mass from Planck scale (in higher dimension) to TeV scale (on the visible brane) for a range of values of $kr$ exceeding the original RS value (where $k=$ Planck mass and $r=$ size of extra dimension). However, it requires an extraordinary suppression of the KR field density, indicating the re-appearence of the fine tuning problem in a different guise. The new spacetime also generates a small negative cosmological constant on the visible brane. These results are particularly relevant for certain string based models, where the KR field is unavoidably present in the bulk. We further show that such a bulk antisymmetric KR field fails to stabilize the braneworld.Comment: 4 Pages, Revtex, 1 figure. Important changes and addition. Version to appear in Class. Quant. Grav. Letter