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Ring relations and mirror map from branes
Received: February
Ring relations and mirror map from branes
Benjamin Assel 0 1
Open Access 0 1
c The Authors. 0 1
0 CH-1211 Geneva 23 , Switzerland
1 Theoretical Physics Department , CERN
We study the space of vacua of three-dimensional N = 4 theories from a novel approach building on the type IIB brane realization of the theory and in which the insertion of local chiral operators in the path integral is obtained from integrating out light modes in appropriate brane setups. Most of our analysis focuses on abelian quiver theories which can be realized as the low-energy theory of D3-D5-NS5 brane arrays. Their space of vacua contains a Higgs branch, parametrized by the vevs of half-BPS meson operators, and a Coulomb branch, parametrized by the vevs of half-BPS monopole operators. operators are inserted by adding D1 strings and D3 branes, with speci c orientations, to the initial brane setup of the theory. This approach has two main advantages. First the ring relations describing the Higgs and Coulomb branches can be derived by looking at speci c brane setups with multiple interpretations in terms of operator insertions. This provides a new derivation of the Coulomb branch quantum relations. Secondly the map between the Higgs and Coulomb operators of mirror dual theories can be derived in a trivial way from IIB S-duality.
mirror; branes; that the Higgs operators are inserted by adding F1 strings and D3 branes; while Coulomb
1 Introduction and discussion
2 Light review on N = 4 theories and their moduli space of vacua
3 The brane realization of half-BPS local operator insertions
3.1 Branes and strings inserting half-BPS local operators
4 Analysis in the T [SU(2)] theory 4.1 4.2 4.3
Higgs branch operators
Coulomb branch operators
Mirror symmetry
4.5 Deformations
5 Abelian generalizations 5.1 SQED
5.2 Abelian Quiver 5.1.1 5.1.2 5.2.1
Mirror symmetry
5.4 Another example
6 Other chiral operators
7 Non-abelian theories
A SQED Higgs branch relations
B Brane setup | operator insertion dictionary
B.1 Higgs branch operators
B.2 Coulomb branch operators
B.3 Mirror map
Introduction and discussion
are fully characterized by a choice of gauge group G, associated to a vector multiplet, and
a pseudo-real representation R in which the hyper-multiplet matter elds transform. This
xes uniquely the Lagrangian of the theory. Their space of vacua splits into
several subspaces or \branches", each of which is a product of hyper-Kahler manifolds, with
two branches playing a special role. The Higgs branch, which is free of quantum
corrections [1], is parametrized by the vacuum expectation values (vevs) of the scalars in the
hyper-multiplet, subject to a triplet of D-term constraints, and modulo gauge
transformations. In a chosen complex structure the Higgs branch can be described as a complex
algebraic variety, with singularities, parametrized by the vevs of gauge invariant operators,
(inherited from the chiral ring relations). The Coulomb branch is parametrized by the vevs
and which form a ring with relations arising from non-trivial quantum e ects [2, 3]. The
monopole operators are de ned in the quantum theory by imposing in the path integral
formulation a Dirac monopole singularity for the gauge eld at a point in Euclidean space
and \dressing" it with a polynomial in the vector multiplet complex scalars. A special case
are monopoles with zero magnetic charges which are simply gauge invariant combinations
of the vector multiplet complex scalars.1
While there is a relatively clear path to study the Higgs branch from the classical
Lagrangian of the theory, the Coulomb branch is more di cult to access, since the ring
relations between monopole operators do not follow from a superpotential, but from the
quantum dynamics of the theory. In abelian theories the Coulomb branch metric receives
corrections only at one-loop and can be computed directly [4, 5]. It is also possible to
rely on mirror symmetry [1], which exchanges the Higgs and Coulomb branches of mirror
dual theories. Much progress has been made recently on deriving the ring relations of the
Coulomb branch of quiver gauge theories from di erent approaches. One approach uses the
Coulomb branch Hilbert series [6{10],2 which is a protected index counting chiral monopole
operators re ned with fugacities keeping track of their charges under Cartan R-symmetry
and global topological symmetries. From the resumed series one is able to extract a set
of generators and to read the Coulomb branch relations, up to coe cients which must be
determined by other methods. Non-abelian quiver theories of ADE types with unitary or
ortho-symplectic gauge nodes have been studied using this method. A di erent construction
was proposed by Bullimore, Dimofte and Gaiotto in [12] to derive the Coulomb branch
relations of non-abelian quiver theories. The construction is based on the embedding on
the non-abelian CB (Coulomb branch) chiral ring into the CB chiral ring of the low-energy
abe (...truncated)