Defects and Quantum Seiberg-Witten Geometry
We study the Nekrasov partition function of the five dimensional gauge theory with maximal supersymmetry on in the presence of codimension two defects. The codimension two defects can be described either as monodromy defects, or by coupling to a certain class of three dimensional quiver gauge theories on . We explain how these computations are connected with both classical and quantum integrable systems. We check, as an expansion in the instanton number, that the aforementioned partition functions are eigenfunctions of an elliptic integrable many-body system, which quantizes the Seiberg-Witten geometry of the five-dimensional gauge theory.
Supersymmetric gauge theories provide a rich source of inspiration for various branches of mathematics. From a practical viewpoint, they can also provide a powerful set of techniques to solve challenging mathematical problems using physics. The interplay between supersymmetric gauge theories and mathematics is enhanced by introducing defects that preserve some amount of supersymmetry.
In this work, we study 5d supersymmetric gauge theories with codimension two and codimension four defects and how they are connected to the quantization of the integrable system associated to its Seiberg-Witten geometry Seiberg:1994rs (); Seiberg:1994aj (). We focus on a class of codimension two defects preserving supersymmetry in three dimensions. They can be described either as Gukov-Witten monodromy defects Gukov:2006jk (); Gukov:2008sn () or by coupling to a class of 3d supersymmetric gauge theories. We will concentrate on the surface defect obtained by introducing the most generic monodromy for the gauge field, or alternatively by coupling to the 3d theory Gaiotto:2008ak (). The codimension four defects are described by supersymmetric Wilson loops.
As a preliminary step towards understanding and computing with surface defects, we will first find a reformulation of the twisted chiral ring of a canonical deformation of the theory on , building on the work of Gaiotto:2013bwa (). We will show that the twisted chiral ring relations are equivalent to the spectral curve of an associated classical -body integrable system, known as the complex trigonometric Ruijsenaars-Schneider (RS) system. In addition, this provides a reformulation of the equivariant quantum K-theory of the cotangent bundle to a complete flag variety, via the Nekrasov-Shatashvili correspondence Nekrasov:2009ui (); Nekrasov:2009uh (). We will also explore the connection with quantum K-theory for more general linear quiver gauge theories.
This classical integrable system can be quantized by turning on an equivariant parameter for rotations in , otherwise known as the three-dimensional Omega background . As it is technically simpler, we will first consider the squashed partition function of theory using results from supersymmetric localization Kapustin:2009kz (); Hama:2011ea (). The partition functions on can then be obtained by factorization of the partition function Pasquetti:2011fj (); Beem:2012mb (). We show that these supersymmetric partition functions are eigenfunctions of the quantized trigonometric Ruijsenaars-Schneider system with the Planck constant proportional to . The corresponding eigenvalues are given by supersymmetric Wilson loops for background flavor symmetries. The twisted chiral ring relations, or equivalently the spectral curve of the classical integrable system, are reproduced in the semi-classical limit .
In coupling the three-dimensional theory theory as a surface defect in 5d gauge theory, the twisted chiral ring relations are deformed by an additional complex parameter , which is related to the 5d holomorphic gauge coupling. According to Gaiotto:2013sma () it is expected that this deformation provides a presentation of the Seiberg-Witten curve of the 5d theory on . The Seiberg-Witten curve of the 5d supersymmetric gauge theory is known to correspond to the spectral curve of the -body elliptic Ruijsenaars-Schneider system Nekrasov:1996cz (). This is indeed a deformation of the trigonometric RS system by an additional complex parameter .
In order to test this relationship, we will compute the 5d Nekrasov partition function of supersymmetric gauge theory on in the presence of surface defects wrapping one of the two-planes . This computation is performed by treating the surface defect as a monodromy defect and applying the orbifolding procedure introduced in Alday:2010vg (); Kanno:2011fw (). In order to check that the Gukov-Witten monodromy defect is reproducing the same surface defect as coupling to theory, we check that this computation reproduces the partition function of theory in the limit where the coupling to the 5d degrees of freedom is turned off. In particular, we note that the Gukov-Witten monodromy parameters are identified with the Fayet-Iliopoulos parameters of the 3d gauge theory supported on the defect.
After performing this preliminary check, we study the full Nekrasov partition function on as an expansion in the parameter . In the the Nekrasov-Shatashvili limit , we will show that the expectation value of the most generic surface defect is formally an eigenfunction of the elliptic RS system. Furthermore, we find that the corresponding eigenvalues are given by the expectation values of supersymmetric Wilson loops wrapping in the 5d gauge theory. This computation provides a quantization of the Seiberg-Witten geometry.
We will also study another type of codimension two defect by coupling directly to 3d hypermultiplets Lamy-Poirier:2014sea (); GK2014 (). We focus on the simplest example where two free hypermultiplets of flavor symmetry are coupled to the bulk gauge field of the 5d gauge theory. We will show that the partition function of this coupled system solves an eigenfunction equation of the so-called two-body dual elliptic Ruijsenaars-Schneider system. Indeed, the S-transformation of the 3d theory in Witten:2003ya () relates this partition function to the partition function of the gauge theory with a monodromy defect.
The paper is organized as follows. In Section 2 we will relate the twisted chiral ring of to the spectral curve of the classical trigonometric Ruijsenaars-Schneider system and discuss connections to equivariant quantum K-theory. In Section 3 we show that partition functions on squashed and are eigenfunctions of the corresponding quantized integrable system. Then in Section 4 we explain how to compute the expectation values of surface defects in gauge theory on and demonstrate that they are eigenfunctions of the elliptic RS system. Later in Section 5 we present a cursory discussion of various limits and degenerations of the results presented in this paper. Finally, in section Section 6 we summarize the connections of this work to integrable systems and discuss areas for further research.
2 Twisted Chiral Rings
In this section, we will study 3d linear quiver gauge theories on , deformed by hypermultiplet masses, FI parameters, and with the canonical mass deformation. It was observed by Nekrasov and Shatashvili Nekrasov:2009ui (); Nekrasov:2009uh () that the equations determining the supersymmetric massive vacua on , or the twisted chiral ring relations, can be identified with Bethe ansatz equations for a quantum XXZ integrable spin chain.
We will focus for the most part on the triangular quiver gauge theory: . We will reformulate the statement of its twisted chiral ring in terms of the spectral curve of a classical -body integrable system known as the complexified trigonometric Ruijsenaars-Schneider system. Alternatively, it can be viewed as a lagrangian correspondence that diagonalizes this classical integrable system. This reformulation will be important when we come to couple this theory as a codimension two defect in five-dimensions.
We will explain how the corresponding statements for more general linear quivers can be obtained by a combination of Higgsing and mirror symmetry, and demonstrate this in a simple example. We will also briefly discuss connections to results in the mathematical literature on the equivariant quantum K-theory of the cotangent bundles to partial flag manifolds.
2.1 The Nekrasov-Shatashvili Correspondence
Three-dimensional theories with supersymmetry have R-symmetry and flavor symmetries acting on the fields parametrizing the Higgs and Coulomb branches respectively. For the purpose of this paper, it is important to turn on a canonical deformation preserving only supersymmetry. The corresponding is the diagonal combination of Cartan generators of . The anti-diagonal combination becomes an additional flavor symmetry with real mass parameter . Furthermore, we turn on real deformation parameters by coupling to vectormultiplets for and giving a vacuum expectation value to the real scalar. In a UV description, these deformation parameters enter as real hypermultiplet masses denoted typically by and FI parameters denoted by . We refer to this setup as supersymmetry. We refer the reader to Gaiotto:2013bwa () for a more complete description of this setup.
Here, we focus on theories with a UV description as a linear quiver with unitary gauge groups. Our notation is summarized in figure 1. It is convenient to introduce a symmetry acting trivially such that the Higgs branch symmetry is given by . The corresponding mass parameters are denoted by . Similarly, we introduce an additional topological so that the Coulomb branch symmetry manifest in the UV description is with corresponding parameters . The physical FI parameter at the -th gauge node is . This symmetry can be enhanced by monopole operators up to a maximum of in the IR.
We will focus on he twisted chiral ring of the effective 2d theory obtained by compactifying on a circle of radius . In this case, the real deformation parameters are complexified by background Wilson lines wrapping the circle and behave as twisted masses in the language of supersymmetry.
For generic deformation parameters there is a discrete set of massive supersymmetric vacua on each with an associated effective twisted superpotential . This is a holomorphic function that is independent of superpotential and gauge couplings. In the UV theory one can integrate out three-dimensional chiral multiplets to find an effective twisted superpotential for the dynamical vectormultiplets. The supersymmetric vacua are then solutions to
which can be identified with the twisted chiral ring relations of the effective two-dimensional theory.
The effective twisted superpotential of the generic linear quiver shown in figure 1 is given by
The first term includes contributions from the FI parameters at each node together with phase 111Sign conventions are slightly altered compared to Gaiotto:2013bwa ().. The second term includes the 1-loop contributions from the KK tower of chiral multiplets. The basic building block of the 1-loop contributions is the contribution from a three-dimensional chiral multiplet of mass , which is a solution of the differential equation . We refer the reader to Gaiotto:2013bwa () for an explicit expression. The final term is included to ensure that mirror symmetry acts straightforwardly in the presence of the spurious symmetries.
As the imaginary components of the twisted mass parameters are periodic, it is convenient to introduce exponentiated parameters
With this notation, the equations for the supersymmetric vacua are
for all . It was observed by Nekrasov and Shatashvili Nekrasov:2009ui (); Nekrasov:2009uh () that these equations can be identified with the Bethe equations of a quantum integrable XXZ spin chain. A complete dictionary for linear quivers can be found in Gaiotto:2013bwa ().
In order to write down the twisted chiral ring it is necessary to introduce a generating function for the gauge invariant combinations of the ’s. For this purpose, we introduce an auxiliary parameter and monic polynomials
The equations for supersymmetric vacua (4) can be expressed in terms of these polynomials as
where the polynomials are understood to be evaluated at for . We defined to be the 1-loop contribution to the scaling dimensions of monopole operators charged under -th gauge group. The superscripts on the polynomials are shorthand for multiplicative shifts of the arguments by , for example , . The twisted chiral ring relations for gauge invariant combinations of ’s are given by expanding equations (6) in .
In what follows, it will be useful to introduce another slightly less familiar reformulation of the twisted chiral ring 222We thank Davide Gaiotto for exhibiting us this calculation in a sample example.. We first rescale the FI parameters by to absorb the dependence on . Then we introduce the polynomial equations
where are auxiliary polynomials of rank . To recover equations (6) for the supersymmetric vacua we shift the argument this polynomial equation by and evaluate both at roots of . Dividing one equation by the other, the combination and the auxiliary polynomials cancel out and we reproduce (4). The twisted chiral ring relations are given by expanding equations (7) in .
Finally, 3d theories have a remarkable duality known as mirror symmetry Intriligator:1996ex (); deBoer1997148 (). This can be understood from brane constructions in Type IIB String theory Hanany:1996ie (), where it is realized as the S-duality. Mirror symmetry acts in quite a non-trivial manner on the data of the linear quiver, which is spelled out in reference Gaiotto:2013bwa (). Mirror symmetry interchanges mass parameters and FI parameters of the quivers and also acts non-trivially on the mass deformation. Schematically, we have 333Here the mirror map for has a different sign compared to Gaiotto:2013bwa (), where was mapped onto ..
where the check symbol designates parameters of the dual theory.
2.2 Quantum/Classical Duality
As mentioned above, the equations for supersymmetric vacua can be identified with the Bethe equations for a quantum integrable spin chain. Remarkably, the same system of equations are related to a second, classical integrable system of interacting relativistic particles in one dimension – the complexified trigonometric Ruijsenaars-Schneider system Gaiotto:2013bwa ().
This correspondence is most straightforward to understand in the case of the triangular quiver (see Fig. 2) – this will be our main example throughout this paper.
The effective twisted superpotential (2) of this theory is given by
where 444In the notations of (4) , so the this definition of is consistent with the above conventions modulo two. and we define and to simplify notation. For this theory, the Higgs and Coulomb branch symmetries are both and we have corresponding exponentiated mass and FI parameters with . This theory is invariant under mirror symmetry with the transformation .
By introducing the conjugate momenta to and
we provide canonical coordinates on two copies and of the cotangent bundle to with the following holomorphic symplectic forms
This is the phase space of our complex classical integrable system. The defining equations for the conjugate momenta (10) sweep out a complex Lagrangian in the product with holomorphic symplectic form and generating function given by the on-shell twisted effective superpotential .
It is straightforward to find an explicit description of the Lagrangian for theory. The supersymmetric vacua equations read as follows
The conjugate momenta for the FI terms are as follows
and for the masses
Given the above definitions of conjugate momenta vacua equation (12) can be presented in two equivalent ways. First, as
and, second, as
In (15) the combinations appearing on the left are the Hamiltonians of the complex trigonometric RS system for two particles with positions and momenta . The right hand sides are independent of the momenta so the Lagrangian correspondence diagonalizes the system. From this perspective, evaluated on supersymmetric vacua are solutions of the relativistic Hamilton-Jacobi equation. The evident symmetry under and means this Lagrangian also diagonalizes the same system with the coupling inverted .
To eliminate dynamical vectormultiplet scalars from the supersymmetric vacuum equations in favor of the conjugate momenta or in the case is rather non-trivial task. Below we demonstrated that one can do this in two equivalent ways. First as
and, second, as
In both relations above
is the Lax matrix for the -body complex trigonometric RS system. One can clearly see that (17) and (18) are related to each other by mirror symmetry map (8). Therefore, somewhat artificially, we can refer to the former relation as written in the ‘electric’ frame, where eigenvalues of are related to masses , and to the latter relation as presented in the ‘magnetic’ frame, in which, using mirror frame variables, the eigenvalues of are identified with FI parameters .
As we mentioned above, in order to understand why (17) and (18) are true it is convenient to re-formulate the supersymmetric vacuum equations arising from this twisted superpotential. We introduce monic degree- polynomials , for each node of the quiver. Note that we treat the matter polynomial in a uniform manner, that is we define and hence . With this definition, we have equations (7)
Note that in this case and there is no need to redefine . The original supersymmetric vacuum equations are obtained by shifting arguments in the above by , evaluating at the roots of and eliminating the auxiliary polynomials . They can be expressed uniformly as
evaluated on the roots .
2.2.1 Electric Frame
Firstly, we explain how to eliminate the in favor of the momenta conjugate to the FI parameters (17). For this purpose, we will set up an inductive procedure to solve the equations recursively node by node. We first note that by the definition (10)
and hence by evaluating the equation at we can immediately solve for the constant terms in the polynomials and as follows
Now, given the polynomials and for we can determine from the equation (20). Then, by shifting in the same equation and evaluating it on the roots of we have just enough data to determine the unknown coefficients in .
To illustrate this process, let us perform the first interation explicitly. It is convenient to introduce the monic degree one polynomials . Then from equation (23) we have and . Then, from equation from (20) with , we find
Now, evaluating equation (20) with on the root of the polynomial it is straightfoward to compute the coefficient of the linear term in and hence find
We can now immediately compute the polynomial using equation (20) with
We have implemented this procedure to many orders in and found experimentally that the solution can be expressed as follows. We introduce the following matrices
where we define and we remind the reader that superscripts are not exponentials but shifts of the argument by . Then the solution is given by a ratio of Vandermonde-like determinants
Solutions of this form for similar functional equations have appeared in the integrability literature. We expect that these techniques could be used to prove the solution we have found (see e.g. Kazakov:2007na ()).
Since all polynomials are monic of degree one the above ratios can be simplified and by inverting the matrix , the ratio of determinants can be reexpressed as a single spectral determinant
is the Lax matrix of the -body complex trigonometric RS system. At the final stage of the recursion, the polynomial becomes the matter polynomial , providing us with the required relation (17). By expanding both sides of (17) in we find explicitly the Hamiltonians
and their eigenvalues
Thus we can explicitly write the full set of conserved charges for trigonometric RS system , or, more explicitly as
2.2.2 Magnetic Frame
Let us now look at the other presentation of twisted chiral ring (18). Now we want to eliminate in favor of the momentum conjugate to the masses, . In this case, it will not be possible to provide an argument that lands directly on the Lax matrix formulation of the complex trigonometric RS model. Instead, we attempt to verify the mirror equations
related to those above by and .
Let us first consider the first independent Hamiltonian with . The momentum conjugate to the masses (10) can be expressed in terms of the polynomials as follows
where we remind the reader that . It is now straightforward to see that the first Hamiltonian can be expressed as a contour integral
where the contour surrounds the roots of i.e. the masses . Our proposition is that this contour integral evaluates to .
We prove this proposition by induction. To perform the inductive step, we contract the contour such that it surrounds the roots of , and , then eliminate the dependence on using the supersymmetric vacuum equations (21), and then express the result once again as contour integral. Performing these steps, we find
as required. In the second line above contour surrounds only roots of .
The argument for the Hamiltonian appearing at order proceeds in a similar manner. We first express the Hamiltonian as a contour integral
where the contour surrounds the poles arising from the denominators . Note the critical role of the numerator factor in ensuring that the non-zero residues are labelled by sets . This contour integral is the path integral of a supersymmetric gauged quantum mechanics on the circle, an observation we explain further below. Our claim is that this contour integral evaluates to
To prove this statement, we again proceed by induction. The inductive step depends on the following contour integral identity
which we have checked in numerous examples.
We expect that this expression as well as (35) can be interpreted as partition functions of quantum mechanics on the 1d supersymmetric defect on coupled to the 3d gauge theory on . The integral relation (39) can be interpreted as an identity between the partition functions of 1d defects coupled to neighboring nodes of the quiver. This observation requires further study.
2.3 Line Operators and Interfaces in Sym
Many of the computations presented in the preceding section, in particular the connections to classical integrable systems, have a useful interpretation in terms of interfaces between copies of four-dimensional theory.
The starting point for this construction is the moduli space of vacua of theory on . There is always a region at infinity in the moduli space where the gauge group is broken to the maximal abelian subgroup and complex coordinates valued in corresponding to complexified electric and magnetic Wilson lines in each abelian factor, complexified by vectormultiplet scalars. Classically, we would have , which is the phase space of the complex trigonometric RS system Gaiotto:2013bwa ().
The quantum corrected moduli space in the appropriate complex structure is given by the space of flat connections on a torus with puncture. This can be described by the holonomies and around the two cycles of the torus, which must obey where has eigenvalues . Remarkably, these equations can be solved in terms of coordinates such that
In particular, there is a choice of gauge where becomes the Lax matrix of the complexified trigonometric RS system. It can be confirmed by localization computations that and correspond respectively to BPS Wilson and ’t Hooft loops in the anti-fundamental representations of wrapping the .
Interfaces between two theories with moduli spaces and correspond to Lagrangian submanifolds . Let us recall that the three-dimensional theory can be identified with the S-duality interface for the theory. In this context the complex parameters of the three-dimensional theory and are identified with the Darboux coordinates for the moduli space on either side of the interface. The corresponding Lagrangian submanifold is then described precisely by the equations (17) and (18). These relations are interpreted as Ward identities for line operators at the interface: a ’t Hooft loop approaching from one side is equivalent to a Wilson loop approaching from the other. This is expected since Wilson loops and ’t Hooft loops are interchanged under S-duality.