4d Theories, Fivebranes, and M-Theory

Abstract

In this section, we study line operators in the 4d $\mathcal {N}=2^*$ theories. A 4d $\mathcal {N}=2$ theory of class $\mathcal {S}$ arises from a compactification of 6d $\mathcal {N}=(2, 0)$ theory on a once-punctured torus C. The spectrum of line operators in the theory depends on additional discrete data, a maximal isotropic lattice $\textsf{L}\subset H^1(C,Z(G))$ where line operators must be invariant under the discrete group $\textsf{L}$. Therefore, we will show that a non-commutative algebra of line operators of a 4d $\mathcal {N}=2^*$ theory on $S^1\times \mathbb {R}\times _q \mathbb {R}^2$ with the $\Omega$-background is the $\textsf{L}$-invariant subalgebra of spherical DAHA.

In this section, we study line operators in the 4d $\mathcal {N}=2^*$ theories. A 4d $\mathcal {N}=2$ theory of class $\mathcal {S}$ arises from a compactification of 6d $\mathcal {N}=(2, 0)$ theory on a once-punctured torus C. The spectrum of line operators in the theory depends on additional discrete data, a maximal isotropic lattice $\textsf{L}\subset H^1(C,Z(G))$ where line operators must be invariant under the discrete group $\textsf{L}$. Therefore, we will show that a non-commutative algebra of line operators of a 4d $\mathcal {N}=2^*$ theory on $S^1\times \mathbb {R}\times _q \mathbb {R}^2$ with the $\Omega$-background is the $\textsf{L}$-invariant subalgebra of spherical DAHA. Also, we give an explicit geometric relation between Hitchin moduli spaces and an elliptic fibration of the Coulomb branch of a 4d $\mathcal {N}=2^*$ theory in the rank-one case. Besides, we include a surface operator of Gukov–Witten type in the story, and consider an algebra of line operators on a surface operator to realize the full DAHA instead of the spherical DAHA. An advantage of the fivebrane system of class $\mathcal {S}$ is that we can relate line operators of a 4d theory to boundary conditions of a 2d sigma-model by a compactification. Taking this advantage, we propose a canonical coisotropic brane $\widehat{\mathfrak {B}}_{cc}$ of higher rank which realizes the full DAHA as ${{\,\textrm{Hom}\,}}(\widehat{\mathfrak {B}}_{cc},\widehat{\mathfrak {B}}_{cc})$.

4.1 Coulomb Branches of 4d $\mathcal {N}=2^*$ Theories of Rank One

In this subsection, we study a stack of M5-branes on $C\times S^1 \times \mathbb {R}^3$. A 4d $\mathcal {N}=2$ theory of class $\mathcal {S}$ is constructed by a compactification of the 6d (2, 0) theory of type G (G is of Cartan type ADE) on a Riemann surface C [62, 70] (generally with punctures) with additional discrete data $\textsf{L}$ [141] (see also [64] where such choice is referred to as a “polarization on C”), denoted by $\mathcal {T}[C,G,\textsf{L}]$. The basic information of a theory of class $\mathcal {S}$ is encoded in a Hitchin system

(4.1)

where $\Sigma$ is a Seiberg–Witten curve. The Coulomb branch of the theory on $\mathbb {R}^4$, called the u-plane, is an affine space $\mathcal {B}_u=\bigoplus _{k=1}^r H^0(C,K_C^{\otimes d_k})$ where the exponents $d_k$ depend on G. Given a point $u\in \mathcal {B}_u$, the Seiberg–Witten curve $\Sigma$ is expressed as the characteristic polynomial $\det (xdz-\varphi )=f(x,u(z))$ where x, z are local coordinates of the fiber and base of $T^*C$. To introduce the additional data, we pick a symplectic basis of $H_1(C)$ of C of genus g in terms of intersection numbers

$$(\alpha _1,\ldots ,\alpha _g,\beta _1,\ldots ,\beta _g)\in H_1(C), \qquad \alpha _i\cdot \alpha _j=0=\beta _i\cdot \beta _j, \ \alpha _i\cdot \beta _j=\delta _{ij}=-\beta _j\cdot \alpha _i,$$

which yields a symplectic lattice $(H^{1}(C,Z(G)),\omega )$ where Z(G) is the center of G. In fact, the additional data are given by a maximal isotropic sublattice $\textsf{L}\subset (H^{1}(C, Z(G)),\omega )$, and they specify an allowed set of charges of line operators that are compatible with the Dirac quantization conditions [6]. Given a maximal isotropic sublattice $\textsf{L} \subset (H^{1}(C,Z(G)),\omega )$, the Coulomb branch $\mathcal {M}_C(C,G,\textsf{L})$ of the $\mathcal {T}[C,G,\textsf{L}]$ theory on $S^1\times \mathbb {R}^3$ admits an elliptic fibration over the u-plane $\mathcal {B}_u$ [69, 141]

$$ \pi : \mathcal {M}_C(C,G,\textsf{L}) \rightarrow \mathcal {B}_u. $$

This is sometimes called the Donagi-Witten integrable system of class $\mathcal {S}$. In fact, $H^{1}(C, Z(G))$ freely acts on the Hitchin fibration $\pi :{\mathcal {M}}_H(C,G)\rightarrow \mathcal {B}_H$ fiberwise, and the Coulomb branch can be obtained by the quotient of $\pi :{\mathcal {M}}_H(C,G)\rightarrow \mathcal {B}_H$ by a fiberwise action of $\textsf{L}$ so that

$$\mathcal {M}_C(C,G,\textsf{L})={\mathcal {M}}_H(C,G)/\textsf{L}.$$

Note that this action can be obtained by twists of a Higgs bundle by a flat Z(G)-bundle over C associated to $\textsf{L}$, and it acts freely on a generic fiber of the Hitchin fibration. Therefore, the Coulomb branch $\mathcal {M}_C(C,G,\textsf{L})$ inherits a hyper-Kähler structure from the Hitchin moduli space ${\mathcal {M}}_H(C,G)$.

Of our interest are certainly the class $\mathcal {S}$ theories of type $A_1$ associated to the once-punctured torus $C_p$, namely 4d $\mathcal {N}=2^*$ theories of rank one [47, 71]. The lattice $H^{1}(C_p,\mathbb {Z}_2)=\mathbb {Z}_2\oplus \mathbb {Z}_2$ with the natural symplectic form can be identified with the electric and magnetic charges of line operators of the $\mathcal {N}=2^*$ theory wrapping $S^1$. Line operators with charges $\lambda =(\lambda _{e}, \lambda _{m})$ and $\nu =(\nu _{e}, \nu _{m})$ must be subject to the Dirac quantization condition

$$ \omega (\lambda ,\nu )=\lambda _e\nu _m-\lambda _m\nu _e \in 2 \mathbb {Z} . $$

There are three ways to pick a maximal isotropic lattice, corresponding to (0, 1) (1, 0) and (1, 1) $\in H^{1}(C_p,\mathbb {Z}_2)=\mathbb {Z}_2\oplus \mathbb {Z}_2$. They are known as ${{\,\textrm{SU}\,}}(2)$, ${{\,\textrm{SO}\,}}(3)_{+}$ and ${{\,\textrm{SO}\,}}(3)_{-}$ gauge theories, respectively, where the theta angles of ${{\,\textrm{SO}\,}}(3)_\pm$ differ by $2\pi$. Under the ${{\,\textrm{SL}\,}}(2,\mathbb {Z})$ transformation on the complexified gauge coupling (electromagnetic duality), these theories are related to each other as follows:

(4.2)

(4.3)

Next, we study the geometry of the Coulomb branches of the 4d $\mathcal {N}=2^*$ theories of rank one on $S^1\times \mathbb {R}^3$. The Coulomb branches can be obtained by $\mathbb {Z}_2$ quotients of the Hitchin moduli space $\mathcal {M}_{H}(C_p,{{\,\textrm{SU}\,}}(2))$ by $\xi _{i}\in \Xi =H^1(C_p,\mathbb {Z}_2)$ ($i=1,2,3$) [69, Sect. 8.4] as in (4.3). The ramification parameters ${\frac{1}{2}}(\beta _p+i\gamma _p)$ at the Higgs field $\varphi$ is indeed equivalent to the complex mass of the adjoint hypermultiplet in the 4d $\mathcal {N}=2^*$ theory. The ramification parameter $\alpha _p$ is the holonomy along $S^1$ for the $\text {U}(1)$ flavor symmetry. Let us investigate the action of $\Xi$ on the Hitchin moduli space $\mathcal {M}_{H}(C_p,{{\,\textrm{SU}\,}}(2))$ at a generic ramification more in detail. As in Fig. 2.2, $\mathcal {M}_{H}(C_p,{{\,\textrm{SU}\,}}(2))$ with a generic ramification has three singular fibers of Kodaira type $I_2$. As described in Sect. 2.1, the action of $\Xi$ on each fiber in the Hitchin fibration ${\mathcal {M}}_H(C_p,{{\,\textrm{SU}\,}}(2))\rightarrow \mathcal {B}_H$ is of order two, and the action is moreover free on a generic fiber. Hence, an interesting part is the action on the singular fibers. Two irreducible components, $\textbf{U}_{2i-1}$ and $\textbf{U}_{2i}$, in the singular fiber $\pi ^{-1}(b_i)$ can be understood as two $\mathbb {C}\textbf{P}^1$’s meeting at the north and south pole as double points. As illustrated in Fig. 2.6, the element $\xi _{1}$ in (2.29) acts on each irreducible component of the singular fiber $\pi ^{-1}(b_1)$ as the $180^{\circ }$ rotation around the polar axis of $\mathbb {C}\textbf{P}^1$. Likewise, $\xi _{1}$ acts on a generic fiber $\textbf{F}$ nearby as the $180^{\circ }$ rotation along the (1, 0)-cycle (meridian) of $\textbf{F}$. As we have seen in Sect. 2.6.2, the singular fiber $\pi ^{-1}(b_1)$ is mapped to $\pi ^{-1}(b_2)$ by the modular S-transformation $\sigma$. Therefore, in the neighborhood of the singular fiber $\pi ^{-1}(b_2)$, $\xi _{1}$ acts on a generic fiber $\textbf{F}$ as the $180^{\circ }$ rotation along the (0, 1)-cycle (longitude) of $\textbf{F}$. Consequently, $\xi _{1}$ exchanges the two irreducible components $\textbf{U}_3$ and $\textbf{U}_4$ by the corresponding rotation on the singular fiber $\pi ^{-1}(b_2)$. In a similar fashion, $\tau _+\in {{\,\textrm{PSL}\,}}(2,\mathbb {Z})$ maps the singular fiber $\pi ^{-1}(b_1)$ to the other fiber $\pi ^{-1}(b_3)$. Therefore, $\xi _{1}$ acts on a generic fiber $\textbf{F}$ as the $180^{\circ }$ rotation along the (1, 1)-cycle of $\textbf{F}$ around the singular fiber $\pi ^{-1}(b_3)$. Moreover, it exchanges the two irreducible components $\textbf{U}_5$ and $\textbf{U}_6$ with additional rotation around the polar axis of $\mathbb {C}\textbf{P}^1$. The actions of $\xi _{2}$ and $\xi _{3}$ are obtained by the cyclic permutations of $b_{i}$ ($i=1,2,3$).

Since $\xi _i$ acts freely on a generic Hitchin fiber with order two, the quotient of the Hitchin fibration ${\mathcal {M}}_H(C_p,{{\,\textrm{SU}\,}}(2))\rightarrow \mathcal {B}_H$ by $\xi _i$ provides the structure of an elliptic fibration of the Coulomb branch. Namely, this double cover is obtained by an isogeny of each elliptic fiber of degree two [1, 2]. As illustrated in Fig. 2.6, $\xi _1$ acts on each irreducible component of the singular fiber $\pi ^{-1}(b_1)$ by the $180^{\circ }$ rotation so that the quotient by its action turns the double points $p_{1,2}$ into the $A_1$ orbifold points. In fact, the quotient can be understood as a particular limit of the fiber of Kodaira type $I_4$. Generically, the fiber of type $I_4$ consists of four $\mathbb {C}\textbf{P}^1$’s joining like a necklace, or the affine $\widehat{A}_3$ Dynkin diagram. The quotient is indeed the zero-volume limit of the two disjoint $\mathbb {C}\textbf{P}^1$’s as in Fig. 4.3. On the other hand, the quotient of the other singular fibers $\pi ^{-1}(b_{2,3})$ by $\xi _1$ identifies the two irreducible components and the two double points by the rotation, yielding the fiber of Kodaira type $I_1$. Again, the quotients of $\xi _{2}$ and $\xi _{3}$ are obtained by the cyclic permutations of $b_{i}$ ($i=1,2,3$). As a result, the quotient of the Hitchin moduli space ${\mathcal {M}}_H(C_p,{{\,\textrm{SU}\,}}(2))$ by $\xi _i$ leads to an elliptic fibration $\mathcal {M}_C\rightarrow \mathcal {B}_u$ of the Coulomb branch with one singular fiber of type $I_4$ at $b_i\in \mathcal {B}_u$ and two singular fibers of type $I_1$ at $b_{i+1},b_{i+2}\in \mathcal {B}_u$  [1, 2] as illustrated in Fig. 4.1. Hence, an $\mathcal {N}=2^*$ theory of rank one enjoys a subgroup of ${{\,\textrm{SL}\,}}(2,\mathbb {Z})$ that fixes the singular fiber of type $I_4$. One can easily read off such a subgroup from (2.38) that is consistent with a duality group (4.2) of an $\mathcal {N}=2^*$ theory. Note that $\tau _+$ and $\tau _-$ correspond to the T and TST elements, respectively, of the electromagnetic duality of the 4d $\mathcal {N}=2^*$ theories of rank one, which is different from the matrix assignment in (2.34).

So far we have studied the Coulomb branches with generic ramification parameters $(\alpha _p,\beta _p,\gamma _p)$. When $\beta _p=0=\gamma _p$, the Hitchin fibration ${\mathcal {M}}_H(C_p,{{\,\textrm{SU}\,}}(2))\rightarrow \mathcal {B}_H$ has one singular fiber of type $I_0^*$ at the global nilpotent cone, and it is easy from Fig. 2.1 to see the quotient by $\xi _i$. For instance, at generic values of $\alpha _p$, the quotient of the global nilpotent cone by $\xi _1$ again leads to the singular fiber of type $I_0^*$, but the volumes of $\textbf{D}_2$ and $\textbf{D}_4$ shrink to zero in this case. Therefore, it has two $A_1$ singularities. Similarly, the quotient by another generator $\xi _i$ can be obtained by exchanging non-trivial exceptional divisor $\textbf{D}_3$ to another one ($\textbf{D}_2$ ($\xi _2$) or $\textbf{D}_4$ ($\xi _3$)).

As in (4.3), the moduli space ${\mathcal {M}}_H(C_p,{{\,\textrm{SO}\,}}(3))$ of ${{\,\textrm{SO}\,}}(3)$-Higgs bundles over $C_p$ can be obtained by the further quotient of the Coulomb branch by the other generator of $\Xi$. By the further quotient, the two irreducible components and the two $A_1$ singular points are identified in the singular fiber of type $I_4$, and the quotient of each singular fiber of type $I_1$ by the $180^{\circ }$ rotation of around the polar axis turns the double point into the $A_1$ singularity. As a result, all the singular fibers of ${\mathcal {M}}_H(C_p,{{\,\textrm{SO}\,}}(3))\rightarrow \mathcal {B}_H$ can be understood as the limit of a singular fiber of type $I_2$ in which one of the irreducible components shrinks to zero as in Fig. 4.2. When $\beta _p=0=\gamma _p$, the global nilpotent cone is again the singular fiber of type $I_0^*$, but it has three $A_1$ singularities for generic values of $\alpha _p$ [82, Fig. 1] (Fig. 4.3).

Fig. 4.1

Schematic illustration of elliptic fibration of Coulomb branch $\mathcal {M}_C(C_p,{{\,\textrm{SO}\,}}(3)_+)\rightarrow \mathcal {B}_u$

Fig. 4.2

Schematic illustration of the Hitchin fibration of ${\mathcal {M}}_H(C_p,{{\,\textrm{SO}\,}}(3))\rightarrow \mathcal {B}_H$

Fig. 4.3

(Left) The Coulomb branch $\mathcal {M}_C$ contains a particular limit of the fiber of type $I_4$ so that there are two $A_1$ singularities. (Right) Each singular fiber of the Hitchin fibration ${\mathcal {M}}_H(C_p,{{\,\textrm{SO}\,}}(3))\rightarrow \mathcal {B}_H$ with a generic ramification is a certain limit of the fibers of type $I_2$

4.2 Algebra of Line Operators

It is known [69, 70] that loop operators along $S^1$ in a 4d $\mathcal {N}=2$ theory $\mathcal {T}(C,G,\textsf{L})$ on $S^1\times \mathbb {R}^3$ form a commutative algebra that is the coordinate ring $\mathscr {O}(\mathcal {M}_C(C,G,\textsf{L}))$ of the Coulomb branch holomorphic in complex structure J. Once we introduce the $\Omega$-background $S^1\times \mathbb {R}\times _q \mathbb {R}^2$, loop operators wrapped on $S^1$ are localized on the axis of the $\Omega$-deformation as depicted in Fig. 4.4. Consequently, they are forced to come across each other as they exchange their positions, which yields non-commutative deformation of the algebra [12, 42, 97, 129, 134, 156]. Thus, an algebra of line operators of a 4d $\mathcal {N}=2$ theory on the $\Omega$-background provides the deformation quantization $\mathscr {O}^q(\mathcal {M}_C)$ of the coordinate ring of its Coulomb branch, a.k.a quantized Coulomb branch.

Fig. 4.4

An algebra of line operators (colored circles) in a 4d $\mathcal {N}=2$ theory becomes non-commutative in the $\Omega$-background $S^1\times \mathbb {R}\times _q \mathbb {R}^2$, which provides deformation quantization of holomorphic coordinate ring of the Coulomb branch. The 4d $\mathcal {N}=2$ theory compactified on $S^1\times S^1_q$ is described by 2d A-model $\Sigma \rightarrow \mathcal {M}_C$ on the Coulomb branch where the boundary condition at $\partial \Sigma$ is given by $\mathfrak {B}_{cc}$. Here $\mathbb {R}^2 \supset S^1_q$ is the circle generating the $\Omega$-deformation

Now we are ready to discuss quantized Coulomb branches of the 4d $\mathcal {N}=2^*$ theories of rank one and their relation to spherical DAHA $S\!\ddot{H}$. Once we specify a maximal isotropic lattice $\textsf{L}\subset H^2(C,Z(G))$, we can read off charges of line operators in a 4d $\mathcal {N}=2$ theory subject to the Dirac quantization condition. The cases of the 4d $\mathcal {N}=2^*$ theories of rank one have been studied in detail [6, 69]. In fact, the generators x, y, z of $S\!\ddot{H}$ in (2.47) correspond to the minimal Wilson (1, 0), ’t Hooft (0, 1) and dyonic (1, 1) line operator, respectively. Therefore, it is natural to expect that the relations of quantized Coulomb branches to $S\!\ddot{H}$ are as follows:

• The ${{\,\textrm{SU}\,}}(2)$ theory has line operators of charge $(\lambda _{e}, \lambda _{m})$ with $\lambda _{e} \in \mathbb {Z}, \lambda _{m} \in 2 \mathbb {Z}$, including a Wilson operator with the fundamental representation. Therefore, the quantized Coulomb branch is isomorphic to the $\xi _2$-invariant subalgebra of $S\!\ddot{H}$

  $$\begin{aligned} \mathscr {O}^q(\mathcal {M}_C(C_p,{{\,\textrm{SU}\,}}(2)))\cong S\!\ddot{H}^{\xi _2}\end{aligned}$$

  (4.4)

  generated by

  $$\begin{aligned} x=(X+X^{-1})\textbf{e}, \qquad y^2-1=(Y^2+1+Y^{-2})\textbf{e}.\end{aligned}$$

  (4.5)

• The ${{\,\textrm{SO}\,}}(3)_+$ theory has line operators of charge $(\lambda _{e}, \lambda _{m})$ with $\lambda _{e} \in 2 \mathbb {Z}, \lambda _{m} \in \mathbb {Z}$, including an ’t Hooft operator with the fundamental representation. Therefore, the quantized Coulomb branch is isomorphic to the $\xi _1$-invariant subalgebra of $S\!\ddot{H}$

  $$\begin{aligned} \mathscr {O}^q(\mathcal {M}_C(C_p,{{\,\textrm{SO}\,}}(3)_+))\cong S\!\ddot{H}^{\xi _1}\end{aligned}$$

  (4.6)

  generated by

  $$\begin{aligned} x^2-1=(X^2+1+X^{-2})\textbf{e}, \qquad y=(Y+Y^{-1})\textbf{e}.\end{aligned}$$

  (4.7)

• The ${{\,\textrm{SO}\,}}(3)_-$ theory has line operators of charge $(\lambda _{e}, \lambda _{m})$ with $\lambda _{e}, \lambda _{m} \in \mathbb {Z}$ such that $\lambda _{e}+\lambda _{m} \in 2 Z$, including a minimal dyonic operator $(\lambda _{e}, \lambda _{m})=(1, 1)$. Therefore, the quantized Coulomb branch is isomorphic to the $\xi _3$-invariant subalgebra of $S\!\ddot{H}$

  $$\begin{aligned} \mathscr {O}^q(\mathcal {M}_C(C_p,{{\,\textrm{SO}\,}}(3)_-))\cong S\!\ddot{H}^{\xi _3}\end{aligned}$$

  (4.8)

  generated by

  $$\begin{aligned} x^2-1=(X^2+1+X^{-2})\textbf{e}, \qquad z=(q^{-1/2} Y^{-1}X + q^{1/2} X^{-1}Y)\textbf{e}.\end{aligned}$$

  (4.9)

To see the connection to a 2d sigma-model in Chap. 2, we can employ a trick similar to Fig. 3.1. Namely, we can compactify a 4d $\mathcal {N}=2$ theory on $T^2\cong S^1\times S^1_q$ as illustrated in Fig. 4.4, which leads to 2d A-model $\mathbb {R}\times \mathbb {R}_+\cong \Sigma \rightarrow \mathcal {M}_{C}(C,G,\textsf{L})$ on the Coulomb branch. Here $S^1_q\subset \mathbb {R}^2$ is the circle around the axis of the $\Omega$-background. As a result, the axis of the $\Omega$-background on which loop operators meet each other becomes the boundary $\partial \Sigma$. Therefore, the boundary $\partial \Sigma$ should give rise the quantized Coulomb branch $\mathscr {O}^q(\mathcal {M}_C)$ so that the canonical coistoropic boundary condition $\mathfrak {B}_{cc}$ naturally shows up at $\partial \Sigma$ [129]. By the state-operator correspondence, a loop operator in the 4d $\mathcal {N}=2$ theory becomes a state in ${{\,\textrm{Hom}\,}}(\mathfrak {B}_\text {cc},\mathfrak {B}_\text {cc})$ up on the compactification. In this way, $\mathfrak {B}_\text {cc}$ arises from “the axis of the $\Omega$-deformation” (or a tip of a cigar as in [129]).

We have seen that the ${{\,\textrm{SU}\,}}(2)$ and ${{\,\textrm{SO}\,}}(3)_\pm$ theories are related by ${{\,\textrm{PSL}\,}}(2,\mathbb {Z})$ so that the quantized Coulomb branches are indeed isomorphic. We expect the conjectural functor (1.3) exists even when $\mathfrak {X}=\mathcal {M}_C(C_p,{{\,\textrm{SU}\,}}(2),\textsf{L})$ are the Coulomb branches of the 4d $\mathcal {N}=2^*$ theories of rank one. Thus, we can compare the representation category ${{\,\mathrm{\textsf{Rep}}\,}}(\mathscr {O}^q(\mathfrak {X}))$ of the quantized Coulomb branch with the A-brane category $\text {A-}{{{\,\mathrm{\textsf{Brane}}\,}}}(\mathfrak {X},\omega _\mathfrak {X})$ of the Coulomb branch as in Chap. 2. In fact, we can construct a polynomial representation of a quantized Coulomb branch by using the two generators, and finite-dimensional modules can be obtained by quotients of the polynomial representation under the corresponding shortening conditions. The geometry of the Coulomb branches is explored in the previous section, and we confirm that there is a one-to-one correspondence between finite-dimensional modules of the quantized Coulomb branch and compact A-branes in the Coulomb branch as in Chap. 2.

For illustrative purposes, let us briefly study representation theory of the quantized Coulomb branch $\mathscr {O}^q(\mathcal {M}_C(C_p,{{\,\textrm{SO}\,}}(3)_+))\cong S\!\ddot{H}^{\xi _1}$. As in (4.7), it is generated by $x^2$ and y. Consequently, the polynomial representation of $S\!\ddot{H}$ splits into the ± eigenspaces of the $\mathbb {Z}_2$ action $\xi _1:X\rightarrow -X$ as the $S\!\ddot{H}^{\xi _1}$-modules for generic (q, t). Therefore, $S\!\ddot{H}^{\xi _1}$ acts on $\widetilde{\mathscr {P}}:=\mathbb {C}_{q,t}[X^{\pm 2}]^{\mathbb {Z}_2}$ (resp. $\widetilde{\mathscr {P}}:=(X+X^{-1})\mathbb {C}_{q,t}[X^{\pm 2}]^{\mathbb {Z}_2}$) under the polynomial representation, which is spanned by Macdonald polynomials of even (resp. odd) degrees. We can define a raising and lowering operator as in (2.79) with these generators

$$\begin{aligned} \begin{aligned} \widetilde{\textsf{R}}_j&:=q^{j-1}t\frac{q^{-1}(x^2-1)y-qy(x^2-1)}{q^2-q^{-2}}+ \frac{q^{2j}(q^2-t^2)(1-t^2)}{1-q^{2+2 j} t^{2}},\\ \widetilde{\textsf{L}}_j&:=q^{-1-j}t^{-1}\frac{q^{-1}y(x^2-1)-q(x^2-1)y}{q^2-q^{-2}}-\frac{(q^2-t^2)(1-t^2)}{t^2(q^2-q^{2j} t^2)}. \end{aligned}\end{aligned}$$

(4.10)

They act on Macdonald polynomials as

$$\begin{aligned} {{\,\textrm{pol}\,}}(\widetilde{\textsf{R}}_j)\cdot P_{j}(X;q,t)&= (1-q^{2 j-1} t^{2}) P_{j+2}(X;q,t), \end{aligned}$$

(4.11)

$$\begin{aligned} {{\,\textrm{pol}\,}}(\widetilde{\textsf{L}}_j)\cdot P_{j}(X;q,t)&= -\frac{q^{-2j}(1-q^{2j})(1-q^{2(j-1)})(1-q^{2(j-2)} t^{4})(1-q^{2(j-1)} t^{4})}{t^2(1-q^{2(j-1)} t^2)^{2}(1-q^{2(j-2)} t^2)}P_{j-2}(X;q,t). \end{aligned}$$

(4.12)

Thus, using these operations, we can study finite-dimensional representations as quotients of the polynomial representation $\widetilde{\mathscr {P}}$ of $S\!\ddot{H}^{\xi _1}$ as in Sect. 2.6. Focusing on the polynomial representation of even degrees, when q is a 2nth root of unity (i.e. $q^{2n}=1$), the ideal $(X^{2n}+X^{-2n}-x_{2n}-x_{2n})$ is invariant under the action of $S\!\ddot{H}^{\xi _1}$ since the q-shift operator $\varpi$ acts trivially on $X^{\pm 2n}$. Consequently, the quotient by this ideal yields an n-dimensional finite-dimensional representation

$$\begin{aligned} \widetilde{\mathscr {F}}^{x_{2n},+}_{n}:=\widetilde{\mathscr {P}}/(X^{2n}+X^{-2n}-x_{2n}-x_{2n}) . \end{aligned}$$

(4.13)

An analogous representation can be obtained from the polynomial representation of odd degrees. They correspond to branes supported on a generic fiber with prescribed holonomy in $\mathcal {M}_C(C_p,{{\,\textrm{SO}\,}}(3)_+)$. Comparing (2.103) at $m=2n$, the dimension is a half because it splits into the ± eigenspaces of the $\mathbb {Z}_2$ action $\xi _1$. Furthermore, when n is even $n=2p$, we have another finite-dimensional representation because one lowering operator (4.12) becomes null:

$$\begin{aligned} \widetilde{\mathscr {U}}_{p}:=\widetilde{\mathscr {P}}/(P_{n}) . \end{aligned}$$

(4.14)

This corresponds to a brane supported on an irreducible component in the singular fiber $\pi ^{-1}(b_1)$ in Fig. 4.1. The dimension is a half of (2.107) due to the $\xi _1$ invariance. Under the condition, we have the short exact sequence analogous to (2.128)

$$\begin{aligned} 0\rightarrow \widetilde{\mathscr {U}}_{p} \rightarrow \widetilde{\mathscr {F}}^{-,+}_{n} \rightarrow \xi _2(\widetilde{\mathscr {U}}_{p}) \rightarrow 0. \end{aligned}$$

(4.15)

This can be interpreted as a bound state formed by the branes supported on the two irreducible components at the singular fiber $\pi ^{-1}(b_1)$ in Fig. 4.1. Similarly, we can obtain finite-dimensional irreducible representations analogous to $\mathscr {V}$ in (2.113) and $\mathscr {D}$ in (2.122) under the same shortening conditions where the dimensions are halved, respectively.

Note that the deformation quantization of the holomorphic coordinate ring of the Hitchin moduli space ${\mathcal {M}}_H(C_p,{{\,\textrm{SO}\,}}(3))$ can be understood as the spherical subalgebra of “${{\,\textrm{SO}\,}}(3)$ DAHA”. It is isomorphic to the $\Xi$-invariant subalgebra of $S\!\ddot{H}$

$$\mathscr {O}^q({\mathcal {M}}_H(C_p,{{\,\textrm{SO}\,}}(3)))\cong S\!\ddot{H}^\Xi $$

generated by $x^2-1$ and $y^2-1$. Since the weight and root lattices of ${{\,\textrm{SU}\,}}(2)$ and ${{\,\textrm{SO}\,}}(3)$ are related by $\textsf{Q}({{\,\textrm{SU}\,}}(2))=\textsf{P}({{\,\textrm{SO}\,}}(3))\subset \textsf{P}({{\,\textrm{SU}\,}}(2))=\textsf{Q}({{\,\textrm{SO}\,}}(3))$, this is consistent with the construction of DAHA from the symplectic lattice $(\textsf{P}\oplus \textsf{P}^\vee ,\omega )$ described at the beginning of Chap. 2 and the deformation quantization of the moduli space $\mathcal {M}_\text {flat}(C_p,{{\,\textrm{PSL}\,}}(2,\mathbb {C}))$ of ${{\,\textrm{PSL}\,}}(2,\mathbb {C})$-flat connections. Again, we can study representation theory of the ${{\,\textrm{SO}\,}}(3)$ DAHA from the perspective of brane quantization though the detail is omitted here.

There is yet another way [115, 117] to connect the 4d $\mathcal {N}=4$ theory to a 2d sigma-model and to see a category of line operators in the 4d $\mathcal {N}=4$ theory. (See also [43] for a similar analysis in 3d.) Let us consider a line operator supported on $\mathbb {R}\times \text {pt} \subset \mathbb {R}\times \mathbb {R}^3$ in the 4d $\mathcal {N}=4$ theory with gauge group G. Then, the neighborhood around the line operator at $\text {pt}\in \mathbb {R}^3$ consists of two disks glued along with a punctured disk

, called a “raviolo” [11, 127]. The “effect” of a line operator is measured by the modification of field configurations from one disk $\mathbb {C}$ to the other disk $\mathbb {C}$, namely a Hecke modification. The compactification of the 4d $\mathcal {N}=4$ theory on the raviolo leads to a 2d sigma-model on the Hitchin moduli space

of the raviolo, and a line operator gives rise to a boundary condition of the worldsheet as depicted in Fig. 4.5. It was shown [117] that a Wilson operator provides a boundary condition of type (B, B, B) whereas an ’t Hooft operator gives a boundary condition of type (B, A, A) in

. Since a boundary condition for a Wilson operator is holomorphic (B, B, B) in every complex structure on

, its fusion with another line operator preserves all supersymmetry. Taking into account the fact that the fusion of a Wilson and an ’t Hooft operator leads to a dyonic operator, a dyonic operator hence gives rise to a brane of type (B, A, A) in 2d sigma-model on

. Thus, upon the compactification, line operators in the 4d $\mathcal {N}=4$ theory all become B-branes of type I on

, and an algebra structure can be defined by the convolution product of B-branes.

Fig. 4.5

(Left) The neighborhood around a line operator at $\text {pt}\in \mathbb {R}^3$ is a “raviolo”. (Right) A line operator (blue) gives rise to a boundary condition in the 2d sigma-model upon the compactification of the 4d $\mathcal {N}=4$ theory on the raviolo

To formulate this idea into a mathematical model [16, 63, 146], let us first consider the moduli space of G-bundles over the raviolo. Since the coordinate ring of $\mathbb {C}$ is the formal power series ring $\mathscr {O}:=\mathbb {C}\llbracket z\rrbracket$ and that of $\mathbb {C}^\times$ is its field $\mathscr {K}:=\mathbb {C}(\!( z )\!)$ of fractions, the moduli space of G-bundles over

can be expressed as a double coset model, namely the space $G_\mathbb {C}^\mathscr {K}:=G_\mathbb {C}(\!( z )\!)$ of transition functions over the punctured disk $\mathbb {C}^\times$ modulo the spaces of gauge transformations $G_\mathbb {C}^\mathscr {O}:=G_\mathbb {C}\llbracket z\rrbracket$ over each $\mathbb {C}$:

(4.16)

In fact, if we take only the right quotient by the gauge transformation, the resulting space $\mathcal {G}r(G_\mathbb {C}):= G_\mathbb {C}^\mathscr {K}/G_\mathbb {C}^\mathscr {O}$ is called the affine Grassmannian.

To consider the Hitchin moduli space, we need to introduce the Higgs field. This can be achieved by considering the affine Grassmannian Steinberg variety

$$\begin{aligned} \mathcal {R}= \{ (x,[g])\in \mathfrak {g}_\mathbb {C}^\mathscr {O}\times \mathcal {G}r(G_\mathbb {C}) \mid {\text {Ad}}_{g^{-1}}(x) \in \mathfrak {g}^\mathscr {O}_\mathbb {C}\}. \end{aligned}$$

(4.17)

The quotient $G_{\mathbb {C}}^{\mathscr {O}}\backslash \mathcal {R}$ is the moduli space of a pair of G-bundles and sections of its adjoint associated bundle over the raviolo, which can be regarded as the mathematical model of

Taking the B-model viewpoint in complex structure I, the category $\textsf{Line}$ of line operators in the 4d $\mathcal {N}=4$ theory with gauge group G and zero theta angle is equivalent to the derived category of $G_{\mathbb {C}}^{\mathscr {O}}$-equivariant coherent sheaves on $\mathcal {R}$

$$ \textsf{Line}\bigl [\mathcal {T}[C,G,\textsf{L}]\bigr ]\cong D^b\textsf{Coh}^{G_\mathbb {C}^\mathscr {O}}(\mathcal {R}), $$

where a maximal isotropic lattice $\textsf{L}$ is chosen in such a way that the theta angle is zero. For instance, it is easy to see that it automatically incorporates the category of Wilson operators as

$$ \textsf{Coh}^{G_\mathbb {C}^\mathscr {O}}(\text {pt})\cong \textsf{Coh}^{G}(\text {pt})\cong \textsf{Rep}(G). $$

By taking the Grothendieck ring of this category, we obtain the algebra of line operators in the 4d $\mathcal {N}=4$ theory [16]

$$\begin{aligned} K^{G_\mathbb {C}^\mathscr {O}}(\mathcal {R})\cong \mathbb {C}[T_\mathbb {C}\times T_\mathbb {C}^\vee ]^W, \end{aligned}$$

(4.18)

which is indeed isomorphic to the coordinate ring of the Coulomb branch

$$\begin{aligned} \mathcal {M}_C(C,G,\textsf{L})=\frac{T_\mathbb {C}\times T_\mathbb {C}^\vee }{W} \end{aligned}$$

(4.19)

of the 4d $\mathcal {N}=4$ theory on $S^1\times \mathbb {R}^3$ holomorphic in complex structure J [16, 63, 146]. The coordinates $\mathbb {C}[T_\mathbb {C}]^W$ and $\mathbb {C}[T_\mathbb {C}^\vee ]^W$ are spanned by Wilson and ’t Hooft line operators, respectively. It is important to note that the Coulomb branch is not isomorphic to the moduli space $\mathcal {M}_{\text {flat}}(C,G_\mathbb {C})$ of $G_\mathbb {C}$ flat connections on a two-torus $C\cong T^2$ in (B.6) as a holomorphic symplectic manifold. It is rather a quotient of $\mathcal {M}_{\text {flat}}(C,G_\mathbb {C})$ by $\textsf{L}$.

To obtain the algebra of line operators in the 4d $\mathcal {N}=2^*$ theory, we turn on the equivariant action $\mathbb {C}^\times _t$ on the cotangent fiber of the affine Grassmannian $\mathcal {G}r(G_\mathbb {C})$, which is equivalent to switching on the ramification parameters (3.12). Moreover, its quantization can be further achieved by introducing the equivariant action $\mathbb {C}^\times _q$of the loop rotation $z\mapsto qz$. In this way, we obtain the quantized Coulomb branch of the 4d $\mathcal {N}=2^*$ theory on $S^1\times \mathbb {R}^3$

$$\begin{aligned} K^{(G_\mathbb {C}^\mathscr {O}\times \mathbb {C}^\times _t)\rtimes \mathbb {C}_q^\times }(\mathcal {R})\cong S\!\ddot{H}(W)^{\textsf{L}}. \end{aligned}$$

(4.20)

As we have seen in the examples of rank one, it is not isomorphic to the spherical subalgebra $S\!\ddot{H}(W)$ of DAHA associated to W. It is rather the $\textsf{L}$-invariant subalgebra of $S\!\ddot{H}(W)$.

Even with the same gauge group G, discrete theta angles provide different theories as in the examples of rank one. Generally, they are distinguished by characteristic classes of Higgs bundles such as the Stiefel-Whitney classes $w_2$ and $w_4$ [6, 61]. Above we consider only the cases in which the theta angle is zero, but we can generalize it to a theory with non-trivial discrete theta angle by constructing the moduli space of Higgs bundles with non-trivial topological classes over the raviolo.

4.3 Including Surface Operator

So far, we focus on physical realizations of the spherical DAHA $S\!\ddot{H}(W)$ and its subalgebras, and it is natural to ask whether we can realize DAHA $\ddot{H}(W)$ itself. To see that, let us consider an algebra of line operators on a surface operator of Gukov–Witten type [84]. A surface operator of Gukov–Witten type arises as an intersection of M5-branes at codimension two locus:

$$\begin{matrix} &{}{\text{ space-time: }} &{} \quad &{}\mathbb {R}^4 &{} \times &{}T^*C &{} \times &{}\mathbb {R}^3 \\ &{}{N\, \text{ M5-branes: }} &{} \quad &{}\mathbb {R}^4 &{} \times &{} C &{} \times &{} \text {pt} \\ {\text{(surface } \text{ operator) }} &{} {\text {M5'-brane:}} &{} \quad &{} \mathbb {R}^2 &{} \times &{}C &{} \times &{}\mathbb {R}^2 \\ \end{matrix}$$

where $C\cong T^2$ is a two-torus. Here a surface operator is supported on $\mathbb {R}^2 \times \text {pt} \subset \mathbb {R}^4$ in the $\mathcal {N}=4$ ${{\,\textrm{SU}\,}}(N)$ theory $\mathcal {T}[C,{{\,\textrm{SU}\,}}(N),\textsf{L}]$. A half-BPS surface operator breaks the gauge group down to a Levi subgroup $L \subset G$, and as in (2.4) the singular behavior of the gauge field around the surface operator is

$$\begin{aligned} \begin{aligned}A&= \alpha d \vartheta + \cdots \,, \end{aligned}\end{aligned}$$

(4.21)

where $z=r e^{i \vartheta }$ is a local coordinate of the plane normal to the surface operator. The singular behavior for one $\Phi$ of the adjoint chiral scalars is described by

$$D_{\bar{z}}\Phi =(\beta +i \gamma ) \delta ^{(2)}(z,\bar{z}).$$

A surface operator can also be microscopically realized as a 2d $\mathcal {N}=(4,4)$ theory coupled to the 4d $\mathcal {N}=4$ theory where the triple $(\alpha ,\beta ,\gamma )$ can be understood as the $\mathcal {N}=(4,4)$ Fayet–Iliopoulos parameters. In addition to the triple $(\alpha ,\beta ,\gamma )$, they are also labeled by the theta angles $\eta$ of the 2d theory. The quadruple $(\alpha ,\beta ,\gamma ,\eta )$ takes a value in the $W_L$-invariant part of $T \times \mathfrak {t}\times \mathfrak {t}\times T^{\vee }$ where we write $W_L$ for the Weyl group of the Levi subgroup L. We remark that surface operators exist in $\mathcal {N}=2$ supersymmetric gauge theories where the parameters $\beta$ and $\gamma$ are absent due to the number of supersymmetry.

In the following, we consider a category and algebra of line operators on a surface operator that breaks the gauge group G to its maximal torus T, which is often called the full surface operator. In this case, the corresponding Weyl group is that of the gauge group $W_T=W$. Since we will eventually consider the 4d $\mathcal {N}=2^*$ theory, we set $\beta =0=\gamma$ and the surface operator is parametrized by the pair $(\alpha ,\eta )$. As in the previous subsection, we can study this by compactifying the 4d theory on the “raviolo”. However, due to the presence of the surface operator, there are ramifications at the centers of the two disks of the raviolo

around a line operator (Fig. 4.6). We write the resulting Hitchin moduli space by

, and we are interested in a category of B-branes of type I on

Again, we first consider the moduli space of G-bundles over the ramified raviolo

to formulate this into a mathematical model [146, 148, 149]. The full surface operator breaks the space of gauge transformations on a disk from the loop group $G_\mathbb {C}^\mathscr {O}$ to the Iwahori subgroup

$$\begin{aligned} \mathscr {I}:=\{ a_0 +a_1 z +a_2 z^2 +\cdots \in G_\mathbb {C}^\mathscr {O}| a_0\in B \}\end{aligned}$$

(4.22)

that is the preimage of a Borel subgroup B under the projection $G_\mathbb {C}^\mathscr {O}\rightarrow G_\mathbb {C}$. Hence, the moduli space of G-bundles over

can be expressed as the double coset space

Actually, $\mathcal {F}l(G_\mathbb {C}):=G_\mathbb {C}^\mathscr {K}/\mathscr {I}$ is called the affine flag variety, which is the fiber bundle over the affine Grassmannian with the full flag variety $G_\mathbb {C}/B$ a fiber

As a mathematical model of the Hitchin moduli space

, we can consider the moduli space $\mathscr {I}\backslash \mathcal {Z}$ of a pair of G-bundles and sections of its adjoint associated bundles on

where $\mathcal {Z}$ is the affine flag Steinberg variety defined as

$$\begin{aligned} \mathcal {Z} = \{ (x,[g])\in \text {Lie}(\mathscr {I})\times \mathcal {F}l(G_\mathbb {C}) \mid {\text {Ad}}_{g^{-1}}(x) \in \text {Lie}(\mathscr {I})\}.\end{aligned}$$

(4.23)

Fig. 4.6

The raviolo around a line operator (blue) on the surface operator (red) has tame ramifications at the centers of the two disks. A line operator (blue) gives rise to a boundary condition in the 2d sigma-model upon the compactification of the 4d $\mathcal {N}=4$ theory on the raviolo

Consequently, the category of line operators on the full surface operator in the 4d $\mathcal {N}=4$ theory with gauge group G and zero theta angle is equivalent to the derived category of ${\mathscr {I}}$-equivariant coherent sheaves on $\mathcal {Z}$

$$\begin{aligned} \textsf{Line}[\mathcal {T}[C,G,\textsf{L}],T]\cong D^b\textsf{Coh}^{\mathscr {I}}(\mathcal {Z}), \end{aligned}$$

(4.24)

where a maximal isotropic lattice $\textsf{L}$ is chosen in such a way that the theta angle is zero. This includes the category of Wilson operators on the full surface operator

$$ \textsf{Coh}^{\mathscr {I}}(\text {pt})\cong \textsf{Coh}^{T}(\text {pt})\cong \textsf{Rep}(T), $$

which sees that the gauge group G is broken to the maximal torus T due to the surface operator. Clearly, the Grothendieck ring of the category (4.24) is the algebra of line operators on the full surface operator in the 4d $\mathcal {N}=4$ theory [146, 148, 149]

$$ K^{\mathscr {I}}(\mathcal {Z})=\mathbb {C}[T_\mathbb {C}\times T_\mathbb {C}^\vee ]\rtimes \mathbb {C}[W]. $$

Unlike (4.18), this ring is non-commutative because line operators on the surface operators know the order of multiplications even without quantization ($\Omega$-deformation).

By introducing the equivariant actions as in (4.20), we obtain an algebra of line operators on the full surface operator in the 4d $\mathcal {N}=2^*$ theory with gauge group G and zero theta angle on the $\Omega$-background

$$\begin{aligned} K^{(\mathscr {I}\times \mathbb {C}^\times _t)\rtimes \mathbb {C}_q^\times }(\mathcal {Z})\cong \ddot{H}(W)^{\textsf{L}}. \end{aligned}$$

(4.25)

This is isomorphic to the $\textsf{L}$-invariant subalgebra of DAHA $\ddot{H}(W)$. In the case of $G={{\,\textrm{SU}\,}}(2)$, this is isomorphic to the $\xi _2$-invariant subalgebra of DAHA $\ddot{H}$ generated by $X,Y^2,T$. For $G={{\,\textrm{SO}\,}}(3)$, it is isomorphic to the $\xi _1$-invariant subalgebra of DAHA $\ddot{H}$ generated by $X^2,Y,T$. (See Sect. 2.2.)

Although we consider the full surface operator that breaks a gauge group ${{\,\textrm{SU}\,}}(N)$ to $\text {S}[\text {U}(1)^N]$ here, we can instead include a surface operator of another type associated to a partition of N. For this, we replace the Borel subgroup B in (4.22) by a parabolic subgroup P associated to a partition of N. In this way, we can obtain a variant of DAHA as an algebra of line operators on a surface operator.

Canonical Coisotropic Brane of Higher Ranks

In the previous subsection, the canonical coisotropic brane $\mathfrak {B}_{cc}$ emerges as the boundary condition at the axis of the $\Omega$-deformation by compactifying the 4d $\mathcal {N}=2^*$ theory on $T^2\cong S^1\times S^1_q$. Moreover, an algebra of line operators can be understood as the algebra of $(\mathfrak {B}_{cc},\mathfrak {B}_{cc})$-strings in 2d A-model on the Coulomb branch. It is natural to ask how to describe the boundary condition at the axis of the $\Omega$-deformation in the presence of the full surface operator up on the same compactification (Fig. 4.7).

Fig. 4.7

The 4d $\mathcal {N}=2^*$ theory with the Gukov–Witten surface operator on $S^1\times S^1_q$ is described by 2d A-model $\Sigma \rightarrow \mathcal {M}_C(C_p,G,\textsf{L})$ where the boundary condition at $\partial \Sigma$ is described by the canonical coisotropic brane $\widehat{\mathfrak {B}}_{cc}$ of higher rank

For this purpose, we refer to the idea [15, 88] employed in the geometric construction of rational Cherednik algebra. Roughly speaking, spherical DAHA $S\!\ddot{H}(W)$ can be interpreted as the subalgebra averaged over the action of the Weyl group W. We want to construct the part of the Weyl group by constructing a brane of higher rank since the algebra of line operators on the full surface operator realizes (4.25). In fact, there is the natural construction of the Weyl group just by taking projection $T_\mathbb {C}\times T_\mathbb {C}^\vee \rightarrow (T_\mathbb {C}\times T_\mathbb {C}^\vee )/W$. If we regard the Coulomb branch of the 4d $\mathcal {N}=2^*$ theory as the resolution $\eta :\mathcal {M}_C(C_p,G,\textsf{L})\rightarrow (T_\mathbb {C}\times T_\mathbb {C}^\vee )/W$, we can define their fiber-product $\mathfrak {Y}$ via

Therefore, $\mathfrak {Y}$ can be understood as the universal family of $\mathcal {M}_C(C_p,G,\textsf{L})$. Following [88], we define the “unusual” tautological bundle $\mathcal {P}:= \rho _*\mathscr {O}(\mathfrak {Y})$ on $\mathcal {M}_C(C_p,G,\textsf{L})$, which is called Procesi bundle, by the push-forward of the sheaf $\mathscr {O}(\mathfrak {Y})$ of regular functions (or the trivial bundle) on $\mathfrak {Y}$. By construction, the Procesi bundle $\mathcal {P}$ has rank |W|, with the regular representation of the Weyl group W on every fiber. Then, the canonical coisotropic brane $\widehat{\mathfrak {B}}_{cc}:=\mathcal {P}\otimes \mathfrak {B}_{cc}$ in the presence of the full surface operator is indeed the tensor product of the original line bundle $\mathcal {L}$ for $\mathfrak {B}_{cc}$ (2.57) with the Procesi bundle $\mathcal {P}$. Consequently, the algebra of $(\widehat{\mathfrak {B}}_{cc},\widehat{\mathfrak {B}}_{cc})$-strings realizes an algebra of line operators on the full surface operator

$$ {{\,\textrm{Hom}\,}}(\widehat{\mathfrak {B}}_{cc},\widehat{\mathfrak {B}}_{cc})\cong \ddot{H}(W)^{\textsf{L}}. $$

If we replace $\mathcal {M}_C(C_p,G,\textsf{L})$ and $T_\mathbb {C}^\vee$ by the Hitchin moduli space ${\mathcal {M}}_H(C_p,G)$ and $T_\mathbb {C}$, respectively, in the construction above, we obtain the full DAHA as ${{\,\textrm{End}\,}}(\widehat{\mathfrak {B}}_{cc})\cong \ddot{H}(W)$. In particular, when $q=1$, the bundle $\mathcal {P}\otimes \mathcal {L}$ is equivalent to the vector bundle E constructed in [130, Corollaries 6.1 and 6.2]. In fact, the space of $(\widehat{\mathfrak {B}}_{cc},\mathfrak {B}_{cc})$-strings can be understood as $\ddot{H}(W)$-left and $S\!\ddot{H}(W)$-right module $\ddot{H}(W)\textbf{e}$, to which the Procesi bundle $\mathcal {P}$ is associated (Fig. 4.8).

Fig. 4.8

$(\widehat{\mathfrak {B}}_{cc},\widehat{\mathfrak {B}}_{cc})$-strings and $(\mathfrak {B}_{cc},\mathfrak {B}_{cc})$-strings form DAHA $\ddot{H}$ and spherical DAHA $S\!\ddot{H}$, respectively. Hence, a $(\widehat{\mathfrak {B}}_{cc},\mathfrak {B}_{cc})$-string leads to $\ddot{H}$-left and $S\!\ddot{H}$-right module $\ddot{H}\textbf{e}$

Fig. 4.9

Joining of a $(\widehat{\mathfrak {B}}_{cc},\mathfrak {B}_{cc})$-string and a $(\mathfrak {B}_{cc},\mathfrak {B}')$-string leads to $(\widehat{\mathfrak {B}}_{cc},\mathfrak {B}')$-string

This has the following remarkable consequence. In Chap. 2, we have seen that given an A-brane $\mathfrak {B}'$, the space of $(\mathfrak {B}_{cc},\mathfrak {B}')$-strings can be understood as a representation of spherical DAHA $S\!\ddot{H}(W)$. In fact, given a $(\mathfrak {B}_{cc},\mathfrak {B}')$-string, its joining with a $(\widehat{\mathfrak {B}}_{cc},\mathfrak {B}_{cc})$-string always yields a $(\widehat{\mathfrak {B}}_{cc},\mathfrak {B}')$-string (Fig. 4.9), which receives the action of the full DAHA $\ddot{H}(W)$. In a similar fashion, by reversing a $(\mathfrak {B}_{cc},\mathfrak {B}')$-string, one can obtain a $(\mathfrak {B}_{cc},\mathfrak {B}')$-string from a $(\widehat{\mathfrak {B}}_{cc},\mathfrak {B}')$-string. This leads to the Morita equivalence of the two representation categories

$$\begin{aligned} {{\,\textrm{Hom}\,}}(\widehat{\mathfrak {B}}_{cc},\mathfrak {B}_{cc}) :\textsf{Rep}(S\!\ddot{H}(W)) \xrightarrow {\sim } \textsf{Rep}(\ddot{H}(W)). \end{aligned}$$

(4.26)

Of course, not every object produces an equivalence of this type. We expect that both $\mathfrak {B}_{cc}$ and $\widehat{\mathfrak {B}}_{cc}$ can be understood as generating objects of the category of A-branes. In general, generating objects are far from unique, and their non-uniqueness is one way that Morita equivalences arise. For example, any free R-module is a generating object, giving rise to the usual Morita equivalences between matrix algebras. Since $\hat{\mathfrak {B}}_{cc}$ is, in a sense, analogous to a higher-rank module over $\mathfrak {B}_{cc}$, we expect a similar story here, but do not pursue this in this paper.

In particular, since the space of $(\widehat{\mathfrak {B}}_{cc},\mathfrak {B}_{cc})$-strings is associated to the Procesi bundle $\mathcal {P}$, the dimension formula for the representation corresponding to the space of $(\widehat{\mathfrak {B}}_{cc},\mathfrak {B}_\textbf{L})$-strings for a compact Lagrangian submanifold $\textbf{L}$ is obtained by just tensoring $\mathcal {P}$ in (2.72)

$$\begin{aligned} \begin{aligned} \dim {{\,\textrm{Hom}\,}}(\widehat{\mathfrak {B}}_{cc},\mathfrak {B}_\textbf{L})&=\dim H^0(\textbf{L},\mathcal {P}\otimes \mathfrak {B}_{cc}\otimes \mathfrak {B}_\textbf{L}^{-1}),\\&=|W| \dim {{\,\textrm{Hom}\,}}(\mathfrak {B}_{cc},\mathfrak {B}_\textbf{L}). \end{aligned} \end{aligned}$$