Nakajima: Relative Langlands

Plan:

1. Relative Langlands (BZSV) from TQFT point of view.
2. Definition of Coulomb brunch and $S$-dual.
3. Examples (quantum symmetric pair).

Langlands Correspondence from TQFT view point (Kapustin-Witten)

Let $G$ be a reductibe group, and $G_ {c}$ be compact Lie group.

Let $\mathscr{A}_ {G}$ and $\mathscr{B}_ {G}$ be two topologically twisted 4$d$ $N=4$ SYM theories.

Atiyah-Segal axiomatic way to understand $\mathscr{T}$:

• $\mathscr{T}(X^{4})$: a number;
• $\mathscr{T}(Y^{3})$: a vector space, $\mathscr{T}(Y_ {1}\sqcup Y_ {2}) = \mathscr{T}(Y_ {1})\otimes \mathscr{T}(Y_ {2})$;
• $\mathscr{T}(\Sigma^{2})$: a $\mathbb{C}$-linear category;
• $\mathscr{T}(\text{1-manifold})$: a $2$-category.

such that

• For $\partial X^{4}=Y^{3}$, get $\mathscr{T}(X)\in \mathscr{T}(Y)$ a vector in a vector space. In particular,

KW claim: (Let $\Sigma$ be a complex Riemman surface)

1. $\mathscr{A}_ {G}(\Sigma)$ category related to automorphic side of GLC $\operatorname{Shv}(\operatorname{Bun}_ {G}(\Sigma))$, corresponds to $\mathscr{B}_ {G}(\Sigma)$ category related to spectral/Galois side of GLC $\operatorname{IndCoh}(\operatorname{Loc}_ {\check{G}}(\Sigma))$.
2. ($S$-duality) $\mathscr{A}_ {G}(-)\cong \mathscr{B}_ {\check{G}}(-)$, where $\check{G}$ is the Langlands dual group. This is an explanation of GLC when evaluated at $\Sigma$.

Operators

Let $M=\Sigma\times[0,1]-B$, where $B$ is a small ball. Then $\partial M = \Sigma\sqcup \Sigma\sqcup S^{2}$. Then one gets $$\mathscr{T}(\Sigma)\times \mathscr{T}(S^{2})\rightarrow \mathscr{T}(\Sigma).$$ In particular, $\mathscr{T}(S^{2})$ is monoidal category and $\mathscr{T}(\Sigma)$ is its module.

• Take $\mathscr{T}=\mathscr{A}_ {G}$, one gets Hecke operator $\in \mathscr{A}_ {G}(S^{2})$.
• Take $\mathscr{T}=\mathscr{B}_ {\check{G}}$, one gets Wilson operator.

Then this explains (derived) geometric Satake. Namely, $$\operatorname{Perv}(L^{+}G\backslash LG/L^{+}G)\cong \operatorname{Rep}(\check{G}),$$ $$\operatorname{D}(L^{+}G\backslash LG/L^{+}G)\cong D(\operatorname{Sym}^{\huge{\text{▱}}}\check{\mathfrak{g}}^{\vee}/\check{G}).$$

Interfaces (Gaiotto-Witten)

This is a proposal of relative Langlands functoriality.

Let $G,H$ be twi reductive groups and one has TQFT $\mathscr{T}_ {G}$ and $\mathscr{T}_ {H}$ respectively.

A interface is a “homomorphism” $\mathscr{I}=\mathscr{I}_ {H,G}$ from $\mathscr{T}_ {H}(-)$ to $\mathscr{T}_ {G}(-).$

Namely, one gets $\mathscr{I}(Y):\mathscr{T}_ {H}(Y^{3})\rightarrow \mathscr{T}_ {G}(Y^{3})$ a homomorphism of vector spaces, and $\mathscr{I}(\Sigma):\mathscr{T}_ {H}(\Sigma^{2})\rightarrow \mathscr{T}_ {G}(\Sigma^{2})$ a functor of categories.

There is a picture:

Special case $H=\{1\}$:

• $\mathscr{I}(Y)\in \mathscr{T}_ {G}(Y)$ a vector (function) in a vector space (vector space of functions).
• $\mathscr{I}(\Sigma)\in \mathscr{T}_ {G}(\Sigma)$ an object in a category.
• $\mathscr{I}(S^{1})\in \mathscr{T}_ {G}(S^{1})$ a category in a 2-category.

Example 3 (old material in differential geometry). Let $G_ {c}$ be a compact Lie group. Then Nahm’s equation is an ODE for $\mathfrak{g}_ {c}$-valued functions on $[0,\frac{1}{2})$ $$\frac{d}{dt}T_ {1}+ [T_ {0},T_ {1}]=[T_ {2},T_ {3}]$$ and cyclic permutations. We want solve it up to gauge transform. $T_ {0}$ has no pole and $T_ {1,2,3}$ has no pole at $t=\frac{1}{2}$.

Let $T_ {i}=\frac{a_ {i}}{t-\frac{1}{2}}+\text{regular}$, Nahm’s equation requires $a_ {1}=[a_ {2},a_ {3}]$ and etc. Then one get $\rho: \mathfrak{su}_ {2}\rightarrow \mathfrak{g}_ {c}$ a Lie algebra homomorphism and one get $\rho:\mathfrak{sl}_ {2}\rightarrow \mathfrak{g}$ a $\mathfrak{sl}_ {2}$-triple $\langle e,f,h \rangle$.

Theorem 4 (Bielawski 1995). The hyperKähler manifold of Solutions of Nahm’s equations $/$ guage transforms $\cong T^{\ast}G/\!/\!/_ {u,\psi}= G\times S_ {e}^{\mathfrak{g}}$ equivariant Slodowy slice.

It should be possible to compose interfaces.

Claim (Gaiotto-Witten): there is a (topologically twisted) 3$d$ $N=4$ SUSY QFT with $G\times H$-symmetry, which gives an interface between $\mathscr{A}_ {H}\xrightarrow{\theta} \mathscr{A}_ {G}$ and $\mathscr{B}_ {H}\xrightarrow{\mathscr{L}} \mathscr{B}_ {G}$.

Then

Geometry of interface

Now consider a $G$-Hamiltonian space $(M,\omega)$, consisting of

• a complex symplectic manifold $M$,
• the action of $G$ on $M$ preserves $\omega$. Let $\mu: M\rightarrow \mathfrak{g}^{\ast}$ be the moment map, which is $G$-equivariant and $d\langle \mu, \xi \rangle =\iota_ {\xi_ {M}}$ for any $\xi\in\mathfrak{g}$, and $\xi_ {M}$ is the vector field on $M$ generated by $\xi$. Then one can consider symplectic reduction (or called Hamiltonian reduction)

$$M/\!\!/\!\!/G:=\mu^{-1}(0)//G:=\operatorname{Spec}(\mathscr{O}[\mu^{-1}(0)])^{G},$$ or the dg-stack $M\times^{G}_ {\mathfrak{g}^{\ast}}\{0\}.$

We have many examples of 3d TQFT defined by $H\times G$-Hamiltonian space $M$. Let $\mathscr{I}^{M}=\mathscr{I}^{M}_ {H,G}$ be a interface. There are two versions corresponding to $\mathscr{A}_ {G},\mathscr{B}_ {G}$ for 4d TQFT.

One have $$(\mathscr{A})\qquad \theta^{M}: \mathscr{A}_ {H}(-)\rightarrow \mathscr{A}_ {G}(-),$$ and $$(\mathscr{B})\qquad \mathscr{L}^{M}: \mathscr{B}_ {H}(-)\rightarrow \mathscr{B}_ {G}(-),$$ where $\theta^{M}$ and $\mathscr{L}^{M}$ are given (very roughly)

Secretly, we replace $M=T^{\ast}N$ by $N$.

Langlands functoriality: what is the dual of $M$?

If $M^{\vee}$ is the $S$-dual of $M$, then the diagram

There is a well-defined mao $\mathscr{T}\xrightarrow{Higgs} \mathrm{Higg}(\mathscr{I})$, where $\mathrm{Higg}(\mathscr{I})$ is an afine algebraic variety with symplectic structure on smooth locus.

If $\mathscr{I}$ has $G$-symmetry, then $G$ acts on $\mathrm{Higg}(G)$. Then composition of interface $\mathscr{I}_ {12}\circ \mathscr{I}_ {23}$ corresponds to $\mathrm{Higgs}(\mathscr{I}_ {12}\times \mathscr{I}_ {23}\bcancel{/\!\!/\!\!/}{}_ {G_ {2}}) = \mathrm{Higgs}(\mathscr{I}_ {12})\times \mathrm{Higgs}(\mathscr{I}_ {23}){/\!\!/\!\!/}G_ {2}.$

Dual of $M$

Definition of $M^{\vee}$

Goal today: Assume that $G$ acts on $M=T^{\ast}N$ for $N$ smooth affine variety, we propose a definition of $\check{M}$ with action of $\check{G}$ approximating $S$-dual.

Hope: if $\check{M}$ is smooth, this $S$-dual is coming from $\check{M}$. For example, this is true when $M$ is hyperspherical in the sense of [BZSV].

Take $\Sigma=S^{2}$ or raviolo space $\mathbb{D}\sqcup_ {\mathbb{D}^{\times}}\mathbb{D}$. Then $$\mathscr{A}_ {G}(S^{2})=D_ {G(\mathbb{O})}(\mathrm{Gr}_ {G})\xrightarrow[\text{BF(M)G}]{\cong} D^{\check{G}}(\mathrm{Sym}^{\huge{\text{▱}}}\mathfrak{g}^{\vee})=\mathscr{B}_ {\check{G}}(S^{2})$$ as monoidal categories.

One considers objects $\theta^{M}(S^{2})\in \mathscr{A}_ {G}(S^{2})$ and $\mathscr{L}^{\check{M}}(S^{2})\in \mathscr{B}_ {\check{G}}(S^{2}).$ They are ring objects in the respective monoidal category. THey are coming from

$$\operatorname{Map}(S^{2}, [M/G])\rightarrow \operatorname{Map}(S^{2},[\mathrm{pt}]/G),$$ where $M=T^{\ast}N$.

• on $\mathscr{B}$-side, $\mathscr{O}(\check{M})$= coordinate ring of $\check{M}^{\vee}$ viewed as $\operatorname{Sym}(\check{\mathfrak{g}})$-module via $\check{M}\xrightarrow{\mu}\check{\mathfrak{g}}^{\ast}$.
• on $\mathscr{A}$-side, $p:\operatorname{Maps(S^{2}),[N/G]}\rightarrow \mathrm{Map}(S^{2},[\text{pt}/G])=L^{+}G\backslash \mathrm{Gr}_ {G}$, and the source is $\{(\mathscr{P}_ {G},s):\mathscr{P}_ {G}\text{ a $G$-bundle on $S^{2}$ with }s\in \Gamma(\mathscr{P_ {G}\times^{G}N})\}$,

Then $\mathscr{O}^{M}(S^{2}) = p_ {\ast}\omega$, where $\omega$ is the dualizing sheaf.

By [BFN], one have $\mathscr{M}_ {c}=\check{M}\times \check{G}\times \check{S}/\!\!/\!\!/G^{\vee}$ Kostant reduction of $S$-dual and $\mathscr{M}_ {c}(G\curvearrowright M)=[\{1\}\curvearrowright M/\!\!/\!\!/G]^{\vee} = \check{M}\circ (\check{G}\times \check{S})$.

Therefore, $[G\curvearrowright\text{pt}]^{\vee} = \check{G}\curvearrowright \check{G}\times S^{\vee},$ where $S^{\vee}$ is silce in $\check{\mathfrak{g}}$ at $e_ {\text{regular}}$.

Lst week universal centralizaer is same as $H^{G(\mathbb{O})}(\mathrm{Gr}_ {G})$. Take $M=\text{pt}$ and $\check{M}=\check{G}\times \check{S}$.

Recall that $G$ acts on $M=T^{\ast}N$ where $N$ os a smooth affine $G$-manifold.

Let $S^{2}= \mathbb{D}\sqcup_ {\mathbb{D}^{\times}}\mathbb{D}$.

There is a map $$p:\operatorname{Map}(S^{2},N/G)\rightarrow \operatorname{Map}(S^{2},\text{pt}/G)\cong \operatorname G(\mathbb{O})\backslash{Gr}_ {G}.$$ Then $$\theta^{M}({S^{2}}):=p_ {\ast}\omega_ {\operatorname{Map}(S^{2},N/G)}.$$ We defined $$M^{\vee}:=\operatorname{Spec}(H^{\ast}(\text{derived Satake})\theta^{M}(S^{2})).$$

This is an approximation of 3d TQFT, $S$-dual to $\theta^{M}$.

Computation of $M^{\vee}$

In order to compute $M^{\vee}$, by [BFN3] $$H^{\ast}(\text{derived Satake})\theta^{M}(S^{2}) = H^{\ast}_ {G(\mathbb{O})}(\theta^{M}(S^{2})\otimes^{!},sA_ {\text{reg}}),$$

where $\mathscr{A}_ {\text{reg}}$ is regular sheaf on $\operatorname{Gr}_ {G}$ and perverse, defined by \begin{align} \mathscr{A} & = (\text{derived Satake})^{-1} (\mathbb{C}[T^{\ast}G^{\vee}]) \\ & = (\text{geometric Satake})^{-1} (\mathbb{C}[G^{\vee}]) \\ & = \bigoplus_ {\lambda}V_ {\check{G}}(\lambda)^{\ast}\otimes \operatorname{IC}(\operatorname{Gr}_ {G}^{\lambda}). \end{align}

Then $$ H^{\ast}(\text{derived Satake})\theta^{M}(S^{2}) =(p_ {\operatorname{Gr}_ {G}\rightarrow \operatorname{Gr}_ {\{1\}}})_ {\ast}(\theta^{M}(S^{2})\otimes^{!}\mathscr{A}_ {\text{reg}}) $$ where the pushforward is $\bcancel{/\!\!/\!\!/}G$, and $\mathscr{A}_ {\text{reg}}$ has $\theta^{\mathscr{A}_ {\text{reg}}}$ as its 3d TQFT lift.

$$\mathscr{L}^{M^{\vee}} = (\theta^{M}\times \theta^{\mathscr{A}_ {\text{reg}}}/\!\!/\!\!/_ {G}(S^{2}))^{!},$$ where $\theta^{M}$ has $G$ symmetry and $\theta^{\mathscr{A}_ {\text{reg}}}$ has $G\times G^{\vee}$-symmetry, and $!$ is the 3d mirror symmetry:

• $\mathfrak{g}$ one has $\operatorname{Higgs}(\mathscr{I})$ and $\operatorname{Coulomb}(\mathscr{I})$,
• one $\mathscr{I}^{!}$ with $\operatorname{Higgs}(\mathscr{I}^{!}) =\operatorname{Coulomb}(\mathscr{I}) $ and $\operatorname{Coulomb}(\mathscr{I}^{!})=\operatorname{Higgs}(\mathscr{I}).$

In this case, $$\mathscr{A}_ {G}\xrightarrow{\theta_ {\mathscr{L}}}\mathscr{B}_ {G} $$ is just $\mathscr{A}_ {\text{reg}}$ and $!$.

Consider $$\operatorname{Coulomb}(\mathscr{L}^{M^{\vee}}) = \operatorname{Higgs}(\theta^{M}\times \theta^{\mathscr{A}_ {\text{reg}}}\bcancel{/\!\!/\!\!/}G) = M\times \mathcal{N}_ {G}/\!\!/\!\!/ G,$$ where $\mathcal{N}_ {G}$ is the nipotent cone.

Expectation: If $\mathscr{L}^{M^{\vee}}$ is really coming from $M^{\vee}$ (e.g. if smooth), we should have $\operatorname{Coulomb}(\mathscr{L}^{M^{\vee}})=“\text{pt}”.$

Its $\operatorname{GL}_ {n}=\operatorname{GL}_ {n}^{vee}$ equivariant sturecutre is not manifest.

Application:

$$M^{\vee} = M_ {c}(M\times 1-2-\cdots-n)\bcancel{/\!\!/\!\!/}\operatorname{GL}_ {n}.$$

Affine closure $$\overline{T^{\ast}(\operatorname{GL}_ {n}/U_ {n_ {1},\dots,n_ {r}})}= [\operatorname{GL}_ {n}\curvearrowright T^{\ast}\operatorname{GL}_ {n}\curvearrowleft \operatorname{GL}_ {n_ {1}\times n_ {2}\times\cdots\times\operatorname{GL}_ {n_ {r}}}]^{\vee}.$$

Moregenreally, we expect (Ginzburg-Riche?) $$[G\curvearrowright T^{\ast G}\curvearrowleft L]^{\vee} = [G^{\vee}\curvearrowright \overline{T^{\ast}(G^{\vee}\times L^{\vee})/P^{\vee}}\curvearrowleft L^{\vee}].$$