DERIVED HAMILTONIAN REDUCTION

Transcription

1 DERIVED HAMILTONIAN REDUCTION PAVEL SAFRONOV 1. Classical definitions 1.1. Motivation. In classical mechanics the main object of study is a symplectic manifold X together with a Hamiltonian function H O(X). X is called the phase space of the system. For instance, classical mechanics of a particle on a Riemannian manifold M has as the classical phase space X = T M. The Hamiltonian is given by H = (p, p)/2, where p is the fiber coordinate on T M called the momentum. Suppose that M has a -action. How to describe classical mechanics on M/? In other words, one wants to compute T (M/) Definitions. Suppose (X, ω) is a symplectic manifold with a group acting by preserving the symplectic form. That is, acts on X by symplectomorphisms. We say that the action of on X is Hamiltonian if there is a moment map µ: X g such that (1) d dr µ(v) = ι a(v) ω v g, where a: g Γ(X, T X ) is the infinitesimal action map. A symplectic manifold (X, ω) with a Hamiltonian action of is called a Hamiltonian -space. Exercise. Show that the moment map is -equivariant. It is clear that if is connected, the existence of the moment map implies that the -action preserves the symplectic form. Indeed, due to connectedness assumption this is equivalent to L a(v) ω = 0 v g. Differentiating the moment map equation (1) we get 0 = d dr ι a(v) ω = L a(v) ω. Here we have used the fact that ω is d dr -closed. Lectures given in April 2015 at Derived algebraic geometry with a focus on derived symplectic structures in Warwick. 1

2 2 PAVEL SAFRONOV 1.3. Existence of moment maps. Let us now describe a converse problem; given a symplectic -action, when is it Hamiltonian? In other words, we have a lifting problem g O(X) Vect symp (X). Here Vect symp (X) is the Lie algebra of symplectic vector fields, O(X) is the Lie algebra under the Poisson bracket and O(X) Vect symp (X) is the so-called symplectic gradient map ω 1 (d dr ( )). If X is connected, we have an exact sequence of Lie algebras (2) 0 k O(X) Vect symp (X). Suppose v, w are two symplectic vector fields. Indeed, Here we have used that ι [v,w] (ω) = L v ι w ω ι w L v ω = d dr ι v ι w ω. ι v d dr ι w ω = ι v L w ω ι v ι w d dr ω = 0. Then [v, w] has a natural Hamiltonian. So, ι v ι w ω is the Hamiltonian for [v, w]. If H 1 (g, k) = 0, then the sequence (2) gives a central extension of g by k, i.e. we get an obstruction class in H 2 (g, k) whose vanishing is equivalent to having a moment map. For instance, if g is semisimple, both (co)homology groups are trivial, so any symplectic action by a connected semisimple group has a moment map. Suppose acts on a manifold M and X = T M. We claim that there is an induced Hamiltonian action of on X. Indeed, the moment map is given by g Γ(M, T M ) Γ(M, Sym(T M )) = O(T M). Exercise. Show that this is the moment map for the natural action of on T M Hamiltonian reduction. iven a Hamiltonian -space X, we can define the reduced space X red X// := µ 1 (0)/. Theorem (Marsden Weinstein). When X red is a manifold, it is symplectic. Note that there are two issues in defining X red : 0 might not be a regular value, so µ 1 (0) might be singular; the action of on µ 1 (0) might not be free and proper. The first of these issues is resolved by going from manifolds (schemes) to derived manifolds (derived schemes); the second of these issues is resolved by going from derived manifolds (derived schemes) to derived stacks. We will do it in the next section, but let us first give some examples. One can slightly generalize the construction. Suppose C g is a coadjoint orbit. Then we can define X// C := µ 1 (C)/. It is also a symplectic manifold if it is a manifold.

3 1.5. Examples. DERIVED HAMILTONIAN REDUCTION 3 (1) Let M = R 3, X = T M and = Aff(3) = R 3 O(3) is the group of affine transformations acting on M. Then we have a moment map µ: X R 3 o(3). The components in R 3 are the linear momenta and the components in o(3) are the angular momenta. This explains the terminology. (2) Suppose M is a manifold with a -action and X = T M. Exercise. The reduced space X red is isomorphic to T (M/). (3) Here is an algebraic example. Suppose M is a scheme and L is a line bundle on M. We denote by L the total space of L minus the zero section; it is the associated m -bundle. By the previous example, T L // m = T M. More generally, we define the cotangent bundle twisted by L to be T L // 1 m = T L M. It is an affine bundle over T M whose class in H 1 (M, Ω 1 ) is the first Chern class of L Why derived reduction? Let us explain why one might be interested in considering Hamiltonian reduction in the setting of derived algebraic (or smooth) geometry. As we have already mentioned, the space X// might not be a smooth manifold; we will show that it always exists as a derived stack and the smooth quotient is simply the underlying coarse scheme of the derived stack. Another source of derived reduction comes from field theories. An n-dimensional classical field theory associates to a closed (n 1)-manifold an ordinary symplectic scheme. For instance, classical mechanics (a 1-dimensional field theory) associated to the point a symplectic manifold which is the phase space of the system. Considering manifolds of different dimensions, we encounter shifted symplectic schemes; systems with a group symmetry then give examples of derived Hamiltonian reductions. For instance, on a closed n-manifold one encounters examples of the form T [ 1](M/). Finally, there is a variant of Hamiltonian reduction, quasi-hamiltonian reduction, where the moment map takes value in the group rather than in the dual Lie algebra. In this case the original manifold is not symplectic (however, the reduced manifold is). Thus, we have to leave the world of symplectic geometry to describe these reductions. On the other hand, in the derived description of Hamiltonian reduction quasi-hamiltonian reduction is defined purely in terms of Lagrangian and symplectic schemes. For instance, this is helpful to understand quantization of such spaces.

4 4 PAVEL SAFRONOV 2. Derived Hamiltonian reduction 2.1. Ordinary Hamiltonian reduction. Let us switch to the algebraic context. That is, all our spaces will be derived stacks locally of finite presentation and we use the language of shifted symplectic structures of [PTVV]. Consider T [1](B) = g /. Since B is an Artin stack, [PTVV, Proposition 1.21] does not apply and so we have to check nondegeneracy by hand. Consider Ω (g /). By descent we can compute this complex as the complex of differential forms on the action groupoid of g by which, by [Be], is isomorphic to the Cartan model of equivariant cohomology. Using the vector space structure on g we can identify Ω 1 (g ) = O(g ) g. Therefore, we have a canonical form θ Ω 1 (g ) g. It is clearly nondegenerate. Exercise. Show that the Liouville one-form on g / is given by the form λ g g O(g ) g. Show that θ = d dr λ. Thus, g / is a 1-shifted symplectic stack. Suppose X is a smooth scheme with a - action and a -equivariant map µ: X g. When is the induced morphism X/ g / Lagrangian? Since X/ is a smooth stack, a Lagrangian structure is given by a degree 0 two-form h such that f θ = ḥ 0 = d dr h. Again using the Cartan model of cohomology of X/, we see that h is a closed two-form on X and the first equation is exactly the moment map equation (1). Exercise. Show that h is nondegenerate if X/ g / is Lagrangian. Thus, we have established the following theorem. Theorem. A Hamiltonian -space is equivalent to a Lagrangian in g /. We have the following basic examples of Lagrangians: (1) pt is a -Hamiltonian space, so pt/ g / is Lagrangian. The Lagrangian intersection is simply the Hamiltonian reduction X/ g / pt/ = (X g pt)/ µ 1 (0)/ X//. (2) Let O g be a coadjoint orbit and O is the stabilizer of a point in O. Then pt/ O = O/ g / has a unique isotropic structure since the space of degree 0 and degree 1 two-forms on pt/ O is contractible. One can easily show that it is nondegenerate. The induced symplectic structure on O is the so-called Kirillov Kostant symplectic structure. (3) Isotropic structures on g g / are the same as closed two-forms on g. One can show that any such two-form is nondegenerate and so defines a Lagrangian structure. Let us choose the zero two-form as the Lagrangian structure.

5 We have a Lagrangian intersection DERIVED HAMILTONIAN REDUCTION 5 X/ g / g = X, which recovers X with its symplectic structure Shifted symplectic reduction. We can repeat the construction of the previous section for shifted symplectic structures. We have an (n + 1)-shifted symplectic scheme The zero section defines a Lagrangian T [n + 1](B) = g [n]/. pt/ g [n]/. We also have a Lagrangian g [n] g [n]/ which is the cotangent fiber at the basepoint of B. Definition. An n-shifted -Hamiltonian space is a Lagrangian in g [n]/. iven a Hamiltonian space L g [n]/ we define its underlying symplectic space to be L symp := L g [n 1]/ g [n]. By the Lagrangian intersection theorem [PTVV, Theorem 2.9] L symp is n-shifted symplectic. We can also define the reduced space L red := L g [n]/ pt/. It is also a Lagrangian intersection, so L red is n-shifted symplectic Reduction of shifted cotangent bundles. iven a map of derived stacks X Y we have a correspondence expressing the pullback of differential forms T [n]x T [n]y Y X T [n]y. Under the same hypotheses that ensure that the cotangent bundles are symplectic, one can show that this is a Lagrangian correspondence. We define the shifted conormal bundle N [n](x/y ) to be N [n](x/y ) := (T [n]y Y X) T [n]x X. As a composition of Lagrangian correspondences, N [n](x/y ) T [n]y is Lagrangian. For instance, the map pt pt/ gives rise to the Lagrangian g g [n]/ that we mentioned previously. Now consider a -space X or, equivalently, a space Y = X/ B. Then is an n-shifted -Hamiltonian space. reduced space is T [n](x/). N [n + 1](Y/B) g [n]/ Its underlying symplectic space is T [n]x and the

6 6 PAVEL SAFRONOV Thus, T [n]x// = T [n](x/). Recall that we ve introduced cotangent bundles twisted by a line bundle as Hamiltonian reductions. What is the analog for shifted cotangent bundles? If X has a m -action, then an n-shifted moment map is a map X k[n], so we cannot take Hamiltonian reduction at a nonzero moment map value. Instead, we have to change the group that acts. Suppose X is a space together with a n-gerbe. Let X be the total space which is a B n m -torsor. Then T [n] carries a Hamiltonian action of B n m and its moment map takes values in k. Therefore, we can define shifted cotangent bundle by an n-gerbe to be T [n](x) := T [n]( )// 1 B n m. 3. Quasi-Hamiltonian reduction 3.1. Definition. Recall that Hamiltonian spaces are simply Lagrangians in g / and Hamiltonian reduction was interpreted as a Lagrangian intersection. To obtain variants of Hamiltonian reduction, one just have to replace g / by another 1-shifted symplectic stack. Let us define a 1-shifted symplectic structure on the adjoint quotient. We will again use a Cartan model of equivariant cohomology. Choose a nondegenerate -invariant bilinear pairing (, ) on g and define and ω 0 = 1 (θ + θ, ) 2 ω 1 = 1 (θ, [θ, θ]), 12 where θ and θ are the left- and right-invariant Maurer-Cartan forms. Then ω 0 + uω 1 is a 1-shifted symplectic structure on. Alternatively, recall that a nondegenerate -invariant bilinear pairing defines a 2-shifted symplectic structure on B. Theorem. The natural isomorphism is a symplectomorphism. = Map(S 1 B, B) As before, pt/ is Lagrangian; moreover, for any conjugacy class C we have a Lagrangian C/. Note, however, that is not Lagrangian, but one can endow it with a coisotropic structure instead. So, let us call Lagrangians in quasi-hamiltonian spaces. iven such a Lagrangian L, we define its reduction L red to be the intersection L red := L pt/. We do not have an underlying symplectic space as is not Lagrangian, but one can still consider this intersection and endow it with a quasi-symplectic structure.

7 DERIVED HAMILTONIAN REDUCTION Character variety. The main examples of quasi-hamiltonian spaces come from character varieties. Endow B with a 2-shifted symplectic structure and consider Σ, a compact oriented surface of genus g. Denote by Σ the same surface minus a disk. One can easily see that Σ deformation retracts onto a wedge of 2g circles. Define the character variety to be Loc ( ) = Map( B, B). Since Σ = Σ S 1 D, we can write Loc (Σ) = Loc (Σ ) pt/. Therefore, Loc (Σ ) is a quasi-hamiltonian space and its Hamiltonian reduction is Loc (Σ). Note that we can identify Loc (Σ ) = 2g /, so this gives a very explicit presentation for Loc (Σ) with its symplectic structure Springer resolution. Let be a reductive group and B a Borel subgroup. Denote by B H its abelianization which is an abstract torus. We get a correspondence BB B BH One can easily check that it is Lagrangian. In fact, this is true if one replaces B by a parabolic subgroup and H by its Levi factor. Taking loop spaces, we obtain a Lagrangian correspondence B B This is a correspondence that allows one to turn quasi-hamiltonian spaces into H quasi- Hamiltonian spaces and vice versa. Indeed, given a Lagrangian L we get a Lagrangian in H H L. This procedure is known as symplectic implosion. Consider B B H/H pt/h. It is a Lagrangian in, so is a quasi-hamiltonian space. If one writes N for the variety of unipotent elements and Ñ for its Springer resolution, then B B H H B B H/H pt/h = Ñ /. In other words, we have shown that Ñ is a quasi-hamiltonian space. Note that in the additive case the Springer resolution is simply T (/B) and the Hamiltonian structure comes from the -action on the flag variety.

8 8 PAVEL SAFRONOV 4. Hamiltonian reduction in field theories 4.1. Field theories. Consider the following three (, n)-categories: The (, n)-category of Lagrangian correspondence LagrCorr n as follows. Its objects are (n 1)-shifted symplectic stacks, morphisms between X 1 and X 2 are Lagrangian correspondences X 1 L X 2 and so on. The precise definition will be given in the upcoming work of Calaque Haugseng. (, n + 1)-category Mor n. Its objects are unital E n -algebras, morphisms from A to B are E n A B bimodules and so on. Here a left E n -module M over an E n -algebra A is an E n 1 -algebra M together with a map of E n algebras A HH E n 1 (M) (note that by the higher Deligne conjecture, HH E n 1 (M) is an E n -algebra). Alternatively, one can define them as E n 1 -algebra objects in the category of left A-modules. (, n)-category Bord n whose objects are finite collections of points, morphisms are cobordisms between the collections and so on. We define a classical theory of observables of dimension n to be a symmetric monoidal functor Z obs : Bord n LagrCorr n. So, on a point Z obs gives an (n 1)-shifted symplectic stack Z obs (pt), to an interval it attaches a Lagrangian correspondence Z obs (pt) Z obs ([0, 1]) Z obs (pt). We can define the trivial classical field theory 1 which attaches pt thought of as an (n 1)- shifted symplectic stack to the point, pt thought of as a Lagrangian correspondence between pt pt to an interval and so on. We define a classical theory of states associated to a theory of observables Z obs to be a homomorphism Z states : 1 Z obs, i.e. a relative field theory. On a point the theory of states gives a Lagrangian correspondence pt Z states (pt) Z obs (pt), i.e. Z state (pt) Z obs (pt) is Lagrangian. In the same way one can define quantum theories of observables and states Z q obs and Zq states by replacing the category LagrCorr n by Mor n. In this case Z q obs (pt) is an E n-algebra and Z q states(pt) is a left E n -module over Z q obs (pt). One can talk about quantizations of given classical field theories using, for instance, the notion of Beilinson-Drinfeld quantization. Example. Take n = 1. Then we have topological classical mechanics. It is defined to be Z obs (pt) = T M, Z states (pt) = M T M.

9 Its quantum version is DERIVED HAMILTONIAN REDUCTION 9 Z q obs (pt) = D(M), Zq states(pt) = O(M) Classical gauge theories. Now we want to make sense of classical mechanics with an action of a group. The easiest way to formulate it is as a theory relative to a certain 2-dimensional theory we are going to define. That is, Z 1 obs : Z 2 obs 1. iven such theory of relative observables, we can forget that it is relative by composing with the theory of states: 1 Z1 obs Zobs 2 Zobs 1 1. This gives an honest 1-dimensional theory. Let us now present the relevant 2-dimensional theories Additive case. In this case Z 2 obs(pt) = T [1](B) = g /, Z 2 states(pt) = B. Classical mechanics relative to this theory is defined by a Lagrangian in g /, i.e. a - Hamiltonian space. The underlying 1-dimensional field theory is simply the theory defined by the reduced space Multiplicative case. In this case Z 2 obs(pt) =, Z2 states(pt) = B. Classical mechanics relative to this field theory is defined by a quasi-hamiltonian space Chern Simons. Notes that the multiplicative theory we ve introduced attaches the loop space of B to the point. So, it can be thought of as a compactification of a 3- dimensional theory Zobs 3 which attaches B to the point. This is simply the classical Chern Simons theory. Surfaces with boundaries give interesting operations on Lagrangians in, i.e. quasi- Hamiltonian spaces. For instance, the disk gives a Lagrangian pt/ and, as we ve mentioned before, a composition of a quasi-hamiltonian space with the disk is simply quasi-hamiltonian reduction. The other interesting example is a pair of pants. This gives a Lagrangian correspondence The map on the left is simply the projection and the map on the right is multiplication. Composing two quasi-hamiltonian spaces with this Lagrangian correspondence gives a single.

10 10 PAVEL SAFRONOV quasi-hamiltonian space; this process is known as fusion. If µ 1, µ 2 are the moment maps for the quasi-hamiltonian spaces, then the moment map for the composite is simply µ 1 µ 2. However, the correspondence has a nontrivial Lagrangian structure which is given by the two-form h = 1 2 (p 1θ, p 2θ) Quantization. Let us finally discuss quantum version of Hamiltonian reductions. A neat way is to quantize the 2d gauge theories and consider quantum Hamiltonian spaces as giving relative theories to the quantum gauge theories. We will perform categorical quantizations, i.e. we will take the categories of quasi-coherent sheaves on the spaces appearing in classical gauge theories and deform them as E n categories Additive case. We have QCoh(g /) = Sym(g) mod Rep. Its quantization is the monoidal category U(g) mod Rep known as the category of Harish-Chandra bimodules. The Lagrangian B g / quantizes to the category Rep which is a module category over the category of Harish-Chandra bimodules. iven a Hamiltonian space, i.e. a Lagrangian L g /, one expects QCoh(L) to quantize to another module category C. The Hamiltonian reduction quantizes to the tensor product C U(g) modrep Rep. Recall that a quantum Hamitlonian space consists of an algebra A with a -action, and a -equivariant moment map µ: U(g) A satsifying the moment map equation [µ(v), ] = a(v), v g, where a(v): g Der(A) is the infinitesimal action map. The reduced algebra is defined to be A// := A (Aµ(g)). Theorem. If is a formal group, the category of modules over quantum Hamitlonian reduction A// is equivalent to A mod Rep Ug modrep Rep Multiplicative case. We have QCoh( ) = O() mod Rep. Its quantization is the monoidal category A q () mod Repq. Here A q () is the so-called reflection equation algebra quantizing O() and Rep q is the category of representations of the quantum group. The Lagrangian B quantizes to Rep q seen as a module category over A q () mod Repq. Proceeding as before, we recover multiplicative moment maps A q () A and the quantum Hamiltonian reduction can be written on the categorical level as C Aq() mod Repq Rep q, where C is the quantization of the category QCoh(L) (for instance, C can be taken to be A mod Repq ).

11 DERIVED HAMILTONIAN REDUCTION 11 References [Be] K. Behrend, Cohomology of stacks, ICTP Lect. Notes XIX (2004) [Ca] D. Calaque, Lagrangian structures on mapping stacks and semi-classical TFTs, arxiv: [PTVV] T. Pantev, B. Toën, M. Vaquié,. Vezzosi, Shifted symplectic structures, Publ. Math. IHES 117 (2013) , arxiv: [Sa1] P. Safronov, Quasi-Hamiltonian reduction via classical Chern Simons theory, arxiv: [Sa2] P. Safronov, Symplectic implosion and the rothendieck-springer resolution, arxiv: