FACTORIZATION ALGEBRAS IN QUANTUM FIELD THEORY LEARNING SEMINAR
SUNGHYUK PARK
Abstract. These are notes for “Factorization Algebras in Quantum Field Theory” learning seminar. We follow the two-volume book by Costello and Gwilliam [CG1, CG2] as well as Costello’s book [C], almost verbatim in many parts.
Contents 1. Free theories ([CG1] 2.1-2.5, 4.1-4.2)1 1.1. General framework of QFTs1 1.2. Divergence operator2 1.3. Divergence complex5 2. Examples of factorization algebras ([CG1] 4.5, 5.4, ··· )6 3. Interacting theories and renormalization ([C, CG2])7 4. Gauge theories and BV formalism ([C, CG2])7 4.1. Classical BV formalism7 4.2. Quantum BV formalism 11 Appendix A. Some background 12 A.1. Chevalley-Eilenberg complex and Koszul resolution 12 A.2. L∞ algebras ([KS] 3.2) 12 References 13
1. Free theories ([CG1] 2.1-2.5, 4.1-4.2) In this section we’ll see how (pre)factorization algebras naturally arise in quantum field theories.
1.1. General framework of QFTs. What is a quantum field theory? A space-time is a (not necessarily compact) manifold M. Often, fields are sections of a bundle E on M. Let E be the corresponding sheaf of sections. For instance, scalar fields, often denoted by φ, are sections of the trivial line bundle. Often, a quantum field theory is described by an action S, which is a function of fields. If this action is quadratic (kinetic term) and has no higher order terms (interaction terms), then this theory is called free. In path integral formalism, the action determines a (not mathematically well-defined) functional measure e−S/~dφ.
Date: October 2019. 1 FACTORIZATION ALGEBRAS IN QUANTUM FIELD THEORY LEARNING SEMINAR 2
Here, the Planck’s constant ~ is a formal parameter that should be regarded “small”. In the limit ~ → 0, the theory becomes classical, and the physical fields satisfy the Euler-Lagrange equation (equation of motion) describing the critical locus of S. Observables O are, roughly, functions on the space of fields. More precisely, classical observables are functions on the critical locus of S, whereas the meaning of quantum observables will be described in later subsections. The expectation value of an observable O is often formally expressed as Z hOi = O e−S/~dφ. Because this expression itself is usually not well-defined in infinite dimensional case, we take another approach, which is described in the next subsection. So far we have described QFT from Lagrangian point of view, but there is also Hamiltonian point of view. Let’s put the theory on M = Σ × R and look at the moduli space MΣ of germs of classical solutions on a space-like slice Σ (i.e. the space of classical boundary conditions). Whenever the equation of motion is given by an elliptic PDE, MΣ is symplectic. The classical field theory is then reduced to a classical mechanics where fields are functions
f : R → MΣ.
The classical observables are functions on MΣ, and it is equipped with a Poisson bracket. We’ll see later in this seminar that deformation quantization replaces classical observables with quantum observables and the Poisson bracket with commutator in such a way that in the classical limit it recovers the classical picture. 1.2. Divergence operator. 1.2.1. Finite dimensional model. Instead of defining expectation values directly, we can define when two observables have the same expectation values. Let’s consider a finite dimensional model of free scalar fields. Let Rn be the space of fields and consider Gaussian integrals of the form Z Z 1 X n f(x) exp − xiAijxj d x = f(x) ωA x∈Rn 2 x∈Rn where A is a symmetric positive definite matrix and f ∈ P (Rn), the space of polynomial functions on Rn. Let Vect(Rn) be the space of vector fields on Rn with polynomial coefficients. Definition 1 (p.25 of [CG1]). The divergence operator associated to the Gaussian measure ωA is the linear map n n DivωA : Vect(R ) → P (R ) satisfying
LV ωA = (DivωA V ) ωA for any V ∈ Vect(Rn).
Applying the Cartan’s formula LV = dιV +ιV d, we can express the divergence in coordinates. X ∂ 1 X Div f ω = d ιP ∂ exp − x A x dx ∧ · · · ∧ dx ωA i A fi i ij j 1 n ∂xi ∂xi 2 X ∂ 1 X = fk exp − xiAijxj dx1 ∧ · · · ∧ dxn ∂xk 2 k FACTORIZATION ALGEBRAS IN QUANTUM FIELD THEORY LEARNING SEMINAR 3 X ∂ X X ∂fi ⇒ Div f = − f A x + ωA i ∂x i ij j ∂x i i,j i i By Stokes’ theorem, Z Z
(DivωA V ) ωA = d(ιV ωA) = 0
for all V ∈ Vect(Rn). Lemma 1 (Lemma 2.1.1 of [CG1]). The quotient map n n ∼ P (R ) → P (R )/Im DivωA = R is the map sending a function f to its expectation value R n f ωA R hfiA := R n ωA R n Proof. It suffices to show that Im DivωA has codimension 1. By changing basis of R , we may assume that A = I. In this basis it is easy to see that we have a splitting n ∼ P (R ) = Im DivωA ⊕ R n where R ⊂ P (R ) denotes constant polynomials. 1.2.2. Free scalar field theory. Now let’s study infinite dimensional case, the free scalar field theory on a Riemannian manifold (M, g). The space of fields is C∞(M). Set Z 1 2 S(φ) = φ(∆g + m )φ dvolg 2 M To define the corresponding divergence operator, we should first define the space of polynomial ∞ functions and of polynomial vector fields. Observe that every element f ∈ Cc (U) ⊂ Dc(U) defines a linear functional on C∞(U) by Z φ 7→ fφ dvolg. U ∞ Similarly each monomial f1 ··· fn defines a function on C (U) by Z φ 7→ f1(x1)φ(x1) ··· fn(xn)φ(xn) dvolg(x1) ∧ · · · ∧ dvolg(xn). U n ∞ ∞ Hence as the space of polynomial functions, we can take Pe(C (U)) = Sym Cc (U), or rather its completion ∞ M ∞ n P (C (U)) = Cc (U )Sn . n≥0 ∞ The subscript indicates coinvariants (i.e. quotients). Similarly we define Vectg c(C (U)) = ∞ ∞ Pe(C (U)) ⊗ Cc (U), or rather its completion ∞ M ∞ n+1 Vectc(C (U)) = Cc (U )Sn n≥0 where Sn acts on the first n factors. FACTORIZATION ALGEBRAS IN QUANTUM FIELD THEORY LEARNING SEMINAR 4
Definition 2 (Definition 2.2.1 in [CG1]). The divergence operator associated to the quadratic form Z 1 2 S(φ) = φ(∆ + m )φ dvolg 2 U is the linear map ∞ ∞ Div : Vectc(C (U)) → P (C (U)) defined by1 n Z ∂ 2 X f1 ··· fn 7→ −f1 ··· fn(∆ + m )φ + f1 ··· fbi ··· fn fi φ dvolg ∂φ i=1 U and more generally n Z 2 X F (x1, ··· , xn+1) 7→ −(∆n+1 +m )F (x1, ··· , xn+1)+ F (x1, ··· , xi, ··· , xn, xi)dvolg. i=1 xi∈U Definition 3 (Definition 2.2.3 in [CG1]). For an open subset U ⊂ M, define the quantum observables of a free field theory to be H0(Obsq(U)) = P (C∞(U))/Im Div. In next subsection we’ll see that the map Div naturally extends to a cochain complex of quantum observables for which the 0-th cohomology is what we have just defined.
1.2.3. Prefactorization algebra structure on observables. The space of polynomial functions forms a commutative algebra, but Im Div is not an ideal, and H0(Obsq(U)) is not an algebra. Still, H0(Obsq(U)) is a prefactorization algebra. Definition 4. A prefactorization algebra F on M is an assignment U 7→ F(U) of vector spaces to each open subset, along with linear maps (“structure maps”)
⊗1≤i≤nF(Ui) → F(V ) whenever U1, ··· ,Un ⊂ V are disjoint. The structure maps should satisfy obvious compati- bility conditions. In other words, it is a functor (of multicategories) from DisjM to Vect. In general Vect can be replaced by any symmetric monoidal category, or even a multicategory.
Lemma 2 (Lemma 2.3.1 in [CG1]). If U1,U2 are disjoint open subsets of V ⊂ M, then we have a map ∞ ∞ ∞ P (C (U1)) ⊗ P (C (U2)) → P (C (V )) obtained by inclusion followed by product. This map does descends to a map2 0 q 0 q 0 q H (Obs (U1)) ⊗ H (Obs (U2)) → H (Obs (V )). Proof. We have the following diagram :
1 ∂ ∂φ denotes the constant-coefficient vector field given by infinitesimal translation in the direction φ in ∞ ∂ R C (U). Hence it is linear in φ, which may be a bit confusing because of the notation. That ∂φ f = U fφ dvolg is clear if we consider Dirac delta functions. 2Note that the product map itself does not descend to the level of observables. FACTORIZATION ALGEBRAS IN QUANTUM FIELD THEORY LEARNING SEMINAR 5
∞ ∞ q12 0 q 0 q ker q12 P (C (U1)) ⊗ P (C (U1)) H (Obs (U1)) ⊗ H (Obs (U2))
? q Im Div P (C∞(V )) H0(Obsq(V ))
0 q It suffices to show that the image of ker q12 in H (Obs (V )) is zero, or equivalently, that its ∞ image in P (C (V )) lies in Im DivV . Note that ker q12 is the image of the following map.
1⊗Div +Div ⊗1 ∞ ∞ ∞ ∞ U2 U1 ∞ ∞ P (C (U1))⊗Vectc(C (U2))⊕Vectc(C (U1))⊗P (C (U2)) −−−−−−−−−−−→ P (C (U1))⊗P (C (U2))
∞ ∞ Let F ∈ P (C (U1)) and X ∈ Vectc(C (U2)). Then F Div X = Div (FX) − XF = Div (FX)
because F and X has disjoint support! This proves the lemma.
Similarly, for disjoint open subsets U1, ··· ,Un of V ⊂ M, the product map ∞ ∞ ∞ P (C (U1)) ⊗ · · · ⊗ P (C (Un)) → P (C (V )) descends to a map
0 q 0 q 0 q H (Obs (U1)) ⊗ · · · ⊗ H (Obs (Un)) → H (Obs (V )), and these structure maps satisfy compatibility conditions to define a prefactorization algebra.
1.2.4. From quantum to classical. These quantum observables should be thought of as deformation of classical observables which are functions on the space of solutions to the
equation of motion. Consider action S/~ instead of S. Let Div~ be the corresponding divergence operator. Note that ~Div~ has the same image as long as ~ 6= 0, but when ~ = 0 the ~Div~ becomes ∂ ∂S f ··· f 7→ −f ··· f . 1 n ∂φ 1 n ∂φ The image is precisely the topological ideal generated by ∂S I (U) = : φ ∈ C∞(U) EL ∂φ c
0 cl ∞ Hence H (Obs (U)) = P (C (U))/IEL(U) is the space of polynomial functions on the critical locus of S, and it forms a commutative prefactorization algebra.
1.3. Divergence complex.
1.3.1. Finite dimensional model. Let’s again start from a finite dimensional model. Let M be a smooth n-manifold (modelling the space of fields) and let ω0 be a smooth measure on −S/~ M. Let S be a function on M. The divergence operator for the measure e ω0 is the map ∞ Div~ : Vect(M) → C (M) 1 X 7→ − XS + Divω0 X. ~ FACTORIZATION ALGEBRAS IN QUANTUM FIELD THEORY LEARNING SEMINAR 6
−S/~ n−1 By contracting with the volume form e ω0, we can identify Vect(M) with Ω (M) and C∞(M) with Ωn(M) as follows.
−S/~ X ↔ ιX e ω0 −S/~ φ ↔ φ e ω0. Under this identification, the divergence operator is simply the de Rham operator Ωn−1(M) −−→ddR Ωn(M). Hence, we can define the divergence complex analogous to the de Rham complex. Definition 5 (p.90 of [CG1]). Let PVi(M) = C∞(M, ∧iT M) be the space of polyvector fields on M. The divergence complex is the complex Div Div · · · → PV2(M) −−→~ PV1(M) −−→~ PV0(M) i i−1 where the differential Div~ : PV (M) → PV (M) is defined so that the diagram Div PVi(M) ~ PVi−1(M)
−S/ −S/ ∨e ~ω0 ∨e ~ω0 d Ωn−i(M) dR Ωn−i+1(M) commutes. Here ∨ denotes contraction. In other words, 1 Div~ = ∨(− dS) + Divω0 . ~
By considering ~Div~ instead, we see that there is a family of cochain complexes over C[~]. The family is isomorphic to the divergence complex when ~ =6 0, and when ~ = 0, it becomes ∨(− 1 dS) ∨(− 1 dS) · · · → PV2(M) −−−−−→~ PV1(M) −−−−−→~ PV0(M).
∨(− 1 dS) Note that the image of PV1(M) −−−−−→~ PV0(M) is the ideal cutting out the critical locus. Indeed, the whole complex is the Koszul complex cutting out the derived critical locus. 1.3.2. Infinite dimensional case. (Chapter 4 of [CG1]) Construction using factorization envelope, more on classical and quantum observables, their prefactorization algebra structure, pushforward and Hamiltonian viewpoint
2. Examples of factorization algebras ([CG1] 4.5, 5.4, ··· )
Definition 6. A cover {Ui}i∈I of U is called a Weiss cover if for any finite subset F ⊂ U, there is i ∈ I such that F ⊂ Ui. Definition 7. A prefactorization algebra F is a factorization algebra if it satisfies the descent condition with respect to Weiss topology. That is, for any Weiss cover {Ui}i∈I the following is exact : ⊕i,j∈I F(Ui ∩ Uj) ⇒ ⊕i∈I F(Ui) → F(U) A fundamental example of a factorization algebra is the Sym of a cosheaf. Sym of a cosheaf, factorization envelope, En-universal enveloping algebra homolorphically translation-invariant factorization algebras and vertex algebras FACTORIZATION ALGEBRAS IN QUANTUM FIELD THEORY LEARNING SEMINAR 7
3. Interacting theories and renormalization ([C, CG2]) Our discussion so far was only on free theories. The divergence operator is not well-defined when there is an interaction term. This is analogous to the fact in physics that the naive Feynman diagram calculation for interacting theories blow up. Renormalization tells us that we should instead consider a family of effective theories for each length scale (or energy scale), and these effective field theories are related to each other via renormalization group flow.[ C] 2-4, φ4 theory as an example, divergence operator for each length scale
4. Gauge theories and BV formalism ([C, CG2]) In this section, we review Batalin-Vilkovisky (BV) formalism, a powerful way of quantizing gauge theories. In the classical setting, the BV formalism tells us that we should take the derived critical locus of the action. As we will work perturbatively (infinitesimally close to a given solution), our main object of study will be a formal (elliptic) moduli problem with a degree −1 symplectic form. The classical observables form a commutative factorization algebra which turns out to be weakly equivalent to a P0 factorization algebra. In the context of deformation quantization, this P0 factorization algebra will be quantized to a BD factorization algebra. 4.1. Classical BV formalism. 4.1.1. Finite dimensional model. In guage theory, one is interested in making sense of functional integrals of the form Z eS/~ V/G The BRST construction tells us that we should replace V/G with the derived quotient; that is, functions on V/G will be derived g-invariant functions on V O(V/G) = C∗(g, O(V )) which is the Chevalley-Eilenberg complex. As graded vector spaces,3 ∗ C∗(g, O(V )) = Symd (g[1] ⊕ V ) and thus V/G can be identified with a differential graded (dg) manifold whose underlying graded manifold is g[1] ⊕ V. The Chevalley-Eilenberg differential is a degree 1 vector field of square 0 on g[1] ⊕ V . Let’s denote this vector field by X. Now that we have the derived quotient, how do we get the critical locus of S? The BV formalism tells us that we should take the derived critical locus. Recall that in derived algebraic geometry, if S is a function on M, the derived critical locus is the derived intersection of Γ(dS) with the 0-section of the cotangent bundle of M. Hence the ring of functions on the derived critical locus of S is the commutative dg algebra h L O(Crit (S)) = O(Γ(dS)) ⊗O(T ∗M) O(M). As an explicit model, we can take the Koszul resolution of O(M) as a module over O(T ∗M): ∨dS ∨dS ∨dS O(Crith(S)) ∼= ··· −−→ Γ(M, ∧2TM) −−→ Γ(M,TM) −−→O(M).
3Hereˆdenotes the dual space. FACTORIZATION ALGEBRAS IN QUANTUM FIELD THEORY LEARNING SEMINAR 8
As graded vector spaces, h ∼ ∗ O(Crit (S)) = SymO(M)(Γ(M,TM)[1]) =: O(T [−1]M). Note that O(T ∗[−1]M) = Γ(M, ∧∗TM) has a degree 1 Poisson bracket called the Schouten- Nijenhuis bracket. The Poisson bracket is given by {X,Y } = [X,Y ] {X, f} = Xf {f, g} = 0 for f, g ∈ O(M) and X,Y ∈ Γ(M,TM), and other brackets are inferred from the Leibniz rule. The differential on O(T ∗[−1]M) corresponding to that on O(Crith(S)) is given by dφ = {S, φ}. The derived critical locus thus has a degree −1 symplectic form. Back to our derived critical locus on a derived quotient, we need to take the shifted cotangent bundle E = T ∗[−1](g[1] ⊕ V )4 so that E = g[1] ⊕ V ⊕ V ∨[−1] ⊕ g∨[−2].5 The (action) function S on g[1]⊕V pulls back to a function E. The vector field X on g[1]⊕V extends to a vector field on E (again, coming from the Chevalley-Eilenberg differential). The identity [X,X] holds on E as well. X continues to preserve S on E. E is a dg manifold equipped with a degree −1 symplectic form, and the degree 1 vector field X preserves the symplectic form. Thus there is a degree 0 function hX on E whose Hamiltonian vector field is X, and which vanishes at the origin in E. The statement [X,X] translates into {hX , hX } = 0, and XS = 0 becomes {hX ,S} = 0. Finally, {S,S} = 0 is automatic. Therefore, the function S + hX satisfies the BV classical master equation :
{S + hX ,S + hX } = 0. The BV action is defined to be
SBV := S + hX .
We can split SBV itno kinetic and interacting parts as 1 S (e) = he, Qei + I (e) BV 2 BV where Q : E → E is a degree 1 linear map, skew self-adjoint for the pairing h−, −i, and IBV consists of terms which are at least cubic. The classical master equation implies Q2 = 0 and 1 QI + {I ,I } = 0. BV 2 BV BV The functional integral we started with now becomes
Z Z 1 1 SBV (e)/~ he,Qei+ IBV (e) e = e 2~ ~ e∈L e∈L
4The notation T ∗[−1]M denotes the manifold M equipped with the graded-commutative algebra
Sym ∞ (Γ(M,TM)[1]) as its ring of functions. CM 5In physics language, g[1] is the space of ghosts, V is the space of fields, V ∨[−1] is the space of antifields, and g∨[−2] is the space of antighosts. FACTORIZATION ALGEBRAS IN QUANTUM FIELD THEORY LEARNING SEMINAR 9 where L ∈ E is a small generic Lagrangian perturbation of the zero section g[1] ⊕ V ⊂ E. This subspace can be described by a choice of a gauge fixing operator. 4.1.2. Classical field theory. Definition 8 (Definition 5.4.0.3-4 in [CG2]). A pre-classical field theory on a manifold M consists of a Z-graded vector bundle E on M, whose global sections will be denoted E, equipped with a degree −1 symplectic pairing, and a local functional
S ∈ Oloc(Ec(M)) of cohomological degree 0, satisfying the following properties. (1) S satisfies the classical master equation {S,S} = 0. (2) S is at least quadratic, so that 0 ∈ Ec(M) is a critical point of S. In this situation we can write S as a sum 1 S(e) = he, Qei + I(e) 2 where Q : E → E is a skew self-adjoint differential operator of cohomological degree 1 and square zero. A pre-classical field theory is a classical field theory if the complex (E,Q) is elliptic. Example 1 (Free massless scalar field theory). Let (M, g) be a Riemannian manifold and let DensM be the density line bundle. The volume form dvolg provides an insomorphism between DensM and the trivial line bundle R. Let 1 Z S(φ) = φDφ 2 M
denote the action functional for the free massless field theory on M. Here Dφ = (∆g)dvolg. The derived critical locus of S is described by the elliptic L∞ algebra L = C∞(M)[−1] −→D Dens(M)[−2]. Example 2 (Interacting scalar field theory). Let’s consider the φ4 theory with action functional 1 Z 1 Z S(φ) = φDφ + φ4. 2 M 4! M The cochain complex underlying the elliptic L∞ algebra is the same as before L = C∞(M)[−1] −→D Dens(M)[−2].
The interacting term gives rise to a higher bracket l3 on L, given by C∞(M)⊗3 → Dens(M)
φ1 ⊗ φ2 ⊗ φ3 7→ φ1φ2φ3dvolg. Example 3 (Chern-Simons theory). Let M be an oriented 3-manifold. The space of fields is Ω1(M) ⊗ g. The Lie algebra of infinitesimal gauge symmetries is Ω0(M) ⊗ g which act by A 7→ [X,A] + dX 1 0 1 where A ∈ Ω (M) ⊗ g and X ∈ Ω (M) ⊗ g. The Chern-Simons action SCS on Ω (M) ⊗ g is defined by Z 1 1 SCS(A) = hA, dAi + hA, [A, A]i. M 2 6 FACTORIZATION ALGEBRAS IN QUANTUM FIELD THEORY LEARNING SEMINAR 10
Here, h−, −i is the pairing on Ω∗(M) ⊗ g defined by Z hω1 ⊗ E1, ω2 ⊗ E2i = ω1 ∧ ω2hE1,E2ig. M The BV space of fields is then E = Ω∗(M) ⊗ g[1] The degree −1 symplectic pairing is given by the pairing above. The BV action is 1 1 S(e) = he, dei + he, [e, e]i = S + S 2 6 CS Gauge where 1 S = h[X,X],X∨i + hdX + [X,A],A∨i. Gauge 2 The first term is due to the Lie bracket on ghosts, and the second term is due to the action of ghosts on fields. Example 4 (Yang-Mills theory). Let M be a 4-manifold equipped with a metric (actually, we only need a conformal structure). Let g be a semi-simple Lie algebra, and fix an invariant 1 pairint h−, −ig. The space of fields is Ω (M) ⊗ g, and the Lie algebra of infinitesimal gauge symmetries is Ω0(M) ⊗ g. The action of this Lie algebra is A 7→ [X,A] + dX where A ∈ Ω1(M) ⊗ g and X ∈ Ω0(M) ⊗ g. 1 The Yang-Mills action SYM on Ω (M) ⊗ g is defined by
SYM (A) = hF (A)+,F (A)+i 1 2 where F (A)+ = 2 (1 + ∗)F (A) ∈ Ω+(M) ⊗ g. We’re interested in the functional integral of the form Z eSYM (A)/~ A∈Ω1(M)⊗g/G Applying the BV formalism, the resulting graded symplectic vector space of BV fields is E = Ω0(M) ⊗ g[1] ⊕ Ω1(M) ⊗ g ⊕ Ω3(M) ⊗ g[−1] ⊕ Ω4(M) ⊗ g[−2] The degree -1 symplectic pairing is the obvious pairing given by Z hω1 ⊗ E1, ω2 ⊗ E2i = ω1 ∧ ω2hE1,E2ig. M The BV procedure gives us an action
S = SYM + SGauge ∈ Oloc(E) which satisfies the classical master equation {S,S} = 0. Explicitly, 1 S = h[X,X],X∨i + hdX + [X,A],A∨i Gauge 2 where X is a ghost, A is a field, A∨ is an anti-field and X∨ is an anti-ghost. The quadratic part of S induces the differential Q of square-zero described by Ω0(M) ⊗ g −→d Ω1(M) ⊗ g −−→d∗d Ω3(M) ⊗ g −→d Ω4(M) ⊗ g. FACTORIZATION ALGEBRAS IN QUANTUM FIELD THEORY LEARNING SEMINAR 11
4.1.3. Observables of of a classical field theory. Definition 9. The classical observables with support in the open subset U is the commutative dg algebra Obscl(U) = O(E(U)) equipped with the differential {S, −}. We denote by the corresponding factorization algebra of classical observables by Obscl.
Theorem 1 (Theorem 6.2.0.2 in [CG2]). For any classical field theory on M, there is a P0 cl factorization algebra Obsg , together with a weak equivalence of commutative factorization algebras cl Obsg ∼= Obscl. 4.2. Quantum BV formalism. 4.2.1. Free theory in BV formalism. We first state a free theory on a compact manifold M in the BV formalism. We’ll later see how BV formalism can be combined with renormalization. Definition 10. A free theory on M is described by the following data : (1) A Z-graded vector bundle E on M, equipped with anti-symmetric map of vector bundles on M of cohomological degree −1,
h−, −iloc : E ⊗ E → Dens(M) that is non-degenerate on each fiber. (2) A degree 1 differential operator Q : E → E, which is of square zero, and is skew self-adjoint for the pairing. Further, (E,Q) should be an elliptic complex. Example 5 (Free massless scalar field theory). The equation of motion in this theory is Dφ = 0 where D : C∞(M) → C∞(M) is the non-negative Laplacian on M. The derived moduli space of solutions to the equations of motion is E = C∞(M) −→D C∞(M)[−1] concentrated in degrees 0 and 1. The degree −1 symplectic pairing on E is Z 0 1 0 1 hφ , φ i = φ φ dvolg M where φ0 ∈ C∞(M)[0] and φ1 ∈ C∞(M)[−1]. Example 6 (Abelian Chern-Simons theory). Let g be an Abelian Lie algebra with a non- degenerate invariant pairing, and let M be a compact 3-manifold. In the Chern-Simons theory with gauge Lie algebra g, the fields are Ω1(M) ⊗ g. The Lie algebra of gauge symmetries is Ω0(M) ⊗ g. It acts on the space of fields in an affine-linear fasion : X(A) = A + dX for A ∈ Ω1(M) ⊗ g and X ∈ Ω0(M) ⊗ g. Hence the derived quotient is the two-term complex Ω0(M) ⊗ g −→d Ω1(M) ⊗ g R concentrated on degrees −1 and 0. The action on the space of fields is M hA, dAig. Thus, the derived moduli space of solutions to the equation of motion is the 4-term complex E = Ω0(M) ⊗ g → Ω1(M) ⊗ g → Ω2(M) ⊗ g → Ω3(M) ⊗ g FACTORIZATION ALGEBRAS IN QUANTUM FIELD THEORY LEARNING SEMINAR 12
concentrated in degrees −1, 0, 1, 2. The differential is the de Rham differential. Hence we can write E = Ω∗(M) ⊗ g[1]. The symplectic pairing on E is given by Z 1 2 1 2 hω ⊗ X1, ω ⊗ X2i = ω ∧ ω hX1,X2ig M 1 1 2 2 where ω ∈ Ω (M), ω ∈ Ω (M) and Xi ∈ g[1].
4.2.2. Quantum field theory. The divergence operator is DivS = ∆ + {S, −}, where the BV n Laplacian ∆ is basically Divω0 in R . It satisfies the quantum master equation 2 DivS = 0 The space of solutions to QME = the space of BD quantizations 4.2.3. Observables in a quantum field theory. BD factorization algebra
Appendix A. Some background A.1. Chevalley-Eilenberg complex and Koszul resolution. The Chevalley-Eilenberg complex for Lie algebra cohomology of a g-module M is C∗(g,M) := (Sym(g∨[−1]) ⊗ M, d) where d acts as k X k i j X k l d(e ⊗ m) = − e ([ei, ej])e ∧ e ⊗ m + e ∧ e ⊗ [el, m] i
A.2. L∞ algebras ([KS] 3.2). Let V be a Z-graded vector space. Let’s denote by C(V ) the cofree cocommutative coassociative coalgebra without counit generated by V .
Definition 11. An L∞ algebra is a pair (V,Q) where V is a Z-graded vector space and Q is a differential on the graded coalgebra C(V [1]). Consider the composition
n Q projection ⊕n≥1S (V [1]) = C(V [1]) −→ C(V )[2] −−−−−→ V [2]. This gives rise to a collection of morphisms of graded vector spaces n Qn : S (V [1]) → V [2], which, under Sn(V [1]) ' ∧n(V )[n], give rise to higher brackets n ln : ∧ (V ) → V [2 − n]. FACTORIZATION ALGEBRAS IN QUANTUM FIELD THEORY LEARNING SEMINAR 13
References [C] Kevin Costello. Renormalization and effective field theory, volume 170 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2011. [CG1] Kevin Costello and Owen Gwilliam. Factorization algebras in quantum field theory. Vol. 1, volume 31 of New Mathematical Monographs. Cambridge University Press, Cambridge, 2017. [CG2] Kevin Costello and Owen Gwilliam. Factorization algebras in quantum field theory. Vol. 2. [KS] Maxim Kontsevich and Yan Soibelman. Deformation theory. I.