Calabi-Yau algebras

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by  Victor Ginzburg

CALABI-YAU ALGEBRAS
7
0 VICTOR GINZBURG
0
2

n
a
Abstract. We introduce some new algebraic structures arising naturally in the geom-
J
etry of CY manifolds and mirror symmetry. We give a universal construction of CY
9
algebras in terms of a noncommutative symplectic DGalgebra resolution.

In dimension 3, the resolution is determined by a noncommutative potential. Repre-
]
G sentation varieties of the CY algebra are intimately related to the set of critical points,
A and to thesheaf of vanishingcycles of thepotential. Numerical invariants, like ranksof
cyclichomologygroups,areexpectedtobegivenby‘matrixintegrals’overrepresentation
.
h varieties.
t We discuss examples of CY algebras involving quivers, 3-dimensional McKay corre-
a
m spondence, crepant resolutions, Sklyanin algebras, hyperbolic 3-manifolds and Chern-
Simons. Examples related to quantumDel Pezzo surfaces are discussed in [EtGi].
[

3
v Table of Contents
9
3 1. Algebras and potentials
1 2. Representation functor, critical points and vanishing cycles
2 3. Calabi-Yau condition
1
4. Quiver algebras and McKay correspondence in dimension 3
6
5. Calabi-Yau algebras of dimension 3
0
6. Fundamental groups, Chern-Simons, and the Weil representation
/
h 7. Calabi-Yau algebras and Calabi-Yau manifolds
t 8. Noncommutative Hessian
a
m 9. Some homological algebra
:
v
1. Algebras and potentials
i
X
1.1. Introduction. Inthispaper,westudysomenewalgebraicstructures,suchasCalabi-
r
a Yau (CY) algebras, arising naturally in the geometry of CY manifolds. The ultimate goal
of introducing CY algebras is to transplant most of conventional CY geometry to the
setting of noncommutative geometry. Some motivation for this, to be explained in §3.1,
comes from mirror symmetry. Besides that, it will be demonstrated by numerousconcrete
examples that CY algebras do arise ‘in nature’. Furthermore, CY algebras enjoy quite
intriguinghomological propertieswhich areclosely related toalgebro-geometric properties
of the corresponding representation schemes, see §2.
In this section 1, we begin with most essential, elementary algebraic constructions, and
introduce them in as ‘ground to earth’ a way as possible. Thus, our exposition won’t be
strictlylogical; thedefinitionsandconstructionsof§1willnotbegiveninfullgenerality. A
moregeneral andmoreconceptual approach, as well as motivation for theseconstructions,
will be provided in subsequent sections 2, 3, and 5.
1
Among our most important results, we would like to mention Theorem 3.4.3, Theorem
3.6.4, Theorem 4.4.6, and Theorem 5.3.1.
ThepointofviewonCalabi-Yaualgebrasadvocatedinthispaperisnottheonlypossible
one. The reader is referred to [Bo], [CR], [IR], and [KR] for alternative approaches.
1.2. Acknowledgements. IamverymuchindebtedtoMaximKontsevichforgenerouslysharing
with me his unpublished ideas. Maxim Kontsevich has read a preliminary version of this paper
and made a number of very interesting comments which are incorporated in the present version.
IhavebenefitedalotfromusefuldiscussionsandE-mailexchangeswithKevinCostello,Raphael
Rouquier, and Michel Van den Bergh. Thanks are also due to Pavel Etingof, Dmitry Kaledin,
Nikita Nekrasov,Viktor Ostrik, Travis Schedler, Amnon Yekutiely, and James Zhang.
1.3. Algebras defined by a potential. Wewillworkwithunitalassociative C-algebras.
Given such an algebra A, let [A,A] denote the C-vector subspace in A spanned by the
commutators and write A = A/[A,A] for the commutator quotient space.
cyc
LetF = Chx ,...,x ibeafree associative algebra withn generators. Thevector space
1 n
F has an obvious basis labelled by cyclic words in the alphabet x ,...,x .
cyc 1 n
For each j = 1,...,n, M. Kontsevich [Ko1] introduced a linear map ∂ : F → F,
∂xj cyc
Φ 7→ ∂Φ, as follows. Given a cyclic word Φ = x x ...x , one finds all occurrences
∂xj i1 i2 ir
of the variable x in Φ. Deleting one such occurrence of x breaks up the cycle Φ, thus
j j
creating an ordinary (not cyclic) word. We carry out this procedure with each occurrence
of the variable x in Φ, one at a time. The element ∂Φ is defined to be the sum of all the
j ∂xj
resulting words. More formally, we put
∂Φ
:= x x ...x x x ...x ∈ Chx ,...,x i, (1.3.1)
∂x is+1 is+2 ir i1 i2 is−1 1 n
j
{s|Xis=j}
and extend this definition to linear combinations of cyclic words by C-linearity.
Thus, given any element Φ ∈ F , to be referred to as potential, we have a well defined
cyc
collection of elements ∂Φ ∈F, i = 1,...,n.
∂xi
A key object of interest for us is an associative algebra
A(F,Φ) := Chx ,...,x i ((∂Φ/∂x )) , (1.3.2)
1 n i i=1,...,n
aquotientofthefreealgebraF bythetwo-sided(cid:14)idealgeneratedbyallnpartialderivatives
of the potential Φ.
Example 1.3.3 (Basic example). Let F = Chx,y,zi, and let Φ = xyz − yxz ∈ F , a
cyc
difference of two cyclic words. We compute
∂Φ ∂Φ ∂Φ
= yz−zy, = zx−xz, =xy−yx.
∂x ∂y ∂z
Therefore, takingthequotient modulotheideal generated bytheaboveelements produces
a polynomial algebra, i.e., we get
A(Chx,y,zi,Φ) = C[x,y,z]. (1.3.4)
2
Akeyfeatureoftheaboveexampleisthatweworkwithapolynomialalgebra, C[x,y,z],
in 3 variables. The algebra C[x,y,z] is, in effect, a basic example of CY algebra of
dimension 3. One of the main messages of this paper is that, roughly speaking:
Any Calabi-Yau algebra of dimension 3 ‘arising in nature’ is defined
by a potential, i.e. has the form A(F,Φ).
We refer to §3.2 for the definition of CY algebras, and to §5.3 for a more precise version
of the claim above.
Remark 1.3.5. (i) Not every algebra of the form A(F,Φ), where F = Chx ,...,x i, is
1 n
a CY algebra of dimension 3. There seems to be no simple characterisation of those
potentials Φ for which A(F,Φ) is a CYalgebra of dimension 3, cf. however Theorem 5.3.1.
(ii) It should be emphasized that, for a given algebra A, its presentation in the form
A = A(F,Φ) is by no means determined by the algebra itself. A given algebra may have
many different presentations involving different free algebras F and different potentials Φ.
(iii)LetAut(F)bethegroupofalgebraautomorphismsf :x 7→ f(x ),...,x 7→ f(x ),
1 1 n n
of the free algebra F = Chx ,...,x i. The group Aut(F), as well as the Lie algebra
1 n
Der(F,F) ofallderivationsF → F,acts naturally onthecommutator quotient spaceF .
cyc
Potentials from the same Aut(F)-orbit clearly give rise to isomorphic algebras A(F,Φ).
A potential Φ ∈ F is said to be isolated if the corresponding Aut(F)-orbit is ‘in-
cyc
finitesimally open’ in the sense that we have Der(F,F)(Φ) = F . It is an interesting
cyc
open problem to describe all (Aut(F)-orbits of) isolated potentials Φ such that A(F,Φ)
is a CY algebra of dimension 3. It seems likely that the CY algebras arising in the theory
of cluster categories, see [IR], [KR], provide examples of such CY algebras of dimension 3
associated to an isolated potential. ♦
Below, we provide a few more interesting examples of algebras defined by a potential.
The philosophy behind them is to search for deformations of the polynomial algebra, like
in Example 1.3.3, obtained by an appropriate deformation of the potential.
Example 1.3.6. Fix q ∈ C and let Φ = xyz − q · yxz − f for some f ∈ F , where
cyc
F = Chx,y,zi, as before. Then, the corresponding algebra A(F,Φ) is a quotient of the
free algebra Chx,y,zi by the relations
xy−q·yx = ∂f/∂z, zx−q·xz = ∂f/∂y, yz−q·zy = ∂f/∂x. (1.3.7)
In particular, put
Φ = xyz−q·yxz+ 1(x2+y2+z2).
q 2
For q = 1, we get A(F,Φ )= U(sl ), the enveloping algebra of the Lie algebra sl .
1 2 2
As another example, in (1.3.7), putf ∈C[x], a polynomial independentof the variables
y andz. Further, take q tobeaprimitiven-th rootofunity. Thenthealgebra A(F,Φ)has
a large center, Z(F,Φ) := Z(A(F,Φ)). Specifically, the elements x := xn, y := yn, z := zn
are central.
Assume in addition that, in the expansion f = d a · xr, the coefficient a ∈
r=1 r r
C vanishes whenever n|r. Then, the center Z(F,Φ) contains an extra element u :=
d r·ar ·xr. One can show that the center is, inPeffect, generated by the elements
r=1 1−qr
x,y,z,u. Moreover, wehaveZ(F,Φ) ∼= C[x,y,z,u]/((x·y·z−φ(x,u))),whereφ∈ C[x,u]
P
3
is a certain (complicated) polynomial in two variables which is determined byf. For more
discussion, cf. Example 3.5.7 and also [EtGi].
Example 1.3.8 (Sklyanin algebras). Keep F = Chx,y,zi. Given a triple a,b,c ∈ C, we put
Φ = axyz+byxz+c(x3+y3+z3).
The corresponding algebra A(F,Φ) is called quadratic Sklyanin algebra. The original
definition in [AS], [ATV], [VdB2] was not given in this form but is equivalent to ours, as
can be readily seen from e.g. [VdB2, formula (8)].
The cubic Sklyanin algebra (of type A) may be defined as an algebra of the form
A(Chx,yi,Φ), Φ = ax2y2+bxyxy+c(x4+y4).
If a6= b, the same algebra has the following different presentation, see [Ma],
A(Chx,y,zi, Φ ), Φ = 1z2+xyz−yxz+p(x2y2+xyxy)+q(x4+y4).
p,q p,q 2
Here, the parameters p,q ∈ C are related to parameters a,b,c by the equations a =
−r(p+1), b= r(2−p), c = −rq, for some r 6= 0.
Note that, for p = q = 0, the algebra A(F,Φ ) degenerates to the enveloping algebra
p,q
of the 3-dimensional HeisenbergLie algebra such that z is a central element and [x,y] =z.
1.4. A DG algebra. We will denote the grading on a DG algebra by either upper or
lower index dependingon whether the differential in the DG algebra has degree +1 or −1,
respectively. From now until the end of §1, we let F = Chx ,...,x i.
1 n
We will view the algebra F as the degree zero component in a free graded algebra
D = Chx ,...,x ,θ ,...,θ ,ti, degt = 2, degθ = 1, j = 1,...,n. (1.4.1)
1 n 1 n j
Any potential Φ ∈ F gives rise to a differential d :Dq → Dq defined as follows
cyc −1
n
d : t 7→ [x ,θ ], θ 7→ ∂Φ/∂x , x 7→ 0, ∀j = 1,...,n. (1.4.2)
j j j j j
j=1
X
This assignment on generators can beuniquely extended to an oddsuper-derivation on D.
Using Proposition 1.5.13(i) below, one checks that d2 = 0. We write Dq(F,Φ), d for
the resulting DG algebra. It is immediate from formulas (1.4.2) that the zeroth homology
(cid:0) (cid:1)
of D(F,Φ) is nothing but the algebra defined by our potential Φ; thus, one has a diagram
D(F,Φ) = r≥0Dr(F,Φ) // // D0(F,Φ) // // H0 D(F,Φ), d = A(F,Φ). (1.4.3)
ThegeometriLcmeaningoftheDGalgebraDq(F,Φ)w(cid:0)illbeclarifi(cid:1)edin§2.8. Specifically,
let G be a Lie group and X a G-manifold. It will explained that any G-invariant smooth
function φ on X gives rise to a complex, called Batalin-Vilkovisky complex, whose coho-
mology arerelated tocritical pointsofthefunctionφ. Wewillshowthattheabovedefined
DG algebra Dq(F,Φ) is nothing but a noncommutative version of the Batalin-Vilkovisky
complex.
One of the main results of this paper says that (a suitable completion of) the DG
algebra Dq(F,Φ), d isacyclicinpositivedegrees,thatis,theprojection(1.4.3)isaquasi-
isomorphism, iff A(F,Φ) is a CY algebra of dimension 3, cf. Theorem 5.3.1. This result
(cid:0) (cid:1)
is at the origin of an especially nice homological behavior of CY algebras of dimension 3.
4
1.5. Noncommutative calculus. To proceed further, we need to introduce a few basic
concepts of noncommutative geometry.
Write⊗ = ⊗C. ForanyalgebraA,thespaceA⊗Ahastwocommuting(andisomorphic)
A-bimodule structures, called the outer, resp., inner, bimodule structure. These two
bimodule structures are given by
a(b′⊗b′′)c := (ab′)⊗(b′′c), resp., a(b′⊗b′′)c := (b′c)⊗(ab′′), ∀b′,b′′,a,c ∈ A. (1.5.1)
We will view A⊗A as a bimodule with respect to the outer structure, unless specified
out
otherwise; thus A⊗A = A ⊗ A is a rank one free A-bimodule.
Let A-Bimod be the category of A-bimodules. There is a contravariant duality functor
Hom (−,A⊗A) : A-Bimod −→ A-Bimod, M 7−→ M∨. (1.5.2)
A-Bimod
Here the target bimodule A⊗A is taken with respect to the outer structure. The inner
structure on A⊗A survives in the Hom-space, making it an A-bimodule again.
ThebimoduleΩ1A, of noncommutative differentials, is definedas thekernel of multipli-
cation map µ : A⊗A → A. The dual, DerA := (Ω1A)∨, is called the bimodule of double
derivations, a noncommutative counterpart of the space of vector fields on a manifold.
Elements of DerA may be identified with derivations A → A⊗A as follows. First of
all, there is a canonically defined 1 distinguished double derivation (de Rham differential)
δ :A → Ω1A ֒→ A⊗A, a7→ da:= 1⊗a−a⊗1 ∈ Ω1A⊂ A⊗A. (1.5.3)
Now, the double derivation corresponding to an element θ ∈ Hom (Ω1A,A⊗A) is
A-Bimod
given by the assignment A ∋ a7→ θ(da).
Associated with each element a′ ⊗ a′′ ∈ A ⊗ A, there is an inner double derivation
ad(a′⊗a′′) :u 7→ ua′⊗a′′−a′⊗a′′u. By definition, one has δ = ad(1⊗1).
Clearly, we have (A o⊗ut A)∨ = A⊗in A. The duality functor, M 7→ M∨, interchanges  ,
A
the tautological A-bimodule imbedding below, with the A-bimodule map ad : a′ ⊗a′′ 7→
ad(a′⊗a′′), as follows
 : Ω1A ֒→ A o⊗ut A duality // ad = ∨ : A⊗in A→ DerA. (1.5.4)
A A
(1.5.2)
Example 1.5.5. The bimodule Ω1F of 1-forms for the free algebra F = Chx ,...,x i is a
1 n
rank n free F-bimodule with basis dx ,...,dx . In this basis, one can write the double
1 n
derivation (1.5.3), for A = F, in the form
n n
∂f ′ ∂f ′′
δ : F −→ Ω1F = F·dx ·F, f 7→ df = ·dx · . (1.5.6)
j ∂x j ∂x
j j
Mj=1 Xj=1(cid:16) (cid:17) (cid:16) (cid:17)
For each j = 1,...,n, the corresponding term in the sum in the RHS of the above
′ ′′
formula determines a certain element ∂f ⊗ ∂f ∈ F ⊗F. Here and elsewhere, we
∂xj ∂xj
use Sweedler’s notation and write u′⊗(cid:16)u′′ in(cid:17)stead(cid:16)of (cid:17) u′ ⊗u′′ for an element in a tensor
i i i
product.
P
1Note that thereis nocanonical ordinary derivation A→A.
5
Thus, one obtains a collection of maps
∂ ∂f ′ ∂f ′′
: F → F ⊗F, f 7→ ⊗ , j = 1,...,n. (1.5.7)
∂x ∂x ∂x
j j j
(cid:16) (cid:17) (cid:16) (cid:17)
It is immediate that each of these maps is a double derivation; moreover, these double
derivations form a basis of DerF, dual to the basis {dx , j = 1,...,n} on 1-forms, i.e.,
j
n
∂ ∂ 1⊗1 if j = i
DerF = F· ·F, (x )= (1.5.8)
i
∂xj ∂xj (0 if j 6= i.
j=1
M
In terms of this basis, the distinguished double derivation (1.5.3) reads
n n
∂ ∂ ∂
δ = ad(1⊗1) = x · − ·x = x , . (1.5.9)
i ∂x ∂x i i ∂x
i i i
Xj=1(cid:16) (cid:17) Xj=1h i
Here, for any F-bimodule M and elements x∈ F, m ∈ M, we write [x,m] := xm−mx.
Remark 1.5.10. We use the same notation ∂ both in (1.5.7) and in (1.3.1) since the
∂xj
effect of the action of double derivations (1.5.7) on a (non cyclic) word is similar to that
of formula (1.3.1). ♦
Higher derivatives. To any potential Φ ∈ F , one associates its Hessian, a noncom-
cyc
mutativeversionoftheHessianofasmoothfunctiononCn. ThenoncommutativeHessian,
k ∂2Φ k, is an n×n-matrixwith entries inF⊗F; its (i,j)-th entry is definedas theimage
∂xi∂xj
of Φ under the composite map, cf. (1.3.1) and (1.5.7),
∂ ∂
Fcyc ∂xj // F ∂xi // F ⊗F , Φ 7→ ∂x∂2∂Φx = (∂x∂2∂Φx )′ ⊗ (∂x∂2∂Φx )′′ ∈ F ⊗F.
i j i j i j
Remark 1.5.11. TheHessianofafunctionφonanarbitrarymanifoldX iswelldefinedonly
on the critical locus crit(φ) ⊂ X, the zero scheme of the 1-form dφ. A noncommutative
analogue of such a construction will be discussed in §8. ♦
Similarly, for each r ≥ 1, there is a map ∂ : F⊗r → F⊗(r+1). Thus, for any Φ ∈ F
∂xi cyc
and any r-tuple of indices, one inductively defines the elements ∂rΦ ∈ F⊗r.
∂xi1∂xi2...∂xir
The classic result on total symmetry of the tensor of r-th derivatives of a function gets
replaced, in noncommutative geometry, by cyclic symmetry. Specifically, in F⊗r, one has
∂rΦ ∂rΦ
σ = , ∀Φ∈ F , r ≥ 1. (1.5.12)
cyc
∂x ∂x ...∂x ∂x ∂x ...∂x
(cid:18) i1 i2 ir(cid:19) σ(i1) σ(i2) σ(ir)
IntheLHSofthisformula,σ denotesthemapF⊗r → F⊗r givenbythecyclicpermutation
of the tensor factors while in the RHS σ stands for a cyclic permutation of indices.
One can also prove the following
Proposition 1.5.13. (i) (Poincar´e lemma). For an n-tuple {f ∈ F} , we have
i i=1,...,n
n
[x ,f ]= 0 ⇐⇒ ∃Φ∈ F such that f = ∂Φ/∂x , i = 1,...,n;
i i cyc i i
i=1
X
6
(ii)(Frobenius theorem). Let kf k be an F ⊗F-valued n×n-matrix such that
i,j
∂f ∂f
i,j σ(i),σ(j)
σ(f )= f , and σ = , ∀1≤ i,j,k ≤ n.
i,j σ(i),σ(j)
∂x ∂x
(cid:18) k (cid:19) σ(k)
Then, there exists a potential Φ ∈F , such that we have kf k = k ∂2Φ k. (cid:3)
cyc i,j ∂xi∂xj
The implication ‘⇐’ in part (i) has been first noticed by M. Kontsevich [Ko1, §6].
1.6. Noncommutative cotangent complex. Write T for the tangent, resp. T∗ for
X X
the cotangent, sheaf of an algebraic variety (or scheme) X. Given a scheme imbedding
Y ֒→ X one gets, by restriction, two sheaves T∗| and T | , and also the conormal
X Y X Y
sheaf, N ∗ . There is a standard short exact sequence of sheaves on Y,
X|Y
0 // NX∗|Y // TX∗|Y pX|Y // TY∗ // 0. (1.6.1)
The geometric setting above can be imitated in algebra. A morphism of schemes cor-
responds to an algebra map B → A. Given such a map, we introduce the following
A-bimodules which are algebraic counterparts of T∗| and T | , respectively,
X Y X Y
Ω1(B|A) := A⊗ Ω1B ⊗ A resp., Der(B|A) := A⊗ DerB ⊗ A. (1.6.2)
B B B B
The map B → A induces a canonical A-bimodule map p :Ω1(B|A) → Ω1A.
B|A
In the special case where A = B/I, one has the following short exact sequence of
A-bimodules, cf. [CQ], which is a noncommutative analogue of (1.6.1),
d p
0 // I/I2 B|A // Ω1(B|A) B|A // Ω1A // 0. (1.6.3)
Here, I ⊂ B is a two-sided ideal, and the map d is induced by restriction to I of the
B|A
de Rham differential d :B → Ω1B, b7→ db, cf. (1.5.3).
Now, fix a potential Φ ∈ F , on F = Chx ,...,x i. We have the algebra A =
cyc 1 n
A(F,Φ) = F/((∂Φ)) , the DG algebra D = D(F,Φ), and the algebra projection
∂xi i=1,...,n
D ։ A, see (1.4.3). Thus, we can form the corresponding A-bimodule Ω1(D|A).
Definition 1.6.4. Define the cotangent complex associated to (F,Φ) to be
LΩ1q(F,Φ) := Ω1(D|A) = A(F,Φ) Ω1D(F,Φ) A(F,Φ).
D(F,Φ) D(F,Φ)
This is a DG A-bimodule, with the grading LNΩ1q(F,Φ) = LNΩ1(F,Φ) and differential
r≥0 r
d :LΩ1q(F,Φ) → LΩ1q (F,Φ), both being induced from those on the DG algebra D.
−1 L
To give a more explicit description of the cotangent complex, consider the composite
pD|A: LΩ10(F,Φ) // // Ω1A(cid:31)(cid:127) A // A⊗A,
an A-bimodule map induced by the projection D։ A and the tautological imbedding  ,
A
cf. (1.5.4), and also a similar map p : Ω1(F|A) ։ Ω1A ֒→ A⊗A.
F|A
7
Further, it follows from (1.5.6)-(1.5.8) that the spaces Ω1(F|A) and Der(F|A) are both
free A-bimodules, with bases {dx } and { ∂ } , respectively. M. Van den
i i=1,...,n ∂xi i=1,...,n
Bergh [VdB3] introduced an important contraction map Der(F|A) → Ω1(F|A) defined by
n ′ ′′
a′⊗ ∂∂xi ⊗a′′ (cid:31) // ia′∂∂xia′′(d2Φ):= j=1 a′ ∂x∂i2∂Φxj ⊗dxj ⊗ ∂x∂i2∂Φxj a′′. (1.6.5)
(cid:16) (cid:17) (cid:16) (cid:17)
P
Proposition 1.6.6. (i) The DG-module LΩ1q(F,Φ) is concentrated in degrees 0,1,2, and
there is an isomorphism between the complexes in two rows of the following diagram
0 // LΩ1(F,Φ) d2 // LΩ1(F,Φ) d1 // LΩ1(F,Φ) pD|A // A⊗A // 0 (1.6.7)
2 1 0
0 // A⊗A ad // Der(F|A) (1.6.5) // Ω1(F|A) pF|A // A⊗A // 0.
(ii)The nontrivial homology groups of the cotangent complex are as follows
Ω1A if r = 0;
H LΩ1q(F,Φ), d = Ext1 (A,A⊗A) if r = 1;
r  A-Bimod
(cid:0) (cid:1) HomA-Bimod(A,A⊗A) if r = 2.
(iii)Each row in (1.6.7) is self-dual, i.e., it goes to itself under the duality (1.5.2).

We will refer to the complex in either row of diagram (1.6.7) as the extended cotangent
complex. The complex in the bottom row has been independently introduced by M. Van
den Bergh [VdB3]. The proof of Proposition 1.6.6 will be given in §8.2.
2. Representation functor, critical points, and vanishing cycles
2.1. Informal outline. Representation functor provides a bridge between noncommuta-
tive and the usual commutative geometry. The main objective of §2 is to describe various
geometric structures which arise once one applies the representation functor either to the
DG algebra Dq(F,Φ) or to the cotangent complex LΩ1q(F,Φ). It may be instructive to
keep in mind the following general setup, cf. §3.5 for more details.
One starts with a data (F,α), where
• F is a smooth algebra, cf. Definition 3.5.1, for instance, F = Chx ,...,x i;
1 n
• α ∈ (Ω1F) is a closed cyclic 1-form, e.g. α= dΦ, for Φ ∈ F .
cyc cyc
Associated with the data (F,α), one defines an algebra A= A(F,α). This is a quotient
of F by an appropriate two sided ideal; in the special case where F = Chx ,...,x i and
1 n
α = dΦ, the algebra A reduces to A(F,Φ), the algebra considered in the previous section.
For any d = 1,2,..., one has the scheme Rep F, of d-dimensional F-representations.
d
Itisasmoothmanifoldandtheclosed cyclic1-formαgives risetoTrα,anordinaryclosed
1-form on X := Rep F, see §2.2. The scheme Rep A turns out to be a closed subscheme
d d
in X which is equal to the zero locus of the 1-form Trα.
b
The crucial geometric feature here is that the zero locus of a closed 1-form may be
identified with an intersection of two Lagrangian submanifolds in T∗X, the total space
b
of the cotangent bundle on X (the first submanifold is the graph of the 1-form and the
8
second submanifold is the zero section of T∗X). This applies, in particular, to the scheme
Rep A.
d
Intheabove setting, one of thegoals thatone would like to achieve is to express various
numerical invariants of the algebra A = A(F,α), such as ranks of cyclic or Hochschild
homology groups of A, or the corresponding Euler characteristics, in terms of certain
integrals, cf. Problems 2.10.7-2.10.8 below. The integrals in question should be similar to
those used in Witten’s nonabelian localization theorem for equivariant cohomology, [JK].
The problem of expressing homological invariants of an algebra A in terms of integrals
has been already studied in [EG] in the case where the scheme Rep A was a complete
d
intersection. For algebras of the form A = A(F,α), however, the scheme Rep A has
d
virtual dimension zero. Such a scheme can never be a complete intersection in Rep F
d
except for the trivial case where it consists of isolated points. Thus, the technique of [EG]
requires a serious modification which is not known at the moment.
It seems very likely that the right approach to the problem is provided by the so-called
Batalin-Vilkovisky (BV) formalism. In general, let X be a manifold, φ a regular function
on X, and L ⊂ T∗ X a Lagrangian submanifold in the ‘odd’ cotangent bundle on X.
odd
Following BV-formalism, one considers integrals of the form
eφ♭, (2.1.1)
ZL
where φ♭ is a function on T∗ X obtained by a slight modification of q∗φ, the pull-back of
odd
the function φ via the bundle projection q :T∗ X → X.
odd
To make sense of the integral in (2.1.1), one needs to specify a manifold L as well as a
volume-formonL. Ingeneral,thereisneitherapreferredchoiceofLagrangiansubmanifold
L nor a canonically defined volume on it. It is known that a choice of nowhere vanishing
volume-form on X itself provides a natural volume-form on any Lagrangian submanifold
in T∗ X. We see that in order to apply BV-formalism, X has to be a CY manifold. Now,
odd
given a CYmanifold X, there is a canonically definedsecond order differential operator ∆
on T∗ X. Furthermore, one proves a version of Stokes theorem saying that the integral
odd
in (2.1.1) depends only on the isotopy class of the Lagrangian submanifold L ⊂ T∗ X,
odd
provided the function φ♭ satisfies the quantum master equation ∆φ♭ + {φ♭,φ♭} = 0, cf.
(2.6.6) and (2.9.3).
Now, let F = Chx ,...,x i. Then, each of the manifolds Rep F, d = 1,2,..., is a
1 n d
vector space, hence, it has a natural Eucledian volume. Any potential Φ ∈ F gives rise
cyc
to a polynomial TrΦ on the vector space Rep F. Further, let A(F,Φ) be the algebra
d
associated with the potential Φ. The corresponding representation scheme, Rep A(F,Φ),
d
may by identified wibth the critical set of the polynomial TrΦ.
Following BV-formalism, one might expect that interesting algebraic invariants of the
algebra A(F,Φ) are provided by the asymptotics of integralsbof the form, cf. (2.1.1),
ed·TrΦb♭ where d → ∞. (2.1.2)
ZLd⊂To∗dd(RepdF)
More generally, let F be an arbitrary, not necessarily free, smooth algebra. In [GS2],
we analyze noncommutative structures on the algebra F that are necessary to insure that
each of the manifolds Rep F, d = 1,2,..., be a CY manifold, so that it makes sense to
d
9
consider integrals like (2.1.2). It turns out that the DG algebra D(F,Φ) plays the role of
T∗ X. Specifically, in [GS2], we introduce the notion of a noncommutative BV structure;
odd
furthermore, we show that giving the algebra D(F,Φ) a noncommutative BV structure
provides each of the manifolds T∗ (Rep F) with a natural BV-operator ∆ .
odd d d
This way, using the formalism developed in [GS2], it is not difficult to extend the con-
structionsdiscussedinthepresentsectionbelowinthespecialcasewhereF = Chx ,...,x i
1 n
to the more general setting of an arbitrary smooth algebra equipped with a noncommu-
tative BV structure.
2.2. Reminder. WewriteMat forthealgebraofcomplexd×d-matrices. Givenanalge-
d
braicvariety(orscheme)X,letO bethestructuresheafofX andputC[X] = Γ(X,O ).
q q X X
We let Λ T(X), resp. Λ T∗(X), denote the graded algebra of regular polyvector fields,
resp. differential forms, on X.
LetAbeafinitelypresentedC-algebra. Foreachintegerd ≥ 1,thesetHom (A,Mat ),
alg d
of all algebra homomorphisms A → Mat , has the natural structure of a (not necessarily
d
reduced) affine scheme of finite type over C, to be denoted Rep A.
d
The group GL acts on Mat by algebra automorphisms via conjugation. This gives a
d d
GL -actionontheschemeRep Abybasechangetransformations. WewriteC[Rep A]GLd
d d d
for the algebra of GL -invariant regular functions on Rep A.
d d
Anyalgebrahomomorphismf :B → AinducesaGL -equivariantmorphismofschemes
d
f∗ :Rep A→ Rep B. This way, we obtain a contravariant functor
d d
Rep : Algebras −→ Affine GL -schemes. (2.2.1)
d d
To any element a ∈ A, one associates naturally a GL -equivariant polynomial map
d
a : Rep A → Mat , ρ 7→ ρ(a). Taking the trace of the matrix ρ(a) ∈ Mat , yields a
d d d
GL -invariant regular function Tra : Rep A → C, ρ 7→ Trρ(a). For a ∈ [A,A], we have
d d
Tra= 0, due to symmetry of the trace. Hence, we obtain a well defined linear map
b
Tr : A = A/b[A,A] −→ C[Rep A]GLd, a 7→ Tra. (2.2.2)
d cyc d
b
q
Notation 2.2.3. Given a graded associative (super) algebra B , the notation B will be
cyc
b
used to denote the super-commutator quotient, B/[B,B] , where [B,B] stands
super super
for the C-linear span of the elements of the form ab−(−1)|a|·|b|ba, for all homogeneous
a,b ∈ B of degrees |a| and |b|, respectively. Given a Z/2Z-graded super-vector space
V = Veven⊕Vodd, we use the notation SymV := (SymVeven)⊗(ΛVodd).
For any positive integer d, we introduce a (super)-commutative algebra
Sym(B/[B,B] )
super
O (B):= Sym(B ) ((1 −d·1 )) = .
d cyc Sym B
((1 −d·1 ))
Sym B
(cid:14)
Here, we write 1 , resp. 1 , for the unit element of B, resp. of Sym(B ), Thus,
B Sym cyc
((1 −d·1 )) stands for the two-sided ideal generated by the element 1 −d·1 ∈
Sym B Sym B
Sym0(B )+Sym1(B ). ♦
cyc cyc
It is clear that the linear map in (2.2.2) sends the unit element 1 to the constant func-
A
tion Tr(1 ) = d. Therefore, the map Tr can be uniquely extended, by multiplicativity,
A d
10
b