Table Of Content1
1 Quantum Sheaf Cohomology, a pr´ecis
0
2
Josh Guffin
n
a
J
6 Abstract. We present a brief introduction to quantum sheaf cohomology,
a generalization of quantum cohomology based on the physics of the (0,2)
] nonlinearsigmamodel.
G
A
.
h
t This paper is based on a talk givenon 13 December, 2010during the Second Latin
a
Congress on Symmetries in Geometry and Physics at the Universidade Federal
m
do Paran´a in Curitiba, Brazil. Throughout, we will consider X to be a K¨ahler
[ manifold of complex dimension n. In addition, we will consider E → X to be a
1 complex Hermitian vector bundle of rank k satisfying
v
5 (i) c (E)=c (T ),
30 (ii) d2etE∨ ∼=2ωXX.
1
. As these conditions imply the usual Green-Schwarz anomaly cancellation condi-
1
0 tions,wewillcallsuchabundleomalous1. Onemayconsider(ii)tobeananalogue
1 of the usual condition for existence of the B-model. A bundle satisfying these con-
1 ditions may be obtained by, for example, selecting a deformation of the tangent
v: bundle when X is a projective toric variety.
i
X
r
a Quantum Cohomology
Ordinary cohomology. We now give some elementary facts about the coho-
mologyofX,statedinawaythatwillfacilitateourpointofviewonquantumsheaf
cohomology. Since X is K¨ahler, there is a Hodge decomposition on H•(X,C),
H•(X,C)∼= Hp(X, qT∨).
X
p,q
M V
By a slight abuse of language, we will refer to elements of the sheaf cohomology
groupsHp(X, qT∨)as(p,q)-forms–clearlyH•(X,C)possessesabasisconsisting
X
of such forms. The cup/wedge product on cohomology furnishes this vector space
V
1991 Mathematics Subject Classification. Primary32L10,81T20;Secondary 14N35.
1Thatis,not anomalous –thisdelightfulterminologyisduetoRonDonagi.
2 JOSHGUFFIN
with the structure of a bigraded C-algebra. Finally, integration of forms induces a
trace on the algebra; in terms of a basis element ω,
ω ω ∈Hn(X, nT∨)
tr(ω)= X
ZX
0 otherwise. V
The pairing (α,β) 7→ tr(α∧β) induced by this trace is a non-degenerate bilinear
form satisfying (α,β ∧γ) = (α∧β,γ), so that H•(X,C) is a bigraded Frobenius
algebra.
Physics. The relationship between ordinary cohomology and quantum coho-
mologymaybeelucidatedbyappealingtophysics–inparticulartoatopologically-
twisted (2,2) nonlinear sigma model of maps P1 →X. Of the many intriguing as-
pectsofthisquantumfieldtheory,wewillbemostinterestedinitsalgebraofmass-
lesssupersymmetricoperators2. Usingelementaryphysicsarguments,oneidentifies
abasisforthesetofsuchoperatorsthatmaybesetintoone-to-onecorrespondence
with (p,q)-forms on X.
The (2,2) supersymmetry of the model forces the product of two massless super-
symmetric operators to be massless and supersymmetric. The particular form of
theproductobtainsbyconsideringthree-pointcorrelationfunctionsinthequantum
field theory: the quantum product of two massless operators O and O is defined
1 2
to be the unique operator (O ∗O ) such that for all massless operators O ,
1 2 3
(1) h1(O ∗O )O i=hO O O i.
1 2 3 1 2 3
Here,1denotestheoperatorcorrespondingto1∈H0(X, 0T∨). Suchacorrelation
X
function is computed using the instanton expansion
V
(2) hO O O i= hO O O i qβ.
1 2 3 1 2 3 β
β∈HX2(X,Z)
Although they have intrinsic meaning in physics, we will consider the expressions
qβ to comprise a set of formal variables endowed with the structure of a monoid
via the product qαqβ = qα+β. We denote by CJqK the ring of formal power series
with complex coefficients in these variables – one sometimes insists on convergence
or other properties, but such subtleties are beyond the scope of this review.
Mathematically, one defines the expression hO O O i as the Gromov-Witten in-
1 2 3 β
variant3 hI i(ω ,ω ,ω ), where ω are the forms corresponding to the operator
0,3,β 1 2 3 i
O . Physically,onesaysthathO O O i qβ denotesthecontributionofinstantons
i 1 2 3 β
of degree β to the correlation function hO O O i. This expression is morally the
1 2 3
integralofinducedformsonsomecompactificationM(X,β)ofthe modulispaceof
holomorphicmapsf :P1 →X ofclassβ =f∗[P1]. We willwritethe inducedforms
schematically using maps
(3) ζ :Hp(X, qT∨)→Hp M(X,β), qT∨ .
β X M(X,β)
(cid:16) (cid:17)
V V
2Moreprecisely,itisthealgebraoflocal,scalar,supersymmetricoperators[Wit91].
3Seeequation7.4of[CK00]foraprecisedefinitionofGromov-Witteninvariants.
QUANTUM SHEAF COHOMOLOGY, A PRE´CIS 3
Ifω aretheformscorrespondingtooperatorsO ,modulothesubtletiesofobstruc-
i i
tion bundles we have that
hO O O i = ζ (ω )∧ζ (ω )∧ζ (ω ).
1 2 3 β β 1 β 2 β 3
ZM(X,β)
Depending onthe compactification,theremaybe morethanonesuchmap–inthe
case of the stable maps compactification, pullbacks via evaluation maps play the
rˆole of ζ . For toric varieties, one often uses the Morrison-Plessercompactification
β
[MRP95] wherein – as indicated in (3) – one map suffices for each β.
Thethree-pointcorrelationfunctionsin(2)induceanon-degeneratebilinearpairing
(ω ,ω ) = h1O O i on the unital algebra Hp(X, qT∨)JqK, leading to the
1 2 1 2 p,q X
following definition.
L V
Definition 1. The quantum cohomology of X is the Frobenius algebra
QH•(X):= Hp(X, qT∨)JqK,
X
p,q
M V
with the product and bilinear pairing induced by (2,2) three-point functions.
Here, the (2,2) correlation functions are defined either via Gromov-Witten invari-
ants or as correlationfunctions in the quantumfield theory,depending on whether
your tastes tend to the mathematical or to the physical.
Example 2. The“classicalsector”is thesetofmapshomotopicto apoint, β =0,
and the moduli space of such maps is simply X itself. Thus, in this sector, the
quantum product reduces to the wedge product on forms; ordinary cohomology is
the“classicallimit”ofquantumcohomology. Thissectormaybeisolatedbysetting
q =0. For example, the ordinary and quantum cohomology of Pn are respectively
C[H]
H•(Pn,C)∼= ,
hHn+1i
C[H]JqK
QH•(Pn)∼= .
hHn+1−qi
Here H denotes the hyperplane class. For Pn×Pm, the equivalent expressions are
C[H ,H ]
H•(Pn×Pm,C)∼= 1 2 ,
Hn+1,Hm+1
1 2
(4)
C[H ,H ]Jq ,q K
QH•(Pn×Pm)∼= (cid:10) 1 2 (cid:11) 1 2 .
Hn+1−q ,Hm+1−q
1 1 2 2
(cid:10) (cid:11)
Quantum Sheaf Cohomology
As in our study of the passagefromordinarycohomologyto quantum cohomology,
wefirstconsiderthe“ordinarysheafcohomology”–inparticularthatofanomalous
bundle E → X. Here, by ordinary sheaf cohomology we mean cohomology valued
in polysections,
(5) Hp(X, qE∨).
p,q
M V
4 JOSHGUFFIN
Again by a slight abuse of language, we will refer to elements of Hp(X, qE∨) as
(p,q)-forms–clearlythevectorspace(5)possessesabasisconsistingofsuchforms,
andthe cup/wedgeproductfurnishesitwiththestructureofabigradedCV-algebra.
ThetraceonthisalgebraisslightlymoresubtleandfollowsfromtheomalityofE. In
particular,one uses the existence ofanisomorphismψ:Hn(X, kE)→Hn(X,ω )
X
to define, for a basis element ω,
V
ψ(ω) ω ∈Hn(X, kE∨)
(6) tr(ω)=
ZX
0 otherwise. V
The pairing (α,β) 7→ tr(α∧β) induced by this trace endows Hp(X, qE∨)
p,q
with the structure of a bigraded Frobenius algebra.
L V
Physics. Tounderstandtherelationshipbetweensheafcohomologyandquan-
tum sheaf cohomology we again appeal to physics – in particular a topologically-
twisted (0,2) nonlinear sigma model of maps P1 → X. A recent physics review of
this and related models may be found in [McO10]. We will again be most inter-
ested in its algebra of massless supersymmetric operators4. The same elementary
physicsargumentsusedforthe (2,2)theoryidentifyabasisforthissetofoperators
that may be placed into one-to-one correspondence with (p,q)-forms (that is, ele-
ments of (5)), and the quantum product of two massless operatorsis defined using
three-point correlation functions of the (0,2) in analogy to (1). Unlike the (2,2)
case, however, there is no mathematical definition of hO O O i in a (0,2) theory
1 2 3 β
so the following definition is purely physical.
Definition 3. The quantum sheaf cohomology of an omalous bundle E → X is
the Frobenius algebra
QH•(X,E):= Hp(X, qE∨)⊗CJqK
p,q
M V
with the product and bilinear pairing induced by (0,2) three-point functions.
As in the case of ordinary quantum cohomology, the classical limit of quantum
sheaf cohomology is precisely the ordinary sheaf cohomology with the Frobenius
structure induced by (6). Unlike the case in (2,2) theories, (0,2) supersymmetry is
not enough to guarantee that the product of massless operators is massless: one
needs to work harder to show that the algebra closes in the set of all operators.
Existence. The (modern) history of quantum sheaf cohomology begins with
theobservationin[ABS04]ofananalogueofQH•(X)for(0,2)theories. Therein,
the quantum sheaf cohomology of a one-parameter family of deformations of the
tangent bundle of P1×P1 was computed using a conjectured form of mirror sym-
metry for (0,2) models. Their calculations were confirmed in a sheaf-cohomology-
basedcomputationby KatzandSharpe[KS06]. Inspiredbythese results, Adams,
4As explained in [ADE06], we are actually interested in local, scalar, supersymmetric op-
eratorswithvanishingholomorphicconformalweight, butforcontinuity wewillrefertothemas
masslesssupersymmetricorsimplymassless.
QUANTUM SHEAF COHOMOLOGY, A PRE´CIS 5
Distler, and Ernebjerg [ADE06] gave a physics definition of quantum sheaf coho-
mology and found a physics proof of two sufficient conditions for its existence. We
restate these conditions here as conjectures.
Conjecture 4. Let E and E′ be omalous elements of a family of bundles U. Let
γ:[0,1] → U continuous with γ(0) = E, γ(1) = E′, γ(t) omalous for all t ∈ [0,1].
Then QH•(X,E) exists iff QH•(X,E′) exists.
Conjecture 5. If E →X is omalous and rk E <8, then QH•(X,E) exists.
Since QH•(X,T )=QH•(X), the formerconditionimplies the existenceofquan-
X
tum sheaf cohomology for all omalous one-parameter families of tangent-bundle
deformations. The latter is likely an artefact of the technique used in the physics
proof–therearenoknownexamplesofomalousbundlesofrankeightorhigherfor
which the massless operators fail to close under the quantum product, and there
are no physical reasons to expect such a bundle to exist.
Computation example. Although there is no definition for the invariants
hO O O i , a number of physics-inspired techniques exist to compute them when
1 2 3 β
the omalous bundle is a deformation of the tangent bundle of a toric variety
[KS06, GK07, MM08] or a complete intersection therein[MM09]. One of the
advantagesofusingatoricvarietyX isthatdeformationsofT areeasilyobtained
X
by deforming the Euler exact sequence:
E
0
0 Or O (D ) T 0.
X X ρ X
ρ∈∆
M
Here,ristherankofthePicardgroup,∆denotesthesetoftorus-invariantdivisors
corresponding to one-cones in the fan of X, and E is a collection of sections of
0
OX(Dρ), which are toric analogues of OPn(1). Taking X = P1×P1, for example,
the sequence becomes
E
0
0 O2 O (1,0)2⊕O (0,1)2 T 0,
X X X X
where the map is
x 0
0
x 0
E0 =01 x .
2
0 x
3
AdeformationofT maybe obtainedby choosinga different collectionofsections
X
for the map. For example, selecting the map to be
x ǫ x +ǫ x
0 1 0 2 1
x ǫ x
(7) E = 1 3 0
γ x +γ x x
1 2 2 3 2
γ x x
3 2 3
as in [GK07] gives a convenient basis for the space of deformations of the tangent
bundle (ǫ ,γ ∈ C). Therein, several of the invariants hO O O i were computed
i i 1 2 3 β
6 JOSHGUFFIN
for the bundle E → P1×P1 defined as the cokernel of (7). These were then used
to deduce the quantum sheaf cohomology of E;
C[ψ,ψ]Jq ,q K
(8) QH•(P1×P1,E)∼= 1 2 .
ψ2+ǫ ψψ−ǫ ǫ ψ2−q ,
1 2 3 1
e
*ψ2+γ1ψψ−γ2γ3ψ2−q2 +
e e
These computations were confirmed in [MM08] using physics techinques. Note
e e
that as ǫ ,γ → 0, the quantum sheaf cohomology in (8) limits to the ordinary
i i
quantum cohomology of P1×P1 in (4).
ThismaterialisbasedonworksupportedbytheNationalScienceFoundationunder
DMS grant no. 0636606.
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Departmentof Mathematics,University of Pennsylvania,Philadelphia, PA19104
E-mail address: [email protected]
