Mutations vs. Seiberg duality

arXiv:0709.3939v2 [math.RA] 25 Nov 2007

JORGEVITÓRIA

Abstra t. For a quiver with potential, Derksen, Weyman and Zelevinsky deneda ombinatorialtransformation-mutations.MukhopadhyayandRay, ontheotherhand,tellushowto omputeSeibergdualquiversforsomequivers withpotentialsthroughatiltingpro edure,thusobtainingderivedequivalent algebras. Inthis text, we ompare mutations withthe on ept of Seiberg duality givenby [10℄, on luding that for a ertain lassof potentials (the good ones)mutations oin ide withSeibergduality,thereforegiving derived equivalen es.

1. Preliminaires

In this se tion we introdu e the material from [7℄ that will be used and re all somedenitions.

Wewill use thefollowingnotation:

K

is aeld;

KQ

is thepath algebraofthe quiver

Q

over

K

( on atenation of paths is written as omposition of fun tions);

P roj(R)

is the full sub ategory of proje tiveright modules overa

K

-algebra

R

;

P (R)

is the full sub ategory of nitely generated proje tive right modules over

R

;

K

b

_(Q)

and

D

b

_(Q)

are, respe tively, the bounded homotopy ategoryand the boundedderived ategoryofrightmodulesover

KQ

.

Denition1.1. Apotentialonaquiverisanelementoftheve torspa espanned bythe y lesofthequiver(denoteitby

KQ

cyc

).

Denition1.2. Let

A =< Q

1

>

,i.e.,theve torspa espannedbyallarrows. For ea h

ξ ∈ A

∗

(the dualof

A

),denea y li derivative: (1.1)

∂/∂ξ :

KQ

cyc

→ KQ

a

1

. . . a

n

7→

P

n

k=1

ξ(a

k

)a

k+1

. . . a

n

a

1

. . . a

k−1

.

If

x ∈ Q

1

, we will denote by

∂/∂x

the y li derivative orrespondent to the elementof

A

∗

whi histhedualof

x

inthedualbasisof

A

.

Denition 1.3. Two potentials are y li ally equivalent if

S − S

′

lies in the span of elements of the form

a

1

. . . a

n−1

a

n

− a

2

. . . a

n

a

1

. A pair

(Q, S)

is said to beaquiverwithpotentialif

Q

hasnoloopsandnotwo y li allyequivalentpaths appearon

S

. Twoquiverswith potentials

(Q, S)

and

( ˜

Q, ˜

S)

aresaid to beright equivalentifthereisisomorphism

φ

between

KQ

and

K ˜

Q

su hthat

φ(S)

is y li- allyequivalentto

S

˜

.

Denition 1.4. Givenaquiverwith potential

(Q, S)

, denetheja obian alge-bra of

(Q, S)

as

J(Q, S) = KQ/ < J(S) >

,where

J(S) = (∂S/∂x)

x∈Q

Remark 1.5. Tworightequivalentquiverswithpotentialshaveisomoprhi ja obian algebras(see[7℄).

Denition 1.6. Dene the trivial partof aquiverwith potential

(Q

triv

, S

triv

)

by taking

S

triv

asthe degreetwohomogeneous omponent of S and

Q

triv

asthe subquiverof

Q

onsistingonlyinthearrowsappearingin

S

triv

. Theredu edpart

(Q

red

, S

red

)

is formedbythenon-trivialpartofthepotential

S

andbythequiver obtainedbytakingthequotientof

A

bythearrowsappearingin

S

triv

.

Thefollowingtheoremwillallowustodenemutationonaquiverwithpotential.

Theorem 1.7 (7). For a quiver with potential

(Q, S)

, there exista trivial quiver with potential

(Q

triv

, S

triv

)

and a redu ed quiver with potential

(Q

red

, S

red

)

su h that

(Q, S)

isright equivalent to

(Q

triv

⊕ Q

red

, S

triv

+ S

red

)

-

Q

triv

⊕ Q

red

stands for the quiver obtainedbytaking the dire t sumof the arrow spans.

Let'snowdes ribethepro edureof mutation ofaquiverwithpotential

(Q, S)

onavertex

k

(denoteitby

µ

k

(Q, S)

).

(1) Suppose

k

doesnotbelongtoany2- y leandthat

S

doesn'thaveany y le startingandnishingon

k

(ifitdoes,substituteitbya y li allyequivalent potentialthatdoesn't).

(2) Changethequiverinthefollowingway:

•

Ree tarrowsstartingorendingat

k

. Denoteree tedarrowsby

(.)

∗

;

•

Createonenewarrowforea hpathoftheform

•

i

α

_{// •}

k

β

_{// •}

j

and denoteitby

[βα]

. Wedenotetheresultingquiverby

Q

˜

.

(3) Changethepotentialinthefollowingway:

•

Substitute fa tors appearing in

S

of the form

βα

by the new arrow

[βα]

anddenoteitby

[S]

;

•

Add

∆

k

=

P

•

i

α

_{// •}

k

β

_{// •}

j

[βα]α

∗

_β

∗

to

[S]

. We denote the resultingpotentialby

S

˜

.

(4) Themutationat

k

of

(Q, S)

is

µ

k

(Q, S) = ( ¯

Q, ¯

S) := ( ˜

Q

red

, ˜

S

red

)

Note that these mutations generalizeree tion fun torson quiverswith no re-lations in the sense that if you do a mutation on either a sink or a sour e, this pro edureredu es toree tarrowsonthat vertex.

Letusre allRi kard'stheorem,startingbydeningtilting omplex([12℄).

Denition 1.8. A tilting omplex over aring

R

is an obje t

T

of

K

b

_{(P (R))}

, su hthat: (1)

∀i 6= 0

,

Hom

K

b

_{(P (R))}

(T, T [i]) = 0

; (2) Tgenerates

K

b

_{(P (R))}

asatriangulated ategory.

Theorem1.9 (Ri kard). Let

R

and

S

betworings. Then

D

b

_(R)

isequivalentto

D

b

_(S)

ithereisatilting omplex

T

over

R

su hthat

S ∼

= End

2. SeibergDuality

Wewill deneSeibergdualityonquiversasatiltingpro edureandthereforeas anequivalen e ofderived ategories. To he kif a omplexis tiltingwewill have to ompute homomorphismsin the derived ategorybetween (nitely generated) proje tivemodules.

Remark 2.1. Notethat

K

b

_{(P (R))}

isafullsub ategoryof

K

b

_(R)

andtherefore,for anobje t

T

in

K

b

_{(P (R))}

(inparti ularforatilting omplex)wehave

End

K

b

_(R)

(T )

op

=

End

K

b

_{(P (R))}

(T )

op

.

Let

(Q, S)

bea quiverwith potentialwith

n

verti essu h that everyvertexis ontainedinsome y le(whi hweshallassumefromnowon)and,forea hvertex

k

, onsider thefollowing omplex:

T

k

_{= ⊕}

n

i=1

T

i

k

where

T

k

i

= 0 → P

i

→ 0, if i 6= k

and

T

k

k

= 0

//

⊕

j→k

P

j

(α

j

)

j

// P

k

// 0

Lemma2.2.

T

k

isatilting omplexovertheja obianalgebraof

(Q, S)

ifandonly if

Hom

K(P (Q))

(T

k

k

, T

s

k

[−1]) = 0

,

∀s

. Proof. (1)

Hom

K

b

_{(P (Q))}

(T

k

, T

k

[i]) = 0 ∀i 6= 0

. Itis learthatif

r, s 6= k

,then

Hom

K(P (Q))

(T

r

k

, T

s

k

[i]) = 0, ∀i 6= 0

. Now,suppose

s = k

and

r 6= k

. Then weonlyhaveto he kthattheset

Hom

K(P (Q))

(T

k

r

, T

k

k

[1])

redu estozero. Note that,sin e ahomomorphismbetween

P

r

to

P

k

isidentied with an elementofthepathalgebrawithea htermbeingapathfrom

r

to

k

,every su h homomorphismfa torsthrough

⊕

j→k

P

j

.

0

// P

i

//

{{



0

0

//

⊕

j→k

P

j

// P

k

// 0

Su hfa torization impliesthatthese mapsof omplexesarehomotopi to zero,thuszeroin thehomotopy ategory.

If

r = k

thenwealsohavesu hahomotopyjustbytakingidentitymaps.

0

//

⊕

j→k

P

j

//

yy



P

k

// 0

0

//

⊕

j→k

P

j

// P

k

// 0

(2) add(

T

k

) generates

K

b

_{(P (Q))}

as atriangulated ategory. It is enough to provethat the stalk omplexes of inde omposable proje tive modules are generatedbythedire tsummandsof

T

k

Considerthedire t summands of

T

k

and takethe one of themap

T

k

k

to

⊕

j→k

T

k

j

denedby:

0

//

⊕

j→k

P

j

//

id



P

k

// 0

0

//

⊕

j→k

P

j

// 0

That one isjust the following omplex(the underlinedterm isin degree zero):

(2.1)

0

//

⊕

j→k

P

j

((α

j

)

j

,0)

// P

k

⊕ (⊕

j→k

P

j

)

// 0

Consider the map from the omplex (2.1) to the stalk omplex of

P

k

in degree zerodened by identity in the rst omponentand

−(α

j

)

j

in the se ond omponent and onsider themap from thissamestalk omplexto (2.1) dened by the in lusion of

P

k

. We will provethat the omposition of thesemaps ishomotopi to theidentitymap, hen eprovingthat these omplexesareisomorphi inthederived ategory. Infa t,thatfollowsfrom thefollowingdiagram:

0

//

⊕

j→k

P

j





((α

j

)

j

,0)

// P

k

⊕ (⊕

j→k

P

j

)

(0,id)



(id,−(α

j

)

j

)



// 0



0



// P

k

(id,0)



// 0

0

//

⊕

j→k

P

j

((α

j

)

j

,0)

// P

k

⊕ (⊕

j→k

P

j

)

// 0

Similarly we an see it for the reverse omposition and therefore (2.1) is isomorphi tothestalk omplex

P

k

in degreezero.

Hen e,the omplexistiltingiwehave

Hom

K(P (Q))

(T

k

k

, T

s

k

[−1]) = 0

,

∀s

.



Denition2.3. Givenaquiverwithpotential

(Q, S)

, dene

δ(Q, S)

astheset of verti esforwhi h the omplexaboveistiltingover

J(Q, S)

,i.e.,

δ(Q, S) =

k ∈ Q

0

: Hom

K(P (Q))

(T

k

k

, T

s

k

[−1]) = 0, ∀s

.

If

δ(Q, S) 6= ∅

,thenwesaythat

(Q, S)

islo allydualisablein

δ(Q, S)

. F urther-more,if

δ(Q, S) = Q

0

thenwesaythat

(Q, S)

isglobally dualisable.

Remark 2.4. Note that to he k whether the omplex is tilting we just need to he kthat thereisnoelement

f

inthepathalgebrasu hthat

(2.2)

⊕

j→k

P

j



(α

j

)

j

// P

k

f



0

// P

s

ommutes. Theexisten eofsu han

f

impliesthatthesetofrelationsmust ontain the set

{f α

j

: j → k}

, whi h is easy to see on e we dierentiate the potential in ordertothearrows.

Theaboveremarkallowsus,givenapotential

S

for

Q

,todetermine

δ(Q, S)

. Fromnowonwe'lldrop thesupers ript on

T

wheneverthevertexwithrespe t to whi hwe're onsidering thetilting omplexisxed.

Denition2.5. TheSeibergdual algebraofaquiver

Q

withpotential

S

(orof itsja obianalgebra)atthevertex

k ∈ δ(Q, S)

istheendomorphismsalgebraof

T

k

asdened above.

Ri kard'stheorem thentells thatseibergdual algebrashavederivedequivalent ategoriesofmodules.

3. Seibergdualityforgood potentials

Let's onsiderthefollowing lassofpotentials:

Denition3.1. Apotentialissaidtobeagoodpotentialifea harrowappears atleasttwi eandnosubpathoflengthtwoappearstwi e.

Note that, in parti ular, a quiverwith agood potentialhas the property that everyarrowis ontainedin atleasttwodistin t y les.

Proposition3.2. A quiverwith goodpotential isglobally dualisable.

Proof. Thisis an immediate onsequen eof thedenition of good potentialsin e all the relations we get from these kind of potentials are of the form

∂S/∂a =

P

d

i=1

λ

i

v

i

= 0

where

λ

i

∈ K

,

d ≥ 2

and the

v

i

's are pathsstartingwith dierent arrowsthankstotherequirementthatnosubpath oflengthtwoshould beshared betweentwotermsofthepotential. Thus, there annoto uranyrelationsofthe

type

uα

j

= 0

andtherefore

δ(Q, S) = Q

0

.



Let

(Q, S)

beaquiverwithgoodpotential. Wewanttogiveapresentationofits Seibergdualalgebraataxedvertex

k

. Wewillseethatthisalgebraisinfa tthe ja obianalgebraof aquiverwith potential. We will allthis quivertheSeiberg dual quiver.

Firstweshould omputethequiver. It hasthesamenumberofverti esasthe initial quiver (sin e we will have that number of inde omposable proje tives in

End

D

b

_(Q)

(T )

orresponds to thenumberof dire t summandsof

T

) and, forea h irredu iblehomomorphismbetweenthe

T

i

's,drawanarrowbetweenthe

orrespon-areofthree types(thisterminology,usedforsimpli ityoflanguage,is inspiredby [10℄):

•

arrowsoftheform

a

,where

a

isalsoanarrowin

Q

,willbe alledinternal arrows

•

arrowsoftheform

α

∗

will be alleddual arrows;

•

arrowsoftheform

[βα]

willbe alledmesoni arrows;

Thetheorembelowshowsthatthis hoi eofnotationisanadequateonesin ethe pro edure to getthe Seiberg dual quiveris the sameas theone that allow us to mutatetheinitialquiver. Alsoasinmutationswewilldothisintwoessentialsteps: obtain aquiver

Q

˜

that may ontain more arrows than theirredu ible homomor-phismsandthen,lookingatrelations,eliminatetheappropriatearrowsthatdonot orrespondtoirredu ibleones (thosewillbethearrowslyingin2- y les).

It turns out that relations on the Seiberg dual quiver an also be en oded in a potential(seeProposition3.4)anditwill bedeterminedasfollows:

(1) Determine

S := [S] +

˜

P

n

i=1

[β

i

α

j

]α

∗

j

β

∗

i

(eventually ontainingsomearrows representingnon-irredu iblehomomorphisms);

(2) Foreveryarrow

a

inatwo y le

ab

,taketherelation

∂ ˜

S/∂a = 0

and substi-tute

b

in

S

˜

usingthisequality(andthuseliminate

b

fromthequiver,sin e

b

isnotirredu ibleasit anbewrittenasa ompostionofarrows). Call

S

¯

tothepotentialthusobtained.

Remark 3.3. Again,forlanguagesimpli ity,arrowsappearingintwo y leswillbe alledmassivearrowsandthepro essdes ribedonitem2ofthealgorithmabove willbe alled integration over massivearrows.

Letus startby omparing themutated quiverandthe Seiberg dual quiver(no relationsonthem, yet).

Theorem 3.4. Let

Q

be aquiver with agoodpotential

S

. The underlying quiver ofthe mutationof

Q

oin ides with theunderlying quiverof theSeibergdualof

Q

. Proof. (1) Firstweprovethat Seibergdualityat

k

invertsin omingarrowsto

k

. The omplex

T

k

hasindegreezeroone opyof

P

j

foreveryarrowfrom

j

to

k

,thereforeforea hsu harrowyouget oneproje tionmap fromthe dire tsumto

P

i

andthereforeanirredu iblehomomorphismfrom

T

k

to

T

j

, hen egettinganarrowfrom

k

to

j

in thedualquiver. Forea h arrow

α

j

from

j

to

k

,denotethe orrespondenthomomorphismfrom

T

k

to

T

j

by

α

∗

j

. There are nomoreirredu ible homomorphims: any otherhomomorphism fa torsthroughsomefa torofthedire tsumrst.

(2) Now we prove that Seiberg duality at

k

inverts outgoing arrows from

k

. Thisrequiresthe ommutativityofadiagramlikethefollowing:

0

// P

i

//

f



0



0

//

⊕

j→k

P

j

(α

j

)

j

// P

k

// 0

anarrow

β

from

i

to

k

and takethe( y li )derivativeofthepotentialin order to

beta

. Sin e

S

is agood potential,

∂S/∂β =

P

d

t=1

λ

t

v

t

where the

v

t

'sarepathsfrom

i

to

k

(sin e

βv

t

isa y leforall

t

)and

d ≥ 2

. Togivea homomorphismfrom

P

i

to

⊕

j→k

P

j

wejustneed togiveahomomorphism from

P

i

to ea h

P

j

bytheuniversalpropertyof thedire tsum. Call

j

t

to theindexoftheproje tivefa torin

⊕

j→k

P

j

su hthat

α

j

t

isonthepath

v

t

.

Observethat

v

t

= α

j

t

v

˜

t

,where

v

˜

t

isapathfrom

i

to

j

t

asin thepi ture.

•

k

β

&&M

M

M

M

M

M

M

M

M

M

M

M

M

•

j

t

α

jt

88p

p

p

p

p

p

p

p

p

p

p

p

p

•

i

˜

v

_jt

oo

Setthehomomorphismfrom

P

i

toea h

P

j

asfollows:

•

zeroif

j 6= j

t

forsome

t

;

• λ

t

v

˜

t

if

j = j

t

forsome

t

; andset

β

∗

tobethehomomorphismindu edbythissetofhomomorphisms tothedire tsumandthereforetothe omplex

T

k

. Clearlythismapmakes thediagramabove ommute. Nowweneedtoprovethatthisisirredu ible. Ifnot,thenitfa torsthroughother

T

r

withthehomomorphismfrom

T

i

to

T

r

beingirredu ible andtherefore omingfrom anarrow

γ : i → r

(

r 6= k

sin ethequiverhasnotwo y les). Buttheexisten eofsu hafa torization would implythat allterms

βv

t

in thepotentialshare asubpath oflength two

γβ

whi h is a ontradi tion sin e, by assumption, the potential is a good one and

d ≥ 2

. Hen e

β

∗

is irredu ible. By onstru tion, these homomorphismsaretheonlyirredu ibleones.

(3) Forea hpathoflengthtwointheinitialquiveroftheform

•

j

α

j

_{// •}

k

β

i

_{// •}

i

wegetahomomorphism

β

i

α

j

from

T

j

to

T

i

. Itwillbeirredu ibleithere isn't ahomomorphismin theoppositedire tion. Infa t this followsfrom the fa t that

S

is a good potential and therefore every arrow appears in S. Thus, if

a

is the arrow going in the opposite dire tion,

∂ ˜

S/∂a

gives an expli it fa torization of the mesoni arrow. On the other hand, if it isn't ontained in a2- y le, then it is irredu ible, sin eit ould only fa -tor through the stalk omplex of

P

k

whi h is not, however, proje tive in

End

D

b

_(Q)

(T )

. Denotethishomomorphismby

[β

i

α

j

]

.

(4) Finally,ifnoneof theprevious asesapply, thenhomomorphismsbetween

T

j

and

T

i

are just arrows from

j

to

i

. Again, these homomorphisms are irredu ible ithey are not ontainedin a2- y leand asimilar argument totheoneaboveappliestothis ase.

Let

Q

˜

be the quiverobtained by taking all the homomorphisms onsidered in the ases above, even if they are not irredu ible. Determining this quiver

Q

˜

is, therefore, learly thesame pro edure viamutations orviaSeiberg duality. Now, sin e both mutation and Seiberg dualityrequire the elimination of 2- y les after thisstep(inthelater asetogetonlytheirredu iblehomomorphisms),thequiver obtainedbymutationat

k

andthequiverobtainedbySeibergdualityon

k

arethe

Atthis point,weshall provethat the algorithm aboveallowsus to obtainthe Seibergdualpotentialofaquiverwithpotential

(Q, S)

onaxedvertex

k

.

Proposition 3.5. The algorithm des ribed above omputes a potential for the Seiberg dualquiver su hthat itsja obianalgebrais

End

D

b

(Q)

(T )

.

Proof. Letthehomomorphismsrepresentedbydual arrowsofoutgoingarrowsbe asit is des ribed onthe proof of (3.4). Wewill rstprove that the relations in-du ed by the potential

S

˜

obtained through the algorithm above are satised by

End

D

b

_(Q)

(T )

. Let

τ (i, j)

bethe oe ientof

[β

i

α

j

]

in

[S]

.

•

Relations omingfrom dierentiatingon

β

∗

i

(dualofanoutgoingarrow):

∂ ˜

S/∂β

∗

i

=

n

X

j=1

[β

i

α

j

]α

∗

j

= β

i

(α

j

)

j

= 0,

sin ethemap inquestionishomotopi to zeroin the omplex ategory.

•

Relations omingfrom dierentiatingon

α

∗

j

(dualofanin omingarrow):

∂ ˜

S/∂α

∗

j

=

X

i

β

∗

i

[β

i

α

j

] = (

X

i

β

∗

i

β

i

)α

j

Let's he kthat

P

i

β

∗

i

β

i

= 0

. Infa t,let's omputethem-thentryofthis ve tor. Forthatwelookto theappearen esof

α

m

in

[S]

. So,ifwehavein

[S]

somesubexpressionoftheform

d

X

t=1

τ (i

t

, m)[β

i

t

α

m

]˜

v

i

t

thenwehavethem-thentryof

P

i

β

∗

i

β

i

givenby

d

X

t=1

τ (i

t

, m)˜

v

i

t

β

i

t

whi h iszerosin e

∂S/∂α

m

= 0

.

•

Relations omingfrom dierentiatingon

a

:

∂ ˜

S/∂a = ∂[S]/∂a = 0,

sin e this is essentially the same as

∂S/∂a

(eventually with some extra squarebra kets).

•

Relations omingfrom dierentiatingon

[β

i

α

j

]

(mesoni arrow): Wejustneedto he kthat

∂[S]/∂[β

i

α

j

] = α

∗

j

β

∗

i

butthisfollowsbydenitionof

α

∗

j

and

β

∗

i

ashomomorphisms(seeproofof (3.3))

Observe now that integration over massive arrowsdoesnot hange the relations indu edbythepotentialsin etheexpressionsobtainedbydierentiatingin order toamassivearrowsarezerointheja obianalgebra,a ordingtotheproofabove. Thelast thing weneed to he k isthat this potential

S

¯

givesall therelations. If

not,supposerstthatthepotentialof

Q

¯

isoftheform

S + W

¯

. Then,foranyarrow

a

inW

∂ ¯

S/∂a + ∂W/∂a = 0

implies that

∂W/∂a = 0

andhen e

W = 0

. Suppose nowthat there is one non-zerorelation

r

su h that

r

is notof theform

∂ ¯

S/∂a

forall

a

in the quiver. This relationisalinear ombinationofhomomorphismssu hthatea htermofthelinear ombination is a map from somexed

T

j

to some xed

T

i

. If this relation does not involve dual arrows, then these homomorphisms an be expressed as linear ombinations of elements of the path algebrafrom

j

to

i

and therefore this is a relation i there is some internal arrow

a

su h that

∂S/∂a

is equal to

r

up to square bra kets. Thus wegeta ontradi tionand therefore

r

has to involvedual arrows. However,the onstru tionofdualarrowsashomomorphismsmakesiteasy toseethatallpossiblerelationsinvolvingthemare ontemplatedinthe asesabove andthusprovingthatin fa talltherelationsareen odedinthepotential

S

¯

.



Denition 3.6. If onemassivearrowappearsin twodierent2- y lesof

S

˜

, that is,wegetanexpressionoftheform:

˜

S =

d

X

i=1

λ

i

ab

i

+

l

X

j=1

au

j

+ W

where,

λ

i

∈ K

;

a

and

b

i

'sarearrows;

d ≥ 2

;the

u

i

'sarepathsoflength

≥ 2

and

a

doesn'tappearin W,thenwesaythatthe

b

i

'sarerelatedarrows.

Given

Q

aquiverwithgoodpotential

S

,supposethat

S

˜

anbewrittenasfollows:

(3.1)

S =

˜

N

X

i=1

(λ

i

a

i

b

i

+

X

j

σ

i,j

a

i

u

i,j

+ b

i

v

i

) + W

where

σ

i,j

,

λ

i

,

µ

i

∈ K

,the

a

i

b

i

'sare2- y les(i.e.,the

a

i

'sandthe

b

i

'saremassive arrows),the

b

i

'saremesoni (thusthe oe ientof

b

i

v

i

is1),Wdoesn'thaveany terminvolvingmassivearrowsand

i 6= j

implies

a

i

6= a

j

(thatis,norelatedarrows o ur). Note that

b

i

6= b

j

be ause of the fa t that

S

, being good, doesn't have repeatedsubpathsoflengthtwo.

Theorem3.7. Let

Q

beaquiverwithagoodpotential

S

. If

k

isavertexsu hthat norelatedarrowso urin themutation, thereisaright equivalen e

φ

from

( ˜

Q, ˜

S)

to

( ˜

Q, S

′

+ ¯

S)

, where

S

′

istrivialand

S

¯

isobtainedbySeibergduality.

Proof. Sin ethere are norelatedarrows,let'sassumethat

S

˜

isof theform (3.1). Taketheautomorphismsgivenby:

φ

i

: K ˜

Q → K ˜

Q

a

i

7→ a

i

−

_λ

1

_i

v

i

b

i

7→ b

i

−

_λ

1

_i

P

j

σ

i,j

u

i,j

z

7→ z

if

z 6= a

i

,

b

i

,

z ∈ Q

1

Computing

φ( ˜

S)

,being

φ

the ompositionofall

φ

i

's,weget:

φ( ˜

S) =

N

X

i=1

(λ

i

a

i

b

i

−

1

λ

i

X

j

σ

i,j

u

i,j

v

i

) + W

whi hredu edpartisexa tly

N

X

i=1

(−

1

λ

i

X

j

σ

i,j

u

i,j

v

i

) + W

Now,ifwedointegrationovermassivearrowsin (3.1),takingina ountthat:

∂ ˜

S/∂a

i

= λ

i

b

i

+

X

j

σ

i,j

u

i,j

∂ ˜

S/∂b

i

= λ

i

a

i

+ v

i

andusingtherelations

∂ ˜

S/∂a

i

= 0

and

∂ ˜

S/∂b

i

= 0

in

S

˜

weget:

N

X

i=1

(−

1

λ

i

X

j

σ

i,j

u

i,j

v

i

) + W

whi histhesameas

φ( ˜

S)

red

.



Corollary 3.8. If

Q

isa quiver with agood potential

S

andif

k

is avertexsu h that no related arrows arise in the mutation pro edure, then mutation at

k

and Seiberg dualityat

k

areisomorphi . In parti ular mutationof goodpotentials give derivedequivalentalgebras.

Proof. We have that

J(Q

triv

⊕ Q

red

, S

triv

+ S

red

) ∼

= J(Q

red

, S

red

)

. Conjugating thisfa t withremark1.5andtheorems3.4and3.7,wegettheresult.



4. An example

Given a Del Pezzo surfa e

S

, we an realise its derived ategory of oherent sheavesas apathalgebrawith relations. This anbedone usingstrongly ex ep-tional olle tions. For the purposeof what follows, letus re allsome resultsand denitions.

Denition4.1. Anex eptional olle tiononaproje tivesurfa eisa olle tion of oherentsheaves

{E

1

, ..., E

n

}

su hthat:

•

Ext

k

(E

i

,E

i

)=0,

∀k > 0

and Hom(E

i

,E

i

)=

K

•

Ext

k

(E

i

,E

j

)=0,

∀1 ≤ j < i ≤ n

,

∀k > 0

•

Thestalk omplexesofthesesheavesgenerateD

b

(Coh(X))asatriangulated ategory.

Itisstrongly ex eptionalif,furthermore,Ext

k

(E

i

,E

j

)=0,

∀1 ≤ i, j ≤ n

,

∀k > 0

Theorem4.2. If

S

isaDel PezzoSurfa e, wehave astronglyex eptional olle -tionsof sheaves given by:

• {O, O(1), O(2)}

if S=

P

2

• {O, O(1, 0), O(0, 1), O(1, 1)}

if S=

P

1

_{× P}

1

• {O, O(E

1

), ..., O(E

r

), O(1), O(2)}

if S is

dP

r

with

r ≤ 8

,whereea h

E

i

is an ex eptional urve of the blow up and

dP

r

is the Del Pezzo obtained by blowingup

1 ≤ r ≤ 8

pointsin

P

2

.

• Ext

k

_{(T, T ) = 0, ∀k > 0}

•

Tgenerates

D

b

_(Coh(X))

astriangulated ategory

•

B=End(T)hasniteglobaldimension

Theorem 4.4. Let

X

be a nonsingular proje tive variety,

T

a oherent sheaf on

X

and

B = End(T )

. Then the following areequivalent: (1) Tistilting;

(2) There is an equivalen e

Φ : D

b

_{(Coh(X)) → D}

b

_(mod(B))

of triangulated ategorieswith

Φ(T ) = B

,wheremod(B)isthe ategoryofnitelygenerated right modules overB.

Now, ifwe takethe dire t someofastrongly ex eptional olle tionover

S

, we getatiltingsheafand,therefore,aderivedequivalen ebetween

Coh(S)

and

KQ/I

forsomequiver

Q

andsomeidealofrelations

I

. Theseare determinedlooking at the irredu ible homomorphismsbetweenthe sheavesin the olle tion and taking relationsbetweenthose homomorphisms.

Example4.5.

dP 1

withex eptional olle tion

{O, O(E

1

), O(1), O(2)}

•

1

a

_//

b

@

@

@

@

@

@

@

@

@

@

@

@

@

@

@

@

•

2

c

1



c

2



•

4

_•

3

d

1

oo

_d

₂

oo

_d

3

oo

withrelations:

d

3

c

1

= d

1

c

2

d

2

c

1

a = d

1

b

d

3

b = d

2

c

2

a

Thisexample, however,doesn'tt in oursettingof quiverswithpotentials. In fa t whatweoughtto onsider isnot

S

itselfbut

X = ω

S

-thetotalspa eofthe anoni al bundle of

S

- instead. This is a lo al Calabi-yau three-fold and if we let

π : X → S

be the natural proje tion, weget

B = End

˜

X

(⊕

i

π

∗

_E

i

)

is derived equivalentto

Coh(X)

,where

(E

i

)

i

isanex eptional olle tionoverS.Thisalgebra

˜

B

analsobeseenasapath algebraof aquiverwhi h anbeobtainedfrom the orrespondentDel Pezzoquiveraddingonearrowforea h relationin theopposite dire tionof the ompositionofarrowsin thatrelation. These willbequiverswith potentials.

Example4.6. The ompletedquiverfor

dP 1

withex eptional olle tion

{O, O(E

1

), O(1), O(2)}

is:

Q =

•

1

a

_//

b

@

@

@

@

@

@

@

@

@

@

@

@

@

@

@

@

•

2

c

1



c

2



•

4

R

3

??

~

~

~

~

~

~

~

~

~

~

~

~

~

~

~

~

R

1

OO

R

2

OO

•

3

d

1

oo

_d

2

oo

_d

3

oo

withpotential:

S = R

3

(d

3

c

1

− d

1

c

2

) + R

1

(d

1

b − d

2

c

1

a) + R

2

(d

2

c

2

a − d

3

b)

Note that this is agood potential. Therefore, using ourresults, mutations on anyvertexof this quiverwill giveus derived equivalent path algebras. Sin e the one above is derived equivalent to

Coh(X)

, so will be

µ

k

(Q, S)

. Let's nish by presenting

µ

1

(Q, S)

.

˜

Q =

•

1

R

∗

1



R

∗

2



•

2

a

∗

oo

c

1



c

2



•

4

[aR

2

]

??

~

~

~

~

~

~

~

~

~

~

~

~

~

~

~

~

[aR

1

]

??

~

~

~

~

~

~

~

~

~

~

~

~

~

~

~

~

R

3

??

~

~

~

~

~

~

~

~

~

~

~

~

~

~

~

~

[bR

1

]

//

[bR

2

]

//

•

3

b

∗

__@

@@

@@

@@

@@

@@

@@

@@

_@

d

1

oo

_d

2

oo

_d

₃

oo

Wetakea y li allyequivalentpotentialsin etherearetermsonitstartingand ending at

1

. Then let'ssubstitute paths oflength twopassingthrough

1

bynew arrowsandadd

∆

1

.

˜

S = R

3

d

3

c

1

− R

3

d

1

c

2

− d

2

c

1

[aR

1

] + d

1

[bR

1

] − d

3

[bR

2

] + d

2

c

2

[aR

2

]

+ [aR

1

]R

∗

1

a

∗

+ [aR

2

]R

∗

2

a

∗

+ [bR

1

]R

1

∗

b

∗

+ [bR

2

]R

∗

2

b

∗

Clearlythispotentialisnotredu ed. Let's ondsiderthefollowingrightequivalen e:

φ : K ˜

Q → K ˜

Q

d

1

7→ d

1

− R

∗

1

b

∗

d

3

7→ −d

3

+ R

∗

2

b

∗

[bR

1

] 7→ [bR

1

] + c

2

R

3

[bR

2

] 7→ [bR

2

] + c

1

R

3

u

7→ u

if

u 6= d

1

, d

3

, [bR

1

], [bR

2

]

,

u ∈ Q

1

.

Ifwe ompute

φ( ˜

S)

,itisoftheform

S

′

thesameresultwhi h is:

¯

Q =

•

1

R

1



R

2



•

2

a

∗

oo

c

1



c

2



•

4

[aR

2

]

??

~

~

~

~

~

~

~

~

~

~

~

~

~

~

~

~

[aR

1

]

??

~

~

~

~

~

~

~

~

~

~

~

~

~

~

~

~

R

3

??

~

~

~

~

~

~

~

~

~

~

~

~

~

~

~

~

•

3

b

∗

__@

@@

@@

@@

@@

@@

@@

@@

_@

d

2

oo

withpotential

¯

S = c

2

R

3

R

∗

1

b

∗

+ c

1

R

3

R

∗

2

b

∗

+ d

2

c

2

[aR

2

] − d

2

c

1

[aR

1

] + [aR

1

]R

∗

1

a

∗

+ [aR

2

]R

∗

2

a

∗

.

Sin ethis newja obianalgebrais derivedequivalentto

J(Q, S)

, itisalso derived equivalentto

Coh(X)

.

Referen es

[1℄ Aspinwall,P.,Melkinov,I.;D-BranesonvanishingdelPezzosurfa es,JournalofHighEnergy Physi s412,42,2004;

[2℄ Aspinwall, P.S., Fidkowski, L.M., Superpotentials for Quiver Gauge Theories, arXiv:hep-th/0506041

[3℄ Baer,D.;Tiltingsheavesinrepresentationtheoryofalgebras,Manus riptaMathemati a,60, 3,1988

[4℄ Bondal,A.;Representationofasso iativealgebrasand oherentsheaves,Mathemati softhe USSR,Izvestiya,Vol.34,1,1990;

[5℄ Braun,V.,OnBerenstein-Douglas-Seibergduality,arxiv:hep-th/0211173v1;

[6℄ Bridgeland, T., T-stru tures on some lo al Calabi-Yau varieties, Journal of Algebra 289 (2005),no2,453-483

[7℄ Derksen,H.,Weyman,J.,Zelevinsky,A.;Quiverswithpotentialsand theirrepresentations I:mutations,arxiv:0704.0649v2[math.RA℄;

[8℄ Gelfand,Manin;MethodsinHomologi alAlgebra,Springer;

[9℄ König,S.,Zimmerman,A.;DerivedCategoriesforGroupRings,Springer,Le ture Notesin Mathemati s1685,1998;

[10℄ Mili i ,D.;Le turenotesonderived ategories,availableon www.math.utah.edu/

˜

mili i /der at.pdf;

[11℄ Mukhopadhyay, S., Ray, K.Seiberg Dualityas derived equivalen e for some quiver gauge theories,arxiv:hep-th/0309191v2;

[12℄ Ri kard,J.; Moritatheoryfor derived ategories,LondonMathemati alSo iety- Journal, 39,1989;