Vielnam J Math (2014)42:393-399 DOI 10.1007/S10013-OI4-0087-X
The Dijkgraaf-Witten Invariants of Circle Bundles
Vu The Khoi
Received: 25 August 2013 / Accepted. 21 April 2014 / Published online: 7 Augusi 2014
© Vietnam Academy of Science and Technology (VAST) and Sponger Science-i-Business Media Singapore 2014
Abstract Turaev (J. Geom. Phys. 57: 2419-2430, 2007) proves a formula for the Dijkgraaf-Witten invariants of surfaces in terms of projective representahons by using the stale sum invariant technique from quantum topology. In Khoi (J. Knot Theory Ramif.
20: 837-846, 2011), the author gives another proof of Turaev's theorem by using classi- cal method of characters and representation theory The purpose of this paper is to prove a formula for the Dijkgraaf-Witten invariants of a circle bundle over a surface by using techniques from Khoi (J. Knot Theory Ramif, 20: 837-846,2011).
Keywords Dijkgraaf-Witten invanants • Circle bundles
Mathematics Subject Classification (2000) Primary 57M27 - Secondary 57R56
1 Introduction
The Dijkgraaf-Witten invariants are homotopy invariants of manifolds which were intro- duced in [2] by using finite gauge group We first recall briefly its definition. Let M be a closed, oriented manifold of dimension n with the fundamental group jr — 7Ti(M,mo). Let F be a field and G be a finite group whose order |G| is invertible in F We denote by X the Eilenberg-MacLane space K(G. I). It is a well-known fact that for every group homo- morphism >- : Tt - * C, there exists a umque up to homotopy map fy : (M.mo) -* X which induces y on the level of fundamental groups. Now for each cohomology class
Dedicated to Professor Mutsuo Oka on Ihe
This research is funded by Vielnam National Foundaiion for Science and Technologj' Development (NAFOSTED) under grant number 101 01-2011,46,
V. T Khoi (El)
Institute of Mathematics (VAST), 18 Hoang Quoc Viei. 10307, Hanoi, Vietnam e-mail: [email protected] vn
a € H"(C, F') = H"(X. F'). where F* is the group of invenible elements of F. we can define the Dijksraaf-Wilten invariant (see [2,5,6]) by:
Z„(M):=|Gr' J^ i{f,)'M.[m}.
yehom(jr,C)
Here, hom(jr, G) is the set of all group homomorphisms from TT to G and ((/y )*(«), [M]}
isihevaiueof (/y)*(a) on the fundamental class {M] e H„(M).
In general, it is hard to wnte down a closed-form formula for the Dijkgraaf- Witten invariant. For the case M is a surface, Turaev gives a nice formula for the Dijkgraaf-Witten invariant.
Theorem 1 ([5, Theorem 1.2]) Let F be an algebraically closed field such that \G\ is invertible in F. For any normalized 2-cocycle c : G x G -^ F" and any closed connected onented surface M,
Z[,|(M) = IGf^"^' J2 (dim/j)^""' • IF, pedc
where / ( M ) is the Euler characteristic of M.
Here are the explanations of the terminology in the statement of the theorem. First of all, a normalized 2-cocycle c is a map c : G x G -^ F* which satisfies the following two conditions:
1. c(g.l) = c{l,g) = 1 for all g e G ;
2, c(gi. g2) c{gig2,g3) = c{g\, g2gi) c(g2. g3) ior all gi. g2, g3 £ G.
We denote by [c] the cohomology class represented by the cocycle c. The reader may consult [I] for backgrounds on group cohomology.
Secondly, Gc is the set of all the equivalent classes of irreducible c-representationsof G.
Let IV be a finite dimensional vector space over F. Recall that a c-representation is a map p :G ^* GL{W) which sadsfies the following conditions:
1, p(\) = \dw;
2. P(g\)p{g2)=^c(gi,g2)p{g\g2) f o r a n y g i , g 2 e G.
Two c-representations y9 : G -> GL(W) and p': G -^ GL{W) aie said lo he equivalent if there exists an isomorphism of vector spaces j W -> W such that p/{g) = j p{g))-' for allg e G. Therefore, the integer dim(p) := dim(H') depends only on the equivalent class of p . A c-representation p : G -^ GL(W) is called irreducible if 0 and W are the only vector subspaces of W invariant under p(G). It is clear diat the irreducibiHty of a c-representation only depends on its equivalent class.
The main result of our paper is Theorem 2 which gives a formula for the Dijkgraaf- Witten invariants of a circle bundle over a closed oriented surface in the style of Turaev In the next section, we will first prove a techmcal lemma and then state and prove the main result.
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The Dijkgraaf-Vi'inen hivanants of Circle Bundles 2 Main Results
Let M bea circle bundle of Euler number e over a closed-onented surface of genus k. It is well-known that the fundamental group of M is given by:
7Ti (M) = iai,bi.. ., ak,bk, T\{a,,T] = [b„T] = f j t a , , feJT"'^ - l)
( = 1
In order to evaluate the Dijkgraaf-Witten invariant, we find a cycle which represents the fundamental class in the third homology group H3{JTI(M), F " ) . To represent a class in Hi{jri{M), F*), we will use the non-homogeneous bar resolution (see [1, II. 3]).
We first prove a lemma which gives us an explicit formula for the fundamental class.
For simplicity, we denote by T x [x\y] the sum t j c | y | r ' ' y " ' ] + [T^^y'^x'^^\x\y] + [xy\T-^y-h-^\x] in C3(^, (M), F').
Lemma 1 Suppose that M is a circle bundle of Euler number e over a closed oriented surface of genus k Assume that e is invertible in the field F, then the fundamental class of M in H3(7T\{M), F*) can be represented by:
Uj^{T'x[L-i\a,Hrx{l,.iai\b,]-\-r'x{L-ia,b,\af^]-\-rx[L_ia,b,a;^\bf^-\)+
T' X [lk-i\ak] -\- T' X {Ik-iak\bk] ^ T' x [Ik-\akbk\a-^]\, where f =
Proof Firstof all, we consider the case where the Euler number of A/ is 1. We represent M as in Fig, 1 (genus ofthe base is 2).
Fig. 1 The circle bundle of Euler number I over a surface of genus 2
€ ! Sprii
Next, we divide M into triangular prisms by drawing diagonal planes which contain a common vertical edge Each prism can be divided into three tetrahedrons as m Fig. -
Therefore, we now can write down the fundamental class using non-honiogeneous bar resolution. Note dial the sum T x [x\y] = {xlylT'h'^] + f?"^'^ -'•' l-^l>'] + [A-y[7-'y-'.i^'|x] in C3(jri(M), F ' ) corresponds to three tetrahedrons of the pnsm.
Wnting down the sum for each pnsm, we get the formula for e = 1.
In die general case of Euler number e, there exists an e-fold covering map (p : M ^ M^, where M is a circle bundle of Euler number 1 Note diat under the map <p, the fiber of M covers the fiber of M e times and the fundamental class of M is mapped to e times the
fundamental class of M So, the general formula follows. ^ We denote by [a] e H^{G, F*) a cohomology class in the third group cohomology of
G widl coefficient in F". Let A' be a semi-simplicial complex that defines die classifying space BG and K„ be its n-skeleton. Following [6], we denote by C „ ( B G , Z ) the rela- tive homology group H„(K„. K„^\) of K. Then the cohomology of the cochain complex Hom(C*(BG, Z), F*) gives the cohomology group / / ' ( C . F*).
As in [2], we take a 3-cocycle a € Z^{BG; F*) representing {a\. So we gtt a normalized 3-cocycle a : G x G x G —* F* which satisfies the following conditions' 1. (cocycle condition) Q;(g, ft, * ) a ( g , / i ^ , O a ( / ' , ' ; , 0 = a(gh, k,l)a{g. h. kl)\
2. (normalized condition) ff(l,g, A) —a(g, \,h) — oi(g,h,\) — \ and ei(g,g~\h) —a(g,h,h~^) = 1 for all g, A e G.
Before giving the proof of the main theorem, we recall some results from the the- ory of twisted representation which will be needed. We denote by Gc the set of all equivalent classes of irreducible c-representation of G. For a representation p e Gc, we denote by dp its dimension and xp its character regarded as a function on F[G] by linear extension Note that equivalent c-representations have the same character.
Fig, 2 Themangulauonof each
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The Dijkgraaf-Witten Invariants of Circle Bundles
We denote by F''[G] the twisted group algebra of G By definition, this is the F-vector space F[G], together with the twisted multiplication given by
gi • g2 ~c{gi,g2)gig2, where gi,g2 e G.
It follows from the defimtion that p(gi gi) = p(g\)p(g2)- Therefore, the character Xp is invanant under twisted conjugation Thatis XpC^"' • g • h) = Xp(g), though it is not a class function on G.
Several results in the theory of representation of finite groups stili hold in the con- text of projective representation [3]. There exists a one-to-one correspondence between c-representations of G and left F'^[G]-modules. Moreover, the regular c-representation of G decomposes into irreducible factors. That is, there exists a decomposition of F'^[G]- modules
F'lG] = © C '
pec.
By taking the trace of the twisted multiplication map by g on both sides of the above decomposinon, we find that
' Y - ^ ^ ^ ( l ' f * - ' - f l In order to state the main result, we need to introduce some notations. Given a normalized
3-cocycle a.GxGxG—fF* and an element / e G, we denote by a, the normalized 2-cocycle defined by
a,{x, y) : - a(t x {x, y]) = a{x,y. t~^y-')a(t-^y-^x-\ x. y)a(xy. t'^y-'x''. x).
In the sequel, we will denote by C(t) the centralizer of an element i in G and by C(/)„ the set of all equivalent classes of irreducible tt,c-representations of C(;)
We now ready to state our main result.
Theorem 2 Let M be a circle bundle of Euler number e over a closed oriented surface of genus k and F be an algebraically closed field. Let a : G x G x G ^- F* be a normalized 3-cocycle. Suppose that \G\ and e are invertible in F.
Then the following formula for the Dijkgraaf-Witten invariant of M holds
z,^iiM)^^j:\^(t)\-^ i: '-f?
' ' f e e p^co), '^P
Proof Note that we define the 2-cocycle a, by taking the value of a at the sum of three tetrahedrons of the pnsm so at is indeed a 2-cocycle (see [2]).
We can identify a representation y ' JTI (M) -^ G with an e]ement ; e G together with a2it-tuple p- - (gi,hi, . ,gk,hk) of elements belonging to C{t) such that fi'' .=
I~[,-^j[g,, h,} — t^ By definition
ZH(M) = ^ E E J»(i;(f'x[',-.1*1+f«[',-,«,!*,]
+1' xlI,-ia,b,\a-^] + t' x[l,_ia,b,a-^\b;^])-^t' x [lk-i\ak] +i' x [h-,ak\hi]
+t' X lh.iakbk\a;']).
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So we may lewrite the Dijkgraaf-Witten invaiiant using the 2-cocycte ff,. as follows
•=«<">'iiiE E V(E(['.-ii«.i+[',-i«.itii
leG i.nClif.'.R^^i' 1 = 1
+H-I0,l., In,-']+ [;,-ia,l),o,-'|i-'l)+ [/,_,la,]+ [ft_io,|ii,] + [4-imiu|oi"'l)
= i E E (n--('-i.«.)".-('-i-.-fc)°.-<'.-i*».."r'>
xii,.(;,-,0,6,0-', i.r'))».'('i-i."n)«>'('i-i«t.i't)«.'('i-i<«''i.»r')' From the fad that a,, is a nonnalized 2-cocycIe and that ri/^ilgj''',] = ^^' we can deduce that
ai.Ui^mbiaj-'.b^') = ai.l.t'bi,. b^') = a,,(bt. b^'l-'T' = a,.(ij"', I"')
= a(b^',r'. l)i»(6i,ij',I-')a(t^'l~',tt,ii"') = 1 and thata,e(4.(-'') = 1
Therefore, we can multiply these terms in the formula for Zia]{M) without changing it:
^ l " l ' " ' = ; i ^ E E {n''i-U,-l.a,)ai,a,-,a„b,-)a,,Ui-li,b,.ar')
xa,,(li_,a,b,a-'.b;')y,.{li, I'').By using identity (*), we deduce that
xt„.(/,-lo,i,,<,-', *,-'))«,•(/*, t - ' ) j ^ 5^ 'i,Xp(«''t-')
P6C(/)„
= ; i H E E Fmj E (n<'.'«-..«,)«,.(/,-ia„i.,) 'eGpeC(0„ rtectri^*, ,=1
(]~[a,'(/,-),o,)o,.(/,-iii,.»,)«,.(;,_io,i,.o,-')a,,(;,_,<,, 4,0-1^ i,-')]
ti,.(/l,l"')x,(X'''"') = X,(«J,, •'"'), where C , := gi •/!! •si' A^'- •-gi • Al • f^'•*;'•
fi Spnnger
The Dijkgraaf-Wiilen Invariants of Circle Bundles So we arrive at
To finish the proof, we will need the following lemma from [4],
Lemma 2 ([4, Lemma 2 2]) For any group elements u, g.h, the following identities hold:
YI''p^P<-" g-^-g'^ • /""') = \G\Xp{u • g)Xpi8~^)- lieG
Yl'^P^P^" • S)Xpig~^) = \G\Xp(u).
gee
Applying this lemma repeatedly, we get
^w<'") = ;i^Eic(')i"-' E ^ -
Remark I Unfortunately, due to the complicated form of the formula, we could not obtain any integrality theorem for the Dijkgraaf-Witten invanants as in the case ofthe surfaces.
Acknowledgments The author would like to thank the VAST-JSPS joint project for the travel support to attend the conference to report on the result of this paper. The author would like to express his grantude to the Vietnam Institute for Advanced Study in Mathematics for the support during the revision of the paper We thank the anonymous referees for the careful reading of our manuscnpt and the valuable c(
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