DEFECTS IN COHOMOLOGICAL GAUGE THEORY AND DONALDSON- THOMAS INVARIANTS
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1 DEFECTS IN COHOMOLOGICAL GAUGE THEORY AND DONALDSON- THOMAS INVARIANTS SISSA - VBAC 2013 Michele Cirafici CAMGSD & LARSyS, IST, Lisbon based on arxiv:
2 OUTLINE Introduction Defects DT as a gauge theory Defects and DT An example Conclusions
3 INTRODUCTION
4 SETTING THE STAGE Quantum theory and geometry are deeply related Use quantum theory to compute interesting invariants: Seiberg-Witten, Gromov-Witten, Donaldson-Thomas Quantum theory can be modified to include defects Broad program: new moduli spaces, new enumerative invariants. Are they interesting? useful? Today: divisor defects in DT from gauge theory perspective
5 DEFECTS
6 DEFECTS Defects are a fundamental modification of a quantum field theory The simplest way to think about a defects is as a boundary condition (on a line, a surface...) In physics the presence of a defect opens up a new BPS sector [Gaiotto-Moore-Neitzke] Mathematically this means that we have new moduli problems
7 SURFACE DEFECTS For example Surface Defects are useful for certain aspects of Geometric Langlands Consider a gauge theory: is a principal G- bundle with connection We define a defect on a surface assume M 4 ' K z = r e i local coordinate on A The theory is ramified. (gauge field). The defect is K E! M 4 2 T G. Locally we F A =2 + Poincaré dual [Gukov-Witten] [Kronheimer-Mrowka] parametrizes the defect
8 SURFACE DEFECTS WITH INSTANTONS One can set up an instanton problem for gauge theory with surface defects The moduli space of ramified instantons defined via the ramified instanton equations The defect has a topological angle [Kronheimer-Mrowka] Instanton counting on the affine space (non compact defect): [L, ] =0 (F 0 A) + =(F A 2 ) + =0 Z( 1, 2, a; L, q, ) = X m X k2z e Tr m q k M 0 k,m Tr m = 1 2 I M 0 k,m 1. Z is Tr F A [Alday-Tachikawa] [Feigin-Finkelberg-Negut-Rybnikov]
9 GAUGE THEORY AND DT
10 DT FROM GAUGE THEORY Donaldson-Thomas program: understand higher dim geometry via gauge theory (in particular CY) Our approach is to reformulate DT theory as an instanton counting problem [C., Sinkovics, Szabo] [Nekrasov] [Nakajima] Define an appropriate 6d Cohomological Gauge Theory and a generalized instanton moduli space Explicit results: enumerative invariant for generic toric CY, NCDT at toric singularities, motivic invariants, cluster algebras.
11 INSTANTON COUNTING Z The gauge theory is a cohomological version of 6d Yang- Mills. Contains a gauge connection + other stuff. (really, coherent sheaf) on a G-bundle It is equivalent to study the generalized instanton moduli space of solutions of F (0,2) A A, F (1,1) A ^ t ^ t +, = lt^ t ^ t, Generating function of DT invariants A = 0. E! X
12 E f o QUIVER QUANTUM MECHANICS Generalized ADHM construction on the affine space via instanton quiver (rep theory) B 2 8 B 1 V I ' W 3X [B,B ]+ 3X B, ' =0, =1 B,B + ', ' + II = &, ' =0 B 3 =1 I ' =0, Use virtual localization wrt maximal torus Very explicit results. I m neglecting a lot of virtual fundamental subtleties
13 DEFECTS AND DT
14 DIVISOR DEFECT Basic idea: study DT with a Divisor Defect Locally we take where is a divisor and the local fiber of the normal bundle. We define the defect by Divisor defects are classified by pairs and, a subgroup of Levi type. with In geometrical terms we have a G-bundle whose structure group is reduced to on.
15 INSTANTON MODULI SPACES Now we consider generalized instantons. Conjectural moduli space The defect is parametrized by a natural set of topological angles which measure the reduction of the structure group on the divisor
16 PARABOLIC SHEAVES Already without the defect one needs coherent sheaves to formulate the problem properly. In our case the proper sheaves to consider are parabolic sheaves. There is a correspondence between Levi subgroups of and parabolic subgroups of We define a parabolic structure on the sheaf over as the flag of subsheaves of At each point this is the flag of vector spaces stabilized by the parabolic group previous example: [Mehta,Shesadri] [Bhosle]
17 PARABOLIC SHEAVES This is the analog of the reduction of the structure group for a sheaf A more convenient definition is The two definitions are related by 0 /F i (E) /E /E D /G i (E) /0. Finally, one is lead to conjecture [Murayama-Yokogawa]
18 AFFINE SPACE
19 ' ' ' ' MODULI SPACE OF PARABOLIC SHEAVES Compactify the affine space to To define a defect on of torsion free sheaves of rank [Feigin-Finkelberg-Negut- Rybnikov],[Negut], [Finkelberg-Rybnikov] consider the flag Framing: F 0 ( D) D1 / F r+1 D1 / / F 0 D1 O r D 1 ( D) / W (1) O D1 O r 1 D 1 ( D) / / W (r) O D1 with Chern classes are specified by the degree
20 PARABOLIC SHEAVES AS ORBIFOLD SHEAVES In this case parabolic sheaves can be understood in terms of orbifold sheaves. Natural action Isotypical decomposition with One can use the covering map to construct an isomorphism In particular: [Okounkov], [Biswas] [Feigin,Finkelberg,Negut,Rybnikov]
21 O O O INSTANTON COUNTING Extend the generalized ADHM construction B a 1 2 B a 1 1 B a 2 3 B / a 1 3 B3 1 V a 1 / a 2 V a / 1 V a+1 ' a 2 ' a 1 ' a I a 1 W a 1 B a 2 I a W a B a 1 B a+1 2 I a+1 B a+1 1 W a+1 B a+1 3 / ' a+1 2 B a 1 B a 2 B a 2 B a 1 =0, B1 a+1 B3 a B3 a B1 a =0, B2 a+1 B3 a B3 a B2 a =0, I a+1 ' a =0. We can set up the DT enumerative problem as an instanton counting problem where equivariant
22 CONCLUSIONS Interplay between quantum theory with defects and geometry Divisor defects: moduli space of sheaves with parabolic condition on a divisor Study enumerative invariants (via localization on toric CY) Extend to other gauge theories (higher and lower dim) and other defects (higher and lower codim)
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