S.G., “Surface Operators and Knot Homologies,” arXiv:0706.2369 S.G., A.Iqbal, C.Kozcaz, C.Vafa,...

• S.G., “Surface Operators and Knot Homologies,” arXiv:0706.2369

• S.G., A.Iqbal, C.Kozcaz, C.Vafa, “Link homologies and the refined topological vertex,” arXiv:0705.1368

• S.G., E.Witten, “Gauge theory, ramification, and the geometric Langlands program,” hep-th/0612073

• S.G., H.Murakami, “SL(2,C) Chern-Simons theory and the asymptotic behavior of the colored Jones polynomial,” math.gt/0608324

Geometric StructuresGeometric Structuresinin

Gauge TheoryGauge Theory

Geometric StructuresGeometric Structuresinin

Gauge TheoryGauge Theory

Sergei GukovSergei Gukov

During the past year I have been mainly working on topological string theory

and topological gauge theory

which, roughly speaking, describe the supersymmetric sector of the physical string theory/gauge theory.

Applications

• Physical Applications•F-terms in string theory compactifications•Black Hole physics•dynamics of SUSY gauge theory

• Mathematical Applications•Enumerative geometry•Homological algebra•Low-dimensional topology•Representation theory•Gauge theory

Chern-Simons Theory• In Chern-Simons theory

• Example: G = SU(2)

[E.Witten]

polynomial in q

Wilson loop operator

Jones polynomial

• Question (M.Atiyah):

Why integer coefficients?

Knot Homologies

• Khovanov homology:

Example:

[M.Khovanov]

j5 7 931

3

012

i

Physical Interpretation

BPS state:membrane ending on the Lagrangian five-brane

space of BPS states [S.G., A.Schwarz, C.Vafa]

M-theory on

M5-brane onEarlier work:[H.Ooguri, C.Vafa][J.Labastida, M.Marino, C.Vafa]

(conifold)

Lagrangian

This Physical Interpretation Leads to Many New Results and Surprising Predictions

• new theory, which unifies all the existing knot homologies [N.Dunfield, S.G., J.Rasmussen]

• generalization to arbitrary groups and representations

[S.G., A.Iqbal, C.Kozcaz, C.Vafa]

• mathematical construction of these is out of reach

[S.G., J.Walcher]

• Codimension 3: Line operators:

• Codimension 4: Local operators

Wilson line ‘t Hooft line

• Codimension 2: Surface operators

much studied in AdS/CFT

Realization in 4D Gauge Theory

Surface operators

• Transform in an interesting way under Electric-Magnetic duality

• OPE algebra of line operators becomes non-commutative

It turns out that many interesting 4D gauge theories admit (supersymmetric) surface operators, which have a number of nice properties:

surfa

ce

operator

S

• Braid group actions on D-branes

[S.G., E.Witten]

It turns out that many interesting 4D gauge theories admit (supersymmetric) surface operators, which have a number of nice properties:

• Mathematical applications– to the so-called ramified case of the

Geometric Langlands program– to Knot Homologies

• Correlation function = vector space

vector space

Open questions and further directions

• New types of surface operators and their transformation under S-duality

• Realization of Chern-Simons invariants in four-dimensional gauge theory

• Geometric construction of representations using surface

operators in gauge theory …