• 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 …
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