**NUS scientist gives plenary talk at prestigious String-Math 2015 conference**

1 Apr 2015 NUS scientist is the FIRST Southeast Asian to be invited as a plenary speaker at prestigious String-Math 2015 conference.

String theorist Prof Meng-Chwan TAN from the Department of Physics in NUS has been invited to give a plenary talk at String-Math 2015, the MOST PRESTIGIOUS conference in all of mathematical string theory. Past and present speakers include Fields medallists Edward Witten, Maxim Kontsevich, Simon Donaldson, Andrei Okounkov and Shing-Tung Yau, Dannie Heineman Prize for Mathematical Physics laureate Greg Moore, Fundamental Physics Prize laureate Ashoke Sen, and other leading string theorists and mathematical physicists.

String theory is the leading candidate for Einstein's dream of a "Theory of Everything" - a unified theory of all the four fundamental forces of the universe which govern every physical and therefore scientific phenomenon in nature. In recent times, it has also inspired new ideas and directions in pure mathematics, and, as such, revived the deep relationship between physics and mathematics first seeded by Isaac Newton who discovered the mathematics of calculus through his efforts to understand the physics of motion. This has resulted in the emergence of a 21st century "third discipline" known as mathematical string theory or "String Mathematics", whereby its practitioners are leading figures in the fields of physics and mathematics who are effectively bilingual in both disciplines. It is at String-Math 2015 where the luminaries of "String Mathematics" will gather to present their latest findings at the cutting edge of physical and mathematical knowledge.

At String-Math 2015, Prof Tan will talk about his work on "M-Theoretic Derivations of 6d/5d/4d-2d Dualities: From a Geometric Langlands Duality for Surfaces, to the 6d/5d/4d AGT Correspondence, to Integrable Systems" [1,2].

In the first [1] of this two-part work, he starts by showing that a mathematically conjectured geometric Langlands duality for complex surfaces [3] and its generalizations -- which relate some cohomology of the moduli space of certain ("ramified") G-instantons to the integrable representations of the Langlands dual of certain affine (sub) G-algebras, where G is any compact Lie group -- can be derived, purely physically, from the principle that the spacetime BPS spectra of string-dual M-theory compactifications ought to be equivalent. Then, by introducing Omega-deformation via fluxbranes and adding half-BPS boundary defects via M9-branes, he goes on to show that the celebrated 4d AGT correspondence [4, 5] and its generalizations -- which essentially relate, among other things, some equivariant cohomology of the moduli space of certain ("ramified") G-instantons to the integrable representations of the Langlands dual of certain affine W-algebras -- can likewise be derived from the principle that the spacetime BPS spectra of string-dual M-theory compactifications ought to be equivalent. Last but not least, by considering various limits of the last setup, he connects the story to chiral fermions and integrable systems. Among other things, he derives the Nekrasov-Okounkov conjecture [6] -- which relates the topological string limit of the dual Nekrasov partition function for pure G to the integrable representations of the Langlands dual of an affine G-algebra -- and also demonstrate that the Nekrasov-Shatashvili limit of the "fully-ramified" Nekrasov instanton partition function for pure G is a simultaneous eigenfunction of the quantum Toda Hamiltonians associated with the Langlands dual of an affine G-algebra. Via the case with matter, he also makes contact with Hitchin systems and the "ramified" geometric Langlands correspondence for curves.

In the second [2] of this two-part work, he generalizes his analysis in the first part, and shows that a 5d and 6d AGT correspondence for SU(N) -- which essentially relates the relevant 5d and 6d Nekrasov instanton partition functions to the integrable representations of a q-deformed and elliptic affine W_N-algebra -- can be derived, purely physically, from the principle that the spacetime BPS spectra of string-dual M-theory compactifications ought to be equivalent. Via an appropriate defect, he also derives a "fully-ramified" version of the 5d and 6d AGT correspondence where integrable representations of a quantum and elliptic affine SU(N)-algebra at the critical level appear on the 2d side, and argues that the relevant "fully-ramified" 5d and 6d Nekrasov instanton partition functions are simultaneous eigenfunctions of commuting operators which define relativistic and elliptized integrable systems. As an offshoot, he also obtains various mathematically novel and interesting relations involving the double loop algebra of SU(N), elliptic Macdonald operators, equivariant elliptic genus of instanton moduli space, and more.

**References**

1. MC Tan. "M-Theoretic Derivations of 4d-2d Dualities: From a Geometric Langlands Duality for Surfaces, to the AGT Correspondence, to Integrable Systems" JHEP, issue 7, no. 171 (2013) [hep-th/1301.1977].

2. MC Tan. "An M-Theoretic Derivation of a 5d and 6d AGT Correspondence, and Relativistic and Elliptized Integrable Systems" JHEP, issue 12, no. 31 (2013) [hep-th/1309.4775].

3. A Braverman, M Finkelberg. "Pursuing the Double Affine Grassmannian I: Transversal Slices via Instantons on A_{k−1} Singularities" Duke Math 152, Number 2 (2010) 175 [math/0711.2083].

4. LF Alday, D Gaiotto, Y Tachikawa. "Liouville Correlation Functions from Four-dimensional Gauge Theories" Lett. Math. Phys. 91 (2010) 167 [hep-th/0906.3219].

5. D Gaiotto. "Asymptotically free N=2 theories and irregular conformal blocks" [hep-th/0908.0307].

6. N Nekrasov, A Okounkov. "Seiberg-Witten Theory and Random Partitions" The Unity of Mathematics, Progress in Mathematics, Volume 244, (2006) 525 [hep-th/0306238].