I am interested in various topics at the intersection of theoretical/mathematical physics and geometry. My most recent work is focused on the geometric aspects of quantum Hall states.

This recent work is about geometry of the integer and fractional quantum Hall states. Main research highlights: derivation of the gravitational anomaly and central charge in integer and fractional quantum Hall states, novel quantized coefficients in geometric adiabatic transport of QH states on moduli spaces of Riemann surfaces, definition and construction of Laughlin states on higher genus Riemann surfaces.

*Laughlin states on higher genus Riemann surfaces*, to appear Commun. Math. Phys., arXiv:1712.09980 [cond-mat.str-el]*Lowest Landau level on a cone and zeta determinants*, J. Phys. A: Math. Theor. 50 (2017) 234003, (Special issue: emerging talents), arXiv:1609.08587 [cond-mat.str-el]*Geometry and large N limits in Laughlin states*, Travaux Math. 24 (2016) 63-127, Lecture notes from the School on Geometry and Quantization, ICMAT, Madrid, 7-11.9.2015, arXiv:1608.02928 [cond-mat.str-el]- (with X. Ma, G. Marinescu and P. Wiegmann)
*Quantum Hall effect and Quillen metric*, Commun. Math. Phys. 349 (2017) 819-855, arXiv:1510.06720 [hep-th] - (with P. Wiegmann)
*Geometric adiabatic transport in quantum Hall states*, Phys. Rev. Lett. 115, 086801 (2015), arXiv:1504.07198 [cond-mat.str-el] - (with F. Ferrari)
*FQHE on curved backgrounds, free fields and large N*, JHEP 12 (2014) 086, arXiv:1410.6802 [hep-th] *Random normal matrices, Bergman kernel and projective embeddings*, JHEP 01 (2014) 133, arXiv:1309.7333 [hep-th]

Work in progress on a novel approach to random metrics in two and higher dimensions, using recent methods in Kähler geometry:

- (with S. Zelditch)
*Heat kernel measures on random surfaces*, Adv. Theor. Math. Phys. v. 20, no. 1 (2016) 135-164, arXiv:1505.05546 [math.PR] - (with S. Zelditch)
*Stability and integration over Bergman metrics*, JHEP 07 (2014) 100, arXiv:1404.0659 [hep-th] - (with A. Bilal and F. Ferrari)
*2D quantum gravity at one loop with Liouville and Mabuchi actions*, Nucl.Phys. B 880 (2014) 203–224, arXiv:1310.1951 [hep-th] - (with F. Ferrari and S. Zelditch)
*Simple matrix models for random Bergman metrics*, J. Stat. Mech. (2012) P04012, arXiv:1112.4382 [hep-th] - (with F. Ferrari and S. Zelditch)
*Gravitational actions in two dimensions and the Mabuchi functional*, Nucl. Phys. B 859 (2012) 341-369, arXiv:1112.1352 [hep-th] - (with F. Ferrari and S. Zelditch)
*Random Kähler metrics*, Nucl. Phys. B 869 (2013) 89-110, arXiv:1107.4575 [hep-th] - (with F. Ferrari and S. Zelditch)
*Random geometry, quantum gravity and the Kähler potential*, Phys. Lett. B 705 (2011) 375-378, arXiv:1107.4022 [hep-th]

My PhD work (my advisor was Michael Douglas) is about physics applications of Bergman kernel and balanced metrics. We show that the Bergman kernel is equivalent to the density matrix of a particle in magnetic field, projected on the lowest Landau level. We use quantum mechanical path integral to derive its asymptotic expansion:

*Bergman kernel, balanced metrics and black holes*, PhD Thesis (2009), Rutgers University Library, [pdf]*Bergman kernel from the lowest Landau level*, Nucl. Phys. B 192-193 (Proc.Suppl.) (2009) 154-155, [pdf]- (with M. Douglas)
*Black holes and balanced metrics*, arXiv:0811.0367 [hep-th] - (with M. Douglas)
*Bergman kernel from path integral*, Commun. Math. Phys. 293 (2010) 205-230, arXiv:0808.2451 [hep-th]

Paper on the connection between Liouville 2d gravity and Stochastic Schramm-Loewner Evolution:

*Connecting SLE and minisuperspace Liouville gravity*, arXiv:0709.3664 [hep-th]

Some earlier papers (written at Lomonosov Moscow State University and at ITEP, Moscow):

- (with D. Gal'tsov, D. Orlov and G. Clement)
*More on general p-brane solutions*, Int. J. Mod. Phys. A 21 (2006) 3575-3604, arXiv:hep-th/0508070 - (with D. Gal'tsov, D. Orlov)
*D instanton on the linear-dilaton background*, Phys. Atom. Nucl. 70 (2007) 1568-1571 - (with D. Gal'tsov, D. Orlov)
*Cylindrical D instantons*, Grav. Cosmol. 11 (2005) 127-131 *Yang-Mills theory from string field theory on D-branes*, Proc. of Cargese School "Progress in string, field and particle theory" (2002) 425-428, [pdf]*On vertex operator construction of quantum affine algebras*, Theor. Math. Phys. 154, 2 (2008) 201-208, arXiv:hep-th/0110148